id sid tid token lemma pos iajs-3461 1 1 367 367 NUM iajs-3461 1 2 Β© Β© ADP iajs-3461 1 3 2025 2025 NUM iajs-3461 1 4 the the DET iajs-3461 1 5 author(s author(s NOUN iajs-3461 1 6 ) ) PUNCT iajs-3461 1 7 . . PUNCT iajs-3461 2 1 published publish VERB iajs-3461 2 2 by by ADP iajs-3461 2 3 college college NOUN iajs-3461 2 4 of of ADP iajs-3461 2 5 education education NOUN iajs-3461 2 6 for for ADP iajs-3461 2 7 pure pure ADJ iajs-3461 2 8 science science NOUN iajs-3461 2 9 ( ( PUNCT iajs-3461 2 10 ibn ibn PROPN iajs-3461 2 11 al al PROPN iajs-3461 2 12 - - PUNCT iajs-3461 2 13 haitham haitham PROPN iajs-3461 2 14 ) ) PUNCT iajs-3461 2 15 , , PUNCT iajs-3461 2 16 university university NOUN iajs-3461 2 17 of of ADP iajs-3461 2 18 baghdad baghdad PROPN iajs-3461 2 19 . . PUNCT iajs-3461 3 1 this this PRON iajs-3461 3 2 is be AUX iajs-3461 3 3 an an DET iajs-3461 3 4 open open ADJ iajs-3461 3 5 - - PUNCT iajs-3461 3 6 access access NOUN iajs-3461 3 7 article article NOUN iajs-3461 3 8 distributed distribute VERB iajs-3461 3 9 under under ADP iajs-3461 3 10 the the DET iajs-3461 3 11 terms term NOUN iajs-3461 3 12 of of ADP iajs-3461 3 13 the the DET iajs-3461 3 14 creative creative ADJ iajs-3461 3 15 commons common NOUN iajs-3461 3 16 attribution attribution NOUN iajs-3461 3 17 4.0 4.0 NUM iajs-3461 3 18 international international ADJ iajs-3461 3 19 license license NOUN iajs-3461 3 20 local local ADJ iajs-3461 3 21 bifurcation bifurcation NOUN iajs-3461 3 22 of of ADP iajs-3461 3 23 the the DET iajs-3461 3 24 dynamic dynamic ADJ iajs-3461 3 25 behavior behavior NOUN iajs-3461 3 26 of of ADP iajs-3461 3 27 predator predator NOUN iajs-3461 3 28 - - PUNCT iajs-3461 3 29 prey prey NOUN iajs-3461 3 30 system system NOUN iajs-3461 3 31 with with ADP iajs-3461 3 32 refuge refuge NOUN iajs-3461 3 33 for for ADP iajs-3461 3 34 both both DET iajs-3461 3 35 species specie NOUN iajs-3461 3 36 intsar intsar VERB iajs-3461 3 37 m. m. NOUN iajs-3461 3 38 kafi1 kafi1 PROPN iajs-3461 4 1 * * PUNCT iajs-3461 5 1 and and CCONJ iajs-3461 5 2 saad saad PROPN iajs-3461 5 3 naji2 naji2 PROPN iajs-3461 5 4 1department 1department NUM iajs-3461 5 5 of of ADP iajs-3461 5 6 mathematics mathematic NOUN iajs-3461 5 7 , , PUNCT iajs-3461 5 8 college college NOUN iajs-3461 5 9 of of ADP iajs-3461 5 10 science science NOUN iajs-3461 5 11 , , PUNCT iajs-3461 5 12 university university NOUN iajs-3461 5 13 of of ADP iajs-3461 5 14 baghdad baghdad PROPN iajs-3461 5 15 , , PUNCT iajs-3461 5 16 iraq iraq PROPN iajs-3461 5 17 . . PUNCT iajs-3461 6 1 2department 2department NUM iajs-3461 6 2 of of ADP iajs-3461 6 3 mathematics mathematic NOUN iajs-3461 6 4 , , PUNCT iajs-3461 6 5 college college NOUN iajs-3461 6 6 of of ADP iajs-3461 6 7 science science NOUN iajs-3461 6 8 for for ADP iajs-3461 6 9 women woman NOUN iajs-3461 6 10 , , PUNCT iajs-3461 6 11 university university NOUN iajs-3461 6 12 of of ADP iajs-3461 6 13 baghdad baghdad PROPN iajs-3461 6 14 , , PUNCT iajs-3461 6 15 iraq iraq PROPN iajs-3461 6 16 . . PUNCT iajs-3461 7 1 * * PUNCT iajs-3461 7 2 corresponding correspond VERB iajs-3461 7 3 author author NOUN iajs-3461 7 4 . . PUNCT iajs-3461 8 1 received receive VERB iajs-3461 8 2 : : PUNCT iajs-3461 8 3 5 5 NUM iajs-3461 8 4 may may PROPN iajs-3461 8 5 2023 2023 NUM iajs-3461 8 6 accepted accept VERB iajs-3461 8 7 : : PUNCT iajs-3461 8 8 4 4 NUM iajs-3461 8 9 september september PROPN iajs-3461 8 10 2023 2023 NUM iajs-3461 8 11 published publish VERB iajs-3461 8 12 : : PUNCT iajs-3461 8 13 20 20 NUM iajs-3461 8 14 january january PROPN iajs-3461 8 15 2025 2025 NUM iajs-3461 8 16 doi.org/10.30526/38.1.3461 doi.org/10.30526/38.1.3461 NOUN iajs-3461 8 17 abstract abstract ADJ iajs-3461 8 18 the the DET iajs-3461 8 19 main main ADJ iajs-3461 8 20 purpose purpose NOUN iajs-3461 8 21 of of ADP iajs-3461 8 22 this this DET iajs-3461 8 23 paper paper NOUN iajs-3461 8 24 is be AUX iajs-3461 8 25 to to PART iajs-3461 8 26 study study VERB iajs-3461 8 27 a a DET iajs-3461 8 28 predator predator NOUN iajs-3461 8 29 – – PUNCT iajs-3461 8 30 prey prey NOUN iajs-3461 8 31 dynamical dynamical ADJ iajs-3461 8 32 system system NOUN iajs-3461 8 33 consisting consist VERB iajs-3461 8 34 of of ADP iajs-3461 8 35 three three NUM iajs-3461 8 36 species specie NOUN iajs-3461 8 37 prey prey NOUN iajs-3461 8 38 , , PUNCT iajs-3461 8 39 specialized specialized ADJ iajs-3461 8 40 predator predator NOUN iajs-3461 8 41 and and CCONJ iajs-3461 8 42 generalist generalist ADJ iajs-3461 8 43 predator predator NOUN iajs-3461 8 44 namely namely ADV iajs-3461 8 45 h h NOUN iajs-3461 8 46 ( ( PUNCT iajs-3461 8 47 t t PROPN iajs-3461 8 48 ) ) PUNCT iajs-3461 8 49 , , PUNCT iajs-3461 9 1 i i PRON iajs-3461 9 2 ( ( PUNCT iajs-3461 9 3 t t PROPN iajs-3461 9 4 ) ) PUNCT iajs-3461 9 5 and and CCONJ iajs-3461 9 6 j j PROPN iajs-3461 9 7 ( ( PUNCT iajs-3461 9 8 t t PROPN iajs-3461 9 9 ) ) PUNCT iajs-3461 9 10 respectively respectively ADV iajs-3461 9 11 , , PUNCT iajs-3461 9 12 w w PROPN iajs-3461 9 13 food food NOUN iajs-3461 9 14 web web NOUN iajs-3461 9 15 and and CCONJ iajs-3461 9 16 refuge refuge NOUN iajs-3461 9 17 for for ADP iajs-3461 9 18 the the DET iajs-3461 9 19 prey prey NOUN iajs-3461 9 20 and and CCONJ iajs-3461 9 21 specialized specialized ADJ iajs-3461 9 22 predator predator NOUN iajs-3461 9 23 population population NOUN iajs-3461 9 24 . . PUNCT iajs-3461 10 1 the the DET iajs-3461 10 2 consider consider VERB iajs-3461 10 3 system system NOUN iajs-3461 10 4 has have VERB iajs-3461 10 5 five five NUM iajs-3461 10 6 equilibrium equilibrium NOUN iajs-3461 10 7 points point NOUN iajs-3461 10 8 𝐴0 𝐴0 PROPN iajs-3461 10 9 = = SYM iajs-3461 10 10 ( ( PUNCT iajs-3461 10 11 0 0 NUM iajs-3461 10 12 , , PUNCT iajs-3461 10 13 0 0 NUM iajs-3461 10 14 , , PUNCT iajs-3461 10 15 0 0 NUM iajs-3461 10 16 ) ) PUNCT iajs-3461 10 17 , , PUNCT iajs-3461 10 18 𝐴1 𝐴1 PROPN iajs-3461 10 19 = = PUNCT iajs-3461 10 20 ( ( PUNCT iajs-3461 10 21 1 1 NUM iajs-3461 10 22 , , PUNCT iajs-3461 10 23 0 0 NUM iajs-3461 10 24 , , PUNCT iajs-3461 10 25 0 0 NUM iajs-3461 10 26 ) ) PUNCT iajs-3461 10 27 , , PUNCT iajs-3461 10 28 𝐴2 𝐴2 PROPN iajs-3461 10 29 = = PUNCT iajs-3461 10 30 ( ( PUNCT iajs-3461 10 31 β„Ž β„Ž X iajs-3461 10 32 βˆ’ βˆ’ PROPN iajs-3461 10 33 , , PUNCT iajs-3461 10 34 𝑖 𝑖 PROPN iajs-3461 10 35 βˆ’ βˆ’ NOUN iajs-3461 10 36 , , PUNCT iajs-3461 10 37 0 0 NUM iajs-3461 10 38 ) ) PUNCT iajs-3461 10 39 , , PUNCT iajs-3461 10 40 𝐴3 𝐴3 PROPN iajs-3461 10 41 = = SYM iajs-3461 10 42 ( ( PUNCT iajs-3461 10 43 β„ŽΜΏ β„ŽΜΏ ADV iajs-3461 10 44 , , PUNCT iajs-3461 10 45 0 0 NUM iajs-3461 10 46 , , PUNCT iajs-3461 10 47 𝑗 𝑗 NOUN iajs-3461 10 48 ΜΏ ΜΏ NOUN iajs-3461 10 49 ) ) PUNCT iajs-3461 10 50 , , PUNCT iajs-3461 10 51 and and CCONJ iajs-3461 10 52 the the DET iajs-3461 10 53 positive positive ADJ iajs-3461 10 54 equilibrium equilibrium NOUN iajs-3461 10 55 point point NOUN iajs-3461 10 56 𝐴4 𝐴4 PROPN iajs-3461 10 57 = = SYM iajs-3461 10 58 ( ( PUNCT iajs-3461 10 59 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 10 60 , , PUNCT iajs-3461 10 61 𝑖̃ 𝑖̃ PROPN iajs-3461 10 62 , , PUNCT iajs-3461 10 63 𝑗̃ 𝑗̃ PROPN iajs-3461 10 64 ) ) PUNCT iajs-3461 10 65 .the .the PRON iajs-3461 10 66 stability stability NOUN iajs-3461 10 67 and and CCONJ iajs-3461 10 68 bifurcation bifurcation NOUN iajs-3461 10 69 of of ADP iajs-3461 10 70 the the DET iajs-3461 10 71 equilibrium equilibrium NOUN iajs-3461 10 72 points point NOUN iajs-3461 10 73 was be AUX iajs-3461 10 74 studied study VERB iajs-3461 10 75 and and CCONJ iajs-3461 10 76 the the DET iajs-3461 10 77 main main ADJ iajs-3461 10 78 influence influence NOUN iajs-3461 10 79 was be AUX iajs-3461 10 80 the the DET iajs-3461 10 81 qualitative qualitative ADJ iajs-3461 10 82 behavior behavior NOUN iajs-3461 10 83 of of ADP iajs-3461 10 84 the the DET iajs-3461 10 85 solution solution NOUN iajs-3461 10 86 . . PUNCT iajs-3461 11 1 it it PRON iajs-3461 11 2 was be AUX iajs-3461 11 3 found find VERB iajs-3461 11 4 that that SCONJ iajs-3461 11 5 𝐴0 𝐴0 PROPN iajs-3461 11 6 was be AUX iajs-3461 11 7 unstable unstable ADJ iajs-3461 11 8 while while SCONJ iajs-3461 11 9 the the DET iajs-3461 11 10 other other ADJ iajs-3461 11 11 equilibrium equilibrium NOUN iajs-3461 11 12 points point NOUN iajs-3461 11 13 are be AUX iajs-3461 11 14 stable stable ADJ iajs-3461 11 15 under under ADP iajs-3461 11 16 condition condition NOUN iajs-3461 11 17 so so SCONJ iajs-3461 11 18 we we PRON iajs-3461 11 19 study study VERB iajs-3461 11 20 their their PRON iajs-3461 11 21 bifurcation bifurcation NOUN iajs-3461 11 22 and and CCONJ iajs-3461 11 23 show show VERB iajs-3461 11 24 that that SCONJ iajs-3461 11 25 𝐴1 𝐴1 PROPN iajs-3461 11 26 , , PUNCT iajs-3461 11 27 𝐴2 𝐴2 PROPN iajs-3461 11 28 and and CCONJ iajs-3461 11 29 𝐴3 𝐴3 PROPN iajs-3461 11 30 are be AUX iajs-3461 11 31 transcritical transcritical ADJ iajs-3461 11 32 while while SCONJ iajs-3461 11 33 𝐴4 𝐴4 PROPN iajs-3461 11 34 is be AUX iajs-3461 11 35 saddle saddle ADJ iajs-3461 11 36 node node ADJ iajs-3461 11 37 bifurcation bifurcation NOUN iajs-3461 11 38 . . PUNCT iajs-3461 12 1 numerical numerical ADJ iajs-3461 12 2 simulations simulation NOUN iajs-3461 12 3 were be AUX iajs-3461 12 4 used use VERB iajs-3461 12 5 to to PART iajs-3461 12 6 illustrate illustrate VERB iajs-3461 12 7 the the DET iajs-3461 12 8 occurrence occurrence NOUN iajs-3461 12 9 of of ADP iajs-3461 12 10 local local ADJ iajs-3461 12 11 bifurcation bifurcation NOUN iajs-3461 12 12 of of ADP iajs-3461 12 13 this this DET iajs-3461 12 14 model model NOUN iajs-3461 12 15 . . PUNCT iajs-3461 13 1 keywords keyword NOUN iajs-3461 13 2 : : PUNCT iajs-3461 13 3 local local ADJ iajs-3461 13 4 bifurcation bifurcation NOUN iajs-3461 13 5 , , PUNCT iajs-3461 13 6 predator predator NOUN iajs-3461 13 7 – – PUNCT iajs-3461 13 8 prey prey NOUN iajs-3461 13 9 , , PUNCT iajs-3461 13 10 stability stability NOUN iajs-3461 13 11 analysis analysis NOUN iajs-3461 13 12 , , PUNCT iajs-3461 13 13 lyapunov lyapunov PROPN iajs-3461 13 14 ’s ’s PART iajs-3461 13 15 function function NOUN iajs-3461 13 16 , , PUNCT iajs-3461 13 17 ecological ecological ADJ iajs-3461 13 18 , , PUNCT iajs-3461 13 19 refuge refuge ADJ iajs-3461 13 20 . . PUNCT iajs-3461 14 1 1 1 X iajs-3461 14 2 . . X iajs-3461 14 3 introduction introduction NOUN iajs-3461 14 4 the the DET iajs-3461 14 5 mathematical mathematical ADJ iajs-3461 14 6 study study NOUN iajs-3461 14 7 of of ADP iajs-3461 14 8 changes change NOUN iajs-3461 14 9 in in ADP iajs-3461 14 10 a a DET iajs-3461 14 11 dynamical dynamical ADJ iajs-3461 14 12 system system NOUN iajs-3461 14 13 's 's PART iajs-3461 14 14 qualitative qualitative ADJ iajs-3461 14 15 asymptotic asymptotic ADJ iajs-3461 14 16 structure structure NOUN iajs-3461 14 17 is be AUX iajs-3461 14 18 known know VERB iajs-3461 14 19 as as ADP iajs-3461 14 20 bifurcation bifurcation NOUN iajs-3461 14 21 theory theory NOUN iajs-3461 14 22 ( ( PUNCT iajs-3461 14 23 1 1 NUM iajs-3461 14 24 , , PUNCT iajs-3461 14 25 2 2 NUM iajs-3461 14 26 ) ) PUNCT iajs-3461 14 27 . . PUNCT iajs-3461 15 1 as as ADV iajs-3461 15 2 well well ADV iajs-3461 15 3 as as ADP iajs-3461 15 4 its its PRON iajs-3461 15 5 attempts attempt NOUN iajs-3461 15 6 to to PART iajs-3461 15 7 explain explain VERB iajs-3461 15 8 various various ADJ iajs-3461 15 9 phenomena phenomenon NOUN iajs-3461 15 10 that that PRON iajs-3461 15 11 have have AUX iajs-3461 15 12 been be AUX iajs-3461 15 13 described describe VERB iajs-3461 15 14 in in ADP iajs-3461 15 15 the the DET iajs-3461 15 16 natural natural ADJ iajs-3461 15 17 sciences science NOUN iajs-3461 15 18 over over ADP iajs-3461 15 19 the the DET iajs-3461 15 20 centuries century NOUN iajs-3461 15 21 . . PUNCT iajs-3461 16 1 where where SCONJ iajs-3461 16 2 performing perform VERB iajs-3461 16 3 bifurcation bifurcation NOUN iajs-3461 16 4 analysis analysis NOUN iajs-3461 16 5 is be AUX iajs-3461 16 6 often often ADV iajs-3461 16 7 a a DET iajs-3461 16 8 powerful powerful ADJ iajs-3461 16 9 way way NOUN iajs-3461 16 10 to to PART iajs-3461 16 11 analyze analyze VERB iajs-3461 16 12 the the DET iajs-3461 16 13 properties property NOUN iajs-3461 16 14 of of ADP iajs-3461 16 15 such such ADJ iajs-3461 16 16 systems system NOUN iajs-3461 16 17 . . PUNCT iajs-3461 17 1 the the DET iajs-3461 17 2 prey prey NOUN iajs-3461 17 3 and and CCONJ iajs-3461 17 4 predator predator NOUN iajs-3461 17 5 model model NOUN iajs-3461 17 6 is be AUX iajs-3461 17 7 an an DET iajs-3461 17 8 important important ADJ iajs-3461 17 9 topic topic NOUN iajs-3461 17 10 at at ADP iajs-3461 17 11 present present NOUN iajs-3461 17 12 as as SCONJ iajs-3461 17 13 it it PRON iajs-3461 17 14 is be AUX iajs-3461 17 15 used use VERB iajs-3461 17 16 to to PART iajs-3461 17 17 solve solve VERB iajs-3461 17 18 many many ADJ iajs-3461 17 19 problems problem NOUN iajs-3461 17 20 in in ADP iajs-3461 17 21 the the DET iajs-3461 17 22 ecological ecological ADJ iajs-3461 17 23 nature nature NOUN iajs-3461 17 24 and and CCONJ iajs-3461 17 25 other other ADJ iajs-3461 17 26 sciences science NOUN iajs-3461 17 27 . . PUNCT iajs-3461 18 1 the the DET iajs-3461 18 2 prey prey NOUN iajs-3461 18 3 system system NOUN iajs-3461 18 4 includes include VERB iajs-3461 18 5 several several ADJ iajs-3461 18 6 interactions interaction NOUN iajs-3461 18 7 , , PUNCT iajs-3461 18 8 such such ADJ iajs-3461 18 9 as as ADP iajs-3461 18 10 competition competition NOUN iajs-3461 18 11 coexistence coexistence NOUN iajs-3461 18 12 and and CCONJ iajs-3461 18 13 stage stage NOUN iajs-3461 18 14 - - PUNCT iajs-3461 18 15 structured structure VERB iajs-3461 18 16 ( ( PUNCT iajs-3461 18 17 3 3 NUM iajs-3461 18 18 ) ) PUNCT iajs-3461 18 19 . . PUNCT iajs-3461 19 1 the the DET iajs-3461 19 2 system system NOUN iajs-3461 19 3 is be AUX iajs-3461 19 4 also also ADV iajs-3461 19 5 impacted impact VERB iajs-3461 19 6 by by ADP iajs-3461 19 7 a a DET iajs-3461 19 8 number number NOUN iajs-3461 19 9 of of ADP iajs-3461 19 10 other other ADJ iajs-3461 19 11 factors factor NOUN iajs-3461 19 12 , , PUNCT iajs-3461 19 13 such such ADJ iajs-3461 19 14 as as ADP iajs-3461 19 15 shelter shelter NOUN iajs-3461 19 16 , , PUNCT iajs-3461 19 17 sickness sickness NOUN iajs-3461 19 18 , , PUNCT iajs-3461 19 19 and and CCONJ iajs-3461 19 20 others other NOUN iajs-3461 19 21 . . PUNCT iajs-3461 20 1 a a DET iajs-3461 20 2 bifurcation bifurcation NOUN iajs-3461 20 3 , , PUNCT iajs-3461 20 4 which which PRON iajs-3461 20 5 is be AUX iajs-3461 20 6 the the DET iajs-3461 20 7 primary primary ADJ iajs-3461 20 8 qualitative qualitative ADJ iajs-3461 20 9 shift shift NOUN iajs-3461 20 10 in in ADP iajs-3461 20 11 the the DET iajs-3461 20 12 behavior behavior NOUN iajs-3461 20 13 of of ADP iajs-3461 20 14 a a DET iajs-3461 20 15 dynamic dynamic ADJ iajs-3461 20 16 system system NOUN iajs-3461 20 17 as as ADP iajs-3461 20 18 a a DET iajs-3461 20 19 result result NOUN iajs-3461 20 20 of of ADP iajs-3461 20 21 changing change VERB iajs-3461 20 22 one one NUM iajs-3461 20 23 of of ADP iajs-3461 20 24 its its PRON iajs-3461 20 25 coefficients coefficient NOUN iajs-3461 20 26 , , PUNCT iajs-3461 20 27 can can AUX iajs-3461 20 28 occasionally occasionally ADV iajs-3461 20 29 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3461 20 30 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3461 20 31 https://doi.org/10.30526/38.1.3462 https://doi.org/10.30526/38.1.3462 PROPN iajs-3461 20 32 https://orcid.org/orcid-search/search?searchquery=0009-0002-4001-8873 https://orcid.org/orcid-search/search?searchquery=0009-0002-4001-8873 VERB iajs-3461 20 33 mailto:intsarmkafi@gmail.com mailto:intsarmkafi@gmail.com X iajs-3461 21 1 https://orcid.org/%200002-0009-6865-9798 https://orcid.org/%200002-0009-6865-9798 X iajs-3461 21 2 mailto:saadnaji58@gmail.com mailto:saadnaji58@gmail.com PROPN iajs-3461 21 3 ihjpas ihjpas PROPN iajs-3461 21 4 . . PUNCT iajs-3461 22 1 2025 2025 NUM iajs-3461 22 2 , , PUNCT iajs-3461 22 3 38 38 NUM iajs-3461 22 4 ( ( PUNCT iajs-3461 22 5 1 1 NUM iajs-3461 22 6 ) ) PUNCT iajs-3461 22 7 368 368 NUM iajs-3461 22 8 emerge emerge VERB iajs-3461 22 9 from from ADP iajs-3461 22 10 variations variation NOUN iajs-3461 22 11 in in ADP iajs-3461 22 12 any any DET iajs-3461 22 13 parameter parameter NOUN iajs-3461 22 14 in in ADP iajs-3461 22 15 the the DET iajs-3461 22 16 system system NOUN iajs-3461 22 17 , , PUNCT iajs-3461 22 18 leading lead VERB iajs-3461 22 19 to to ADP iajs-3461 22 20 complex complex ADJ iajs-3461 22 21 behavior behavior NOUN iajs-3461 22 22 that that PRON iajs-3461 22 23 leads lead VERB iajs-3461 22 24 to to ADP iajs-3461 22 25 system system NOUN iajs-3461 22 26 instability instability NOUN iajs-3461 22 27 . . PUNCT iajs-3461 23 1 local local ADJ iajs-3461 23 2 and and CCONJ iajs-3461 23 3 global global ADJ iajs-3461 23 4 bifurcations bifurcation NOUN iajs-3461 23 5 were be AUX iajs-3461 23 6 the the DET iajs-3461 23 7 two two NUM iajs-3461 23 8 main main ADJ iajs-3461 23 9 classes class NOUN iajs-3461 23 10 that that PRON iajs-3461 23 11 made make VERB iajs-3461 23 12 up up ADP iajs-3461 23 13 the the DET iajs-3461 23 14 bifurcation bifurcation NOUN iajs-3461 23 15 . . PUNCT iajs-3461 24 1 changes change NOUN iajs-3461 24 2 in in ADP iajs-3461 24 3 the the DET iajs-3461 24 4 local local ADJ iajs-3461 24 5 stability stability NOUN iajs-3461 24 6 parameters parameter NOUN iajs-3461 24 7 of of ADP iajs-3461 24 8 equilibria equilibrium NOUN iajs-3461 24 9 or or CCONJ iajs-3461 24 10 periodic periodic ADJ iajs-3461 24 11 orbits orbit NOUN iajs-3461 24 12 can can AUX iajs-3461 24 13 be be AUX iajs-3461 24 14 used use VERB iajs-3461 24 15 to to PART iajs-3461 24 16 evaluate evaluate VERB iajs-3461 24 17 local local ADJ iajs-3461 24 18 bifurcation bifurcation NOUN iajs-3461 24 19 . . PUNCT iajs-3461 25 1 global global ADJ iajs-3461 25 2 bifurcation bifurcation NOUN iajs-3461 25 3 , , PUNCT iajs-3461 25 4 on on ADP iajs-3461 25 5 the the DET iajs-3461 25 6 other other ADJ iajs-3461 25 7 hand hand NOUN iajs-3461 25 8 , , PUNCT iajs-3461 25 9 happens happen VERB iajs-3461 25 10 when when SCONJ iajs-3461 25 11 periodic periodic ADJ iajs-3461 25 12 orbits orbit NOUN iajs-3461 25 13 run run VERB iajs-3461 25 14 into into ADP iajs-3461 25 15 equilibrium equilibrium NOUN iajs-3461 25 16 . . PUNCT iajs-3461 26 1 this this PRON iajs-3461 26 2 leads lead VERB iajs-3461 26 3 to to ADP iajs-3461 26 4 changes change NOUN iajs-3461 26 5 in in ADP iajs-3461 26 6 the the DET iajs-3461 26 7 topology topology NOUN iajs-3461 26 8 of of ADP iajs-3461 26 9 the the DET iajs-3461 26 10 trajectories trajectory NOUN iajs-3461 26 11 in in ADP iajs-3461 26 12 phase phase NOUN iajs-3461 26 13 space space NOUN iajs-3461 26 14 that that PRON iajs-3461 26 15 can can AUX iajs-3461 26 16 not not PART iajs-3461 26 17 be be AUX iajs-3461 26 18 contained contain VERB iajs-3461 26 19 within within ADP iajs-3461 26 20 a a DET iajs-3461 26 21 limited limited ADJ iajs-3461 26 22 region region NOUN iajs-3461 26 23 as as SCONJ iajs-3461 26 24 is be AUX iajs-3461 26 25 the the DET iajs-3461 26 26 case case NOUN iajs-3461 26 27 with with ADP iajs-3461 26 28 local local ADJ iajs-3461 26 29 bifurcation bifurcation NOUN iajs-3461 26 30 . . PUNCT iajs-3461 27 1 these these DET iajs-3461 27 2 bifurcations bifurcation NOUN iajs-3461 27 3 occur occur VERB iajs-3461 27 4 when when SCONJ iajs-3461 27 5 a a DET iajs-3461 27 6 single single ADJ iajs-3461 27 7 parameter parameter NOUN iajs-3461 27 8 is be AUX iajs-3461 27 9 changed change VERB iajs-3461 27 10 ( ( PUNCT iajs-3461 27 11 4 4 NUM iajs-3461 27 12 - - SYM iajs-3461 27 13 9 9 NUM iajs-3461 27 14 ) ) PUNCT iajs-3461 27 15 . . PUNCT iajs-3461 28 1 perko perko NOUN iajs-3461 28 2 ( ( PUNCT iajs-3461 28 3 10 10 NUM iajs-3461 28 4 ) ) PUNCT iajs-3461 28 5 , , PUNCT iajs-3461 28 6 on on ADP iajs-3461 28 7 the the DET iajs-3461 28 8 other other ADJ iajs-3461 28 9 hand hand NOUN iajs-3461 28 10 , , PUNCT iajs-3461 28 11 identified identify VERB iajs-3461 28 12 the the DET iajs-3461 28 13 prerequisites prerequisite NOUN iajs-3461 28 14 for for ADP iajs-3461 28 15 local local ADJ iajs-3461 28 16 bifurcation bifurcation NOUN iajs-3461 28 17 , , PUNCT iajs-3461 28 18 including include VERB iajs-3461 28 19 saddle saddle NOUN iajs-3461 28 20 - - PUNCT iajs-3461 28 21 node node NOUN iajs-3461 28 22 , , PUNCT iajs-3461 28 23 transcritical transcritical ADJ iajs-3461 28 24 , , PUNCT iajs-3461 28 25 pitchfork pitchfork NOUN iajs-3461 28 26 , , PUNCT iajs-3461 28 27 and and CCONJ iajs-3461 28 28 period period NOUN iajs-3461 28 29 - - PUNCT iajs-3461 28 30 doubling doubling NOUN iajs-3461 28 31 . . PUNCT iajs-3461 29 1 finally finally ADV iajs-3461 29 2 , , PUNCT iajs-3461 29 3 sotomayor sotomayor PROPN iajs-3461 29 4 's 's PART iajs-3461 29 5 theorem(11 theorem(11 NUM iajs-3461 29 6 ) ) PUNCT iajs-3461 29 7 for for ADP iajs-3461 29 8 local local ADJ iajs-3461 29 9 bifurcation bifurcation NOUN iajs-3461 29 10 was be AUX iajs-3461 29 11 applied apply VERB iajs-3461 29 12 in in ADP iajs-3461 29 13 this this DET iajs-3461 29 14 work work NOUN iajs-3461 29 15 to to PART iajs-3461 29 16 examine examine VERB iajs-3461 29 17 the the DET iajs-3461 29 18 occurrence occurrence NOUN iajs-3461 29 19 of of ADP iajs-3461 29 20 local local ADJ iajs-3461 29 21 bifurcation bifurcation NOUN iajs-3461 29 22 at at ADP iajs-3461 29 23 equilibrium equilibrium NOUN iajs-3461 29 24 sites site NOUN iajs-3461 29 25 local local ADJ iajs-3461 29 26 bifurcation bifurcation NOUN iajs-3461 29 27 methods method NOUN iajs-3461 29 28 close close ADJ iajs-3461 29 29 to to ADP iajs-3461 29 30 all all DET iajs-3461 29 31 equilibrium equilibrium NOUN iajs-3461 29 32 points point NOUN iajs-3461 29 33 , , PUNCT iajs-3461 29 34 and and CCONJ iajs-3461 29 35 a a DET iajs-3461 29 36 number number NOUN iajs-3461 29 37 of of ADP iajs-3461 29 38 fundamental fundamental ADJ iajs-3461 29 39 results result NOUN iajs-3461 29 40 ( ( PUNCT iajs-3461 29 41 12 12 NUM iajs-3461 29 42 - - SYM iajs-3461 29 43 22 22 NUM iajs-3461 29 44 ) ) PUNCT iajs-3461 29 45 . . PUNCT iajs-3461 30 1 2 2 X iajs-3461 30 2 . . X iajs-3461 30 3 model model NOUN iajs-3461 30 4 formulation formulation NOUN iajs-3461 30 5 an an DET iajs-3461 30 6 ecological ecological ADJ iajs-3461 30 7 model model NOUN iajs-3461 30 8 was be AUX iajs-3461 30 9 suggested suggest VERB iajs-3461 30 10 for for ADP iajs-3461 30 11 investigation investigation NOUN iajs-3461 30 12 in in ADP iajs-3461 30 13 this this DET iajs-3461 30 14 section section NOUN iajs-3461 30 15 . . PUNCT iajs-3461 31 1 a a DET iajs-3461 31 2 prey prey NOUN iajs-3461 31 3 was be AUX iajs-3461 31 4 included include VERB iajs-3461 31 5 in in ADP iajs-3461 31 6 the the DET iajs-3461 31 7 model model NOUN iajs-3461 31 8 , , PUNCT iajs-3461 31 9 and and CCONJ iajs-3461 31 10 its its PRON iajs-3461 31 11 overall overall ADJ iajs-3461 31 12 population population NOUN iajs-3461 31 13 density density NOUN iajs-3461 31 14 at at ADP iajs-3461 31 15 time time NOUN iajs-3461 31 16 t t PROPN iajs-3461 31 17 is be AUX iajs-3461 31 18 represented represent VERB iajs-3461 31 19 by by ADP iajs-3461 31 20 the the DET iajs-3461 31 21 symbol symbol NOUN iajs-3461 31 22 h(t h(t PROPN iajs-3461 31 23 ) ) PUNCT iajs-3461 31 24 , , PUNCT iajs-3461 31 25 engaging engage VERB iajs-3461 31 26 with with ADP iajs-3461 31 27 a a DET iajs-3461 31 28 specialized specialized ADJ iajs-3461 31 29 predator predator NOUN iajs-3461 31 30 i(t i(t NOUN iajs-3461 31 31 ) ) PUNCT iajs-3461 31 32 is be AUX iajs-3461 31 33 the the DET iajs-3461 31 34 population population NOUN iajs-3461 31 35 density density NOUN iajs-3461 31 36 at at ADP iajs-3461 31 37 time time NOUN iajs-3461 31 38 t t PROPN iajs-3461 31 39 and and CCONJ iajs-3461 31 40 generalist generalist NOUN iajs-3461 31 41 s s VERB iajs-3461 31 42 assumed assume VERB iajs-3461 31 43 that that SCONJ iajs-3461 31 44 the the DET iajs-3461 31 45 prey prey NOUN iajs-3461 31 46 wa wa PROPN iajs-3461 31 47 . . PUNCT iajs-3461 32 1 it it PRON iajs-3461 32 2 j(t)by j(t)by PRON iajs-3461 32 3 4is 4is NOUN iajs-3461 32 4 denoted denote VERB iajs-3461 32 5 tat tat NOUN iajs-3461 32 6 time time NOUN iajs-3461 32 7 4population 4population NUM iajs-3461 32 8 density density NOUN iajs-3461 32 9 whosepredator whosepredator NOUN iajs-3461 32 10 with with ADP iajs-3461 32 11 refuge refuge NOUN iajs-3461 32 12 and and CCONJ iajs-3461 32 13 specialist specialist ADJ iajs-3461 32 14 predator predator NOUN iajs-3461 32 15 is be AUX iajs-3461 32 16 with with ADP iajs-3461 32 17 the the DET iajs-3461 32 18 prey prey NOUN iajs-3461 32 19 refuge refuge NOUN iajs-3461 32 20 . . PUNCT iajs-3461 33 1 the the DET iajs-3461 33 2 following follow VERB iajs-3461 33 3 presumptions presumption NOUN iajs-3461 33 4 are be AUX iajs-3461 33 5 now now ADV iajs-3461 33 6 used use VERB iajs-3461 33 7 to to PART iajs-3461 33 8 create create VERB iajs-3461 33 9 the the DET iajs-3461 33 10 fundamental fundamental ADJ iajs-3461 33 11 ecological ecological ADJ iajs-3461 33 12 model model NOUN iajs-3461 33 13 shown show VERB iajs-3461 33 14 in in ADP iajs-3461 33 15 table table NOUN iajs-3461 33 16 1 1 NUM iajs-3461 33 17 : : PUNCT iajs-3461 33 18 𝑑𝐻 𝑑𝐻 ADJ iajs-3461 33 19 𝑑𝑇 𝑑𝑇 PROPN iajs-3461 33 20 = = PRON iajs-3461 33 21 𝜌h 𝜌h PROPN iajs-3461 33 22 ( ( PUNCT iajs-3461 33 23 1 1 NUM iajs-3461 33 24 𝐻 𝐻 PROPN iajs-3461 33 25 𝐾 𝐾 PROPN iajs-3461 33 26 ) ) PUNCT iajs-3461 33 27 – – PUNCT iajs-3461 33 28 Ξ± Ξ± PROPN iajs-3461 33 29 ( ( PUNCT iajs-3461 33 30 1-π‘š1 1-π‘š1 NUM iajs-3461 33 31 ) ) PUNCT iajs-3461 33 32 h h NOUN iajs-3461 34 1 i i PRON iajs-3461 34 2 – – PUNCT iajs-3461 34 3 Ξ² Ξ² X iajs-3461 34 4 ( ( PUNCT iajs-3461 34 5 1-π‘š2 1-π‘š2 NUM iajs-3461 34 6 ) ) PUNCT iajs-3461 34 7 h h NOUN iajs-3461 35 1 j j NOUN iajs-3461 35 2 𝑑𝐼 𝑑𝐼 ADJ iajs-3461 36 1 𝑑𝑇 𝑑𝑇 ADJ iajs-3461 36 2 = = SYM iajs-3461 36 3 𝛼1 𝛼1 NOUN iajs-3461 36 4 ( ( PUNCT iajs-3461 36 5 1 1 NUM iajs-3461 36 6 π‘š1 π‘š1 NOUN iajs-3461 36 7 ) ) PUNCT iajs-3461 36 8 h h NOUN iajs-3461 36 9 i i PRON iajs-3461 36 10 – – PUNCT iajs-3461 36 11 𝛾 𝛾 X iajs-3461 36 12 ( ( PUNCT iajs-3461 36 13 1-π‘š3 1-π‘š3 NUM iajs-3461 36 14 ) ) PUNCT iajs-3461 36 15 ij ij NOUN iajs-3461 36 16 – – PUNCT iajs-3461 36 17 𝑑1 𝑑1 NOUN iajs-3461 37 1 i i PRON iajs-3461 37 2 ( ( PUNCT iajs-3461 37 3 1 1 X iajs-3461 37 4 ) ) PUNCT iajs-3461 37 5 𝑑𝐽 𝑑𝐽 NOUN iajs-3461 37 6 𝑑𝑇 𝑑𝑇 PROPN iajs-3461 37 7 = = SYM iajs-3461 37 8 Ξ²1 Ξ²1 NOUN iajs-3461 37 9 ( ( PUNCT iajs-3461 37 10 1 1 NUM iajs-3461 37 11 -π‘š2 -π‘š2 NOUN iajs-3461 37 12 ) ) PUNCT iajs-3461 37 13 h h NOUN iajs-3461 37 14 j j PROPN iajs-3461 37 15 – – PUNCT iajs-3461 37 16 𝛾1 𝛾1 PROPN iajs-3461 37 17 ( ( PUNCT iajs-3461 37 18 1-π‘š3 1-π‘š3 NUM iajs-3461 37 19 ) ) PUNCT iajs-3461 37 20 ij ij NOUN iajs-3461 37 21 – – PUNCT iajs-3461 37 22 𝑑2 𝑑2 NOUN iajs-3461 37 23 j j PROPN iajs-3461 37 24 table table NOUN iajs-3461 37 25 1 1 NUM iajs-3461 37 26 . . PUNCT iajs-3461 38 1 the the DET iajs-3461 38 2 parameters parameter NOUN iajs-3461 38 3 of of ADP iajs-3461 38 4 model model NOUN iajs-3461 38 5 ( ( PUNCT iajs-3461 38 6 1 1 X iajs-3461 38 7 ) ) PUNCT iajs-3461 38 8 parameters parameter NOUN iajs-3461 38 9 biological biological ADJ iajs-3461 38 10 meaning meaning NOUN iajs-3461 38 11 𝜌 𝜌 ADP iajs-3461 38 12 > > X iajs-3461 38 13 0 0 NUM iajs-3461 38 14 intrinsic intrinsic ADJ iajs-3461 38 15 growth growth NOUN iajs-3461 38 16 π‘˜ π‘˜ ADP iajs-3461 38 17 > > X iajs-3461 38 18 0 0 NUM iajs-3461 38 19 carrying carry VERB iajs-3461 38 20 capacity capacity NOUN iajs-3461 38 21 ( ( PUNCT iajs-3461 38 22 in in ADP iajs-3461 38 23 logistic logistic ADJ iajs-3461 38 24 growth growth NOUN iajs-3461 38 25 ) ) PUNCT iajs-3461 39 1 Ξ± Ξ± X iajs-3461 39 2 > > X iajs-3461 39 3 0 0 NUM iajs-3461 39 4 maximum maximum ADJ iajs-3461 39 5 attack attack NOUN iajs-3461 39 6 rate rate NOUN iajs-3461 39 7 by by ADP iajs-3461 39 8 specialist specialist ADJ iajs-3461 39 9 predator predator NOUN iajs-3461 39 10 Ξ² Ξ² X iajs-3461 39 11 > > X iajs-3461 39 12 0 0 NUM iajs-3461 39 13 maximum maximum ADJ iajs-3461 39 14 attack attack NOUN iajs-3461 39 15 rate rate NOUN iajs-3461 39 16 by by ADP iajs-3461 39 17 generalist generalist ADJ iajs-3461 39 18 predator predator NOUN iajs-3461 39 19 𝑑1π‘Žπ‘›π‘‘ 𝑑1π‘Žπ‘›π‘‘ NOUN iajs-3461 39 20 𝑑2 𝑑2 VERB iajs-3461 39 21 natural natural ADJ iajs-3461 39 22 death death NOUN iajs-3461 39 23 rate rate NOUN iajs-3461 39 24 of of ADP iajs-3461 39 25 specialist specialist NOUN iajs-3461 39 26 and and CCONJ iajs-3461 39 27 generalist generalist ADJ iajs-3461 39 28 predator predator NOUN iajs-3461 39 29 𝛼1 𝛼1 NOUN iajs-3461 39 30 > > SYM iajs-3461 39 31 0 0 PUNCT iajs-3461 40 1 maximum maximum ADJ iajs-3461 40 2 predation predation NOUN iajs-3461 40 3 rate rate NOUN iajs-3461 40 4 of of ADP iajs-3461 40 5 the the DET iajs-3461 40 6 specialist specialist ADJ iajs-3461 40 7 predator predator NOUN iajs-3461 40 8 over over ADP iajs-3461 40 9 the the DET iajs-3461 40 10 prey prey NOUN iajs-3461 40 11 Ξ²1 Ξ²1 PROPN iajs-3461 40 12 > > X iajs-3461 40 13 0 0 PUNCT iajs-3461 41 1 maximum maximum ADJ iajs-3461 41 2 predation predation NOUN iajs-3461 41 3 rate rate NOUN iajs-3461 41 4 of of ADP iajs-3461 41 5 the the DET iajs-3461 41 6 generalist generalist ADJ iajs-3461 41 7 predator predator NOUN iajs-3461 41 8 over over ADP iajs-3461 41 9 the the DET iajs-3461 41 10 prey prey NOUN iajs-3461 41 11 𝛾 𝛾 PROPN iajs-3461 41 12 > > X iajs-3461 41 13 0 0 NUM iajs-3461 41 14 maximum maximum ADJ iajs-3461 41 15 attack attack NOUN iajs-3461 41 16 rate rate NOUN iajs-3461 41 17 by by ADP iajs-3461 41 18 generalist generalist ADJ iajs-3461 41 19 predator predator NOUN iajs-3461 41 20 on on ADP iajs-3461 41 21 specialist specialist ADJ iajs-3461 41 22 predator predator NOUN iajs-3461 41 23 𝛾1 𝛾1 PROPN iajs-3461 41 24 > > SYM iajs-3461 41 25 0 0 PUNCT iajs-3461 42 1 maximum maximum ADJ iajs-3461 42 2 predation predation NOUN iajs-3461 42 3 rate rate NOUN iajs-3461 42 4 of of ADP iajs-3461 42 5 the the DET iajs-3461 42 6 generalist generalist ADJ iajs-3461 42 7 predator predator NOUN iajs-3461 42 8 over over ADP iajs-3461 42 9 the the DET iajs-3461 42 10 specialist specialist ADJ iajs-3461 42 11 predator predator NOUN iajs-3461 42 12 0 0 NUM iajs-3461 42 13 < < X iajs-3461 42 14 π‘š1 π‘š1 X iajs-3461 42 15 < < X iajs-3461 42 16 1 1 NUM iajs-3461 42 17 , , PUNCT iajs-3461 42 18 the the DET iajs-3461 42 19 refuge refuge NOUN iajs-3461 42 20 rates rate VERB iajs-3461 42 21 constants constant NOUN iajs-3461 42 22 of of ADP iajs-3461 42 23 the the DET iajs-3461 42 24 prey prey NOUN iajs-3461 42 25 from from ADP iajs-3461 42 26 the the DET iajs-3461 42 27 specialist specialist ADJ iajs-3461 42 28 predator predator NOUN iajs-3461 42 29 0 0 NUM iajs-3461 42 30 < < X iajs-3461 42 31 π‘š2 π‘š2 NOUN iajs-3461 42 32 < < X iajs-3461 42 33 1 1 NUM iajs-3461 42 34 the the DET iajs-3461 42 35 refuge refuge NOUN iajs-3461 42 36 rates rate NOUN iajs-3461 42 37 constants constant NOUN iajs-3461 42 38 of of ADP iajs-3461 42 39 the the DET iajs-3461 42 40 prey prey NOUN iajs-3461 42 41 from from ADP iajs-3461 42 42 the the DET iajs-3461 42 43 generalist generalist ADJ iajs-3461 42 44 predator predator NOUN iajs-3461 42 45 0 0 NUM iajs-3461 42 46 < < X iajs-3461 42 47 π‘š3 π‘š3 NOUN iajs-3461 42 48 < < X iajs-3461 42 49 1 1 NUM iajs-3461 42 50 the the DET iajs-3461 42 51 refuge refuge NOUN iajs-3461 42 52 rates rate NOUN iajs-3461 42 53 constants constant NOUN iajs-3461 42 54 of of ADP iajs-3461 42 55 the the DET iajs-3461 42 56 specialist specialist ADJ iajs-3461 42 57 predator predator NOUN iajs-3461 42 58 from from ADP iajs-3461 42 59 the the DET iajs-3461 42 60 generalist generalist ADJ iajs-3461 42 61 predator predator NOUN iajs-3461 42 62 ihjpas ihjpa NOUN iajs-3461 42 63 . . PUNCT iajs-3461 43 1 2025 2025 NUM iajs-3461 43 2 , , PUNCT iajs-3461 43 3 38 38 NUM iajs-3461 43 4 ( ( PUNCT iajs-3461 43 5 1 1 NUM iajs-3461 43 6 ) ) PUNCT iajs-3461 43 7 369 369 NUM iajs-3461 43 8 the the DET iajs-3461 43 9 above above ADJ iajs-3461 43 10 model model NOUN iajs-3461 43 11 has have VERB iajs-3461 43 12 13 13 NUM iajs-3461 43 13 parameters parameter NOUN iajs-3461 43 14 so so ADV iajs-3461 43 15 , , PUNCT iajs-3461 43 16 it it PRON iajs-3461 43 17 is be AUX iajs-3461 43 18 difficult difficult ADJ iajs-3461 43 19 to to PART iajs-3461 43 20 study study VERB iajs-3461 43 21 all all PRON iajs-3461 43 22 of of ADP iajs-3461 43 23 them they PRON iajs-3461 43 24 , , PUNCT iajs-3461 43 25 there there ADV iajs-3461 43 26 for for SCONJ iajs-3461 43 27 reduce reduce VERB iajs-3461 43 28 them they PRON iajs-3461 43 29 a a DET iajs-3461 43 30 dimensionless dimensionless NOUN iajs-3461 43 31 variables variable NOUN iajs-3461 43 32 and and CCONJ iajs-3461 43 33 parameters parameter NOUN iajs-3461 43 34 are be AUX iajs-3461 43 35 defined define VERB iajs-3461 43 36 : : PUNCT iajs-3461 43 37 π‘‘βˆ— π‘‘βˆ— NOUN iajs-3461 43 38 = = PUNCT iajs-3461 43 39 𝜌 𝜌 ADP iajs-3461 43 40 t t NOUN iajs-3461 43 41 , , PUNCT iajs-3461 43 42 h h NOUN iajs-3461 44 1 = = NOUN iajs-3461 44 2 h h PROPN iajs-3461 45 1 k k NOUN iajs-3461 45 2 , , PUNCT iajs-3461 45 3 i i PRON iajs-3461 45 4 = = VERB iajs-3461 46 1 i i PRON iajs-3461 46 2 k k PROPN iajs-3461 46 3 , , PUNCT iajs-3461 47 1 j j PROPN iajs-3461 48 1 = = SYM iajs-3461 48 2 j j PROPN iajs-3461 48 3 k k PROPN iajs-3461 48 4 , , PUNCT iajs-3461 48 5 r1 r1 PROPN iajs-3461 48 6 = = PUNCT iajs-3461 48 7 Ξ±k(1βˆ’ Ξ±k(1βˆ’ PROPN iajs-3461 48 8 π‘š1 π‘š1 PROPN iajs-3461 48 9 ) ) PUNCT iajs-3461 48 10 𝜌 𝜌 X iajs-3461 48 11 , , PUNCT iajs-3461 48 12 r2 r2 NOUN iajs-3461 48 13 = = SYM iajs-3461 48 14 Ξ²k(1βˆ’ Ξ²k(1βˆ’ NUM iajs-3461 48 15 π‘š2 π‘š2 NOUN iajs-3461 48 16 ) ) PUNCT iajs-3461 48 17 𝜌 𝜌 NOUN iajs-3461 48 18 , , PUNCT iajs-3461 48 19 r4 r4 NOUN iajs-3461 48 20 = = SYM iajs-3461 48 21 Ξ³k(1βˆ’ Ξ³k(1βˆ’ PROPN iajs-3461 48 22 π‘š3 π‘š3 NOUN iajs-3461 48 23 ) ) PUNCT iajs-3461 48 24 𝜌 𝜌 X iajs-3461 48 25 , , PUNCT iajs-3461 48 26 , , PUNCT iajs-3461 48 27 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 48 28 = = SYM iajs-3461 48 29 d1 d1 PROPN iajs-3461 48 30 𝜌 𝜌 X iajs-3461 48 31 , , PUNCT iajs-3461 48 32 r6 r6 NOUN iajs-3461 48 33 = = SYM iajs-3461 48 34 Ξ²1k(1βˆ’ Ξ²1k(1βˆ’ SYM iajs-3461 48 35 π‘š2 π‘š2 PROPN iajs-3461 48 36 ) ) PUNCT iajs-3461 48 37 𝜌 𝜌 X iajs-3461 48 38 , , PUNCT iajs-3461 48 39 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 48 40 = = SYM iajs-3461 48 41 𝛾1π‘˜(1βˆ’π‘š1 𝛾1π‘˜(1βˆ’π‘š1 NOUN iajs-3461 48 42 ) ) PUNCT iajs-3461 48 43 𝜌 𝜌 X iajs-3461 48 44 , , PUNCT iajs-3461 48 45 r3 r3 PROPN iajs-3461 48 46 = = SYM iajs-3461 48 47 Ξ±1k(1βˆ’ Ξ±1k(1βˆ’ X iajs-3461 48 48 π‘š1 π‘š1 PROPN iajs-3461 48 49 ) ) PUNCT iajs-3461 48 50 𝜌 𝜌 ADP iajs-3461 48 51 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 48 52 = = SYM iajs-3461 48 53 d2 d2 PROPN iajs-3461 48 54 𝜌 𝜌 X iajs-3461 48 55 . . PUNCT iajs-3461 49 1 now now ADV iajs-3461 49 2 , , PUNCT iajs-3461 49 3 for for ADP iajs-3461 49 4 simplicity simplicity NOUN iajs-3461 49 5 rename rename ADJ iajs-3461 49 6 π‘‘βˆ— π‘‘βˆ— NOUN iajs-3461 49 7 = = PUNCT iajs-3461 49 8 𝑑. 𝑑. NOUN iajs-3461 49 9 so so ADV iajs-3461 49 10 , , PUNCT iajs-3461 49 11 the the DET iajs-3461 49 12 dimensional dimensional ADJ iajs-3461 49 13 system system NOUN iajs-3461 49 14 ( ( PUNCT iajs-3461 49 15 1 1 X iajs-3461 49 16 ) ) PUNCT iajs-3461 49 17 can can AUX iajs-3461 49 18 be be AUX iajs-3461 49 19 formulated formulate VERB iajs-3461 49 20 as as ADP iajs-3461 49 21 : : PUNCT iajs-3461 49 22 π‘‘β„Ž π‘‘β„Ž NOUN iajs-3461 49 23 𝑑𝑑 𝑑𝑑 ADP iajs-3461 49 24 = = PROPN iajs-3461 49 25 β„Ž[1 β„Ž[1 PROPN iajs-3461 49 26 βˆ’ βˆ’ PROPN iajs-3461 49 27 β„Ž β„Ž NOUN iajs-3461 49 28 βˆ’ βˆ’ NOUN iajs-3461 49 29 r1𝑖 r1𝑖 NOUN iajs-3461 49 30 βˆ’ βˆ’ NOUN iajs-3461 49 31 r2j r2j X iajs-3461 49 32 ] ] X iajs-3461 49 33 = = SYM iajs-3461 49 34 𝑓1 𝑓1 PROPN iajs-3461 49 35 ( ( PUNCT iajs-3461 49 36 β„Ž β„Ž PROPN iajs-3461 49 37 , , PUNCT iajs-3461 49 38 𝑖 𝑖 SYM iajs-3461 49 39 , , PUNCT iajs-3461 49 40 𝑗 𝑗 PROPN iajs-3461 49 41 ) ) PUNCT iajs-3461 49 42 𝑑𝑖 𝑑𝑖 VERB iajs-3461 49 43 𝑑𝑑 𝑑𝑑 ADP iajs-3461 49 44 = = SYM iajs-3461 49 45 𝑖 𝑖 PROPN iajs-3461 49 46 [ [ X iajs-3461 49 47 r3β„Ž r3β„Ž NOUN iajs-3461 49 48 βˆ’ βˆ’ NOUN iajs-3461 49 49 r4𝑗 r4𝑗 NOUN iajs-3461 49 50 βˆ’ βˆ’ NOUN iajs-3461 49 51 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 49 52 ] ] X iajs-3461 49 53 = = SYM iajs-3461 49 54 𝑓2 𝑓2 PROPN iajs-3461 49 55 ( ( PUNCT iajs-3461 49 56 β„Ž β„Ž PROPN iajs-3461 49 57 , , PUNCT iajs-3461 49 58 𝑖 𝑖 SYM iajs-3461 49 59 , , PUNCT iajs-3461 49 60 𝑗 𝑗 NOUN iajs-3461 49 61 ) ) PUNCT iajs-3461 49 62 𝑑𝑗 𝑑𝑗 NOUN iajs-3461 49 63 𝑑𝑑 𝑑𝑑 ADP iajs-3461 49 64 = = SYM iajs-3461 49 65 𝑗 𝑗 PROPN iajs-3461 49 66 [ [ PUNCT iajs-3461 49 67 r6β„Ž r6β„Ž PROPN iajs-3461 49 68 + + NOUN iajs-3461 49 69 r7𝑖 r7𝑖 NOUN iajs-3461 49 70 βˆ’ βˆ’ NOUN iajs-3461 49 71 r8 r8 PROPN iajs-3461 49 72 ] ] X iajs-3461 49 73 = = SYM iajs-3461 49 74 𝑓3 𝑓3 PROPN iajs-3461 49 75 ( ( PUNCT iajs-3461 49 76 β„Ž β„Ž PROPN iajs-3461 49 77 , , PUNCT iajs-3461 49 78 𝑖 𝑖 SYM iajs-3461 49 79 , , PUNCT iajs-3461 49 80 𝑗 𝑗 NOUN iajs-3461 49 81 ) ) PUNCT iajs-3461 49 82 } } PUNCT iajs-3461 49 83 ( ( PUNCT iajs-3461 49 84 2 2 X iajs-3461 49 85 ) ) PUNCT iajs-3461 49 86 with with ADP iajs-3461 49 87 β„Ž β„Ž PROPN iajs-3461 49 88 ( ( PUNCT iajs-3461 49 89 0 0 NUM iajs-3461 49 90 ) ) PUNCT iajs-3461 49 91 β‰₯ β‰₯ NOUN iajs-3461 49 92 0 0 NUM iajs-3461 49 93 , , PUNCT iajs-3461 49 94 𝑖 𝑖 CCONJ iajs-3461 49 95 ( ( PUNCT iajs-3461 49 96 0 0 NUM iajs-3461 49 97 ) ) PUNCT iajs-3461 49 98 β‰₯ β‰₯ NOUN iajs-3461 49 99 0 0 NUM iajs-3461 49 100 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ PROPN iajs-3461 49 101 𝑗 𝑗 PROPN iajs-3461 49 102 ( ( PUNCT iajs-3461 49 103 0 0 NUM iajs-3461 49 104 ) ) PUNCT iajs-3461 49 105 β‰₯ β‰₯ NOUN iajs-3461 49 106 0 0 NUM iajs-3461 49 107 . . PUNCT iajs-3461 50 1 in in ADP iajs-3461 50 2 system system NOUN iajs-3461 50 3 ( ( PUNCT iajs-3461 50 4 2 2 X iajs-3461 50 5 ) ) PUNCT iajs-3461 50 6 there there PRON iajs-3461 50 7 are be VERB iajs-3461 50 8 8 8 NUM iajs-3461 50 9 parameters parameter NOUN iajs-3461 50 10 . . PUNCT iajs-3461 51 1 all all PRON iajs-3461 51 2 of of ADP iajs-3461 51 3 the the DET iajs-3461 51 4 functions function NOUN iajs-3461 51 5 on on ADP iajs-3461 51 6 system system NOUN iajs-3461 51 7 ( ( PUNCT iajs-3461 51 8 2 2 NUM iajs-3461 51 9 ) ) PUNCT iajs-3461 51 10 right right ADJ iajs-3461 51 11 side side NOUN iajs-3461 51 12 are be AUX iajs-3461 51 13 𝐢2 𝐢2 ADJ iajs-3461 51 14 ( ( PUNCT iajs-3461 51 15 ℝ3 ℝ3 PROPN iajs-3461 51 16 , , PUNCT iajs-3461 51 17 ℝ+ ℝ+ NOUN iajs-3461 51 18 ) ) PUNCT iajs-3461 51 19 . . PUNCT iajs-3461 52 1 𝑅+ 𝑅+ PROPN iajs-3461 52 2 3 3 NUM iajs-3461 52 3 = = SYM iajs-3461 52 4 { { PUNCT iajs-3461 52 5 ( ( PUNCT iajs-3461 52 6 β„Ž β„Ž PROPN iajs-3461 52 7 , , PUNCT iajs-3461 52 8 𝑖 𝑖 SYM iajs-3461 52 9 , , PUNCT iajs-3461 52 10 𝑗 𝑗 INTJ iajs-3461 52 11 ) ) PUNCT iajs-3461 52 12 ∈ ∈ PROPN iajs-3461 52 13 𝑅3 𝑅3 PROPN iajs-3461 52 14 ∢ ∢ PROPN iajs-3461 52 15 β„Ž β„Ž PROPN iajs-3461 52 16 ( ( PUNCT iajs-3461 52 17 0 0 NUM iajs-3461 52 18 ) ) PUNCT iajs-3461 52 19 β‰₯ β‰₯ NOUN iajs-3461 52 20 0 0 NUM iajs-3461 52 21 , , PUNCT iajs-3461 52 22 𝑖 𝑖 CCONJ iajs-3461 52 23 ( ( PUNCT iajs-3461 52 24 0 0 NUM iajs-3461 52 25 ) ) PUNCT iajs-3461 52 26 β‰₯ β‰₯ NOUN iajs-3461 52 27 0 0 NUM iajs-3461 52 28 , , PUNCT iajs-3461 52 29 𝑗 𝑗 PROPN iajs-3461 52 30 ( ( PUNCT iajs-3461 52 31 0 0 NUM iajs-3461 52 32 ) ) PUNCT iajs-3461 52 33 β‰₯ β‰₯ NOUN iajs-3461 52 34 0 0 NUM iajs-3461 52 35 } } PUNCT iajs-3461 52 36 . . PUNCT iajs-3461 53 1 therefore therefore ADV iajs-3461 53 2 , , PUNCT iajs-3461 53 3 these these DET iajs-3461 53 4 functions function NOUN iajs-3461 53 5 are be AUX iajs-3461 53 6 lipschitzian lipschitzian ADJ iajs-3461 53 7 on on ADP iajs-3461 53 8 𝑅+ 𝑅+ PROPN iajs-3461 53 9 3 3 NUM iajs-3461 53 10 , , PUNCT iajs-3461 53 11 and and CCONJ iajs-3461 53 12 as as ADP iajs-3461 53 13 a a DET iajs-3461 53 14 result result NOUN iajs-3461 53 15 , , PUNCT iajs-3461 53 16 system system NOUN iajs-3461 53 17 ( ( PUNCT iajs-3461 53 18 2 2 X iajs-3461 53 19 ) ) PUNCT iajs-3461 53 20 has have VERB iajs-3461 53 21 a a DET iajs-3461 53 22 unique unique ADJ iajs-3461 53 23 and and CCONJ iajs-3461 53 24 existing existing ADJ iajs-3461 53 25 solution solution NOUN iajs-3461 53 26 . . PUNCT iajs-3461 54 1 theorem theorem VERB iajs-3461 54 2 1 1 NUM iajs-3461 54 3 [ [ X iajs-3461 54 4 uniformly uniformly ADV iajs-3461 54 5 boundedness boundedness NOUN iajs-3461 54 6 ] ] X iajs-3461 54 7 : : PUNCT iajs-3461 54 8 5 5 NUM iajs-3461 54 9 all all DET iajs-3461 54 10 the the DET iajs-3461 54 11 solutions solution NOUN iajs-3461 54 12 of of ADP iajs-3461 54 13 system system NOUN iajs-3461 54 14 ( ( PUNCT iajs-3461 54 15 2 2 NUM iajs-3461 54 16 ) ) PUNCT iajs-3461 54 17 with with ADP iajs-3461 54 18 nonnegative nonnegative ADJ iajs-3461 54 19 initial initial ADJ iajs-3461 54 20 conditions condition NOUN iajs-3461 54 21 are be AUX iajs-3461 54 22 uniformly uniformly ADV iajs-3461 54 23 bounded bound VERB iajs-3461 54 24 . . PUNCT iajs-3461 55 1 proof proof NOUN iajs-3461 55 2 : : PUNCT iajs-3461 55 3 let let VERB iajs-3461 55 4 the the DET iajs-3461 55 5 solution solution NOUN iajs-3461 55 6 of of ADP iajs-3461 55 7 ( ( PUNCT iajs-3461 55 8 2 2 X iajs-3461 55 9 ) ) PUNCT iajs-3461 55 10 be be AUX iajs-3461 55 11 [ [ X iajs-3461 55 12 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-3461 55 13 ) ) PUNCT iajs-3461 55 14 , , PUNCT iajs-3461 55 15 𝑖(𝑑 𝑖(𝑑 PROPN iajs-3461 55 16 ) ) PUNCT iajs-3461 55 17 , , PUNCT iajs-3461 55 18 𝑗(𝑑 𝑗(𝑑 PROPN iajs-3461 55 19 ) ) PUNCT iajs-3461 55 20 ] ] PUNCT iajs-3461 56 1 the the DET iajs-3461 56 2 initial initial ADJ iajs-3461 56 3 condition condition NOUN iajs-3461 56 4 [ [ X iajs-3461 56 5 β„Ž(0 β„Ž(0 NOUN iajs-3461 56 6 ) ) PUNCT iajs-3461 56 7 , , PUNCT iajs-3461 56 8 𝑖(0 𝑖(0 PROPN iajs-3461 56 9 ) ) PUNCT iajs-3461 56 10 , , PUNCT iajs-3461 56 11 𝑗(0 𝑗(0 PROPN iajs-3461 56 12 ) ) PUNCT iajs-3461 56 13 ] ] PUNCT iajs-3461 56 14 ∈ ∈ PROPN iajs-3461 56 15 𝑅+ 𝑅+ NUM iajs-3461 56 16 3 3 NUM iajs-3461 56 17 are be AUX iajs-3461 56 18 nonnegative nonnegative ADJ iajs-3461 56 19 . . PUNCT iajs-3461 57 1 now now ADV iajs-3461 57 2 , , PUNCT iajs-3461 57 3 let let VERB iajs-3461 57 4 hΜ‡(𝑑 hΜ‡(𝑑 NOUN iajs-3461 57 5 ) ) PUNCT iajs-3461 57 6 = = SYM iajs-3461 57 7 β„Ž(𝑑 β„Ž(𝑑 PROPN iajs-3461 57 8 ) ) PUNCT iajs-3461 57 9 + + CCONJ iajs-3461 57 10 𝑖(𝑑 𝑖(𝑑 NUM iajs-3461 57 11 ) ) PUNCT iajs-3461 58 1 + + NUM iajs-3461 58 2 𝑗(𝑑 𝑗(𝑑 PROPN iajs-3461 58 3 ) ) PUNCT iajs-3461 58 4 , , PUNCT iajs-3461 58 5 𝑑hΜ‡ 𝑑hΜ‡ PROPN iajs-3461 58 6 𝑑𝑑 𝑑𝑑 ADP iajs-3461 58 7 < < X iajs-3461 58 8 2β„Ž 2β„Ž NUM iajs-3461 58 9 βˆ’ βˆ’ PROPN iajs-3461 58 10 ( ( PUNCT iajs-3461 58 11 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 58 12 βˆ’ βˆ’ NUM iajs-3461 58 13 π‘Ÿ3)β„Žπ‘– π‘Ÿ3)β„Žπ‘– NUM iajs-3461 59 1 βˆ’ βˆ’ PROPN iajs-3461 59 2 ( ( PUNCT iajs-3461 59 3 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 59 4 βˆ’ βˆ’ PROPN iajs-3461 59 5 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 59 6 ) ) PUNCT iajs-3461 60 1 β„Žπ‘— β„Žπ‘— PRON iajs-3461 60 2 βˆ’ βˆ’ PROPN iajs-3461 60 3 ( ( PUNCT iajs-3461 60 4 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 60 5 βˆ’ βˆ’ PROPN iajs-3461 60 6 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 60 7 ) ) PUNCT iajs-3461 60 8 𝑖𝑗 𝑖𝑗 ADP iajs-3461 60 9 βˆ’ βˆ’ PROPN iajs-3461 60 10 β„Ž β„Ž PROPN iajs-3461 60 11 βˆ’ βˆ’ PROPN iajs-3461 60 12 π‘Ÿ5𝑖 π‘Ÿ5𝑖 NOUN iajs-3461 60 13 βˆ’ βˆ’ NOUN iajs-3461 60 14 π‘Ÿ8 π‘Ÿ8 ADV iajs-3461 60 15 𝑗. 𝑗. NOUN iajs-3461 60 16 now now ADV iajs-3461 60 17 , , PUNCT iajs-3461 60 18 ecologically ecologically ADV iajs-3461 60 19 r3 r3 PROPN iajs-3461 60 20 < < X iajs-3461 60 21 r1 r1 PROPN iajs-3461 60 22 , , PUNCT iajs-3461 60 23 r6 r6 NOUN iajs-3461 60 24 < < X iajs-3461 60 25 r2 r2 PROPN iajs-3461 60 26 and and CCONJ iajs-3461 60 27 r7 r7 NOUN iajs-3461 60 28 < < X iajs-3461 60 29 r4 r4 PROPN iajs-3461 60 30 . . PUNCT iajs-3461 61 1 𝑑hΜ‡ 𝑑hΜ‡ VERB iajs-3461 61 2 𝑑𝑑 𝑑𝑑 ADP iajs-3461 61 3 < < X iajs-3461 61 4 2 2 NUM iajs-3461 61 5 βˆ’ βˆ’ PROPN iajs-3461 61 6 Ξ΄ Ξ΄ NOUN iajs-3461 61 7 hΜ‡ hΜ‡ NOUN iajs-3461 61 8 , , PUNCT iajs-3461 61 9 π‘€β„Žπ‘’π‘Ÿπ‘’ π‘€β„Žπ‘’π‘Ÿπ‘’ ADJ iajs-3461 61 10 𝛿 𝛿 ADJ iajs-3461 61 11 = = X iajs-3461 61 12 π‘šπ‘–π‘› π‘šπ‘–π‘› NOUN iajs-3461 61 13 { { PUNCT iajs-3461 61 14 1 1 NUM iajs-3461 61 15 , , PUNCT iajs-3461 61 16 r5 r5 PROPN iajs-3461 61 17 , , PUNCT iajs-3461 61 18 r8 r8 PROPN iajs-3461 61 19 } } PUNCT iajs-3461 61 20 . . PUNCT iajs-3461 62 1 now now ADV iajs-3461 62 2 , , PUNCT iajs-3461 62 3 by by ADP iajs-3461 62 4 solving solve VERB iajs-3461 62 5 this this DET iajs-3461 62 6 differential differential ADJ iajs-3461 62 7 inequality inequality NOUN iajs-3461 62 8 for for ADP iajs-3461 62 9 the the DET iajs-3461 62 10 initial initial ADJ iajs-3461 62 11 value value NOUN iajs-3461 62 12 h h NOUN iajs-3461 62 13 ( ( PUNCT iajs-3461 62 14 0 0 NUM iajs-3461 62 15 ) ) PUNCT iajs-3461 62 16 = = VERB iajs-3461 62 17 𝐻0 𝐻0 PROPN iajs-3461 62 18 , , PUNCT iajs-3461 62 19 we we PRON iajs-3461 62 20 get get VERB iajs-3461 62 21 that that PRON iajs-3461 62 22 : : PUNCT iajs-3461 62 23 hΜ‡(t hΜ‡(t X iajs-3461 62 24 ) ) PUNCT iajs-3461 62 25 ≀ ≀ NOUN iajs-3461 62 26 2 2 NUM iajs-3461 62 27 Ξ΄ Ξ΄ NOUN iajs-3461 62 28 + + CCONJ iajs-3461 62 29 ( ( PUNCT iajs-3461 62 30 hΜ‡(0 hΜ‡(0 X iajs-3461 62 31 ) ) PUNCT iajs-3461 63 1 βˆ’ βˆ’ PROPN iajs-3461 63 2 2 2 NUM iajs-3461 63 3 Ξ΄ Ξ΄ NOUN iajs-3461 63 4 ) ) PUNCT iajs-3461 64 1 eβˆ’Ξ΄ eβˆ’Ξ΄ ADP iajs-3461 64 2 t t NOUN iajs-3461 64 3 thus thus ADV iajs-3461 64 4 0 0 NUM iajs-3461 64 5 ≀ ≀ NUM iajs-3461 64 6 hΜ‡(𝑑 hΜ‡(𝑑 NOUN iajs-3461 64 7 ) ) PUNCT iajs-3461 64 8 ≀ ≀ NUM iajs-3461 64 9 2 2 NUM iajs-3461 64 10 Ξ΄ Ξ΄ NOUN iajs-3461 64 11 as as ADP iajs-3461 64 12 𝑑 𝑑 PROPN iajs-3461 64 13 β†’ β†’ SYM iajs-3461 64 14 ∞. ∞. PROPN iajs-3461 64 15 hence hence ADV iajs-3461 64 16 system system NOUN iajs-3461 64 17 ( ( PUNCT iajs-3461 64 18 2 2 X iajs-3461 64 19 ) ) PUNCT iajs-3461 64 20 has have AUX iajs-3461 64 21 uniformly uniformly ADV iajs-3461 64 22 bounded bound VERB iajs-3461 64 23 . . PUNCT iajs-3461 65 1 3 3 X iajs-3461 65 2 . . X iajs-3461 65 3 equilibrium equilibrium NOUN iajs-3461 65 4 points point NOUN iajs-3461 65 5 are be AUX iajs-3461 65 6 existence existence NOUN iajs-3461 65 7 and and CCONJ iajs-3461 65 8 stable stable ADJ iajs-3461 65 9 : : PUNCT iajs-3461 65 10 there there PRON iajs-3461 65 11 are be VERB iajs-3461 65 12 maximum maximum ADJ iajs-3461 65 13 of of ADP iajs-3461 65 14 five five NUM iajs-3461 65 15 equilibrium equilibrium NOUN iajs-3461 65 16 points point NOUN iajs-3461 65 17 in in ADP iajs-3461 65 18 system system NOUN iajs-3461 65 19 ( ( PUNCT iajs-3461 65 20 2 2 NUM iajs-3461 65 21 ) ) PUNCT iajs-3461 65 22 , , PUNCT iajs-3461 65 23 which which PRON iajs-3461 65 24 are be AUX iajs-3461 65 25 listed list VERB iajs-3461 65 26 below below ADP iajs-3461 65 27 : : PUNCT iajs-3461 65 28 β¦Ώ β¦Ώ PUNCT iajs-3461 65 29 the the DET iajs-3461 65 30 point point NOUN iajs-3461 65 31 of of ADP iajs-3461 65 32 equilibrium equilibrium NOUN iajs-3461 65 33 𝐴0 𝐴0 PROPN iajs-3461 66 1 = = PUNCT iajs-3461 66 2 ( ( PUNCT iajs-3461 66 3 0 0 NUM iajs-3461 66 4 , , PUNCT iajs-3461 66 5 0 0 NUM iajs-3461 66 6 , , PUNCT iajs-3461 66 7 0 0 NUM iajs-3461 66 8 ) ) PUNCT iajs-3461 66 9 , , PUNCT iajs-3461 66 10 it it PRON iajs-3461 66 11 is be AUX iajs-3461 66 12 always always ADV iajs-3461 66 13 present present ADJ iajs-3461 66 14 it it PRON iajs-3461 66 15 is be AUX iajs-3461 66 16 referred refer VERB iajs-3461 66 17 to to ADP iajs-3461 66 18 as as ADP iajs-3461 66 19 the the DET iajs-3461 66 20 vanishing vanishing NOUN iajs-3461 66 21 point point NOUN iajs-3461 66 22 , , PUNCT iajs-3461 66 23 which which PRON iajs-3461 66 24 is be AUX iajs-3461 66 25 unstable unstable ADJ iajs-3461 66 26 and and CCONJ iajs-3461 66 27 always always ADV iajs-3461 66 28 existing exist VERB iajs-3461 66 29 . . PUNCT iajs-3461 67 1 β¦Ώ β¦Ώ PROPN iajs-3461 67 2 the the DET iajs-3461 67 3 axial axial ADJ iajs-3461 67 4 equilibrium5 equilibrium5 NOUN iajs-3461 67 5 point point NOUN iajs-3461 67 6 𝐴1 𝐴1 PROPN iajs-3461 68 1 = = PUNCT iajs-3461 68 2 ( ( PUNCT iajs-3461 68 3 1 1 NUM iajs-3461 68 4 , , PUNCT iajs-3461 68 5 0 0 NUM iajs-3461 68 6 , , PUNCT iajs-3461 68 7 0 0 NUM iajs-3461 68 8 ) ) PUNCT iajs-3461 68 9 , , PUNCT iajs-3461 68 10 existence existence NOUN iajs-3461 68 11 without without ADP iajs-3461 68 12 conditions condition NOUN iajs-3461 68 13 additionally additionally ADV iajs-3461 68 14 . . PUNCT iajs-3461 69 1 ihjpas ihjpas PROPN iajs-3461 69 2 . . PUNCT iajs-3461 70 1 2025 2025 NUM iajs-3461 70 2 , , PUNCT iajs-3461 70 3 38 38 NUM iajs-3461 70 4 ( ( PUNCT iajs-3461 70 5 1 1 NUM iajs-3461 70 6 ) ) PUNCT iajs-3461 70 7 370 370 NUM iajs-3461 70 8 therefore therefore ADV iajs-3461 70 9 , , PUNCT iajs-3461 70 10 the the DET iajs-3461 70 11 characteristic characteristic ADJ iajs-3461 70 12 equation equation NOUN iajs-3461 70 13 of of ADP iajs-3461 70 14 j j PROPN iajs-3461 70 15 ( ( PUNCT iajs-3461 70 16 𝐴1 𝐴1 PROPN iajs-3461 70 17 ) ) PUNCT iajs-3461 70 18 is be AUX iajs-3461 70 19 as as SCONJ iajs-3461 70 20 follows follow VERB iajs-3461 70 21 : : PUNCT iajs-3461 70 22 𝐽1 𝐽1 PROPN iajs-3461 70 23 = = SYM iajs-3461 70 24 𝐽(𝐴1 𝐽(𝐴1 NOUN iajs-3461 70 25 ) ) PUNCT iajs-3461 70 26 = = PUNCT iajs-3461 70 27 [ [ PUNCT iajs-3461 70 28 βˆ’1 βˆ’1 NOUN iajs-3461 70 29 βˆ’ βˆ’ PROPN iajs-3461 70 30 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 70 31 βˆ’ βˆ’ NOUN iajs-3461 70 32 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 70 33 0 0 NUM iajs-3461 70 34 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 70 35 βˆ’ βˆ’ NOUN iajs-3461 70 36 π‘Ÿ5 π‘Ÿ5 ADV iajs-3461 70 37 0 0 NUM iajs-3461 70 38 0 0 NUM iajs-3461 70 39 0 0 NUM iajs-3461 71 1 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 71 2 βˆ’ βˆ’ NOUN iajs-3461 71 3 π‘Ÿ8 π‘Ÿ8 ADV iajs-3461 71 4 ] ] PUNCT iajs-3461 71 5 . . PUNCT iajs-3461 72 1 ( ( PUNCT iajs-3461 72 2 1 1 X iajs-3461 72 3 . . X iajs-3461 72 4 π‘Ž π‘Ž X iajs-3461 72 5 ) ) PUNCT iajs-3461 72 6 ( ( PUNCT iajs-3461 72 7 βˆ’1 βˆ’1 NOUN iajs-3461 72 8 βˆ’ βˆ’ NOUN iajs-3461 72 9 πœ† πœ† NOUN iajs-3461 72 10 ) ) PUNCT iajs-3461 73 1 [ [ X iajs-3461 73 2 ( ( PUNCT iajs-3461 73 3 r3 r3 PROPN iajs-3461 73 4 βˆ’ βˆ’ PROPN iajs-3461 73 5 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 73 6 ) ) PUNCT iajs-3461 73 7 βˆ’ βˆ’ PROPN iajs-3461 74 1 πœ† πœ† X iajs-3461 74 2 ] ] X iajs-3461 74 3 [ [ X iajs-3461 74 4 ( ( PUNCT iajs-3461 74 5 r6 r6 NOUN iajs-3461 74 6 βˆ’ βˆ’ PROPN iajs-3461 74 7 π‘Ÿ8 π‘Ÿ8 PROPN iajs-3461 74 8 ) ) PUNCT iajs-3461 74 9 βˆ’ βˆ’ PUNCT iajs-3461 74 10 πœ† πœ† X iajs-3461 74 11 ] ] X iajs-3461 74 12 = = SYM iajs-3461 74 13 0 0 X iajs-3461 74 14 . . NOUN iajs-3461 74 15 which which PRON iajs-3461 74 16 gives give VERB iajs-3461 74 17 the the DET iajs-3461 74 18 eigenvalues eigenvalue NOUN iajs-3461 74 19 of of ADP iajs-3461 74 20 𝐽1 𝐽1 NOUN iajs-3461 74 21 𝑏𝑦 𝑏𝑦 PROPN iajs-3461 74 22 : : PUNCT iajs-3461 74 23 πœ†1β„Ž πœ†1β„Ž PUNCT iajs-3461 74 24 = = PUNCT iajs-3461 74 25 βˆ’1 βˆ’1 NOUN iajs-3461 74 26 < < X iajs-3461 74 27 0 0 PROPN iajs-3461 74 28 , , PUNCT iajs-3461 74 29 πœ†1𝑖 πœ†1𝑖 NOUN iajs-3461 74 30 = = SYM iajs-3461 74 31 r3 r3 PROPN iajs-3461 75 1 βˆ’ βˆ’ NOUN iajs-3461 75 2 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 75 3 < < X iajs-3461 75 4 0 0 PUNCT iajs-3461 75 5 and and CCONJ iajs-3461 75 6 πœ†1𝑗 πœ†1𝑗 PROPN iajs-3461 75 7 = = SYM iajs-3461 75 8 ( ( PUNCT iajs-3461 75 9 r6 r6 NOUN iajs-3461 75 10 βˆ’ βˆ’ PROPN iajs-3461 75 11 π‘Ÿ8 π‘Ÿ8 PROPN iajs-3461 75 12 ) ) PUNCT iajs-3461 75 13 < < X iajs-3461 75 14 0 0 NUM iajs-3461 76 1 the the DET iajs-3461 76 2 equilibrium equilibrium NOUN iajs-3461 76 3 point point NOUN iajs-3461 76 4 𝐴1 𝐴1 PROPN iajs-3461 76 5 then then ADV iajs-3461 76 6 becomes become VERB iajs-3461 76 7 asymptotically asymptotically ADV iajs-3461 76 8 stable stable ADJ iajs-3461 76 9 under under ADP iajs-3461 76 10 the the DET iajs-3461 76 11 following following ADJ iajs-3461 76 12 conditions condition NOUN iajs-3461 76 13 : : PUNCT iajs-3461 76 14 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 76 15 > > X iajs-3461 76 16 π‘Ÿ5 π‘Ÿ5 INTJ iajs-3461 76 17 , , PUNCT iajs-3461 76 18 ( ( PUNCT iajs-3461 76 19 3 3 X iajs-3461 76 20 ) ) PUNCT iajs-3461 76 21 π‘Ÿ8 π‘Ÿ8 ADV iajs-3461 76 22 > > X iajs-3461 76 23 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 76 24 . . PUNCT iajs-3461 77 1 ( ( PUNCT iajs-3461 77 2 4 4 X iajs-3461 77 3 ) ) PUNCT iajs-3461 77 4 otherwise otherwise ADV iajs-3461 77 5 , , PUNCT iajs-3461 77 6 𝐴1 𝐴1 PROPN iajs-3461 77 7 is be AUX iajs-3461 77 8 unstable unstable ADJ iajs-3461 77 9 . . PUNCT iajs-3461 78 1 however however ADV iajs-3461 78 2 , , PUNCT iajs-3461 78 3 it it PRON iajs-3461 78 4 is be AUX iajs-3461 78 5 a a DET iajs-3461 78 6 saddle saddle NOUN iajs-3461 78 7 point point NOUN iajs-3461 78 8 . . PUNCT iajs-3461 79 1 β¦Ώ β¦Ώ VERB iajs-3461 79 2 the the DET iajs-3461 79 3 equilibrium7 equilibrium7 NOUN iajs-3461 79 4 point point NOUN iajs-3461 79 5 𝐴2 𝐴2 PROPN iajs-3461 79 6 ( ( PUNCT iajs-3461 79 7 β„Ž β„Ž X iajs-3461 79 8 βˆ’ βˆ’ PROPN iajs-3461 79 9 , , PUNCT iajs-3461 79 10 𝑖 𝑖 PROPN iajs-3461 79 11 βˆ’ βˆ’ NOUN iajs-3461 79 12 , , PUNCT iajs-3461 79 13 0 0 NUM iajs-3461 79 14 ) ) PUNCT iajs-3461 80 1 exists7 exists7 PROPN iajs-3461 80 2 uniquely7 uniquely7 NOUN iajs-3461 81 1 in in ADP iajs-3461 81 2 7𝐼𝑛𝑑. 7𝐼𝑛𝑑. NUM iajs-3461 81 3 𝑅+ 𝑅+ SYM iajs-3461 81 4 2 2 NUM iajs-3461 81 5 ( ( PUNCT iajs-3461 81 6 interior interior NOUN iajs-3461 81 7 of of ADP iajs-3461 81 8 𝑅+ 𝑅+ PROPN iajs-3461 81 9 2 2 NUM iajs-3461 81 10 ) ) PUNCT iajs-3461 81 11 of of ADP iajs-3461 81 12 β„Ži β„Ži ADJ iajs-3461 81 13 βˆ’ βˆ’ PROPN iajs-3461 81 14 plane plane NOUN iajs-3461 81 15 provided provide VERB iajs-3461 81 16 that that PRON iajs-3461 81 17 : : PUNCT iajs-3461 81 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 81 19 > > X iajs-3461 81 20 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 81 21 . . PUNCT iajs-3461 82 1 ( ( PUNCT iajs-3461 82 2 5 5 NUM iajs-3461 82 3 ) ) PUNCT iajs-3461 82 4 where where SCONJ iajs-3461 82 5 : : PUNCT iajs-3461 83 1 β„Ž β„Ž X iajs-3461 83 2 βˆ’ βˆ’ NOUN iajs-3461 83 3 = = SYM iajs-3461 84 1 π‘Ÿ5 π‘Ÿ5 ADJ iajs-3461 84 2 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 84 3 > > X iajs-3461 84 4 0 0 X iajs-3461 84 5 . . PUNCT iajs-3461 85 1 ( ( PUNCT iajs-3461 85 2 6 6 NUM iajs-3461 85 3 ) ) PUNCT iajs-3461 85 4 𝑖 𝑖 NOUN iajs-3461 85 5 βˆ’ βˆ’ NOUN iajs-3461 86 1 = = NOUN iajs-3461 86 2 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 86 3 βˆ’ βˆ’ NOUN iajs-3461 86 4 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 86 5 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 86 6 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 86 7 . . PUNCT iajs-3461 87 1 ( ( PUNCT iajs-3461 87 2 7 7 NUM iajs-3461 87 3 ) ) PUNCT iajs-3461 87 4 and and CCONJ iajs-3461 87 5 the the DET iajs-3461 87 6 jacobian jacobian ADJ iajs-3461 87 7 matrix matrix NOUN iajs-3461 87 8 of of ADP iajs-3461 87 9 system system NOUN iajs-3461 87 10 ( ( PUNCT iajs-3461 87 11 2 2 NUM iajs-3461 87 12 ) ) PUNCT iajs-3461 87 13 at at ADP iajs-3461 87 14 𝐴2 𝐴2 PROPN iajs-3461 87 15 can can AUX iajs-3461 87 16 be be AUX iajs-3461 87 17 written write VERB iajs-3461 87 18 as as ADP iajs-3461 87 19 𝐽2 𝐽2 NOUN iajs-3461 87 20 = = SYM iajs-3461 87 21 𝐽(𝐴2 𝐽(𝐴2 NOUN iajs-3461 87 22 ) ) PUNCT iajs-3461 88 1 = = PUNCT iajs-3461 88 2 [ [ PUNCT iajs-3461 88 3 πœ‡π‘–π‘— πœ‡π‘–π‘— NOUN iajs-3461 88 4 ] ] X iajs-3461 88 5 3Γ—3 3Γ—3 NUM iajs-3461 88 6 , , PUNCT iajs-3461 88 7 ( ( PUNCT iajs-3461 88 8 2 2 X iajs-3461 88 9 . . X iajs-3461 88 10 π‘Ž π‘Ž X iajs-3461 88 11 ) ) PUNCT iajs-3461 88 12 where where SCONJ iajs-3461 88 13 : : PUNCT iajs-3461 88 14 πœ‡11 πœ‡11 NOUN iajs-3461 88 15 = = SYM iajs-3461 88 16 1 1 NUM iajs-3461 88 17 βˆ’ βˆ’ NUM iajs-3461 88 18 2β„Ž 2β„Ž NUM iajs-3461 88 19 βˆ’ βˆ’ PROPN iajs-3461 88 20 βˆ’ βˆ’ PROPN iajs-3461 88 21 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 88 22 𝑖 𝑖 NOUN iajs-3461 88 23 βˆ’ βˆ’ PROPN iajs-3461 88 24 , , PUNCT iajs-3461 88 25 πœ‡12 πœ‡12 NOUN iajs-3461 88 26 = = PUNCT iajs-3461 88 27 βˆ’π‘Ÿ1β„Ž βˆ’π‘Ÿ1β„Ž PROPN iajs-3461 89 1 βˆ’ βˆ’ NOUN iajs-3461 89 2 < < X iajs-3461 89 3 0 0 PROPN iajs-3461 89 4 , , PUNCT iajs-3461 89 5 πœ‡13 πœ‡13 NOUN iajs-3461 89 6 = = SYM iajs-3461 89 7 βˆ’π‘Ÿ2β„Ž βˆ’π‘Ÿ2β„Ž NUM iajs-3461 89 8 βˆ’ βˆ’ NUM iajs-3461 89 9 , , PUNCT iajs-3461 89 10 πœ‡21 πœ‡21 ADJ iajs-3461 89 11 = = NOUN iajs-3461 89 12 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 89 13 𝑖 𝑖 NOUN iajs-3461 90 1 βˆ’ βˆ’ PROPN iajs-3461 90 2 > > X iajs-3461 90 3 0 0 NUM iajs-3461 90 4 , , PUNCT iajs-3461 90 5 πœ‡22 πœ‡22 NOUN iajs-3461 90 6 = = SYM iajs-3461 90 7 π‘Ÿ3β„Ž π‘Ÿ3β„Ž PROPN iajs-3461 90 8 βˆ’ βˆ’ PROPN iajs-3461 90 9 βˆ’π‘Ÿ5 βˆ’π‘Ÿ5 PROPN iajs-3461 90 10 , , PUNCT iajs-3461 90 11 πœ‡23 πœ‡23 NOUN iajs-3461 90 12 = = PUNCT iajs-3461 91 1 βˆ’π‘Ÿ4 βˆ’π‘Ÿ4 ADV iajs-3461 91 2 𝑖 𝑖 X iajs-3461 91 3 βˆ’ βˆ’ PROPN iajs-3461 91 4 , , PUNCT iajs-3461 91 5 , , PUNCT iajs-3461 91 6 πœ‡31 πœ‡31 PROPN iajs-3461 91 7 = = SYM iajs-3461 91 8 0 0 NUM iajs-3461 91 9 , , PUNCT iajs-3461 91 10 πœ‡32 πœ‡32 NOUN iajs-3461 91 11 = = SYM iajs-3461 91 12 0 0 NUM iajs-3461 91 13 , , PUNCT iajs-3461 91 14 πœ‡33 πœ‡33 PROPN iajs-3461 91 15 = = SYM iajs-3461 91 16 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 91 17 βˆ’ βˆ’ PROPN iajs-3461 92 1 + + CCONJ iajs-3461 92 2 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 92 3 𝑖 𝑖 NOUN iajs-3461 92 4 βˆ’ βˆ’ NOUN iajs-3461 92 5 βˆ’ βˆ’ NOUN iajs-3461 92 6 π‘Ÿ8 π‘Ÿ8 PROPN iajs-3461 92 7 . . PUNCT iajs-3461 93 1 consequently consequently ADV iajs-3461 93 2 , , PUNCT iajs-3461 93 3 the the DET iajs-3461 93 4 characteristic characteristic ADJ iajs-3461 93 5 equation equation NOUN iajs-3461 93 6 of of ADP iajs-3461 93 7 j j PROPN iajs-3461 93 8 ( ( PUNCT iajs-3461 93 9 𝐴2 𝐴2 PROPN iajs-3461 93 10 ) ) PUNCT iajs-3461 93 11 is be AUX iajs-3461 93 12 as as SCONJ iajs-3461 93 13 follows follow VERB iajs-3461 93 14 : : PUNCT iajs-3461 93 15 ( ( PUNCT iajs-3461 93 16 πœ‡33 πœ‡33 PROPN iajs-3461 93 17 -Ξ» -Ξ» NUM iajs-3461 93 18 ) ) PUNCT iajs-3461 93 19 [ [ PUNCT iajs-3461 93 20 πœ†2 πœ†2 NOUN iajs-3461 93 21 βˆ’ βˆ’ X iajs-3461 93 22 Θƒ)Ξ» Θƒ)Ξ» PUNCT iajs-3461 94 1 + + CCONJ iajs-3461 94 2 det det X iajs-3461 94 3 ( ( PUNCT iajs-3461 94 4 Θƒ Θƒ PROPN iajs-3461 94 5 ) ) PUNCT iajs-3461 94 6 ] ] PUNCT iajs-3461 95 1 = = PUNCT iajs-3461 95 2 0 0 NUM iajs-3461 95 3 where where SCONJ iajs-3461 95 4 ∢ ∢ NOUN iajs-3461 95 5 Θƒ Θƒ X iajs-3461 95 6 = = X iajs-3461 95 7 [ [ PUNCT iajs-3461 95 8 1 1 NUM iajs-3461 95 9 βˆ’ βˆ’ NUM iajs-3461 95 10 2β„Ž 2β„Ž NUM iajs-3461 95 11 βˆ’ βˆ’ PROPN iajs-3461 95 12 βˆ’ βˆ’ PROPN iajs-3461 95 13 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 95 14 𝑖 𝑖 SYM iajs-3461 95 15 βˆ’ βˆ’ PROPN iajs-3461 95 16 βˆ’π‘Ÿ1β„Ž βˆ’π‘Ÿ1β„Ž PUNCT iajs-3461 95 17 βˆ’ βˆ’ PROPN iajs-3461 95 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 95 19 𝑖 𝑖 NUM iajs-3461 95 20 βˆ’ βˆ’ PROPN iajs-3461 95 21 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 95 22 βˆ’ βˆ’ PROPN iajs-3461 95 23 βˆ’π‘Ÿ5 βˆ’π‘Ÿ5 PROPN iajs-3461 95 24 ] ] PUNCT iajs-3461 95 25 , , PUNCT iajs-3461 95 26 then then ADV iajs-3461 95 27 [ [ X iajs-3461 95 28 ( ( PUNCT iajs-3461 95 29 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 95 30 βˆ’ βˆ’ NOUN iajs-3461 95 31 + + CCONJ iajs-3461 95 32 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 95 33 𝑖 𝑖 NOUN iajs-3461 95 34 βˆ’ βˆ’ NOUN iajs-3461 95 35 βˆ’ βˆ’ PROPN iajs-3461 95 36 π‘Ÿ8 π‘Ÿ8 PROPN iajs-3461 95 37 ) ) PUNCT iajs-3461 95 38 βˆ’ βˆ’ ADP iajs-3461 96 1 Ξ» Ξ» SYM iajs-3461 96 2 ] ] X iajs-3461 97 1 [ [ X iajs-3461 97 2 ( ( PUNCT iajs-3461 97 3 1 1 NUM iajs-3461 97 4 βˆ’ βˆ’ NUM iajs-3461 97 5 2β„Ž 2β„Ž NUM iajs-3461 97 6 βˆ’ βˆ’ PROPN iajs-3461 97 7 βˆ’ βˆ’ PROPN iajs-3461 97 8 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 97 9 𝑖 𝑖 NOUN iajs-3461 97 10 βˆ’ βˆ’ NOUN iajs-3461 97 11 βˆ’ βˆ’ PROPN iajs-3461 97 12 Ξ»)(π‘Ÿ3β„Ž Ξ»)(π‘Ÿ3β„Ž NUM iajs-3461 97 13 βˆ’ βˆ’ PROPN iajs-3461 97 14 βˆ’π‘Ÿ5 βˆ’π‘Ÿ5 PROPN iajs-3461 97 15 βˆ’ βˆ’ PROPN iajs-3461 97 16 Ξ» Ξ» NOUN iajs-3461 97 17 ) ) PUNCT iajs-3461 98 1 + + CCONJ iajs-3461 98 2 π‘Ÿ1π‘Ÿ3 π‘Ÿ1π‘Ÿ3 ADP iajs-3461 98 3 β„Ž β„Ž X iajs-3461 98 4 βˆ’ βˆ’ VERB iajs-3461 98 5 𝑖 𝑖 SYM iajs-3461 98 6 βˆ’ βˆ’ PROPN iajs-3461 98 7 ] ] PUNCT iajs-3461 99 1 = = SYM iajs-3461 99 2 0 0 PUNCT iajs-3461 99 3 either either ADV iajs-3461 99 4 , , PUNCT iajs-3461 99 5 πœ†2β„Ž πœ†2β„Ž PUNCT iajs-3461 99 6 = = PUNCT iajs-3461 99 7 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 99 8 βˆ’ βˆ’ PROPN iajs-3461 100 1 + + CCONJ iajs-3461 100 2 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 100 3 𝑖 𝑖 NOUN iajs-3461 100 4 βˆ’ βˆ’ NOUN iajs-3461 100 5 βˆ’ βˆ’ NUM iajs-3461 100 6 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 100 7 which which PRON iajs-3461 100 8 , , PUNCT iajs-3461 100 9 as as ADP iajs-3461 100 10 a a DET iajs-3461 100 11 result result NOUN iajs-3461 100 12 of of ADP iajs-3461 100 13 the the DET iajs-3461 100 14 following follow VERB iajs-3461 100 15 condition condition NOUN iajs-3461 100 16 , , PUNCT iajs-3461 100 17 produces produce VERB iajs-3461 100 18 the the DET iajs-3461 100 19 first first ADJ iajs-3461 100 20 eigenvalues eigenvalue NOUN iajs-3461 100 21 𝐽2with 𝐽2with VERB iajs-3461 100 22 negative negative ADJ iajs-3461 100 23 real real ADJ iajs-3461 100 24 parts part NOUN iajs-3461 100 25 : : PUNCT iajs-3461 100 26 π‘Ÿ8 π‘Ÿ8 ADV iajs-3461 100 27 > > X iajs-3461 100 28 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 100 29 βˆ’ βˆ’ PROPN iajs-3461 101 1 + + CCONJ iajs-3461 101 2 π‘Ÿ7 π‘Ÿ7 PROPN iajs-3461 101 3 ( ( PUNCT iajs-3461 101 4 8) 8) NUM iajs-3461 101 5 or or CCONJ iajs-3461 101 6 , , PUNCT iajs-3461 101 7 πœ†2 πœ†2 PROPN iajs-3461 101 8 βˆ’ βˆ’ PROPN iajs-3461 102 1 tr(Δ…)Ξ» tr(Δ…)Ξ» PROPN iajs-3461 102 2 + + NUM iajs-3461 102 3 det det PROPN iajs-3461 102 4 ( ( PUNCT iajs-3461 102 5 Δ… Δ… PROPN iajs-3461 102 6 ) ) PUNCT iajs-3461 103 1 = = PUNCT iajs-3461 103 2 0 0 NUM iajs-3461 104 1 where where SCONJ iajs-3461 104 2 : : PUNCT iajs-3461 104 3 tr tr ADV iajs-3461 104 4 ( ( PUNCT iajs-3461 104 5 Θƒ Θƒ X iajs-3461 104 6 ) ) PUNCT iajs-3461 104 7 = = SYM iajs-3461 104 8 πœ†2𝑖 πœ†2𝑖 PUNCT iajs-3461 104 9 + + PUNCT iajs-3461 104 10 πœ†2𝑗 πœ†2𝑗 NOUN iajs-3461 104 11 = = SYM iajs-3461 104 12 1 1 NUM iajs-3461 104 13 βˆ’ βˆ’ NUM iajs-3461 104 14 2β„Ž 2β„Ž NUM iajs-3461 104 15 βˆ’ βˆ’ PROPN iajs-3461 104 16 βˆ’ βˆ’ PROPN iajs-3461 104 17 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 104 18 𝑖 𝑖 ADP iajs-3461 104 19 βˆ’ βˆ’ NOUN iajs-3461 105 1 + + PUNCT iajs-3461 105 2 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 105 3 βˆ’ βˆ’ NOUN iajs-3461 105 4 βˆ’π‘Ÿ5 βˆ’π‘Ÿ5 PROPN iajs-3461 105 5 = = SYM iajs-3461 105 6 ( ( PUNCT iajs-3461 105 7 1 1 NUM iajs-3461 105 8 + + NUM iajs-3461 105 9 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 105 10 βˆ’ βˆ’ NUM iajs-3461 105 11 ) ) PUNCT iajs-3461 105 12 βˆ’ βˆ’ PROPN iajs-3461 106 1 ( ( PUNCT iajs-3461 106 2 2β„Ž 2β„Ž NOUN iajs-3461 106 3 βˆ’ βˆ’ PROPN iajs-3461 107 1 + + CCONJ iajs-3461 107 2 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 107 3 𝑖 𝑖 NOUN iajs-3461 108 1 βˆ’ βˆ’ PROPN iajs-3461 108 2 + + ADV iajs-3461 108 3 π‘Ÿ5 π‘Ÿ5 ADJ iajs-3461 108 4 ) ) PUNCT iajs-3461 108 5 > > X iajs-3461 108 6 0 0 NUM iajs-3461 108 7 , , PUNCT iajs-3461 108 8 ihjpas ihjpa NOUN iajs-3461 108 9 . . PUNCT iajs-3461 109 1 2025 2025 NUM iajs-3461 109 2 , , PUNCT iajs-3461 109 3 38 38 NUM iajs-3461 109 4 ( ( PUNCT iajs-3461 109 5 1 1 NUM iajs-3461 109 6 ) ) PUNCT iajs-3461 109 7 371 371 NUM iajs-3461 109 8 det det NOUN iajs-3461 109 9 ( ( PUNCT iajs-3461 109 10 Θƒ Θƒ PROPN iajs-3461 109 11 ) ) PUNCT iajs-3461 109 12 = = SYM iajs-3461 109 13 πœ†2π‘–πœ†2𝑗 πœ†2π‘–πœ†2𝑗 NOUN iajs-3461 109 14 = = SYM iajs-3461 109 15 ( ( PUNCT iajs-3461 109 16 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 109 17 βˆ’ βˆ’ PROPN iajs-3461 109 18 βˆ’π‘Ÿ5 βˆ’π‘Ÿ5 PROPN iajs-3461 109 19 ) ) PUNCT iajs-3461 109 20 ( ( PUNCT iajs-3461 109 21 1 1 NUM iajs-3461 109 22 βˆ’ βˆ’ NUM iajs-3461 109 23 2β„Ž 2β„Ž NUM iajs-3461 109 24 βˆ’ βˆ’ PROPN iajs-3461 109 25 βˆ’ βˆ’ PROPN iajs-3461 109 26 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 109 27 𝑖 𝑖 SYM iajs-3461 109 28 βˆ’ βˆ’ PROPN iajs-3461 109 29 ) ) PUNCT iajs-3461 110 1 + + CCONJ iajs-3461 111 1 π‘Ÿ1π‘Ÿ3 π‘Ÿ1π‘Ÿ3 PRON iajs-3461 111 2 2π‘Ÿ5 2π‘Ÿ5 NUM iajs-3461 111 3 𝑖 𝑖 SYM iajs-3461 111 4 βˆ’ βˆ’ NOUN iajs-3461 111 5 = = SYM iajs-3461 111 6 ( ( PUNCT iajs-3461 111 7 2π‘Ÿ5 2π‘Ÿ5 NUM iajs-3461 111 8 + + CCONJ iajs-3461 111 9 π‘Ÿ3)β„Ž π‘Ÿ3)β„Ž NOUN iajs-3461 111 10 βˆ’ βˆ’ PROPN iajs-3461 112 1 + + CCONJ iajs-3461 112 2 π‘Ÿ1π‘Ÿ5 π‘Ÿ1π‘Ÿ5 PROPN iajs-3461 112 3 𝑖 𝑖 PRON iajs-3461 112 4 βˆ’ βˆ’ PROPN iajs-3461 112 5 βˆ’ βˆ’ PROPN iajs-3461 112 6 ( ( PUNCT iajs-3461 112 7 2π‘Ÿ3 2π‘Ÿ3 NUM iajs-3461 112 8 β„Ž β„Ž PART iajs-3461 112 9 βˆ’ βˆ’ PROPN iajs-3461 112 10 2 2 NUM iajs-3461 112 11 + + CCONJ iajs-3461 112 12 π‘Ÿ5 π‘Ÿ5 ADJ iajs-3461 112 13 ) ) PUNCT iajs-3461 112 14 > > X iajs-3461 112 15 0 0 X iajs-3461 112 16 . . NOUN iajs-3461 113 1 which which PRON iajs-3461 113 2 , , PUNCT iajs-3461 113 3 as as ADP iajs-3461 113 4 a a DET iajs-3461 113 5 result result NOUN iajs-3461 113 6 of of ADP iajs-3461 113 7 the the DET iajs-3461 113 8 following follow VERB iajs-3461 113 9 criteria criterion NOUN iajs-3461 113 10 , , PUNCT iajs-3461 113 11 produces produce VERB iajs-3461 113 12 the the DET iajs-3461 113 13 second second ADJ iajs-3461 113 14 two two NUM iajs-3461 113 15 eigenvalues eigenvalue NOUN iajs-3461 113 16 of of ADP iajs-3461 113 17 j2with j2with NOUN iajs-3461 113 18 negative negative ADJ iajs-3461 113 19 real real ADJ iajs-3461 113 20 parts part NOUN iajs-3461 113 21 : : PUNCT iajs-3461 113 22 2β„Ž 2β„Ž NUM iajs-3461 113 23 βˆ’ βˆ’ PROPN iajs-3461 114 1 + + CCONJ iajs-3461 114 2 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 114 3 𝑖 𝑖 NOUN iajs-3461 114 4 βˆ’ βˆ’ PROPN iajs-3461 115 1 + + ADJ iajs-3461 115 2 π‘Ÿ5 π‘Ÿ5 ADJ iajs-3461 115 3 < < X iajs-3461 115 4 1 1 NUM iajs-3461 115 5 + + NUM iajs-3461 115 6 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 115 7 βˆ’ βˆ’ NOUN iajs-3461 115 8 , , PUNCT iajs-3461 115 9 ( ( PUNCT iajs-3461 115 10 9 9 NUM iajs-3461 115 11 ) ) PUNCT iajs-3461 115 12 2π‘Ÿ3 2π‘Ÿ3 NUM iajs-3461 116 1 β„Ž β„Ž PART iajs-3461 116 2 βˆ’ βˆ’ PROPN iajs-3461 116 3 2 2 NUM iajs-3461 116 4 + + CCONJ iajs-3461 116 5 π‘Ÿ5 π‘Ÿ5 ADV iajs-3461 116 6 < < X iajs-3461 116 7 ( ( PUNCT iajs-3461 116 8 2π‘Ÿ5 2π‘Ÿ5 NUM iajs-3461 116 9 + + CCONJ iajs-3461 116 10 π‘Ÿ3)β„Ž π‘Ÿ3)β„Ž NOUN iajs-3461 116 11 βˆ’ βˆ’ PROPN iajs-3461 117 1 + + CCONJ iajs-3461 117 2 π‘Ÿ1π‘Ÿ5 π‘Ÿ1π‘Ÿ5 PROPN iajs-3461 117 3 𝑖 𝑖 PRON iajs-3461 117 4 βˆ’ βˆ’ PROPN iajs-3461 117 5 . . PUNCT iajs-3461 118 1 ( ( PUNCT iajs-3461 118 2 10 10 NUM iajs-3461 118 3 ) ) PUNCT iajs-3461 118 4 therefore therefore ADV iajs-3461 118 5 , , PUNCT iajs-3461 118 6 𝐴2 𝐴2 PROPN iajs-3461 118 7 is be AUX iajs-3461 118 8 stable stable ADJ iajs-3461 118 9 equilibrium equilibrium NOUN iajs-3461 118 10 point point NOUN iajs-3461 118 11 if if SCONJ iajs-3461 118 12 conditions condition NOUN iajs-3461 118 13 ( ( PUNCT iajs-3461 118 14 8) 8) NUM iajs-3461 118 15 , , PUNCT iajs-3461 118 16 ( ( PUNCT iajs-3461 118 17 9 9 NUM iajs-3461 118 18 ) ) PUNCT iajs-3461 118 19 and and CCONJ iajs-3461 118 20 ( ( PUNCT iajs-3461 118 21 10 10 NUM iajs-3461 118 22 ) ) PUNCT iajs-3461 118 23 are be AUX iajs-3461 118 24 satisfied satisfied ADJ iajs-3461 118 25 . . PUNCT iajs-3461 119 1 however however ADV iajs-3461 119 2 otherwise otherwise ADV iajs-3461 119 3 , , PUNCT iajs-3461 119 4 it it PRON iajs-3461 119 5 is be AUX iajs-3461 119 6 unstable unstable ADJ iajs-3461 119 7 . . PUNCT iajs-3461 120 1 β¦Ώ β¦Ώ VERB iajs-3461 120 2 the the DET iajs-3461 120 3 specialist specialist ADJ iajs-3461 120 4 predator predator NOUN iajs-3461 120 5 free free ADJ iajs-3461 120 6 equilibrium equilibrium NOUN iajs-3461 120 7 point point NOUN iajs-3461 120 8 𝐴3 𝐴3 PROPN iajs-3461 120 9 =( =( PROPN iajs-3461 120 10 β„ŽΜΏ β„ŽΜΏ PROPN iajs-3461 120 11 , , PUNCT iajs-3461 120 12 0 0 NUM iajs-3461 120 13 , , PUNCT iajs-3461 120 14 𝑗 𝑗 PROPN iajs-3461 120 15 ΜΏ ΜΏ NOUN iajs-3461 120 16 ) ) PUNCT iajs-3461 120 17 exists exist VERB iajs-3461 120 18 if if SCONJ iajs-3461 120 19 the the DET iajs-3461 120 20 solutions solution NOUN iajs-3461 120 21 to to ADP iajs-3461 120 22 the the DET iajs-3461 120 23 following follow VERB iajs-3461 120 24 set set NOUN iajs-3461 120 25 of of ADP iajs-3461 120 26 equations equation NOUN iajs-3461 120 27 are be AUX iajs-3461 120 28 positive positive ADJ iajs-3461 120 29 : : PUNCT iajs-3461 120 30 β„ŽΜΏ β„ŽΜΏ ADV iajs-3461 120 31 = = SYM iajs-3461 121 1 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 121 2 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 121 3 > > X iajs-3461 121 4 0 0 PUNCT iajs-3461 121 5 ( ( PUNCT iajs-3461 121 6 11 11 NUM iajs-3461 121 7 ) ) PUNCT iajs-3461 121 8 𝑗̿ 𝑗̿ NOUN iajs-3461 122 1 = = PUNCT iajs-3461 122 2 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 122 3 βˆ’ βˆ’ NOUN iajs-3461 122 4 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 122 5 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 122 6 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 122 7 ( ( PUNCT iajs-3461 122 8 12 12 NUM iajs-3461 122 9 ) ) PUNCT iajs-3461 122 10 the the DET iajs-3461 122 11 equation equation NOUN iajs-3461 122 12 ( ( PUNCT iajs-3461 122 13 12 12 NUM iajs-3461 122 14 ) ) PUNCT iajs-3461 122 15 is be AUX iajs-3461 122 16 positive positive ADJ iajs-3461 122 17 , , PUNCT iajs-3461 122 18 provided provide VERB iajs-3461 122 19 that that SCONJ iajs-3461 122 20 : : PUNCT iajs-3461 122 21 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 122 22 > > X iajs-3461 122 23 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 122 24 . . PUNCT iajs-3461 123 1 ( ( PUNCT iajs-3461 123 2 13 13 NUM iajs-3461 123 3 ) ) PUNCT iajs-3461 123 4 the the DET iajs-3461 123 5 jacobian jacobian ADJ iajs-3461 123 6 matrix matrix NOUN iajs-3461 123 7 of of ADP iajs-3461 123 8 system system NOUN iajs-3461 123 9 ( ( PUNCT iajs-3461 123 10 2 2 NUM iajs-3461 123 11 ) ) PUNCT iajs-3461 123 12 at at ADP iajs-3461 123 13 𝐴2 𝐴2 PROPN iajs-3461 123 14 can can AUX iajs-3461 123 15 be be AUX iajs-3461 123 16 written write VERB iajs-3461 123 17 as as ADP iajs-3461 123 18 : : PUNCT iajs-3461 123 19 𝐽3 𝐽3 PROPN iajs-3461 123 20 = = SYM iajs-3461 123 21 𝐽(𝐴3 𝐽(𝐴3 PUNCT iajs-3461 123 22 ) ) PUNCT iajs-3461 123 23 = = PUNCT iajs-3461 124 1 [ [ PUNCT iajs-3461 124 2 πœ‚π‘–π‘— πœ‚π‘–π‘— X iajs-3461 124 3 ] ] X iajs-3461 124 4 3Γ—3 3Γ—3 NUM iajs-3461 124 5 , , PUNCT iajs-3461 124 6 ( ( PUNCT iajs-3461 124 7 3 3 X iajs-3461 124 8 . . X iajs-3461 124 9 π‘Ž π‘Ž X iajs-3461 124 10 ) ) PUNCT iajs-3461 125 1 where where SCONJ iajs-3461 125 2 : : PUNCT iajs-3461 125 3 πœ‚11 πœ‚11 NOUN iajs-3461 125 4 = = SYM iajs-3461 125 5 1 1 NUM iajs-3461 125 6 βˆ’ βˆ’ NUM iajs-3461 125 7 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 125 8 βˆ’ βˆ’ PROPN iajs-3461 125 9 π‘Ÿ2𝑗,ΜΏ π‘Ÿ2𝑗,ΜΏ PROPN iajs-3461 125 10 πœ‚12 πœ‚12 NOUN iajs-3461 125 11 = = PUNCT iajs-3461 125 12 βˆ’π‘Ÿ1β„ŽΜΏ βˆ’π‘Ÿ1β„ŽΜΏ X iajs-3461 125 13 < < X iajs-3461 125 14 0 0 NUM iajs-3461 125 15 , , PUNCT iajs-3461 125 16 πœ‚13 πœ‚13 NOUN iajs-3461 125 17 = = SYM iajs-3461 125 18 βˆ’π‘Ÿ2β„ŽΜΏ βˆ’π‘Ÿ2β„ŽΜΏ ADJ iajs-3461 125 19 < < X iajs-3461 125 20 0 0 PUNCT iajs-3461 125 21 πœ‚21 πœ‚21 PROPN iajs-3461 125 22 = = SYM iajs-3461 125 23 0 0 NUM iajs-3461 125 24 , , PUNCT iajs-3461 125 25 πœ‚22 πœ‚22 NOUN iajs-3461 125 26 = = SYM iajs-3461 125 27 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 PUNCT iajs-3461 126 1 𝑗 𝑗 NOUN iajs-3461 126 2 ΜΏ ΜΏ X iajs-3461 126 3 βˆ’ βˆ’ PROPN iajs-3461 126 4 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 126 5 , , PUNCT iajs-3461 126 6 πœ‚23 πœ‚23 NOUN iajs-3461 126 7 = = SYM iajs-3461 126 8 0 0 NUM iajs-3461 126 9 , , PUNCT iajs-3461 126 10 πœ‚31 πœ‚31 VERB iajs-3461 126 11 = = SYM iajs-3461 126 12 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 126 13 𝑗,ΜΏ 𝑗,ΜΏ NUM iajs-3461 126 14 πœ‚32 πœ‚32 NOUN iajs-3461 126 15 = = SYM iajs-3461 126 16 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 126 17 𝑗 𝑗 NOUN iajs-3461 126 18 ΜΏ ΜΏ X iajs-3461 126 19 πœ‚33 πœ‚33 PUNCT iajs-3461 127 1 = = PUNCT iajs-3461 127 2 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 127 3 βˆ’ βˆ’ PROPN iajs-3461 127 4 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 127 5 . . PUNCT iajs-3461 127 6 characteristic characteristic ADJ iajs-3461 127 7 equation equation NOUN iajs-3461 127 8 for for ADP iajs-3461 127 9 j j PROPN iajs-3461 127 10 ( ( PUNCT iajs-3461 127 11 𝐴3 𝐴3 PROPN iajs-3461 127 12 ) ) PUNCT iajs-3461 127 13 is be AUX iajs-3461 127 14 then then ADV iajs-3461 127 15 provided provide VERB iajs-3461 127 16 by by ADP iajs-3461 127 17 : : PUNCT iajs-3461 127 18 ( ( PUNCT iajs-3461 127 19 πœ‚22 πœ‚22 NOUN iajs-3461 127 20 Ξ» Ξ» NOUN iajs-3461 127 21 ) ) PUNCT iajs-3461 127 22 [ [ PUNCT iajs-3461 127 23 πœ†2 πœ†2 NOUN iajs-3461 127 24 βˆ’ βˆ’ PROPN iajs-3461 127 25 tr(Γ₯)Ξ» tr(Γ₯)Ξ» SYM iajs-3461 127 26 + + NUM iajs-3461 127 27 det det X iajs-3461 127 28 ( ( PUNCT iajs-3461 127 29 Γ₯ Γ₯ PROPN iajs-3461 127 30 ) ) PUNCT iajs-3461 127 31 ] ] PUNCT iajs-3461 128 1 = = PUNCT iajs-3461 128 2 0 0 NUM iajs-3461 128 3 where where SCONJ iajs-3461 128 4 ∢ ∢ NOUN iajs-3461 128 5 Γ₯ Γ₯ X iajs-3461 128 6 = = PUNCT iajs-3461 128 7 [ [ PUNCT iajs-3461 128 8 1 1 NUM iajs-3461 128 9 βˆ’ βˆ’ NUM iajs-3461 128 10 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 128 11 βˆ’ βˆ’ PROPN iajs-3461 128 12 π‘Ÿ2𝑗 π‘Ÿ2𝑗 PROPN iajs-3461 128 13 ΜΏ ΜΏ NOUN iajs-3461 128 14 βˆ’π‘Ÿ2β„ŽΜΏ βˆ’π‘Ÿ2β„ŽΜΏ ADJ iajs-3461 128 15 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 128 16 𝑗 𝑗 PROPN iajs-3461 128 17 ΜΏ ΜΏ NOUN iajs-3461 128 18 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 128 19 βˆ’ βˆ’ NOUN iajs-3461 128 20 π‘Ÿ8 π‘Ÿ8 ADV iajs-3461 128 21 ] ] PUNCT iajs-3461 128 22 . . PUNCT iajs-3461 129 1 then then ADV iajs-3461 129 2 [ [ X iajs-3461 129 3 ( ( PUNCT iajs-3461 129 4 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 NUM iajs-3461 129 5 𝑗̿ 𝑗̿ NUM iajs-3461 130 1 βˆ’ βˆ’ PROPN iajs-3461 130 2 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 130 3 ) ) PUNCT iajs-3461 130 4 βˆ’ βˆ’ ADP iajs-3461 131 1 Ξ» Ξ» SYM iajs-3461 131 2 ] ] X iajs-3461 132 1 [ [ X iajs-3461 132 2 ( ( PUNCT iajs-3461 132 3 1 1 NUM iajs-3461 132 4 βˆ’ βˆ’ NUM iajs-3461 132 5 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 132 6 βˆ’ βˆ’ NOUN iajs-3461 132 7 π‘Ÿ2𝑗̿ π‘Ÿ2𝑗̿ NOUN iajs-3461 132 8 βˆ’ βˆ’ PROPN iajs-3461 132 9 Ξ» Ξ» NOUN iajs-3461 132 10 ) ) PUNCT iajs-3461 132 11 ] ] X iajs-3461 132 12 ( ( PUNCT iajs-3461 132 13 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 132 14 βˆ’ βˆ’ NOUN iajs-3461 132 15 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 132 16 βˆ’ βˆ’ PROPN iajs-3461 132 17 Ξ» Ξ» PROPN iajs-3461 132 18 ) ) PUNCT iajs-3461 132 19 + + X iajs-3461 132 20 π‘Ÿ2π‘Ÿ6 π‘Ÿ2π‘Ÿ6 X iajs-3461 132 21 β„ŽΜΏ β„ŽΜΏ X iajs-3461 132 22 𝑗 𝑗 X iajs-3461 132 23 ΜΏ ΜΏ X iajs-3461 132 24 = = SYM iajs-3461 132 25 0 0 NUM iajs-3461 132 26 either either ADV iajs-3461 132 27 , , PUNCT iajs-3461 132 28 πœ†3β„Ž πœ†3β„Ž PRON iajs-3461 132 29 = = PUNCT iajs-3461 132 30 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 π‘Ÿ3β„ŽΜΏβˆ’π‘Ÿ4 PUNCT iajs-3461 133 1 𝑗 𝑗 VERB iajs-3461 133 2 ΜΏ ΜΏ X iajs-3461 133 3 βˆ’ βˆ’ NOUN iajs-3461 133 4 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 133 5 because because SCONJ iajs-3461 133 6 of of ADP iajs-3461 133 7 the the DET iajs-3461 133 8 following follow VERB iajs-3461 133 9 circumstance circumstance NOUN iajs-3461 133 10 the the DET iajs-3461 133 11 first first ADJ iajs-3461 133 12 eigenvalues eigenvalue NOUN iajs-3461 133 13 of of ADP iajs-3461 133 14 j3 j3 PROPN iajs-3461 133 15 have have VERB iajs-3461 133 16 negative negative ADJ iajs-3461 133 17 real real ADJ iajs-3461 133 18 portions portion NOUN iajs-3461 133 19 : : PUNCT iajs-3461 133 20 π‘Ÿ3β„ŽΜΏ π‘Ÿ3β„ŽΜΏ X iajs-3461 133 21 > > X iajs-3461 133 22 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 133 23 𝑗 𝑗 PROPN iajs-3461 133 24 ΜΏ ΜΏ X iajs-3461 134 1 + + CCONJ iajs-3461 134 2 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 134 3 ( ( PUNCT iajs-3461 134 4 14 14 NUM iajs-3461 134 5 ) ) PUNCT iajs-3461 134 6 or or CCONJ iajs-3461 134 7 , , PUNCT iajs-3461 134 8 πœ†2 πœ†2 NOUN iajs-3461 134 9 βˆ’ βˆ’ NOUN iajs-3461 134 10 tr(Γ₯ tr(Γ₯ NOUN iajs-3461 134 11 ) ) PUNCT iajs-3461 134 12 Ξ» Ξ» PROPN iajs-3461 134 13 + + NUM iajs-3461 134 14 det det X iajs-3461 134 15 ( ( PUNCT iajs-3461 134 16 Γ₯ Γ₯ PROPN iajs-3461 134 17 ) ) PUNCT iajs-3461 135 1 = = SYM iajs-3461 135 2 0 0 NUM iajs-3461 135 3 where where SCONJ iajs-3461 135 4 , , PUNCT iajs-3461 135 5 tr tr PUNCT iajs-3461 135 6 ( ( PUNCT iajs-3461 135 7 Γ₯ Γ₯ NOUN iajs-3461 135 8 ) ) PUNCT iajs-3461 135 9 = = SYM iajs-3461 135 10 πœ†3𝑖 πœ†3𝑖 NOUN iajs-3461 136 1 + + NUM iajs-3461 137 1 πœ†3𝑗 πœ†3𝑗 X iajs-3461 137 2 = = SYM iajs-3461 137 3 1 1 NUM iajs-3461 137 4 βˆ’ βˆ’ NUM iajs-3461 137 5 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 137 6 βˆ’ βˆ’ NOUN iajs-3461 137 7 π‘Ÿ2𝑗̿ π‘Ÿ2𝑗̿ NOUN iajs-3461 137 8 + + CCONJ iajs-3461 137 9 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 137 10 βˆ’ βˆ’ NOUN iajs-3461 137 11 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 137 12 > > SYM iajs-3461 137 13 0 0 NUM iajs-3461 138 1 det det X iajs-3461 138 2 ( ( PUNCT iajs-3461 138 3 Γ₯ Γ₯ PROPN iajs-3461 138 4 ) ) PUNCT iajs-3461 138 5 = = PRON iajs-3461 138 6 πœ†3π‘–πœ†3𝑗 πœ†3π‘–πœ†3𝑗 PUNCT iajs-3461 138 7 = = SYM iajs-3461 138 8 ( ( PUNCT iajs-3461 138 9 1 1 NUM iajs-3461 138 10 βˆ’ βˆ’ NUM iajs-3461 138 11 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 138 12 βˆ’ βˆ’ NOUN iajs-3461 138 13 π‘Ÿ2𝑗)ΜΏ π‘Ÿ2𝑗)ΜΏ PUNCT iajs-3461 138 14 ( ( PUNCT iajs-3461 138 15 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 138 16 βˆ’ βˆ’ PROPN iajs-3461 138 17 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 138 18 ) ) PUNCT iajs-3461 138 19 + + CCONJ iajs-3461 138 20 π‘Ÿ2π‘Ÿ6 π‘Ÿ2π‘Ÿ6 X iajs-3461 138 21 β„ŽΜΏ β„ŽΜΏ ADP iajs-3461 138 22 𝑗 𝑗 PROPN iajs-3461 138 23 ΜΏ ΜΏ PROPN iajs-3461 138 24 ihjpas ihjpas PROPN iajs-3461 138 25 . . PUNCT iajs-3461 139 1 2025 2025 NUM iajs-3461 139 2 , , PUNCT iajs-3461 139 3 38 38 NUM iajs-3461 139 4 ( ( PUNCT iajs-3461 139 5 1 1 NUM iajs-3461 139 6 ) ) PUNCT iajs-3461 139 7 372 372 NUM iajs-3461 139 8 = = SYM iajs-3461 139 9 ( ( PUNCT iajs-3461 139 10 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 139 11 + + CCONJ iajs-3461 139 12 2 2 NUM iajs-3461 139 13 π‘Ÿ8)β„ŽΜΏ π‘Ÿ8)β„ŽΜΏ NUM iajs-3461 139 14 + + CCONJ iajs-3461 139 15 π‘Ÿ2π‘Ÿ8 π‘Ÿ2π‘Ÿ8 X iajs-3461 139 16 𝑗 𝑗 X iajs-3461 139 17 ΜΏ ΜΏ X iajs-3461 139 18 βˆ’ βˆ’ PROPN iajs-3461 139 19 ( ( PUNCT iajs-3461 139 20 2π‘Ÿ6 2π‘Ÿ6 NUM iajs-3461 139 21 β„ŽΜΏ β„ŽΜΏ ADP iajs-3461 139 22 2 2 NUM iajs-3461 139 23 + + NUM iajs-3461 139 24 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 139 25 ) ) PUNCT iajs-3461 139 26 > > X iajs-3461 139 27 0 0 X iajs-3461 139 28 . . PUNCT iajs-3461 140 1 therefore therefore ADV iajs-3461 140 2 , , PUNCT iajs-3461 140 3 as as ADP iajs-3461 140 4 a a DET iajs-3461 140 5 result result NOUN iajs-3461 140 6 of of ADP iajs-3461 140 7 the the DET iajs-3461 140 8 following follow VERB iajs-3461 140 9 requirements requirement NOUN iajs-3461 140 10 , , PUNCT iajs-3461 140 11 results result NOUN iajs-3461 140 12 in in ADP iajs-3461 140 13 the the DET iajs-3461 140 14 second second ADJ iajs-3461 140 15 two two NUM iajs-3461 140 16 eigenvalues eigenvalue NOUN iajs-3461 140 17 of of ADP iajs-3461 140 18 j3 j3 PROPN iajs-3461 140 19 having have VERB iajs-3461 140 20 negative negative ADJ iajs-3461 140 21 real real ADJ iajs-3461 140 22 portions portion NOUN iajs-3461 140 23 : : PUNCT iajs-3461 140 24 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 140 25 + + CCONJ iajs-3461 140 26 π‘Ÿ2𝑗̿ π‘Ÿ2𝑗̿ NOUN iajs-3461 140 27 βˆ’ βˆ’ NOUN iajs-3461 141 1 π‘Ÿ6β„ŽΜΏ π‘Ÿ6β„ŽΜΏ ADJ iajs-3461 141 2 < < X iajs-3461 141 3 1 1 NUM iajs-3461 141 4 , , PUNCT iajs-3461 141 5 ( ( PUNCT iajs-3461 141 6 15 15 NUM iajs-3461 141 7 ) ) PUNCT iajs-3461 141 8 2π‘Ÿ3 2π‘Ÿ3 NUM iajs-3461 141 9 β„Ž β„Ž PART iajs-3461 141 10 βˆ’ βˆ’ PROPN iajs-3461 141 11 2 2 NUM iajs-3461 141 12 + + CCONJ iajs-3461 141 13 π‘Ÿ5 π‘Ÿ5 ADV iajs-3461 141 14 < < X iajs-3461 141 15 ( ( PUNCT iajs-3461 141 16 2π‘Ÿ5 2π‘Ÿ5 NUM iajs-3461 141 17 + + CCONJ iajs-3461 141 18 π‘Ÿ3)β„Ž π‘Ÿ3)β„Ž NOUN iajs-3461 141 19 βˆ’ βˆ’ PROPN iajs-3461 142 1 + + CCONJ iajs-3461 142 2 π‘Ÿ1π‘Ÿ5 π‘Ÿ1π‘Ÿ5 PROPN iajs-3461 142 3 𝑖 𝑖 PRON iajs-3461 142 4 βˆ’ βˆ’ PROPN iajs-3461 142 5 . . PUNCT iajs-3461 143 1 ( ( PUNCT iajs-3461 143 2 16 16 NUM iajs-3461 143 3 ) ) PUNCT iajs-3461 143 4 therefore therefore ADV iajs-3461 143 5 , , PUNCT iajs-3461 143 6 𝐴3 𝐴3 PROPN iajs-3461 143 7 is be AUX iajs-3461 143 8 stable stable ADJ iajs-3461 143 9 equilibrium equilibrium NOUN iajs-3461 143 10 point point NOUN iajs-3461 143 11 if if SCONJ iajs-3461 143 12 conditions condition NOUN iajs-3461 143 13 ( ( PUNCT iajs-3461 143 14 14 14 NUM iajs-3461 143 15 ) ) PUNCT iajs-3461 143 16 , , PUNCT iajs-3461 143 17 ( ( PUNCT iajs-3461 143 18 15 15 NUM iajs-3461 143 19 ) ) PUNCT iajs-3461 143 20 and and CCONJ iajs-3461 143 21 ( ( PUNCT iajs-3461 143 22 16 16 NUM iajs-3461 143 23 ) ) PUNCT iajs-3461 143 24 are be AUX iajs-3461 143 25 satisfied satisfied ADJ iajs-3461 143 26 . . PUNCT iajs-3461 144 1 on on ADP iajs-3461 144 2 the the DET iajs-3461 144 3 other other ADJ iajs-3461 144 4 hand hand NOUN iajs-3461 144 5 , , PUNCT iajs-3461 144 6 it it PRON iajs-3461 144 7 is be AUX iajs-3461 144 8 unstable unstable ADJ iajs-3461 144 9 . . PUNCT iajs-3461 145 1 β¦Ώ β¦Ώ ADJ iajs-3461 145 2 finally finally ADV iajs-3461 145 3 , , PUNCT iajs-3461 145 4 the the DET iajs-3461 145 5 positive positive ADJ iajs-3461 145 6 ( ( PUNCT iajs-3461 145 7 coexistence coexistence NOUN iajs-3461 145 8 ) ) PUNCT iajs-3461 145 9 equilibrium equilibrium NOUN iajs-3461 145 10 point point NOUN iajs-3461 145 11 𝐴4 𝐴4 PROPN iajs-3461 145 12 = = SYM iajs-3461 145 13 ( ( PUNCT iajs-3461 145 14 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 145 15 , , PUNCT iajs-3461 145 16 𝑖̃ 𝑖̃ PROPN iajs-3461 145 17 , , PUNCT iajs-3461 145 18 𝑗̃ 𝑗̃ PROPN iajs-3461 145 19 ) ) PUNCT iajs-3461 145 20 exists exist VERB iajs-3461 145 21 if if SCONJ iajs-3461 145 22 the the DET iajs-3461 145 23 following follow VERB iajs-3461 145 24 system system NOUN iajs-3461 145 25 of of ADP iajs-3461 145 26 equations equation NOUN iajs-3461 145 27 has have VERB iajs-3461 145 28 a a DET iajs-3461 145 29 positive positive ADJ iajs-3461 145 30 solution solution NOUN iajs-3461 145 31 : : PUNCT iajs-3461 146 1 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 146 2 = = SYM iajs-3461 146 3 ( ( PUNCT iajs-3461 146 4 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 146 5 + + CCONJ iajs-3461 146 6 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 146 7 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 146 8 ) ) PUNCT iajs-3461 146 9 βˆ’ βˆ’ PROPN iajs-3461 146 10 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 146 11 π‘Ÿ4 π‘Ÿ4 VERB iajs-3461 146 12 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 146 13 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 146 14 ( ( PUNCT iajs-3461 146 15 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 146 16 + + CCONJ iajs-3461 146 17 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 146 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 146 19 ) ) PUNCT iajs-3461 146 20 βˆ’ βˆ’ PROPN iajs-3461 146 21 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 146 22 π‘Ÿ4 π‘Ÿ4 VERB iajs-3461 146 23 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 147 1 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 147 2 𝑖̃ 𝑖̃ PROPN iajs-3461 147 3 = = SYM iajs-3461 147 4 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 147 5 βˆ’ βˆ’ PRON iajs-3461 147 6 π‘Ÿ6β„ŽΜƒ π‘Ÿ6β„ŽΜƒ NOUN iajs-3461 147 7 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 147 8 , , PUNCT iajs-3461 147 9 𝑗̃ 𝑗̃ PROPN iajs-3461 147 10 = = SYM iajs-3461 147 11 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 147 12 β„ŽΜƒβˆ’ β„ŽΜƒβˆ’ NOUN iajs-3461 147 13 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 147 14 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 147 15 . . PUNCT iajs-3461 148 1 note note VERB iajs-3461 148 2 that that SCONJ iajs-3461 148 3 β„Ž β„Ž PROPN iajs-3461 148 4 = = PRON iajs-3461 148 5 is be AUX iajs-3461 148 6 positive positive ADJ iajs-3461 148 7 , , PUNCT iajs-3461 148 8 provided provide VERB iajs-3461 148 9 that that SCONJ iajs-3461 148 10 : : PUNCT iajs-3461 148 11 ( ( PUNCT iajs-3461 148 12 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 148 13 + + CCONJ iajs-3461 148 14 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 148 15 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 148 16 ) ) PUNCT iajs-3461 148 17 < < X iajs-3461 149 1 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 149 2 π‘Ÿ4 π‘Ÿ4 NOUN iajs-3461 149 3 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 149 4 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 149 5 and and CCONJ iajs-3461 149 6 ( ( PUNCT iajs-3461 149 7 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 149 8 + + CCONJ iajs-3461 149 9 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 149 10 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 149 11 ) ) PUNCT iajs-3461 149 12 < < X iajs-3461 149 13 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 149 14 π‘Ÿ4 π‘Ÿ4 VERB iajs-3461 149 15 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 149 16 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 149 17 . . PUNCT iajs-3461 150 1 or or CCONJ iajs-3461 150 2 ( ( PUNCT iajs-3461 150 3 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 150 4 + + CCONJ iajs-3461 150 5 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 150 6 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 150 7 ) ) PUNCT iajs-3461 150 8 > > PUNCT iajs-3461 150 9 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 150 10 π‘Ÿ4 π‘Ÿ4 NUM iajs-3461 150 11 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 150 12 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 150 13 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3461 150 14 ( ( PUNCT iajs-3461 150 15 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 150 16 + + CCONJ iajs-3461 150 17 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 150 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 150 19 ) ) PUNCT iajs-3461 150 20 > > PUNCT iajs-3461 150 21 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 150 22 π‘Ÿ4 π‘Ÿ4 VERB iajs-3461 150 23 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 150 24 π‘Ÿ7⁄ π‘Ÿ7⁄ PROPN iajs-3461 150 25 so so ADV iajs-3461 150 26 , , PUNCT iajs-3461 150 27 𝑖̃ 𝑖̃ PROPN iajs-3461 150 28 and and CCONJ iajs-3461 150 29 𝑗̃ 𝑗̃ PROPN iajs-3461 150 30 are be AUX iajs-3461 150 31 positive5 positive5 ADJ iajs-3461 150 32 , , PUNCT iajs-3461 150 33 provided provide VERB iajs-3461 150 34 that that SCONJ iajs-3461 150 35 : : PUNCT iajs-3461 150 36 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 150 37 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 150 38 > > X iajs-3461 150 39 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 150 40 and and CCONJ iajs-3461 151 1 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 151 2 > > X iajs-3461 151 3 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 151 4 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 151 5 respectively respectively ADV iajs-3461 151 6 . . PUNCT iajs-3461 152 1 for for ADP iajs-3461 152 2 𝐴4 𝐴4 NOUN iajs-3461 152 3 = = SYM iajs-3461 152 4 ( ( PUNCT iajs-3461 152 5 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 152 6 , , PUNCT iajs-3461 152 7 𝑖̃ 𝑖̃ PROPN iajs-3461 152 8 , , PUNCT iajs-3461 152 9 𝑗̃ 𝑗̃ PROPN iajs-3461 152 10 ) ) PUNCT iajs-3461 152 11 , , PUNCT iajs-3461 152 12 can can AUX iajs-3461 152 13 be be AUX iajs-3461 152 14 expressed express VERB iajs-3461 152 15 as as ADP iajs-3461 152 16 : : PUNCT iajs-3461 152 17 𝐽4 𝐽4 PROPN iajs-3461 152 18 = = SYM iajs-3461 152 19 𝐽(𝐴4 𝐽(𝐴4 PROPN iajs-3461 152 20 ) ) PUNCT iajs-3461 152 21 = = NOUN iajs-3461 153 1 [ [ PUNCT iajs-3461 153 2 ñ𝑖𝑗 ñ𝑖𝑗 NOUN iajs-3461 153 3 ] ] PUNCT iajs-3461 153 4 3Γ—3 3Γ—3 NUM iajs-3461 153 5 , , PUNCT iajs-3461 153 6 ( ( PUNCT iajs-3461 153 7 4 4 NUM iajs-3461 153 8 . . X iajs-3461 153 9 π‘Ž π‘Ž X iajs-3461 153 10 ) ) PUNCT iajs-3461 153 11 where where SCONJ iajs-3461 153 12 : : PUNCT iajs-3461 153 13 Γ±11 Γ±11 NOUN iajs-3461 153 14 = = SYM iajs-3461 153 15 1 1 NUM iajs-3461 153 16 βˆ’ βˆ’ PROPN iajs-3461 153 17 2β„ŽΜƒ 2β„ŽΜƒ NOUN iajs-3461 153 18 βˆ’ βˆ’ PROPN iajs-3461 153 19 π‘Ÿ1𝑖̃ π‘Ÿ1𝑖̃ NOUN iajs-3461 153 20 βˆ’ βˆ’ PROPN iajs-3461 153 21 π‘Ÿ2 π‘Ÿ2 PROPN iajs-3461 153 22 𝑗̃ 𝑗̃ PROPN iajs-3461 153 23 , , PUNCT iajs-3461 153 24 Γ±12 Γ±12 NOUN iajs-3461 153 25 = = SYM iajs-3461 153 26 βˆ’ βˆ’ NOUN iajs-3461 153 27 π‘Ÿ1β„ŽΜƒ π‘Ÿ1β„ŽΜƒ NOUN iajs-3461 153 28 < < X iajs-3461 153 29 0 0 NUM iajs-3461 153 30 , , PUNCT iajs-3461 153 31 Γ±13 Γ±13 NOUN iajs-3461 153 32 = = SYM iajs-3461 153 33 βˆ’ βˆ’ PROPN iajs-3461 153 34 π‘Ÿ2β„ŽΜƒ π‘Ÿ2β„ŽΜƒ NOUN iajs-3461 153 35 < < X iajs-3461 153 36 0 0 NUM iajs-3461 153 37 , , PUNCT iajs-3461 153 38 Γ±21 Γ±21 NOUN iajs-3461 153 39 = = PUNCT iajs-3461 153 40 π‘Ÿ3𝑖̃ π‘Ÿ3𝑖̃ NOUN iajs-3461 153 41 > > X iajs-3461 153 42 0 0 NUM iajs-3461 153 43 , , PUNCT iajs-3461 153 44 𝑛22 𝑛22 NOUN iajs-3461 153 45 = = SYM iajs-3461 153 46 π‘Ÿ3β„ŽΜƒ π‘Ÿ3β„ŽΜƒ NOUN iajs-3461 154 1 βˆ’ βˆ’ NOUN iajs-3461 154 2 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 155 1 𝑗̃ 𝑗̃ PROPN iajs-3461 156 1 βˆ’ βˆ’ NOUN iajs-3461 156 2 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 156 3 , , PUNCT iajs-3461 156 4 Γ±23 Γ±23 NOUN iajs-3461 156 5 = = SYM iajs-3461 156 6 βˆ’ βˆ’ NOUN iajs-3461 156 7 π‘Ÿ4𝑖̃ π‘Ÿ4𝑖̃ NOUN iajs-3461 156 8 , , PUNCT iajs-3461 156 9 Γ±31 Γ±31 NOUN iajs-3461 156 10 = = SYM iajs-3461 156 11 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 157 1 𝑗̃ 𝑗̃ PROPN iajs-3461 157 2 > > X iajs-3461 157 3 0 0 NUM iajs-3461 157 4 , , PUNCT iajs-3461 157 5 Γ±32 Γ±32 NUM iajs-3461 157 6 = = SYM iajs-3461 157 7 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 157 8 𝑗 𝑗 PROPN iajs-3461 157 9 Μƒ Μƒ PROPN iajs-3461 157 10 > > X iajs-3461 157 11 0 0 NUM iajs-3461 157 12 , , PUNCT iajs-3461 157 13 Γ±33 Γ±33 NOUN iajs-3461 157 14 = = SYM iajs-3461 157 15 π‘Ÿ6β„ŽΜƒ π‘Ÿ6β„ŽΜƒ NOUN iajs-3461 157 16 + + NOUN iajs-3461 157 17 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 157 18 𝑖 𝑖 SYM iajs-3461 157 19 Μƒ Μƒ NOUN iajs-3461 157 20 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 157 21 a a DET iajs-3461 157 22 characteristic characteristic ADJ iajs-3461 157 23 equation equation NOUN iajs-3461 157 24 for for ADP iajs-3461 157 25 j j PROPN iajs-3461 157 26 ( ( PUNCT iajs-3461 157 27 𝐴4 𝐴4 PROPN iajs-3461 157 28 ) ) PUNCT iajs-3461 157 29 is be AUX iajs-3461 157 30 then then ADV iajs-3461 157 31 provided provide VERB iajs-3461 157 32 by by ADP iajs-3461 157 33 : : PUNCT iajs-3461 157 34 πœ†3 πœ†3 NOUN iajs-3461 157 35 + + CCONJ iajs-3461 157 36 Ε™1πœ† Ε™1πœ† PROPN iajs-3461 157 37 2 2 NUM iajs-3461 157 38 + + CCONJ iajs-3461 157 39 Ε™2πœ† Ε™2πœ† PROPN iajs-3461 157 40 + + PUNCT iajs-3461 157 41 Ε™3 Ε™3 PROPN iajs-3461 157 42 = = SYM iajs-3461 157 43 0 0 NUM iajs-3461 157 44 , , PUNCT iajs-3461 157 45 ( ( PUNCT iajs-3461 157 46 4.b 4.b NOUN iajs-3461 157 47 ) ) PUNCT iajs-3461 158 1 where where SCONJ iajs-3461 158 2 : : PUNCT iajs-3461 158 3 Ε™1 Ε™1 NOUN iajs-3461 158 4 = = SYM iajs-3461 158 5 βˆ’(Γ±11 βˆ’(Γ±11 PROPN iajs-3461 158 6 + + CCONJ iajs-3461 158 7 Γ±22 Γ±22 NOUN iajs-3461 158 8 + + CCONJ iajs-3461 158 9 Γ±33 Γ±33 NOUN iajs-3461 158 10 ) ) PUNCT iajs-3461 158 11 , , PUNCT iajs-3461 158 12 Ε™2 Ε™2 NOUN iajs-3461 158 13 = = PUNCT iajs-3461 158 14 βˆ’[Γ±23Γ±32 βˆ’[Γ±23Γ±32 PUNCT iajs-3461 158 15 βˆ’ βˆ’ NUM iajs-3461 158 16 Γ±22Γ±33 Γ±22Γ±33 NOUN iajs-3461 158 17 βˆ’ βˆ’ PROPN iajs-3461 158 18 Γ±11(Γ±22 Γ±11(Γ±22 NOUN iajs-3461 158 19 + + CCONJ iajs-3461 158 20 Γ±33 Γ±33 NOUN iajs-3461 158 21 ) ) PUNCT iajs-3461 158 22 + + CCONJ iajs-3461 158 23 Γ±21Γ±12 Γ±21Γ±12 NOUN iajs-3461 158 24 + + CCONJ iajs-3461 158 25 Γ±13Γ±31 Γ±13Γ±31 ADJ iajs-3461 158 26 ] ] PUNCT iajs-3461 158 27 Ε™3 Ε™3 NOUN iajs-3461 158 28 = = NOUN iajs-3461 158 29 βˆ’Γ±11(Γ±22Γ±33 βˆ’Γ±11(Γ±22Γ±33 NOUN iajs-3461 158 30 βˆ’ βˆ’ NOUN iajs-3461 158 31 Γ±23Γ±32 Γ±23Γ±32 ADJ iajs-3461 158 32 ) ) PUNCT iajs-3461 159 1 + + NUM iajs-3461 160 1 Γ±12Γ±21Γ±33 Γ±12Γ±21Γ±33 NUM iajs-3461 160 2 βˆ’ βˆ’ PROPN iajs-3461 160 3 Γ±12Γ±31Γ±23 Γ±12Γ±31Γ±23 PROPN iajs-3461 160 4 βˆ’ βˆ’ PROPN iajs-3461 160 5 Γ±13Γ±21Γ±32 Γ±13Γ±21Γ±32 PART iajs-3461 160 6 βˆ’ βˆ’ PROPN iajs-3461 160 7 Γ±13Γ±31Γ±22 Γ±13Γ±31Γ±22 NOUN iajs-3461 160 8 . . PUNCT iajs-3461 161 1 now now ADV iajs-3461 161 2 , , PUNCT iajs-3461 161 3 Ε™1 Ε™1 NOUN iajs-3461 161 4 > > X iajs-3461 161 5 0 0 PUNCT iajs-3461 161 6 and and CCONJ iajs-3461 161 7 Ε™2 Ε™2 VERB iajs-3461 161 8 > > SYM iajs-3461 161 9 0 0 PUNCT iajs-3461 161 10 provided provide VERB iajs-3461 161 11 that that SCONJ iajs-3461 161 12 : : PUNCT iajs-3461 161 13 1 1 NUM iajs-3461 161 14 < < X iajs-3461 161 15 2β„ŽΜƒ 2β„ŽΜƒ NUM iajs-3461 161 16 + + NUM iajs-3461 161 17 π‘Ÿ1𝑖̃ π‘Ÿ1𝑖̃ NOUN iajs-3461 161 18 + + CCONJ iajs-3461 161 19 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 161 20 𝑗,Μƒ 𝑗,Μƒ X iajs-3461 162 1 ( ( PUNCT iajs-3461 162 2 17 17 NUM iajs-3461 162 3 ) ) PUNCT iajs-3461 162 4 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 162 5 < < X iajs-3461 162 6 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 162 7 𝑗̃ 𝑗̃ PROPN iajs-3461 162 8 + + CCONJ iajs-3461 162 9 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 162 10 , , PUNCT iajs-3461 162 11 ( ( PUNCT iajs-3461 162 12 18 18 NUM iajs-3461 162 13 ) ) PUNCT iajs-3461 162 14 π‘Ÿ6β„ŽΜƒ π‘Ÿ6β„ŽΜƒ NOUN iajs-3461 162 15 + + CCONJ iajs-3461 162 16 π‘Ÿ7 π‘Ÿ7 PROPN iajs-3461 162 17 𝑖̃ 𝑖̃ PROPN iajs-3461 162 18 < < X iajs-3461 162 19 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 162 20 ( ( PUNCT iajs-3461 162 21 19 19 NUM iajs-3461 162 22 ) ) PUNCT iajs-3461 162 23 also also ADV iajs-3461 162 24 , , PUNCT iajs-3461 162 25 βˆ† βˆ† PROPN iajs-3461 162 26 = = SYM iajs-3461 162 27 Ε™1 Ε™1 PROPN iajs-3461 162 28 Ε™2 Ε™2 VERB iajs-3461 162 29 Ε™3 Ε™3 PROPN iajs-3461 162 30 > > X iajs-3461 162 31 0 0 X iajs-3461 162 32 . . PUNCT iajs-3461 163 1 by by ADP iajs-3461 163 2 the the DET iajs-3461 163 3 following follow VERB iajs-3461 163 4 condition condition NOUN iajs-3461 163 5 : : PUNCT iajs-3461 163 6 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 163 7 < < X iajs-3461 163 8 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 163 9 π‘Ÿ1π‘Ÿ3 π‘Ÿ1π‘Ÿ3 ADP iajs-3461 163 10 + + NOUN iajs-3461 163 11 2 2 NUM iajs-3461 163 12 π‘Ÿ4 π‘Ÿ4 NOUN iajs-3461 163 13 ( ( PUNCT iajs-3461 163 14 1 1 NUM iajs-3461 163 15 βˆ’ βˆ’ PROPN iajs-3461 163 16 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 163 17 𝑖 𝑖 SYM iajs-3461 163 18 Μƒ Μƒ NOUN iajs-3461 163 19 βˆ’ βˆ’ NOUN iajs-3461 163 20 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 163 21 𝑗 𝑗 PROPN iajs-3461 163 22 Μƒ Μƒ NUM iajs-3461 163 23 ) ) PUNCT iajs-3461 163 24 , , PUNCT iajs-3461 163 25 ( ( PUNCT iajs-3461 163 26 20 20 NUM iajs-3461 163 27 ) ) PUNCT iajs-3461 163 28 π‘Ÿ2π‘Ÿ6 π‘Ÿ2π‘Ÿ6 NOUN iajs-3461 163 29 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 163 30 𝑗 𝑗 PROPN iajs-3461 163 31 Μƒ Μƒ PROPN iajs-3461 163 32 < < X iajs-3461 163 33 𝑀1 𝑀1 VERB iajs-3461 163 34 𝑀2 𝑀2 NOUN iajs-3461 163 35 , , PUNCT iajs-3461 163 36 ( ( PUNCT iajs-3461 163 37 21 21 NUM iajs-3461 163 38 ) ) PUNCT iajs-3461 163 39 π‘Ÿ4π‘Ÿ6 π‘Ÿ4π‘Ÿ6 SYM iajs-3461 163 40 𝑖 𝑖 SYM iajs-3461 163 41 Μƒ Μƒ NOUN iajs-3461 163 42 𝑗 𝑗 VERB iajs-3461 163 43 Μƒ Μƒ NOUN iajs-3461 163 44 < < X iajs-3461 163 45 π‘Ÿ1 π‘Ÿ1 X iajs-3461 163 46 β„ŽΜƒ(π‘Ÿ6 β„ŽΜƒ(π‘Ÿ6 NOUN iajs-3461 163 47 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 163 48 + + CCONJ iajs-3461 163 49 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 163 50 𝑖 𝑖 SYM iajs-3461 163 51 Μƒ Μƒ NOUN iajs-3461 163 52 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 163 53 ) ) PUNCT iajs-3461 163 54 . . PUNCT iajs-3461 164 1 ( ( PUNCT iajs-3461 164 2 22 22 NUM iajs-3461 164 3 ) ) PUNCT iajs-3461 164 4 where where SCONJ iajs-3461 164 5 , , PUNCT iajs-3461 164 6 𝑀1 𝑀1 VERB iajs-3461 164 7 = = SYM iajs-3461 164 8 2β„ŽΜƒ 2β„ŽΜƒ PROPN iajs-3461 164 9 + + NUM iajs-3461 164 10 π‘Ÿ1𝑖̃ π‘Ÿ1𝑖̃ NOUN iajs-3461 164 11 + + CCONJ iajs-3461 164 12 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 164 13 𝑗̃ 𝑗̃ PROPN iajs-3461 164 14 -1 -1 NOUN iajs-3461 164 15 , , PUNCT iajs-3461 164 16 𝑀2 𝑀2 NOUN iajs-3461 164 17 = = SYM iajs-3461 164 18 π‘Ÿ6β„ŽΜƒ π‘Ÿ6β„ŽΜƒ NOUN iajs-3461 164 19 + + CCONJ iajs-3461 164 20 π‘Ÿ7 π‘Ÿ7 PROPN iajs-3461 164 21 οΏ½ οΏ½ PROPN iajs-3461 164 22 Μƒ Μƒ PROPN iajs-3461 164 23 οΏ½ οΏ½ PROPN iajs-3461 164 24 π‘Ÿ8 π‘Ÿ8 PROPN iajs-3461 164 25 . . PUNCT iajs-3461 165 1 ihjpas ihjpas PROPN iajs-3461 165 2 . . PUNCT iajs-3461 166 1 2025 2025 NUM iajs-3461 166 2 , , PUNCT iajs-3461 166 3 38 38 NUM iajs-3461 166 4 ( ( PUNCT iajs-3461 166 5 1 1 NUM iajs-3461 166 6 ) ) PUNCT iajs-3461 166 7 373 373 NUM iajs-3461 166 8 using use VERB iajs-3461 166 9 the the DET iajs-3461 166 10 routh routh PROPN iajs-3461 166 11 - - PUNCT iajs-3461 166 12 hurwitz hurwitz PROPN iajs-3461 166 13 criterion criterion NOUN iajs-3461 166 14 , , PUNCT iajs-3461 166 15 however however ADV iajs-3461 166 16 , , PUNCT iajs-3461 166 17 allows allow VERB iajs-3461 166 18 each each PRON iajs-3461 166 19 of of ADP iajs-3461 166 20 the the DET iajs-3461 166 21 additional additional ADJ iajs-3461 166 22 eigenvalues eigenvalue NOUN iajs-3461 166 23 of of ADP iajs-3461 166 24 eq eq NOUN iajs-3461 166 25 . . PUNCT iajs-3461 167 1 ( ( PUNCT iajs-3461 167 2 4 4 X iajs-3461 167 3 . . X iajs-3461 167 4 𝑏 𝑏 NOUN iajs-3461 167 5 ) ) PUNCT iajs-3461 167 6 , , PUNCT iajs-3461 167 7 have have VERB iajs-3461 167 8 negative negative ADJ iajs-3461 167 9 real real ADJ iajs-3461 167 10 parts part NOUN iajs-3461 167 11 if if SCONJ iajs-3461 167 12 and and CCONJ iajs-3461 167 13 only only ADV iajs-3461 167 14 if if SCONJ iajs-3461 167 15 𝑅1 𝑅1 PROPN iajs-3461 167 16 > > X iajs-3461 167 17 0 0 PROPN iajs-3461 167 18 , , PUNCT iajs-3461 167 19 𝑅3 𝑅3 PROPN iajs-3461 167 20 > > X iajs-3461 167 21 0 0 PUNCT iajs-3461 168 1 and and CCONJ iajs-3461 168 2 𝑅1𝑅2 𝑅1𝑅2 PRON iajs-3461 168 3 βˆ’ βˆ’ PROPN iajs-3461 168 4 𝑅3 𝑅3 PROPN iajs-3461 168 5 > > X iajs-3461 168 6 0 0 X iajs-3461 168 7 . . PUNCT iajs-3461 169 1 therefore therefore ADV iajs-3461 169 2 , , PUNCT iajs-3461 169 3 all all PRON iajs-3461 169 4 of of ADP iajs-3461 169 5 𝐽(𝐴4 𝐽(𝐴4 NOUN iajs-3461 169 6 ) ) PUNCT iajs-3461 169 7 eigenvalues eigenvalue VERB iajs-3461 169 8 have have VERB iajs-3461 169 9 a a DET iajs-3461 169 10 negative negative ADJ iajs-3461 169 11 real real ADJ iajs-3461 169 12 portion portion NOUN iajs-3461 169 13 if if SCONJ iajs-3461 169 14 the the DET iajs-3461 169 15 additional additional ADJ iajs-3461 169 16 criteria criterion NOUN iajs-3461 169 17 from from ADP iajs-3461 169 18 ( ( PUNCT iajs-3461 169 19 17 17 NUM iajs-3461 169 20 ) ) PUNCT iajs-3461 169 21 ( ( PUNCT iajs-3461 169 22 22 22 NUM iajs-3461 169 23 ) ) PUNCT iajs-3461 169 24 hence hence ADV iajs-3461 169 25 𝐴4 𝐴4 PROPN iajs-3461 169 26 is be AUX iajs-3461 169 27 asymptotically asymptotically ADV iajs-3461 169 28 stable stable ADJ iajs-3461 169 29 locally locally ADV iajs-3461 169 30 . . PUNCT iajs-3461 170 1 in in ADP iajs-3461 170 2 contrast contrast NOUN iajs-3461 170 3 , , PUNCT iajs-3461 170 4 it it PRON iajs-3461 170 5 is be AUX iajs-3461 170 6 unstable unstable ADJ iajs-3461 170 7 . . PUNCT iajs-3461 171 1 4 4 X iajs-3461 171 2 . . X iajs-3461 171 3 local local ADJ iajs-3461 171 4 bifurcation bifurcation NOUN iajs-3461 171 5 analysis analysis NOUN iajs-3461 171 6 this this DET iajs-3461 171 7 section section NOUN iajs-3461 171 8 investigates investigate VERB iajs-3461 171 9 the the DET iajs-3461 171 10 dynamical dynamical ADJ iajs-3461 171 11 behavior behavior NOUN iajs-3461 171 12 of of ADP iajs-3461 171 13 system system NOUN iajs-3461 171 14 ( ( PUNCT iajs-3461 171 15 2 2 NUM iajs-3461 171 16 ) ) PUNCT iajs-3461 171 17 around around ADP iajs-3461 171 18 each each DET iajs-3461 171 19 equilibrium equilibrium NOUN iajs-3461 171 20 point point NOUN iajs-3461 171 21 as as ADP iajs-3461 171 22 a a DET iajs-3461 171 23 result result NOUN iajs-3461 171 24 of of ADP iajs-3461 171 25 altering alter VERB iajs-3461 171 26 the the DET iajs-3461 171 27 parameter parameter NOUN iajs-3461 171 28 values value NOUN iajs-3461 171 29 . . PUNCT iajs-3461 172 1 remember remember VERB iajs-3461 172 2 that that SCONJ iajs-3461 172 3 the the DET iajs-3461 172 4 existence existence NOUN iajs-3461 172 5 of of ADP iajs-3461 172 6 the the DET iajs-3461 172 7 system system NOUN iajs-3461 172 8 ( ( PUNCT iajs-3461 172 9 2 2 NUM iajs-3461 172 10 ) ) PUNCT iajs-3461 172 11 's 's PART iajs-3461 172 12 non non ADJ iajs-3461 172 13 - - ADJ iajs-3461 172 14 hyperbolic hyperbolic ADJ iajs-3461 172 15 equilibrium equilibrium NOUN iajs-3461 172 16 point point NOUN iajs-3461 172 17 is be AUX iajs-3461 172 18 a a DET iajs-3461 172 19 required require VERB iajs-3461 172 20 , , PUNCT iajs-3461 172 21 but but CCONJ iajs-3461 172 22 not not PART iajs-3461 172 23 sufficient sufficient ADJ iajs-3461 172 24 , , PUNCT iajs-3461 172 25 need need VERB iajs-3461 172 26 for for ADP iajs-3461 172 27 bifurcation bifurcation NOUN iajs-3461 172 28 . . PUNCT iajs-3461 173 1 as as ADP iajs-3461 173 2 a a DET iajs-3461 173 3 result result NOUN iajs-3461 173 4 , , PUNCT iajs-3461 173 5 it it PRON iajs-3461 173 6 is be AUX iajs-3461 173 7 appropriate appropriate ADJ iajs-3461 173 8 to to PART iajs-3461 173 9 apply apply VERB iajs-3461 173 10 the the DET iajs-3461 173 11 sotomayor sotomayor NOUN iajs-3461 173 12 's 's PART iajs-3461 173 13 theorem theorem NOUN iajs-3461 173 14 for for ADP iajs-3461 173 15 local local ADJ iajs-3461 173 16 bifurcation bifurcation NOUN iajs-3461 173 17 in in ADP iajs-3461 173 18 the the DET iajs-3461 173 19 following follow VERB iajs-3461 173 20 theorems theorem NOUN iajs-3461 173 21 . . PUNCT iajs-3461 174 1 currently currently ADV iajs-3461 174 2 , , PUNCT iajs-3461 174 3 in in ADP iajs-3461 174 4 accordance accordance NOUN iajs-3461 174 5 with with ADP iajs-3461 174 6 the the DET iajs-3461 174 7 jacobian jacobian ADJ iajs-3461 174 8 matrix9 matrix9 NOUN iajs-3461 174 9 of of ADP iajs-3461 174 10 system system NOUN iajs-3461 174 11 ( ( PUNCT iajs-3461 174 12 2 2 NUM iajs-3461 174 13 ) ) PUNCT iajs-3461 174 14 . . PUNCT iajs-3461 175 1 j j X iajs-3461 175 2 = = PUNCT iajs-3461 176 1 [ [ PUNCT iajs-3461 176 2 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 176 3 πœ•β„Ž πœ•β„Ž NOUN iajs-3461 176 4 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 176 5 πœ•π‘– πœ•π‘– NOUN iajs-3461 176 6 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 176 7 πœ•π‘— πœ•π‘— VERB iajs-3461 176 8 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3461 176 9 πœ•β„Ž πœ•β„Ž ADV iajs-3461 176 10 πœ•π‘“2 πœ•π‘“2 PROPN iajs-3461 176 11 πœ•π‘– πœ•π‘– NOUN iajs-3461 176 12 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3461 176 13 πœ•π‘— πœ•π‘— VERB iajs-3461 176 14 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3461 176 15 πœ•β„Ž πœ•β„Ž NOUN iajs-3461 176 16 πœ•π‘“3 πœ•π‘“3 PROPN iajs-3461 176 17 πœ•π‘– πœ•π‘– PROPN iajs-3461 176 18 πœ•π‘“3 πœ•π‘“3 PROPN iajs-3461 176 19 πœ•π‘— πœ•π‘— VERB iajs-3461 176 20 ] ] PUNCT iajs-3461 176 21 ( ( PUNCT iajs-3461 176 22 5 5 NUM iajs-3461 176 23 . . X iajs-3461 176 24 π‘Ž π‘Ž X iajs-3461 176 25 ) ) PUNCT iajs-3461 176 26 where where SCONJ iajs-3461 176 27 𝑓𝑖 𝑓𝑖 NOUN iajs-3461 176 28 ; ; PUNCT iajs-3461 176 29 𝑖 𝑖 SYM iajs-3461 176 30 = = NOUN iajs-3461 176 31 1 1 NUM iajs-3461 176 32 , , PUNCT iajs-3461 176 33 2 2 NUM iajs-3461 176 34 , , PUNCT iajs-3461 176 35 3 3 NUM iajs-3461 176 36 are be AUX iajs-3461 176 37 displayed display VERB iajs-3461 176 38 on on ADP iajs-3461 176 39 the the DET iajs-3461 176 40 system system NOUN iajs-3461 176 41 's 's PART iajs-3461 176 42 right right ADJ iajs-3461 176 43 side side NOUN iajs-3461 176 44 ( ( PUNCT iajs-3461 176 45 2 2 NUM iajs-3461 176 46 ) ) PUNCT iajs-3461 176 47 and and CCONJ iajs-3461 176 48 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 176 49 πœ•β„Ž πœ•β„Ž NOUN iajs-3461 176 50 = = SYM iajs-3461 176 51 1 1 NUM iajs-3461 176 52 βˆ’ βˆ’ NUM iajs-3461 176 53 2h 2h NUM iajs-3461 176 54 βˆ’ βˆ’ ADP iajs-3461 176 55 π‘Ÿ1𝑖 π‘Ÿ1𝑖 NOUN iajs-3461 176 56 βˆ’ βˆ’ NOUN iajs-3461 176 57 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 176 58 , , PUNCT iajs-3461 176 59 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 176 60 πœ•π‘– πœ•π‘– NOUN iajs-3461 176 61 = = PUNCT iajs-3461 177 1 βˆ’ βˆ’ PROPN iajs-3461 177 2 π‘Ÿ1β„Ž π‘Ÿ1β„Ž NOUN iajs-3461 177 3 , , PUNCT iajs-3461 177 4 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 177 5 πœ•π‘— πœ•π‘— ADJ iajs-3461 177 6 = = PUNCT iajs-3461 177 7 βˆ’ βˆ’ PROPN iajs-3461 177 8 π‘Ÿ2β„Ž π‘Ÿ2β„Ž NOUN iajs-3461 177 9 , , PUNCT iajs-3461 177 10 πœ•π‘“2 πœ•π‘“2 NOUN iajs-3461 177 11 πœ•β„Ž πœ•β„Ž NOUN iajs-3461 177 12 = = SYM iajs-3461 177 13 π‘Ÿ3𝑖 π‘Ÿ3𝑖 NOUN iajs-3461 177 14 , , PUNCT iajs-3461 177 15 πœ•π‘“2 πœ•π‘“2 PROPN iajs-3461 177 16 πœ•π‘– πœ•π‘– NOUN iajs-3461 177 17 = = PUNCT iajs-3461 177 18 π‘Ÿ3β„Ž π‘Ÿ3β„Ž PROPN iajs-3461 177 19 βˆ’ βˆ’ NOUN iajs-3461 177 20 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 177 21 𝑗 𝑗 PROPN iajs-3461 177 22 π‘Ÿ5 π‘Ÿ5 NOUN iajs-3461 177 23 , , PUNCT iajs-3461 177 24 πœ•π‘“2 πœ•π‘“2 PROPN iajs-3461 177 25 πœ•π‘— πœ•π‘— VERB iajs-3461 177 26 = = SYM iajs-3461 177 27 π‘Ÿ4𝑖 π‘Ÿ4𝑖 PROPN iajs-3461 177 28 , , PUNCT iajs-3461 177 29 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3461 177 30 πœ•β„Ž πœ•β„Ž NOUN iajs-3461 177 31 = = PUNCT iajs-3461 177 32 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 177 33 𝑗 𝑗 PROPN iajs-3461 177 34 , , PUNCT iajs-3461 177 35 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3461 177 36 πœ•π‘– πœ•π‘– NOUN iajs-3461 177 37 = = PUNCT iajs-3461 178 1 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 178 2 𝑗 𝑗 PROPN iajs-3461 178 3 , , PUNCT iajs-3461 178 4 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3461 178 5 πœ•π‘— πœ•π‘— VERB iajs-3461 178 6 = = PUNCT iajs-3461 178 7 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 178 8 + + NUM iajs-3461 178 9 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 178 10 𝑗 𝑗 ADJ iajs-3461 178 11 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 178 12 . . PUNCT iajs-3461 179 1 it it PRON iajs-3461 179 2 is be AUX iajs-3461 179 3 clear clear ADJ iajs-3461 179 4 to to ADP iajs-3461 179 5 7verify 7verify NUM iajs-3461 179 6 that that PRON iajs-3461 179 7 for for ADP iajs-3461 179 8 any any DET iajs-3461 179 9 nonzero nonzero NOUN iajs-3461 179 10 vector7 vector7 PROPN iajs-3461 179 11 οΏ½ οΏ½ PROPN iajs-3461 179 12 Μ‡ Μ‡ PROPN iajs-3461 179 13 οΏ½ οΏ½ PROPN iajs-3461 179 14 = = SYM iajs-3461 179 15 ( ( PUNCT iajs-3461 179 16 οΏ½ οΏ½ PROPN iajs-3461 179 17 Μ‡ Μ‡ PROPN iajs-3461 179 18 οΏ½ οΏ½ PROPN iajs-3461 179 19 1 1 NUM iajs-3461 179 20 , , PUNCT iajs-3461 179 21 οΏ½ οΏ½ PROPN iajs-3461 179 22 Μ‡ Μ‡ PROPN iajs-3461 179 23 οΏ½ οΏ½ PROPN iajs-3461 179 24 2 2 NUM iajs-3461 179 25 , , PUNCT iajs-3461 179 26 οΏ½ οΏ½ PROPN iajs-3461 179 27 Μ‡ Μ‡ VERB iajs-3461 179 28 οΏ½ οΏ½ NOUN iajs-3461 179 29 3 3 NUM iajs-3461 179 30 ) ) PUNCT iajs-3461 179 31 𝑇 𝑇 NOUN iajs-3461 179 32 we we PRON iajs-3461 179 33 have have VERB iajs-3461 179 34 : : PUNCT iajs-3461 179 35 j j PROPN iajs-3461 179 36 οΏ½ οΏ½ PROPN iajs-3461 179 37 Μ‡ Μ‡ PROPN iajs-3461 179 38 οΏ½ οΏ½ PROPN iajs-3461 179 39 =[ =[ PROPN iajs-3461 179 40 πœπ‘–π‘— πœπ‘–π‘— PROPN iajs-3461 179 41 ] ] PUNCT iajs-3461 179 42 3Γ—1 3Γ—1 NUM iajs-3461 179 43 where where SCONJ iajs-3461 179 44 : : PUNCT iajs-3461 179 45 𝜏11 𝜏11 NOUN iajs-3461 179 46 = = SYM iajs-3461 179 47 ( ( PUNCT iajs-3461 179 48 1 1 NUM iajs-3461 179 49 βˆ’ βˆ’ NUM iajs-3461 179 50 2β„Ž 2β„Ž NUM iajs-3461 179 51 βˆ’ βˆ’ PROPN iajs-3461 179 52 π‘Ÿ1𝑖 π‘Ÿ1𝑖 X iajs-3461 179 53 βˆ’ βˆ’ PROPN iajs-3461 179 54 π‘Ÿ2𝑗) π‘Ÿ2𝑗) NUM iajs-3461 179 55 οΏ½ οΏ½ PROPN iajs-3461 179 56 Μ‡ Μ‡ NOUN iajs-3461 179 57 οΏ½ οΏ½ PROPN iajs-3461 179 58 1 1 NUM iajs-3461 179 59 βˆ’ βˆ’ PROPN iajs-3461 179 60 π‘Ÿ1β„Ž π‘Ÿ1β„Ž PROPN iajs-3461 179 61 οΏ½ οΏ½ PROPN iajs-3461 179 62 Μ‡ Μ‡ NOUN iajs-3461 179 63 οΏ½ οΏ½ PROPN iajs-3461 179 64 2 2 NUM iajs-3461 179 65 βˆ’ βˆ’ PROPN iajs-3461 179 66 π‘Ÿ2β„Ž π‘Ÿ2β„Ž NOUN iajs-3461 179 67 οΏ½ οΏ½ PROPN iajs-3461 179 68 Μ‡ Μ‡ VERB iajs-3461 179 69 οΏ½ οΏ½ PROPN iajs-3461 179 70 3 3 NUM iajs-3461 179 71 , , PUNCT iajs-3461 179 72 𝜏21 𝜏21 ADJ iajs-3461 179 73 = = SYM iajs-3461 179 74 π‘Ÿ3𝑖 π‘Ÿ3𝑖 NOUN iajs-3461 179 75 οΏ½ οΏ½ PROPN iajs-3461 179 76 Μ‡ Μ‡ NOUN iajs-3461 179 77 οΏ½ οΏ½ PROPN iajs-3461 179 78 1 1 NUM iajs-3461 179 79 + + CCONJ iajs-3461 179 80 ( ( PUNCT iajs-3461 179 81 π‘Ÿ3β„Ž π‘Ÿ3β„Ž PROPN iajs-3461 179 82 βˆ’ βˆ’ PROPN iajs-3461 179 83 π‘Ÿ4𝑗 π‘Ÿ4𝑗 PROPN iajs-3461 179 84 βˆ’ βˆ’ PROPN iajs-3461 179 85 π‘Ÿ5) π‘Ÿ5) NOUN iajs-3461 179 86 οΏ½ οΏ½ PROPN iajs-3461 179 87 Μ‡ Μ‡ NOUN iajs-3461 179 88 οΏ½ οΏ½ PROPN iajs-3461 179 89 2 2 NUM iajs-3461 179 90 βˆ’ βˆ’ PROPN iajs-3461 179 91 π‘Ÿ4𝑖 π‘Ÿ4𝑖 PROPN iajs-3461 179 92 οΏ½ οΏ½ PROPN iajs-3461 179 93 Μ‡ Μ‡ VERB iajs-3461 179 94 οΏ½ οΏ½ PROPN iajs-3461 179 95 3 3 NUM iajs-3461 179 96 , , PUNCT iajs-3461 179 97 𝜏31 𝜏31 NOUN iajs-3461 179 98 = = SYM iajs-3461 179 99 π‘Ÿ6𝑗 π‘Ÿ6𝑗 X iajs-3461 179 100 οΏ½ οΏ½ NOUN iajs-3461 179 101 Μ‡ Μ‡ NOUN iajs-3461 179 102 οΏ½ οΏ½ PROPN iajs-3461 179 103 1 1 NUM iajs-3461 179 104 + + NUM iajs-3461 179 105 π‘Ÿ7𝑗 π‘Ÿ7𝑗 VERB iajs-3461 179 106 οΏ½ οΏ½ PROPN iajs-3461 179 107 Μ‡ Μ‡ NOUN iajs-3461 179 108 οΏ½ οΏ½ PROPN iajs-3461 179 109 2 2 NUM iajs-3461 179 110 + + CCONJ iajs-3461 179 111 ( ( PUNCT iajs-3461 179 112 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 179 113 + + NUM iajs-3461 179 114 π‘Ÿ7𝑖 π‘Ÿ7𝑖 PROPN iajs-3461 179 115 βˆ’ βˆ’ PROPN iajs-3461 179 116 π‘Ÿ8) π‘Ÿ8) PROPN iajs-3461 179 117 οΏ½ οΏ½ PROPN iajs-3461 179 118 Μ‡ Μ‡ NOUN iajs-3461 179 119 οΏ½ οΏ½ PROPN iajs-3461 179 120 3 3 NUM iajs-3461 179 121 . . PUNCT iajs-3461 179 122 d2 d2 PROPN iajs-3461 179 123 ℉ ℉ PROPN iajs-3461 179 124 πœ‡(Γ½ πœ‡(Γ½ NOUN iajs-3461 179 125 , , PUNCT iajs-3461 179 126 πœ‡)( πœ‡)( NOUN iajs-3461 179 127 οΏ½ οΏ½ PROPN iajs-3461 179 128 Μ‡ Μ‡ PROPN iajs-3461 179 129 οΏ½ οΏ½ PROPN iajs-3461 179 130 , , PUNCT iajs-3461 179 131 οΏ½ οΏ½ PROPN iajs-3461 179 132 Μ‡ Μ‡ PROPN iajs-3461 179 133 οΏ½ οΏ½ PROPN iajs-3461 179 134 ) ) PUNCT iajs-3461 179 135 = = PUNCT iajs-3461 179 136 [ [ PUNCT iajs-3461 179 137 οΏ½ οΏ½ PROPN iajs-3461 179 138 ̈ ̈ X iajs-3461 179 139 οΏ½ οΏ½ NOUN iajs-3461 179 140 𝑖𝑗 𝑖𝑗 NOUN iajs-3461 179 141 ] ] X iajs-3461 179 142 3Γ—1 3Γ—1 NUM iajs-3461 179 143 . . PUNCT iajs-3461 180 1 ( ( PUNCT iajs-3461 180 2 23 23 NUM iajs-3461 180 3 ) ) PUNCT iajs-3461 180 4 where where SCONJ iajs-3461 180 5 : : PUNCT iajs-3461 180 6 οΏ½ οΏ½ PROPN iajs-3461 180 7 ̈ ̈ SYM iajs-3461 180 8 οΏ½ οΏ½ NOUN iajs-3461 180 9 11 11 NUM iajs-3461 180 10 = = SYM iajs-3461 180 11 ( ( PUNCT iajs-3461 180 12 βˆ’2 βˆ’2 PROPN iajs-3461 180 13 οΏ½ οΏ½ PROPN iajs-3461 180 14 Μ‡ Μ‡ NOUN iajs-3461 180 15 οΏ½ οΏ½ PROPN iajs-3461 180 16 1 1 NUM iajs-3461 180 17 βˆ’ βˆ’ PROPN iajs-3461 180 18 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 180 19 οΏ½ οΏ½ PROPN iajs-3461 180 20 Μ‡ Μ‡ NOUN iajs-3461 180 21 οΏ½ οΏ½ PROPN iajs-3461 180 22 2 2 NUM iajs-3461 180 23 βˆ’ βˆ’ NOUN iajs-3461 180 24 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 180 25 οΏ½ οΏ½ NOUN iajs-3461 180 26 Μ‡ Μ‡ VERB iajs-3461 180 27 οΏ½ οΏ½ PROPN iajs-3461 180 28 3) 3) NUM iajs-3461 180 29 οΏ½ οΏ½ PROPN iajs-3461 180 30 Μ‡ Μ‡ NOUN iajs-3461 180 31 οΏ½ οΏ½ PROPN iajs-3461 180 32 1 1 NUM iajs-3461 180 33 βˆ’ βˆ’ PROPN iajs-3461 180 34 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 180 35 οΏ½ οΏ½ PROPN iajs-3461 180 36 Μ‡ Μ‡ PROPN iajs-3461 180 37 οΏ½ οΏ½ PROPN iajs-3461 180 38 1 1 NUM iajs-3461 180 39 οΏ½ οΏ½ PROPN iajs-3461 180 40 Μ‡ Μ‡ NOUN iajs-3461 180 41 οΏ½ οΏ½ ADP iajs-3461 180 42 2 2 NUM iajs-3461 180 43 βˆ’ βˆ’ NOUN iajs-3461 180 44 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 180 45 οΏ½ οΏ½ NOUN iajs-3461 180 46 Μ‡ Μ‡ VERB iajs-3461 180 47 οΏ½ οΏ½ PROPN iajs-3461 180 48 1 1 NUM iajs-3461 180 49 οΏ½ οΏ½ PROPN iajs-3461 180 50 Μ‡ Μ‡ NOUN iajs-3461 180 51 οΏ½ οΏ½ PROPN iajs-3461 180 52 3 3 NUM iajs-3461 180 53 , , PUNCT iajs-3461 180 54 οΏ½ οΏ½ PROPN iajs-3461 180 55 ̈ ̈ SYM iajs-3461 180 56 οΏ½ οΏ½ NOUN iajs-3461 180 57 21 21 NUM iajs-3461 180 58 = = PUNCT iajs-3461 180 59 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 180 60 οΏ½ οΏ½ PROPN iajs-3461 180 61 Μ‡ Μ‡ PROPN iajs-3461 180 62 οΏ½ οΏ½ PROPN iajs-3461 180 63 1 1 NUM iajs-3461 180 64 οΏ½ οΏ½ PROPN iajs-3461 180 65 Μ‡ Μ‡ NOUN iajs-3461 180 66 οΏ½ οΏ½ ADP iajs-3461 180 67 2 2 NUM iajs-3461 180 68 + + CCONJ iajs-3461 180 69 ( ( PUNCT iajs-3461 180 70 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 180 71 οΏ½ οΏ½ NOUN iajs-3461 180 72 Μ‡ Μ‡ NOUN iajs-3461 180 73 οΏ½ οΏ½ PROPN iajs-3461 180 74 1 1 NUM iajs-3461 180 75 βˆ’ βˆ’ PROPN iajs-3461 180 76 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 180 77 οΏ½ οΏ½ PROPN iajs-3461 180 78 Μ‡ Μ‡ PROPN iajs-3461 180 79 οΏ½ οΏ½ PROPN iajs-3461 180 80 3) 3) NUM iajs-3461 180 81 οΏ½ οΏ½ PROPN iajs-3461 180 82 Μ‡ Μ‡ NOUN iajs-3461 180 83 οΏ½ οΏ½ PROPN iajs-3461 180 84 2 2 NUM iajs-3461 180 85 βˆ’ βˆ’ PROPN iajs-3461 180 86 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 180 87 οΏ½ οΏ½ PROPN iajs-3461 180 88 Μ‡ Μ‡ PROPN iajs-3461 180 89 οΏ½ οΏ½ PROPN iajs-3461 180 90 2 2 NUM iajs-3461 180 91 οΏ½ οΏ½ PROPN iajs-3461 180 92 Μ‡ Μ‡ NOUN iajs-3461 180 93 οΏ½ οΏ½ PROPN iajs-3461 180 94 3 3 NUM iajs-3461 180 95 , , PUNCT iajs-3461 180 96 οΏ½ οΏ½ NOUN iajs-3461 180 97 ̈ ̈ X iajs-3461 180 98 οΏ½ οΏ½ NOUN iajs-3461 180 99 31 31 NUM iajs-3461 180 100 = = SYM iajs-3461 180 101 π‘Ÿ6 π‘Ÿ6 PROPN iajs-3461 180 102 οΏ½ οΏ½ PROPN iajs-3461 180 103 Μ‡ Μ‡ PROPN iajs-3461 180 104 οΏ½ οΏ½ PROPN iajs-3461 180 105 1 1 NUM iajs-3461 180 106 οΏ½ οΏ½ PROPN iajs-3461 180 107 Μ‡ Μ‡ NOUN iajs-3461 180 108 οΏ½ οΏ½ ADP iajs-3461 180 109 3 3 NUM iajs-3461 180 110 + + CCONJ iajs-3461 180 111 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 180 112 οΏ½ οΏ½ NOUN iajs-3461 180 113 Μ‡ Μ‡ VERB iajs-3461 180 114 οΏ½ οΏ½ PROPN iajs-3461 180 115 2 2 NUM iajs-3461 180 116 οΏ½ οΏ½ PROPN iajs-3461 180 117 Μ‡ Μ‡ NOUN iajs-3461 180 118 οΏ½ οΏ½ ADP iajs-3461 180 119 3 3 NUM iajs-3461 180 120 + + CCONJ iajs-3461 180 121 ( ( PUNCT iajs-3461 180 122 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 180 123 οΏ½ οΏ½ PROPN iajs-3461 180 124 Μ‡ Μ‡ NOUN iajs-3461 180 125 οΏ½ οΏ½ PROPN iajs-3461 180 126 1 1 NUM iajs-3461 180 127 + + CCONJ iajs-3461 180 128 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 180 129 οΏ½ οΏ½ NOUN iajs-3461 180 130 Μ‡ Μ‡ PROPN iajs-3461 180 131 οΏ½ οΏ½ PROPN iajs-3461 180 132 2) 2) NUM iajs-3461 180 133 οΏ½ οΏ½ PROPN iajs-3461 180 134 Μ‡ Μ‡ NOUN iajs-3461 180 135 οΏ½ οΏ½ PROPN iajs-3461 180 136 3 3 NUM iajs-3461 180 137 . . PUNCT iajs-3461 180 138 where where SCONJ iajs-3461 180 139 Γ½ Γ½ PROPN iajs-3461 180 140 = = SYM iajs-3461 180 141 ( ( PUNCT iajs-3461 180 142 β„Ž β„Ž PROPN iajs-3461 180 143 , , PUNCT iajs-3461 180 144 𝑖 𝑖 ADP iajs-3461 180 145 , , PUNCT iajs-3461 180 146 𝑗)𝑇 𝑗)𝑇 ADJ iajs-3461 180 147 and and CCONJ iajs-3461 180 148 πœ‡ πœ‡ ADV iajs-3461 180 149 is be AUX iajs-3461 180 150 any any DET iajs-3461 180 151 bifurcation bifurcation NOUN iajs-3461 180 152 parameter parameter NOUN iajs-3461 180 153 . . PUNCT iajs-3461 181 1 theorems theorem NOUN iajs-3461 181 2 in in ADP iajs-3461 181 3 the the DET iajs-3461 181 4 following follow VERB iajs-3461 181 5 the the DET iajs-3461 181 6 local local ADJ iajs-3461 181 7 bifurcation bifurcation NOUN iajs-3461 181 8 conditions condition NOUN iajs-3461 181 9 near near ADP iajs-3461 181 10 equilibrium equilibrium NOUN iajs-3461 181 11 points point NOUN iajs-3461 181 12 are be AUX iajs-3461 181 13 established establish VERB iajs-3461 181 14 . . PUNCT iajs-3461 182 1 4.1 4.1 NUM iajs-3461 182 2 7local 7local NUM iajs-3461 182 3 7bifurcation 7bifurcation NUM iajs-3461 182 4 analysis7 analysis7 NOUN iajs-3461 182 5 near near ADP iajs-3461 182 6 π‘¨πŸ π‘¨πŸ PROPN iajs-3461 182 7 : : PUNCT iajs-3461 182 8 theorem theorem ADJ iajs-3461 182 9 ( ( PUNCT iajs-3461 182 10 2 2 NUM iajs-3461 182 11 ): ): PUNCT iajs-3461 182 12 if if SCONJ iajs-3461 182 13 the the DET iajs-3461 182 14 value value NOUN iajs-3461 182 15 of of ADP iajs-3461 182 16 the the DET iajs-3461 182 17 parameter parameter NOUN iajs-3461 182 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 182 19 passes pass VERB iajs-3461 182 20 through through ADP iajs-3461 182 21 π‘Ÿ3̈ π‘Ÿ3̈ PROPN iajs-3461 182 22 = = NOUN iajs-3461 182 23 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 182 24 then then ADV iajs-3461 182 25 , , PUNCT iajs-3461 182 26 system system NOUN iajs-3461 182 27 ( ( PUNCT iajs-3461 182 28 2 2 NUM iajs-3461 182 29 ) ) PUNCT iajs-3461 182 30 at at ADP iajs-3461 182 31 the the DET iajs-3461 182 32 axial axial ADJ iajs-3461 182 33 equilibrium equilibrium NOUN iajs-3461 182 34 point point NOUN iajs-3461 182 35 𝐴1 𝐴1 PROPN iajs-3461 183 1 = = PUNCT iajs-3461 183 2 ( ( PUNCT iajs-3461 183 3 1 1 NUM iajs-3461 183 4 , , PUNCT iajs-3461 183 5 0 0 NUM iajs-3461 183 6 , , PUNCT iajs-3461 183 7 0 0 NUM iajs-3461 183 8 ) ) PUNCT iajs-3461 183 9 possesses possesse NOUN iajs-3461 183 10 : : PUNCT iajs-3461 183 11 β€’ β€’ NUM iajs-3461 183 12 no no DET iajs-3461 183 13 saddle saddle NOUN iajs-3461 183 14 - - PUNCT iajs-3461 183 15 node node NOUN iajs-3461 183 16 bifurcation9 bifurcation9 NOUN iajs-3461 183 17 . . PUNCT iajs-3461 184 1 β€’ β€’ NOUN iajs-3461 184 2 7transcritical 7transcritical NUM iajs-3461 184 3 bifurcation7 bifurcation7 NOUN iajs-3461 184 4 . . PUNCT iajs-3461 185 1 proof proof NOUN iajs-3461 185 2 : : PUNCT iajs-3461 185 3 according accord VERB iajs-3461 185 4 to to ADP iajs-3461 185 5 the the DET iajs-3461 185 6 jacobian jacobian ADJ iajs-3461 185 7 matrix matrix NOUN iajs-3461 185 8 𝐽(𝐴1 𝐽(𝐴1 NOUN iajs-3461 185 9 ) ) PUNCT iajs-3461 185 10 given give VERB iajs-3461 185 11 by by ADP iajs-3461 185 12 eq.(1 eq.(1 ADJ iajs-3461 185 13 . . PUNCT iajs-3461 186 1 π‘Ž π‘Ž X iajs-3461 186 2 ): ): PUNCT iajs-3461 186 3 zero zero NUM iajs-3461 186 4 eigenvalue eigenvalue NOUN iajs-3461 186 5 exists exist VERB iajs-3461 186 6 for for ADP iajs-3461 186 7 system system NOUN iajs-3461 186 8 ( ( PUNCT iajs-3461 186 9 2 2 NUM iajs-3461 186 10 ) ) PUNCT iajs-3461 186 11 at at ADP iajs-3461 186 12 equilibrium equilibrium NOUN iajs-3461 186 13 point point NOUN iajs-3461 186 14 𝐴1= 𝐴1= NOUN iajs-3461 186 15 ( ( PUNCT iajs-3461 186 16 1,0,0 1,0,0 NUM iajs-3461 186 17 ) ) PUNCT iajs-3461 186 18 . . PUNCT iajs-3461 187 1 ( ( PUNCT iajs-3461 187 2 say say VERB iajs-3461 187 3 πœ†1𝑖 πœ†1𝑖 PROPN iajs-3461 187 4 = = NOUN iajs-3461 187 5 0 0 NUM iajs-3461 187 6 ) ) PUNCT iajs-3461 187 7 at at ADP iajs-3461 187 8 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 187 9 = = PUNCT iajs-3461 187 10 π‘Ÿ3̈ π‘Ÿ3̈ NOUN iajs-3461 187 11 , , PUNCT iajs-3461 187 12 and and CCONJ iajs-3461 187 13 the the DET iajs-3461 187 14 jacobian jacobian ADJ iajs-3461 187 15 matrix matrix NOUN iajs-3461 187 16 𝐽1 𝐽1 NOUN iajs-3461 187 17 with with ADP iajs-3461 187 18 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 187 19 = = PUNCT iajs-3461 187 20 π‘Ÿ3̈ π‘Ÿ3̈ PROPN iajs-3461 187 21 becomes become VERB iajs-3461 187 22 : : PUNCT iajs-3461 187 23 ihjpas ihjpas PROPN iajs-3461 187 24 . . PUNCT iajs-3461 188 1 2025 2025 NUM iajs-3461 188 2 , , PUNCT iajs-3461 188 3 38 38 NUM iajs-3461 188 4 ( ( PUNCT iajs-3461 188 5 1 1 NUM iajs-3461 188 6 ) ) PUNCT iajs-3461 188 7 374 374 NUM iajs-3461 188 8 𝐽1̈ 𝐽1̈ NOUN iajs-3461 188 9 = = SYM iajs-3461 188 10 𝐽1 𝐽1 PROPN iajs-3461 188 11 ( ( PUNCT iajs-3461 188 12 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 188 13 = = PUNCT iajs-3461 188 14 π‘Ÿ3̈ π‘Ÿ3̈ PROPN iajs-3461 188 15 ) ) PUNCT iajs-3461 188 16 = = PUNCT iajs-3461 189 1 [ [ PUNCT iajs-3461 189 2 βˆ’1 βˆ’1 NOUN iajs-3461 189 3 βˆ’ βˆ’ PROPN iajs-3461 189 4 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 189 5 βˆ’ βˆ’ NOUN iajs-3461 189 6 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 189 7 0 0 NUM iajs-3461 189 8 0 0 NUM iajs-3461 189 9 0 0 NUM iajs-3461 189 10 0 0 NUM iajs-3461 189 11 0 0 NUM iajs-3461 189 12 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 189 13 βˆ’ βˆ’ NOUN iajs-3461 189 14 π‘Ÿ5 π‘Ÿ5 INTJ iajs-3461 189 15 ] ] X iajs-3461 189 16 , , PUNCT iajs-3461 189 17 now now ADV iajs-3461 189 18 , , PUNCT iajs-3461 189 19 let let VERB iajs-3461 189 20 οΏ½ οΏ½ SYM iajs-3461 189 21 Μ‡ Μ‡ VERB iajs-3461 189 22 οΏ½ οΏ½ PROPN iajs-3461 189 23 [1 [1 X iajs-3461 189 24 ] ] X iajs-3461 189 25 = = SYM iajs-3461 189 26 ( ( PUNCT iajs-3461 189 27 οΏ½ οΏ½ PROPN iajs-3461 189 28 Μ‡ Μ‡ NOUN iajs-3461 189 29 οΏ½ οΏ½ PROPN iajs-3461 189 30 1 1 NUM iajs-3461 190 1 [ [ X iajs-3461 190 2 1 1 NUM iajs-3461 190 3 ] ] PUNCT iajs-3461 190 4 , , PUNCT iajs-3461 190 5 οΏ½ οΏ½ PROPN iajs-3461 190 6 Μ‡ Μ‡ PROPN iajs-3461 190 7 οΏ½ οΏ½ PROPN iajs-3461 190 8 2 2 NUM iajs-3461 190 9 [ [ X iajs-3461 190 10 1 1 NUM iajs-3461 190 11 ] ] PUNCT iajs-3461 190 12 , , PUNCT iajs-3461 190 13 οΏ½ οΏ½ PROPN iajs-3461 190 14 Μ‡ Μ‡ PROPN iajs-3461 190 15 οΏ½ οΏ½ PROPN iajs-3461 190 16 3 3 NUM iajs-3461 190 17 [ [ X iajs-3461 190 18 1 1 NUM iajs-3461 190 19 ] ] PUNCT iajs-3461 190 20 ) ) PUNCT iajs-3461 190 21 𝑇 𝑇 PROPN iajs-3461 190 22 be be VERB iajs-3461 190 23 the the DET iajs-3461 190 24 eigenvector7 eigenvector7 NOUN iajs-3461 190 25 corresponding7 corresponding7 NOUN iajs-3461 190 26 to to ADP iajs-3461 190 27 the the DET iajs-3461 190 28 7eigenvalue 7eigenvalue NUM iajs-3461 190 29 πœ†1𝑖 πœ†1𝑖 NOUN iajs-3461 190 30 = = SYM iajs-3461 190 31 0 0 NUM iajs-3461 190 32 . . PUNCT iajs-3461 191 1 thus thus ADV iajs-3461 191 2 ( ( PUNCT iajs-3461 191 3 𝐽1̈ 𝐽1̈ PROPN iajs-3461 191 4 βˆ’ βˆ’ PROPN iajs-3461 191 5 πœ†1𝑖𝐼 πœ†1𝑖𝐼 PROPN iajs-3461 191 6 ) ) PUNCT iajs-3461 191 7 οΏ½ οΏ½ PROPN iajs-3461 191 8 Μ‡ Μ‡ VERB iajs-3461 191 9 οΏ½ οΏ½ PROPN iajs-3461 192 1 [ [ X iajs-3461 192 2 1 1 NUM iajs-3461 192 3 ] ] X iajs-3461 192 4 = = SYM iajs-3461 192 5 0 0 NUM iajs-3461 192 6 , , PUNCT iajs-3461 192 7 which which PRON iajs-3461 192 8 gives give VERB iajs-3461 192 9 : : PUNCT iajs-3461 192 10 𝜌[1 𝜌[1 NOUN iajs-3461 192 11 ] ] X iajs-3461 193 1 = = SYM iajs-3461 193 2 ( ( PUNCT iajs-3461 193 3 βˆ’π‘Ÿ1 βˆ’π‘Ÿ1 PROPN iajs-3461 193 4 οΏ½ οΏ½ NOUN iajs-3461 193 5 Μ‡ Μ‡ VERB iajs-3461 193 6 οΏ½ οΏ½ NOUN iajs-3461 193 7 2 2 NUM iajs-3461 193 8 [ [ X iajs-3461 193 9 1 1 NUM iajs-3461 193 10 ] ] PUNCT iajs-3461 193 11 , , PUNCT iajs-3461 193 12 οΏ½ οΏ½ PROPN iajs-3461 193 13 Μ‡ Μ‡ PROPN iajs-3461 193 14 οΏ½ οΏ½ PROPN iajs-3461 193 15 2 2 NUM iajs-3461 193 16 [ [ X iajs-3461 193 17 1 1 NUM iajs-3461 193 18 ] ] PUNCT iajs-3461 193 19 , , PUNCT iajs-3461 193 20 0 0 NUM iajs-3461 193 21 ) ) PUNCT iajs-3461 193 22 where where SCONJ iajs-3461 193 23 οΏ½ οΏ½ PROPN iajs-3461 193 24 Μ‡ Μ‡ VERB iajs-3461 193 25 οΏ½ οΏ½ PROPN iajs-3461 193 26 2 2 NUM iajs-3461 193 27 [ [ X iajs-3461 193 28 1 1 NUM iajs-3461 193 29 ] ] PUNCT iajs-3461 193 30 and and CCONJ iajs-3461 193 31 οΏ½ οΏ½ PROPN iajs-3461 193 32 Μ‡ Μ‡ VERB iajs-3461 193 33 οΏ½ οΏ½ PROPN iajs-3461 193 34 2 2 NUM iajs-3461 193 35 [ [ X iajs-3461 193 36 1 1 NUM iajs-3461 193 37 ] ] PUNCT iajs-3461 193 38 are be AUX iajs-3461 193 39 any any DET iajs-3461 193 40 7nonzero 7nonzero NUM iajs-3461 193 41 real real ADJ iajs-3461 193 42 number number NOUN iajs-3461 193 43 . . PUNCT iajs-3461 194 1 let let VERB iajs-3461 194 2 Γ§[1 Γ§[1 PRON iajs-3461 194 3 ] ] X iajs-3461 194 4 = = SYM iajs-3461 194 5 ( ( PUNCT iajs-3461 194 6 Γ§1 Γ§1 NOUN iajs-3461 194 7 [ [ X iajs-3461 194 8 1 1 NUM iajs-3461 194 9 ] ] PUNCT iajs-3461 194 10 , , PUNCT iajs-3461 194 11 Γ§2 Γ§2 PROPN iajs-3461 194 12 [ [ X iajs-3461 194 13 1 1 NUM iajs-3461 194 14 ] ] PUNCT iajs-3461 194 15 , , PUNCT iajs-3461 194 16 Γ§3 Γ§3 PROPN iajs-3461 194 17 [ [ X iajs-3461 194 18 1 1 NUM iajs-3461 194 19 ] ] PUNCT iajs-3461 194 20 ) ) PUNCT iajs-3461 194 21 𝑇 𝑇 PROPN iajs-3461 194 22 be be AUX iajs-3461 194 23 the the DET iajs-3461 194 24 7eigenvector 7eigenvector NUM iajs-3461 194 25 7associated 7associate VERB iajs-3461 194 26 with with ADP iajs-3461 194 27 the the DET iajs-3461 194 28 7eigenvalue 7eigenvalue NUM iajs-3461 194 29 πœ†1𝑖 πœ†1𝑖 NOUN iajs-3461 195 1 = = SYM iajs-3461 195 2 0 0 NUM iajs-3461 195 3 of of ADP iajs-3461 195 4 the the DET iajs-3461 195 5 7matrix 7matrix PROPN iajs-3461 195 6 [ [ PUNCT iajs-3461 195 7 j̈1 j̈1 NOUN iajs-3461 195 8 ] ] PUNCT iajs-3461 195 9 𝑇 𝑇 PROPN iajs-3461 195 10 . . PUNCT iajs-3461 196 1 then then ADV iajs-3461 196 2 we we PRON iajs-3461 196 3 have have VERB iajs-3461 196 4 , , PUNCT iajs-3461 196 5 ( ( PUNCT iajs-3461 196 6 𝐽1̈ 𝐽1̈ PROPN iajs-3461 196 7 𝑇 𝑇 PROPN iajs-3461 196 8 βˆ’ βˆ’ PROPN iajs-3461 196 9 πœ†1𝑖𝐼 πœ†1𝑖𝐼 PROPN iajs-3461 196 10 ) ) PUNCT iajs-3461 196 11 Γ§ Γ§ X iajs-3461 197 1 [ [ X iajs-3461 197 2 1 1 NUM iajs-3461 197 3 ] ] X iajs-3461 197 4 = = SYM iajs-3461 197 5 0 0 X iajs-3461 197 6 . . PUNCT iajs-3461 197 7 by by ADP iajs-3461 197 8 solving7 solving7 NOUN iajs-3461 197 9 this this DET iajs-3461 197 10 7equation 7equation NUM iajs-3461 197 11 Γ§[1 Γ§[1 NOUN iajs-3461 197 12 ] ] PUNCT iajs-3461 197 13 , , PUNCT iajs-3461 197 14 we we PRON iajs-3461 197 15 obtain obtain VERB iajs-3461 197 16 , , PUNCT iajs-3461 197 17 Γ§[1 Γ§[1 NOUN iajs-3461 197 18 ] ] X iajs-3461 197 19 = = SYM iajs-3461 197 20 ( ( PUNCT iajs-3461 197 21 0 0 NUM iajs-3461 197 22 , , PUNCT iajs-3461 197 23 Γ§2 Γ§2 PROPN iajs-3461 197 24 [ [ X iajs-3461 197 25 1 1 NUM iajs-3461 197 26 ] ] PUNCT iajs-3461 197 27 , , PUNCT iajs-3461 197 28 0 0 X iajs-3461 197 29 ) ) PUNCT iajs-3461 197 30 𝑇 𝑇 NOUN iajs-3461 197 31 where where SCONJ iajs-3461 197 32 Γ§2 Γ§2 PROPN iajs-3461 197 33 [ [ X iajs-3461 197 34 1 1 X iajs-3461 197 35 ] ] SYM iajs-3461 197 36 7any 7any NUM iajs-3461 197 37 7nonzero 7nonzero NUM iajs-3461 197 38 real real ADJ iajs-3461 197 39 number number NOUN iajs-3461 197 40 . . PUNCT iajs-3461 198 1 now now ADV iajs-3461 198 2 , , PUNCT iajs-3461 198 3 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 198 4 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 PROPN iajs-3461 198 5 = = SYM iajs-3461 198 6 π‘“π‘Ÿ3(𝐴1 π‘“π‘Ÿ3(𝐴1 PROPN iajs-3461 198 7 , , PUNCT iajs-3461 198 8 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 198 9 ) ) PUNCT iajs-3461 198 10 = = SYM iajs-3461 199 1 ( ( PUNCT iajs-3461 199 2 βˆ‚π‘“1 βˆ‚π‘“1 PROPN iajs-3461 199 3 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 NOUN iajs-3461 199 4 , , PUNCT iajs-3461 199 5 βˆ‚π‘“2 βˆ‚π‘“2 NOUN iajs-3461 199 6 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 NOUN iajs-3461 199 7 , , PUNCT iajs-3461 199 8 βˆ‚π‘“3 βˆ‚π‘“3 PROPN iajs-3461 199 9 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 ADJ iajs-3461 199 10 ) ) PUNCT iajs-3461 199 11 t t NOUN iajs-3461 199 12 = = SYM iajs-3461 199 13 ( ( PUNCT iajs-3461 199 14 0 0 NUM iajs-3461 199 15 , , PUNCT iajs-3461 199 16 h𝑖 h𝑖 NOUN iajs-3461 199 17 , , PUNCT iajs-3461 199 18 0)t 0)t NOUN iajs-3461 199 19 . . PUNCT iajs-3461 200 1 so so ADV iajs-3461 200 2 , , PUNCT iajs-3461 200 3 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 200 4 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 NOUN iajs-3461 200 5 ( ( PUNCT iajs-3461 200 6 𝐴1 𝐴1 PROPN iajs-3461 200 7 , , PUNCT iajs-3461 200 8 οΏ½ οΏ½ PROPN iajs-3461 200 9 ̈ ̈ X iajs-3461 200 10 οΏ½ οΏ½ NOUN iajs-3461 200 11 3 3 NUM iajs-3461 200 12 ) ) PUNCT iajs-3461 200 13 = = SYM iajs-3461 200 14 ( ( PUNCT iajs-3461 200 15 0 0 NUM iajs-3461 200 16 , , PUNCT iajs-3461 200 17 0 0 NUM iajs-3461 200 18 , , PUNCT iajs-3461 200 19 0 0 NUM iajs-3461 200 20 ) ) PUNCT iajs-3461 200 21 𝑇 𝑇 NOUN iajs-3461 200 22 that that PRON iajs-3461 200 23 's be AUX iajs-3461 200 24 why why SCONJ iajs-3461 200 25 ( ( PUNCT iajs-3461 200 26 Γ§)𝑇 Γ§)𝑇 ADJ iajs-3461 200 27 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 200 28 βˆ‚π‘Ÿ3 βˆ‚π‘Ÿ3 NOUN iajs-3461 200 29 ( ( PUNCT iajs-3461 200 30 𝐴1 𝐴1 PROPN iajs-3461 200 31 , , PUNCT iajs-3461 200 32 οΏ½ οΏ½ PROPN iajs-3461 200 33 ̈ ̈ X iajs-3461 200 34 οΏ½ οΏ½ NOUN iajs-3461 200 35 3 3 NUM iajs-3461 200 36 ) ) PUNCT iajs-3461 200 37 = = SYM iajs-3461 200 38 0 0 X iajs-3461 200 39 . . PUNCT iajs-3461 201 1 therefore therefore ADV iajs-3461 201 2 , , PUNCT iajs-3461 201 3 7according 7according PROPN iajs-3461 201 4 to to ADP iajs-3461 201 5 7sotomayor 7sotomayor NUM iajs-3461 201 6 ’s ’s PART iajs-3461 201 7 theorem7 theorem7 NOUN iajs-3461 202 1 the the DET iajs-3461 202 2 saddle7 saddle7 PROPN iajs-3461 202 3 - - PUNCT iajs-3461 202 4 7node 7node PROPN iajs-3461 202 5 7bifurcation 7bifurcation PROPN iajs-3461 202 6 can can AUX iajs-3461 202 7 not not PART iajs-3461 202 8 occur occur VERB iajs-3461 202 9 . . PUNCT iajs-3461 203 1 while while SCONJ iajs-3461 203 2 the the DET iajs-3461 203 3 first first ADJ iajs-3461 203 4 7condition 7condition NUM iajs-3461 203 5 of of ADP iajs-3461 203 6 7transcritical 7transcritical PROPN iajs-3461 203 7 7bifurcation 7bifurcation PROPN iajs-3461 203 8 is be AUX iajs-3461 203 9 satisfied7 satisfied7 NOUN iajs-3461 203 10 . . PUNCT iajs-3461 204 1 now now ADV iajs-3461 204 2 , , PUNCT iajs-3461 204 3 since since SCONJ iajs-3461 204 4 π·π‘“π‘Ÿ3 π·π‘“π‘Ÿ3 PROPN iajs-3461 204 5 ( ( PUNCT iajs-3461 204 6 𝐴1 𝐴1 PROPN iajs-3461 204 7 , , PUNCT iajs-3461 204 8 οΏ½ οΏ½ PROPN iajs-3461 204 9 ̈ ̈ X iajs-3461 204 10 οΏ½ οΏ½ PROPN iajs-3461 204 11 3) 3) NUM iajs-3461 204 12 οΏ½ οΏ½ NOUN iajs-3461 204 13 Μ‡ Μ‡ VERB iajs-3461 204 14 οΏ½ οΏ½ PROPN iajs-3461 205 1 [ [ X iajs-3461 205 2 1 1 NUM iajs-3461 205 3 ] ] X iajs-3461 205 4 = = PUNCT iajs-3461 205 5 ( ( PUNCT iajs-3461 205 6 0 0 NUM iajs-3461 205 7 0 0 NUM iajs-3461 205 8 0 0 NUM iajs-3461 205 9 0 0 NUM iajs-3461 205 10 1 1 NUM iajs-3461 205 11 0 0 NUM iajs-3461 205 12 0 0 NUM iajs-3461 205 13 0 0 NUM iajs-3461 205 14 0 0 NUM iajs-3461 205 15 ) ) PUNCT iajs-3461 205 16 ( ( PUNCT iajs-3461 205 17 βˆ’π‘Ÿ1 βˆ’π‘Ÿ1 PROPN iajs-3461 205 18 οΏ½ οΏ½ NOUN iajs-3461 205 19 Μ‡ Μ‡ VERB iajs-3461 205 20 οΏ½ οΏ½ NOUN iajs-3461 205 21 2 2 NUM iajs-3461 205 22 [ [ SYM iajs-3461 205 23 1 1 NUM iajs-3461 205 24 ] ] PUNCT iajs-3461 205 25 οΏ½ οΏ½ PROPN iajs-3461 205 26 Μ‡ Μ‡ NOUN iajs-3461 205 27 οΏ½ οΏ½ PROPN iajs-3461 205 28 2 2 NUM iajs-3461 205 29 [ [ X iajs-3461 205 30 1 1 NUM iajs-3461 205 31 ] ] PUNCT iajs-3461 205 32 0 0 NUM iajs-3461 205 33 ) ) PUNCT iajs-3461 206 1 = = SYM iajs-3461 206 2 ( ( PUNCT iajs-3461 206 3 0 0 NUM iajs-3461 206 4 βˆ’ βˆ’ NOUN iajs-3461 206 5 οΏ½ οΏ½ NOUN iajs-3461 206 6 Μ‡ Μ‡ NOUN iajs-3461 206 7 οΏ½ οΏ½ NOUN iajs-3461 206 8 2 2 NUM iajs-3461 206 9 [ [ X iajs-3461 206 10 1 1 NUM iajs-3461 206 11 ] ] SYM iajs-3461 206 12 0 0 NUM iajs-3461 206 13 ) ) PUNCT iajs-3461 206 14 , , PUNCT iajs-3461 206 15 ( ( PUNCT iajs-3461 206 16 Γ§[1 Γ§[1 NOUN iajs-3461 206 17 ] ] SYM iajs-3461 206 18 ) ) PUNCT iajs-3461 206 19 𝑇 𝑇 PROPN iajs-3461 207 1 [ [ X iajs-3461 207 2 π·π‘“π‘Ÿ3 π·π‘“π‘Ÿ3 PROPN iajs-3461 207 3 ( ( PUNCT iajs-3461 207 4 𝐴1 𝐴1 PROPN iajs-3461 207 5 , , PUNCT iajs-3461 207 6 οΏ½ οΏ½ PROPN iajs-3461 207 7 ̈ ̈ X iajs-3461 207 8 οΏ½ οΏ½ PROPN iajs-3461 207 9 3) 3) NUM iajs-3461 207 10 οΏ½ οΏ½ NOUN iajs-3461 207 11 Μ‡ Μ‡ VERB iajs-3461 207 12 οΏ½ οΏ½ PROPN iajs-3461 208 1 [ [ X iajs-3461 208 2 1 1 NUM iajs-3461 208 3 ] ] X iajs-3461 208 4 ] ] X iajs-3461 208 5 = = SYM iajs-3461 208 6 ( ( PUNCT iajs-3461 208 7 0 0 NUM iajs-3461 208 8 , , PUNCT iajs-3461 208 9 Γ§2 Γ§2 PROPN iajs-3461 208 10 [ [ X iajs-3461 208 11 1 1 NUM iajs-3461 208 12 ] ] PUNCT iajs-3461 208 13 , , PUNCT iajs-3461 208 14 0 0 X iajs-3461 208 15 ) ) PUNCT iajs-3461 208 16 𝑇 𝑇 PROPN iajs-3461 208 17 ( ( PUNCT iajs-3461 208 18 0 0 NUM iajs-3461 208 19 οΏ½ οΏ½ PROPN iajs-3461 208 20 Μ‡ Μ‡ NOUN iajs-3461 208 21 οΏ½ οΏ½ NOUN iajs-3461 208 22 2 2 NUM iajs-3461 208 23 [ [ X iajs-3461 208 24 1 1 NUM iajs-3461 208 25 ] ] PUNCT iajs-3461 208 26 0 0 NUM iajs-3461 208 27 ) ) PUNCT iajs-3461 209 1 = = SYM iajs-3461 209 2 Γ§2 Γ§2 PROPN iajs-3461 209 3 [ [ X iajs-3461 209 4 1]𝜌 1]𝜌 X iajs-3461 209 5 Μ‡ Μ‡ NOUN iajs-3461 209 6 2 2 NUM iajs-3461 209 7 [ [ X iajs-3461 209 8 1 1 NUM iajs-3461 209 9 ] ] PUNCT iajs-3461 209 10 β‰  β‰  PROPN iajs-3461 209 11 0 0 X iajs-3461 209 12 . . PUNCT iajs-3461 210 1 now now ADV iajs-3461 210 2 , , PUNCT iajs-3461 210 3 by by ADP iajs-3461 210 4 7substituting 7substituting PROPN iajs-3461 210 5 οΏ½ οΏ½ PROPN iajs-3461 210 6 Μ‡ Μ‡ PROPN iajs-3461 210 7 οΏ½ οΏ½ PROPN iajs-3461 210 8 [1 [1 X iajs-3461 210 9 ] ] PUNCT iajs-3461 210 10 in in ADP iajs-3461 210 11 ( ( PUNCT iajs-3461 210 12 23 23 NUM iajs-3461 210 13 ) ) PUNCT iajs-3461 210 14 , , PUNCT iajs-3461 210 15 we we PRON iajs-3461 210 16 get get VERB iajs-3461 210 17 : : PUNCT iajs-3461 210 18 𝐷2𝑓 𝐷2𝑓 PROPN iajs-3461 210 19 ( ( PUNCT iajs-3461 210 20 𝐴1 𝐴1 PROPN iajs-3461 210 21 , , PUNCT iajs-3461 210 22 οΏ½ οΏ½ PROPN iajs-3461 210 23 ̈ ̈ SYM iajs-3461 210 24 οΏ½ οΏ½ PROPN iajs-3461 210 25 3)( 3)( NUM iajs-3461 210 26 οΏ½ οΏ½ PROPN iajs-3461 210 27 Μ‡ Μ‡ NOUN iajs-3461 210 28 οΏ½ οΏ½ PROPN iajs-3461 211 1 [ [ X iajs-3461 211 2 1 1 NUM iajs-3461 211 3 ] ] PUNCT iajs-3461 211 4 , , PUNCT iajs-3461 211 5 οΏ½ οΏ½ PROPN iajs-3461 211 6 Μ‡ Μ‡ PROPN iajs-3461 211 7 οΏ½ οΏ½ PROPN iajs-3461 211 8 [1 [1 NUM iajs-3461 211 9 ] ] X iajs-3461 211 10 ) ) PUNCT iajs-3461 211 11 = = SYM iajs-3461 211 12 ( ( PUNCT iajs-3461 211 13 4π‘Ÿ1 4π‘Ÿ1 NUM iajs-3461 211 14 2[ 2[ NUM iajs-3461 211 15 οΏ½ οΏ½ NOUN iajs-3461 211 16 Μ‡ Μ‡ VERB iajs-3461 211 17 οΏ½ οΏ½ NOUN iajs-3461 211 18 2 2 NUM iajs-3461 211 19 [ [ SYM iajs-3461 211 20 1]]2 1]]2 NUM iajs-3461 211 21 βˆ’ βˆ’ ADP iajs-3461 211 22 π‘Ÿ1π‘Ÿ3[ π‘Ÿ1π‘Ÿ3[ NUM iajs-3461 211 23 οΏ½ οΏ½ NOUN iajs-3461 211 24 Μ‡ Μ‡ VERB iajs-3461 211 25 οΏ½ οΏ½ NOUN iajs-3461 211 26 2 2 NUM iajs-3461 211 27 [ [ X iajs-3461 211 28 1]]2 1]]2 NUM iajs-3461 211 29 0 0 NUM iajs-3461 211 30 ) ) PUNCT iajs-3461 211 31 . . PUNCT iajs-3461 212 1 hence hence ADV iajs-3461 212 2 , , PUNCT iajs-3461 212 3 it it PRON iajs-3461 212 4 is be AUX iajs-3461 212 5 7obtained 7obtained NUM iajs-3461 212 6 that that SCONJ iajs-3461 212 7 : : PUNCT iajs-3461 212 8 ( ( PUNCT iajs-3461 212 9 Γ§[1 Γ§[1 NOUN iajs-3461 212 10 ] ] SYM iajs-3461 212 11 ) ) PUNCT iajs-3461 212 12 𝑇 𝑇 PROPN iajs-3461 212 13 𝐷2𝑓 𝐷2𝑓 PROPN iajs-3461 212 14 ( ( PUNCT iajs-3461 212 15 𝐴1 𝐴1 PROPN iajs-3461 212 16 , , PUNCT iajs-3461 212 17 οΏ½ οΏ½ PROPN iajs-3461 212 18 ̈ ̈ SYM iajs-3461 212 19 οΏ½ οΏ½ PROPN iajs-3461 212 20 3)( 3)( NUM iajs-3461 212 21 οΏ½ οΏ½ PROPN iajs-3461 212 22 Μ‡ Μ‡ NOUN iajs-3461 212 23 οΏ½ οΏ½ PROPN iajs-3461 212 24 [ [ X iajs-3461 212 25 1 1 NUM iajs-3461 212 26 ] ] PUNCT iajs-3461 212 27 , , PUNCT iajs-3461 212 28 οΏ½ οΏ½ PROPN iajs-3461 212 29 Μ‡ Μ‡ PROPN iajs-3461 212 30 οΏ½ οΏ½ PROPN iajs-3461 212 31 [1 [1 NUM iajs-3461 212 32 ] ] X iajs-3461 212 33 ) ) PUNCT iajs-3461 212 34 = = SYM iajs-3461 213 1 ( ( PUNCT iajs-3461 213 2 0 0 NUM iajs-3461 213 3 , , PUNCT iajs-3461 213 4 Γ§2 Γ§2 PROPN iajs-3461 213 5 [ [ X iajs-3461 213 6 1 1 NUM iajs-3461 213 7 ] ] PUNCT iajs-3461 213 8 , , PUNCT iajs-3461 213 9 0 0 X iajs-3461 213 10 ) ) PUNCT iajs-3461 213 11 𝑇 𝑇 PROPN iajs-3461 213 12 ( ( PUNCT iajs-3461 213 13 4π‘Ÿ1 4π‘Ÿ1 NUM iajs-3461 213 14 2[ 2[ NUM iajs-3461 213 15 οΏ½ οΏ½ NOUN iajs-3461 213 16 Μ‡ Μ‡ VERB iajs-3461 213 17 οΏ½ οΏ½ NOUN iajs-3461 213 18 2 2 NUM iajs-3461 213 19 [ [ SYM iajs-3461 213 20 1]]2 1]]2 NUM iajs-3461 213 21 βˆ’ βˆ’ ADP iajs-3461 213 22 π‘Ÿ1π‘Ÿ3[ π‘Ÿ1π‘Ÿ3[ NUM iajs-3461 213 23 οΏ½ οΏ½ NOUN iajs-3461 213 24 Μ‡ Μ‡ VERB iajs-3461 213 25 οΏ½ οΏ½ NOUN iajs-3461 213 26 2 2 NUM iajs-3461 213 27 [ [ X iajs-3461 213 28 1]]2 1]]2 NUM iajs-3461 213 29 0 0 NUM iajs-3461 213 30 ) ) PUNCT iajs-3461 213 31 = = SYM iajs-3461 214 1 βˆ’ βˆ’ PROPN iajs-3461 214 2 π‘Ÿ1π‘Ÿ3[ π‘Ÿ1π‘Ÿ3[ NUM iajs-3461 214 3 οΏ½ οΏ½ NOUN iajs-3461 214 4 Μ‡ Μ‡ VERB iajs-3461 214 5 οΏ½ οΏ½ PROPN iajs-3461 214 6 2 2 NUM iajs-3461 214 7 [ [ SYM iajs-3461 214 8 1]]2 1]]2 NUM iajs-3461 214 9 Γ§2 Γ§2 NOUN iajs-3461 214 10 [ [ X iajs-3461 214 11 1 1 X iajs-3461 214 12 ] ] PUNCT iajs-3461 214 13 β‰  β‰  X iajs-3461 214 14 0 0 NUM iajs-3461 214 15 thus thus ADV iajs-3461 214 16 , , PUNCT iajs-3461 214 17 7according 7according NOUN iajs-3461 214 18 to to ADP iajs-3461 214 19 7sotomayor 7sotomayor NUM iajs-3461 214 20 ’s ’s PART iajs-3461 214 21 7theorem 7theorem NUM iajs-3461 214 22 system system NOUN iajs-3461 214 23 ( ( PUNCT iajs-3461 214 24 2 2 NUM iajs-3461 214 25 ) ) PUNCT iajs-3461 214 26 has have VERB iajs-3461 214 27 7transcrirtical 7transcrirtical PROPN iajs-3461 214 28 7bifurcation 7bifurcation NUM iajs-3461 214 29 but but CCONJ iajs-3461 214 30 not not PART iajs-3461 214 31 7experience 7experience PROPN iajs-3461 214 32 a a DET iajs-3461 214 33 7saddl 7saddl NUM iajs-3461 214 34 node node NOUN iajs-3461 214 35 7bifurcation 7bifurcation NUM iajs-3461 214 36 at at ADP iajs-3461 214 37 𝐴1 𝐴1 PROPN iajs-3461 214 38 with with ADP iajs-3461 214 39 the the DET iajs-3461 214 40 7parameter 7parameter NUM iajs-3461 214 41 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 214 42 , , PUNCT iajs-3461 214 43 where where SCONJ iajs-3461 214 44 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 214 45 = = PUNCT iajs-3461 214 46 π‘Ÿ3̈ π‘Ÿ3̈ PROPN iajs-3461 214 47 . . PUNCT iajs-3461 215 1 4.2 4.2 NUM iajs-3461 215 2 near near ADV iajs-3461 215 3 by by ADP iajs-3461 215 4 local local ADJ iajs-3461 215 5 bifurcation bifurcation NOUN iajs-3461 215 6 analysisπ‘¨πŸ analysisπ‘¨πŸ NOUN iajs-3461 215 7 ( ( PUNCT iajs-3461 215 8 𝐑 𝐑 NOUN iajs-3461 215 9 βˆ’ βˆ’ PROPN iajs-3461 215 10 , , PUNCT iajs-3461 215 11 π’Š π’Š X iajs-3461 215 12 βˆ’ βˆ’ NOUN iajs-3461 215 13 , , PUNCT iajs-3461 215 14 𝟎 𝟎 NUM iajs-3461 215 15 ) ) PUNCT iajs-3461 215 16 : : PUNCT iajs-3461 215 17 theorem theorem NOUN iajs-3461 215 18 ( ( PUNCT iajs-3461 215 19 3 3 NUM iajs-3461 215 20 ): ): PUNCT iajs-3461 215 21 assume assume VERB iajs-3461 215 22 that that SCONJ iajs-3461 215 23 the the DET iajs-3461 215 24 following follow VERB iajs-3461 215 25 conditions condition NOUN iajs-3461 215 26 are be AUX iajs-3461 215 27 met meet VERB iajs-3461 215 28 : : PUNCT iajs-3461 215 29 € € SYM iajs-3461 215 30 2 2 NUM iajs-3461 215 31 β‰  β‰  PROPN iajs-3461 215 32 0 0 NUM iajs-3461 215 33 , , PUNCT iajs-3461 215 34 ( ( PUNCT iajs-3461 215 35 24 24 NUM iajs-3461 215 36 ) ) PUNCT iajs-3461 215 37 € € SYM iajs-3461 215 38 4 4 NUM iajs-3461 215 39 β‰  β‰  NOUN iajs-3461 215 40 0 0 NUM iajs-3461 215 41 , , PUNCT iajs-3461 215 42 ( ( PUNCT iajs-3461 215 43 25 25 NUM iajs-3461 215 44 ) ) PUNCT iajs-3461 215 45 πœ— πœ— NOUN iajs-3461 215 46 = = PUNCT iajs-3461 215 47 βˆ’2€1€3(€1 βˆ’2€1€3(€1 NOUN iajs-3461 215 48 + + CCONJ iajs-3461 215 49 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 215 50 + + CCONJ iajs-3461 215 51 π‘Ÿ2€2 π‘Ÿ2€2 NUM iajs-3461 215 52 ) ) PUNCT iajs-3461 216 1 + + CCONJ iajs-3461 216 2 π‘Ÿ3€1 π‘Ÿ3€1 PROPN iajs-3461 216 3 βˆ’ βˆ’ ADP iajs-3461 216 4 π‘Ÿ4€2 π‘Ÿ4€2 NUM iajs-3461 216 5 + + CCONJ iajs-3461 216 6 € € SYM iajs-3461 216 7 2€4(π‘Ÿ6€1 2€4(π‘Ÿ6€1 NUM iajs-3461 216 8 + + CCONJ iajs-3461 216 9 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 216 10 ) ) PUNCT iajs-3461 216 11 β‰  β‰  PROPN iajs-3461 216 12 0 0 NUM iajs-3461 216 13 ( ( PUNCT iajs-3461 216 14 26 26 NUM iajs-3461 216 15 ) ) PUNCT iajs-3461 216 16 ihjpas ihjpa NOUN iajs-3461 216 17 . . PUNCT iajs-3461 217 1 2025 2025 NUM iajs-3461 217 2 , , PUNCT iajs-3461 217 3 38 38 NUM iajs-3461 217 4 ( ( PUNCT iajs-3461 217 5 1 1 NUM iajs-3461 217 6 ) ) PUNCT iajs-3461 218 1 375 375 NUM iajs-3461 218 2 € € SYM iajs-3461 218 3 1 1 NUM iajs-3461 218 4 = = NUM iajs-3461 218 5 h h NOUN iajs-3461 218 6 βˆ’ βˆ’ NOUN iajs-3461 219 1 [ [ X iajs-3461 219 2 π‘Ÿ1π‘Ÿ4 π‘Ÿ1π‘Ÿ4 X iajs-3461 219 3 h h NOUN iajs-3461 219 4 βˆ’ βˆ’ NOUN iajs-3461 219 5 𝑖 𝑖 SYM iajs-3461 219 6 βˆ’ βˆ’ PROPN iajs-3461 219 7 βˆ’ βˆ’ ADP iajs-3461 219 8 π‘Ÿ2(π‘Ÿ3β„Ž π‘Ÿ2(π‘Ÿ3β„Ž PROPN iajs-3461 219 9 βˆ’ βˆ’ NOUN iajs-3461 219 10 βˆ’ βˆ’ PROPN iajs-3461 219 11 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 219 12 ) ) PUNCT iajs-3461 219 13 ] ] PUNCT iajs-3461 220 1 𝑖 𝑖 X iajs-3461 221 1 [ [ X iajs-3461 221 2 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 221 3 ( ( PUNCT iajs-3461 221 4 1 1 NUM iajs-3461 221 5 βˆ’ βˆ’ PROPN iajs-3461 221 6 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 221 7 𝑖 𝑖 SYM iajs-3461 221 8 βˆ’ βˆ’ PROPN iajs-3461 221 9 ) ) PUNCT iajs-3461 221 10 βˆ’ βˆ’ PROPN iajs-3461 221 11 ( ( PUNCT iajs-3461 221 12 2π‘Ÿ4 2π‘Ÿ4 NUM iajs-3461 221 13 + + CCONJ iajs-3461 221 14 π‘Ÿ2π‘Ÿ3)h π‘Ÿ2π‘Ÿ3)h NUM iajs-3461 221 15 βˆ’ βˆ’ NOUN iajs-3461 221 16 ] ] PUNCT iajs-3461 222 1 βˆ’ βˆ’ PROPN iajs-3461 222 2 , , PUNCT iajs-3461 222 3 € € SYM iajs-3461 222 4 2 2 NUM iajs-3461 222 5 = = SYM iajs-3461 222 6 ( ( PUNCT iajs-3461 222 7 1 1 NUM iajs-3461 222 8 βˆ’ βˆ’ NUM iajs-3461 222 9 2 2 NUM iajs-3461 222 10 h h NOUN iajs-3461 222 11 βˆ’ βˆ’ NOUN iajs-3461 222 12 βˆ’ βˆ’ PROPN iajs-3461 222 13 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 222 14 𝑖 𝑖 SYM iajs-3461 222 15 βˆ’ βˆ’ PROPN iajs-3461 222 16 ) ) PUNCT iajs-3461 222 17 𝑖 𝑖 SYM iajs-3461 222 18 βˆ’ βˆ’ NOUN iajs-3461 222 19 βˆ’ βˆ’ PROPN iajs-3461 223 1 π‘Ÿ1β„Ž π‘Ÿ1β„Ž PROPN iajs-3461 223 2 βˆ’ βˆ’ PROPN iajs-3461 223 3 π‘Ÿ2β„Ž π‘Ÿ2β„Ž ADP iajs-3461 223 4 βˆ’ βˆ’ PROPN iajs-3461 223 5 € € SYM iajs-3461 223 6 3 3 NUM iajs-3461 223 7 = = SYM iajs-3461 223 8 π‘Ÿ3β„Ž π‘Ÿ3β„Ž NOUN iajs-3461 223 9 βˆ’ βˆ’ NOUN iajs-3461 223 10 βˆ’ βˆ’ PROPN iajs-3461 223 11 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 223 12 π‘Ÿ1β„Ž π‘Ÿ1β„Ž PROPN iajs-3461 223 13 βˆ’ βˆ’ PUNCT iajs-3461 223 14 βˆ’ βˆ’ PROPN iajs-3461 223 15 , , PUNCT iajs-3461 223 16 € € SYM iajs-3461 223 17 4 4 NUM iajs-3461 223 18 = = SYM iajs-3461 223 19 π‘Ÿ2β„Ž π‘Ÿ2β„Ž ADP iajs-3461 223 20 βˆ’ βˆ’ NOUN iajs-3461 223 21 € € SYM iajs-3461 223 22 3 3 NUM iajs-3461 223 23 + + CCONJ iajs-3461 223 24 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 223 25 𝑖 𝑖 SYM iajs-3461 223 26 βˆ’ βˆ’ PROPN iajs-3461 223 27 π‘Ÿ6β„Ž π‘Ÿ6β„Ž X iajs-3461 223 28 βˆ’ βˆ’ PROPN iajs-3461 224 1 + + CCONJ iajs-3461 224 2 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 224 3 𝑖 𝑖 NOUN iajs-3461 224 4 βˆ’ βˆ’ NOUN iajs-3461 224 5 βˆ’ βˆ’ NUM iajs-3461 224 6 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 224 7 . . PUNCT iajs-3461 225 1 then then ADV iajs-3461 225 2 system system NOUN iajs-3461 225 3 ( ( PUNCT iajs-3461 225 4 2 2 NUM iajs-3461 225 5 ) ) PUNCT iajs-3461 225 6 at at ADP iajs-3461 225 7 the the DET iajs-3461 225 8 equilibrium5 equilibrium5 PROPN iajs-3461 225 9 point point NOUN iajs-3461 225 10 𝐴2 𝐴2 PROPN iajs-3461 225 11 = = PUNCT iajs-3461 225 12 ( ( PUNCT iajs-3461 225 13 h h NOUN iajs-3461 225 14 βˆ’ βˆ’ PROPN iajs-3461 225 15 , , PUNCT iajs-3461 225 16 𝑖 𝑖 PROPN iajs-3461 225 17 βˆ’ βˆ’ NOUN iajs-3461 225 18 , , PUNCT iajs-3461 225 19 0 0 NUM iajs-3461 225 20 ) ) PUNCT iajs-3461 225 21 with with ADP iajs-3461 225 22 the the DET iajs-3461 225 23 parameter5 parameter5 NOUN iajs-3461 225 24 π‘Ÿ8 π‘Ÿ8 ADP iajs-3461 225 25 ̈ ̈ PUNCT iajs-3461 225 26 = = SYM iajs-3461 225 27 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 225 28 βˆ’ βˆ’ PROPN iajs-3461 226 1 + + CCONJ iajs-3461 226 2 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 226 3 𝑖 𝑖 NOUN iajs-3461 226 4 βˆ’ βˆ’ PROPN iajs-3461 226 5 5possesses 5possesses NUM iajs-3461 226 6 : : PUNCT iajs-3461 226 7 β€’ β€’ ADP iajs-3461 226 8 no no DET iajs-3461 226 9 5saddle-5node 5saddle-5node PROPN iajs-3461 226 10 bifurcation bifurcation NOUN iajs-3461 226 11 . . PUNCT iajs-3461 227 1 β€’ β€’ NUM iajs-3461 227 2 transcritical5 transcritical5 NOUN iajs-3461 228 1 5bifurcation 5bifurcation NUM iajs-3461 228 2 . . PUNCT iajs-3461 229 1 proof proof NOUN iajs-3461 229 2 : : PUNCT iajs-3461 229 3 according5 according5 PROPN iajs-3461 229 4 to to ADP iajs-3461 229 5 the the DET iajs-3461 229 6 jacobian5 jacobian5 PROPN iajs-3461 229 7 matrix matrix NOUN iajs-3461 229 8 𝐽(𝐴2 𝐽(𝐴2 NOUN iajs-3461 229 9 ) ) PUNCT iajs-3461 229 10 given give VERB iajs-3461 229 11 by by ADP iajs-3461 229 12 eq.(2 eq.(2 NOUN iajs-3461 229 13 . . PUNCT iajs-3461 230 1 π‘Ž π‘Ž X iajs-3461 230 2 ) ) PUNCT iajs-3461 230 3 of of ADP iajs-3461 230 4 system system NOUN iajs-3461 230 5 ( ( PUNCT iajs-3461 230 6 2 2 NUM iajs-3461 230 7 ) ) PUNCT iajs-3461 230 8 at at ADP iajs-3461 230 9 the the DET iajs-3461 230 10 5equilibrium 5equilibrium NUM iajs-3461 230 11 point point NOUN iajs-3461 230 12 𝐴2 𝐴2 NOUN iajs-3461 230 13 = = PUNCT iajs-3461 230 14 ( ( PUNCT iajs-3461 230 15 h h NOUN iajs-3461 231 1 βˆ’ βˆ’ PROPN iajs-3461 231 2 , , PUNCT iajs-3461 231 3 𝑖 𝑖 PROPN iajs-3461 231 4 βˆ’ βˆ’ NOUN iajs-3461 231 5 , , PUNCT iajs-3461 231 6 0 0 NUM iajs-3461 231 7 ) ) PUNCT iajs-3461 231 8 has have VERB iajs-3461 231 9 zero zero NUM iajs-3461 231 10 eigenvalue eigenvalue X iajs-3461 231 11 ( ( PUNCT iajs-3461 231 12 say say INTJ iajs-3461 231 13 πœ†2𝑗 πœ†2𝑗 PUNCT iajs-3461 231 14 = = SYM iajs-3461 231 15 0 0 X iajs-3461 231 16 ) ) PUNCT iajs-3461 231 17 at at ADP iajs-3461 231 18 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 231 19 = = SYM iajs-3461 231 20 οΏ½ οΏ½ PROPN iajs-3461 231 21 ̈ ̈ X iajs-3461 231 22 οΏ½ οΏ½ NOUN iajs-3461 231 23 8 8 NUM iajs-3461 231 24 , , PUNCT iajs-3461 231 25 and and CCONJ iajs-3461 231 26 the the DET iajs-3461 231 27 5jacobian 5jacobian ADJ iajs-3461 231 28 matrix matrix NOUN iajs-3461 231 29 𝐽2 𝐽2 NOUN iajs-3461 231 30 with with ADP iajs-3461 231 31 οΏ½ οΏ½ PROPN iajs-3461 231 32 ̈ ̈ X iajs-3461 231 33 οΏ½ οΏ½ NOUN iajs-3461 231 34 8 8 NUM iajs-3461 231 35 = = SYM iajs-3461 231 36 π‘Ÿ6β„Ž π‘Ÿ6β„Ž X iajs-3461 231 37 βˆ’ βˆ’ NOUN iajs-3461 232 1 + + CCONJ iajs-3461 232 2 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 232 3 𝑖 𝑖 PRON iajs-3461 232 4 βˆ’ βˆ’ NOUN iajs-3461 232 5 becomes become VERB iajs-3461 232 6 : : PUNCT iajs-3461 232 7 𝐽2̈ 𝐽2̈ NOUN iajs-3461 232 8 = = SYM iajs-3461 232 9 𝐽2(π‘Ÿ8 𝐽2(π‘Ÿ8 NUM iajs-3461 232 10 = = SYM iajs-3461 232 11 οΏ½ οΏ½ PROPN iajs-3461 232 12 ̈ ̈ X iajs-3461 232 13 οΏ½ οΏ½ PROPN iajs-3461 232 14 8) 8) NUM iajs-3461 232 15 = = SYM iajs-3461 233 1 [ [ X iajs-3461 233 2 πœ‡π‘–π‘—π‘–π‘—]3x3 πœ‡π‘–π‘—π‘–π‘—]3x3 NUM iajs-3461 233 3 , , PUNCT iajs-3461 233 4 where where SCONJ iajs-3461 233 5 , , PUNCT iajs-3461 233 6 πœ‡π‘–π‘—Μ‡ πœ‡π‘–π‘—Μ‡ PROPN iajs-3461 233 7 ij ij X iajs-3461 233 8 = = NOUN iajs-3461 233 9 πœ‡π‘–π‘— πœ‡π‘–π‘— NOUN iajs-3461 233 10 for for ADP iajs-3461 233 11 all all DET iajs-3461 233 12 i i PROPN iajs-3461 233 13 , , PUNCT iajs-3461 233 14 j j PROPN iajs-3461 234 1 = = PROPN iajs-3461 234 2 1,2,3 1,2,3 NUM iajs-3461 234 3 except5 except5 NOUN iajs-3461 235 1 πœ‡π‘–π‘—Μ‡ πœ‡π‘–π‘—Μ‡ NOUN iajs-3461 235 2 33 33 NUM iajs-3461 235 3 = = SYM iajs-3461 235 4 π‘Ÿ6β„Ž π‘Ÿ6β„Ž VERB iajs-3461 235 5 βˆ’ βˆ’ PROPN iajs-3461 236 1 + + CCONJ iajs-3461 236 2 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 236 3 𝑖 𝑖 NUM iajs-3461 236 4 βˆ’ βˆ’ NOUN iajs-3461 236 5 βˆ’ βˆ’ PROPN iajs-3461 236 6 π‘Ÿ8=0 π‘Ÿ8=0 PROPN iajs-3461 236 7 . . PUNCT iajs-3461 237 1 let let VERB iajs-3461 237 2 οΏ½ οΏ½ PROPN iajs-3461 237 3 Μ‡ Μ‡ VERB iajs-3461 237 4 οΏ½ οΏ½ PROPN iajs-3461 237 5 [2 [2 X iajs-3461 237 6 ] ] X iajs-3461 237 7 = = SYM iajs-3461 237 8 ( ( PUNCT iajs-3461 237 9 οΏ½ οΏ½ PROPN iajs-3461 237 10 Μ‡ Μ‡ NOUN iajs-3461 237 11 οΏ½ οΏ½ PROPN iajs-3461 237 12 1 1 NUM iajs-3461 237 13 [ [ X iajs-3461 237 14 2 2 NUM iajs-3461 237 15 ] ] PUNCT iajs-3461 237 16 , , PUNCT iajs-3461 237 17 οΏ½ οΏ½ PROPN iajs-3461 237 18 Μ‡ Μ‡ PROPN iajs-3461 237 19 οΏ½ οΏ½ PROPN iajs-3461 237 20 2 2 NUM iajs-3461 237 21 [ [ X iajs-3461 237 22 2 2 NUM iajs-3461 237 23 ] ] PUNCT iajs-3461 237 24 , , PUNCT iajs-3461 237 25 οΏ½ οΏ½ PROPN iajs-3461 237 26 Μ‡ Μ‡ PROPN iajs-3461 237 27 οΏ½ οΏ½ PROPN iajs-3461 237 28 3 3 NUM iajs-3461 237 29 [ [ X iajs-3461 237 30 2 2 NUM iajs-3461 237 31 ] ] PUNCT iajs-3461 237 32 ) ) PUNCT iajs-3461 237 33 𝑇 𝑇 PROPN iajs-3461 237 34 be be AUX iajs-3461 237 35 the the DET iajs-3461 237 36 5eigenvector 5eigenvector NUM iajs-3461 237 37 corresponding5 corresponding5 NOUN iajs-3461 237 38 to to ADP iajs-3461 237 39 the the DET iajs-3461 237 40 eigenvalue eigenvalue PROPN iajs-3461 237 41 πœ†2𝑗 πœ†2𝑗 SYM iajs-3461 237 42 = = SYM iajs-3461 237 43 0 0 X iajs-3461 237 44 . . PUNCT iajs-3461 238 1 thus thus ADV iajs-3461 238 2 ( ( PUNCT iajs-3461 238 3 𝐽2̈ 𝐽2̈ VERB iajs-3461 238 4 βˆ’ βˆ’ NOUN iajs-3461 238 5 πœ†2𝑗𝐼 πœ†2𝑗𝐼 NOUN iajs-3461 238 6 ) ) PUNCT iajs-3461 238 7 οΏ½ οΏ½ PROPN iajs-3461 238 8 Μ‡ Μ‡ VERB iajs-3461 238 9 οΏ½ οΏ½ PROPN iajs-3461 239 1 [ [ X iajs-3461 239 2 2 2 NUM iajs-3461 239 3 ] ] X iajs-3461 239 4 = = SYM iajs-3461 239 5 0 0 NUM iajs-3461 239 6 , , PUNCT iajs-3461 239 7 which which PRON iajs-3461 239 8 gives5 gives5 PROPN iajs-3461 239 9 : : PUNCT iajs-3461 239 10 οΏ½ οΏ½ PROPN iajs-3461 239 11 Μ‡ Μ‡ PROPN iajs-3461 239 12 οΏ½ οΏ½ PROPN iajs-3461 239 13 1 1 NUM iajs-3461 239 14 [ [ X iajs-3461 239 15 2 2 NUM iajs-3461 239 16 ] ] X iajs-3461 239 17 = = SYM iajs-3461 239 18 € € SYM iajs-3461 239 19 1 1 NUM iajs-3461 239 20 οΏ½ οΏ½ NOUN iajs-3461 239 21 Μ‡ Μ‡ NOUN iajs-3461 239 22 οΏ½ οΏ½ NOUN iajs-3461 239 23 2 2 NUM iajs-3461 239 24 [ [ X iajs-3461 239 25 2 2 NUM iajs-3461 239 26 ] ] PUNCT iajs-3461 239 27 and and CCONJ iajs-3461 239 28 οΏ½ οΏ½ PROPN iajs-3461 239 29 Μ‡ Μ‡ PROPN iajs-3461 239 30 οΏ½ οΏ½ PROPN iajs-3461 239 31 3 3 NUM iajs-3461 239 32 [ [ X iajs-3461 239 33 2 2 NUM iajs-3461 239 34 ] ] X iajs-3461 239 35 = = SYM iajs-3461 239 36 € € SYM iajs-3461 239 37 2 2 NUM iajs-3461 239 38 οΏ½ οΏ½ NOUN iajs-3461 239 39 Μ‡ Μ‡ NOUN iajs-3461 239 40 οΏ½ οΏ½ NOUN iajs-3461 239 41 2 2 NUM iajs-3461 239 42 [ [ X iajs-3461 239 43 2 2 NUM iajs-3461 239 44 ] ] PUNCT iajs-3461 239 45 , , PUNCT iajs-3461 239 46 where where SCONJ iajs-3461 239 47 οΏ½ οΏ½ PROPN iajs-3461 239 48 Μ‡ Μ‡ VERB iajs-3461 239 49 οΏ½ οΏ½ PROPN iajs-3461 239 50 2 2 NUM iajs-3461 239 51 [ [ X iajs-3461 239 52 2 2 NUM iajs-3461 239 53 ] ] PUNCT iajs-3461 239 54 any any DET iajs-3461 239 55 nonzero nonzero ADJ iajs-3461 239 56 , , PUNCT iajs-3461 239 57 real real ADJ iajs-3461 239 58 number number NOUN iajs-3461 239 59 with with ADP iajs-3461 239 60 € € SYM iajs-3461 239 61 1 1 NUM iajs-3461 239 62 , , PUNCT iajs-3461 239 63 € € SYM iajs-3461 239 64 2 2 NUM iajs-3461 239 65 and and CCONJ iajs-3461 239 66 which which PRON iajs-3461 239 67 and and CCONJ iajs-3461 239 68 which which PRON iajs-3461 239 69 are be AUX iajs-3461 239 70 mentioned mention VERB iajs-3461 239 71 in in ADP iajs-3461 239 72 the the DET iajs-3461 239 73 state state NOUN iajs-3461 239 74 of of ADP iajs-3461 239 75 the the DET iajs-3461 239 76 theorem theorem NOUN iajs-3461 239 77 . . PUNCT iajs-3461 239 78 let let VERB iajs-3461 239 79 Γ§[2 Γ§[2 NOUN iajs-3461 239 80 ] ] X iajs-3461 239 81 = = SYM iajs-3461 239 82 ( ( PUNCT iajs-3461 239 83 Γ§1 Γ§1 NOUN iajs-3461 239 84 [ [ X iajs-3461 239 85 2 2 NUM iajs-3461 239 86 ] ] PUNCT iajs-3461 239 87 , , PUNCT iajs-3461 239 88 Γ§2 Γ§2 PROPN iajs-3461 239 89 [ [ X iajs-3461 239 90 2 2 NUM iajs-3461 239 91 ] ] PUNCT iajs-3461 239 92 , , PUNCT iajs-3461 239 93 Γ§3 Γ§3 PROPN iajs-3461 240 1 [ [ X iajs-3461 240 2 2 2 NUM iajs-3461 240 3 ] ] PUNCT iajs-3461 240 4 ) ) PUNCT iajs-3461 240 5 𝑇 𝑇 PROPN iajs-3461 240 6 be be AUX iajs-3461 240 7 the the DET iajs-3461 240 8 4eigenvector 4eigenvector NUM iajs-3461 240 9 4associated 4associated NUM iajs-3461 240 10 with with ADP iajs-3461 240 11 the the DET iajs-3461 240 12 4eigenvalue 4eigenvalue NUM iajs-3461 240 13 πœ†2𝑀 πœ†2𝑀 NUM iajs-3461 240 14 = = SYM iajs-3461 240 15 0 0 NUM iajs-3461 240 16 of of ADP iajs-3461 240 17 the the DET iajs-3461 240 18 4matrix 4matrix NUM iajs-3461 240 19 j j NOUN iajs-3461 240 20 ̈ ̈ PUNCT iajs-3461 240 21 2 2 NUM iajs-3461 240 22 . . PUNCT iajs-3461 241 1 then then ADV iajs-3461 241 2 we we PRON iajs-3461 241 3 have have VERB iajs-3461 241 4 ( ( PUNCT iajs-3461 241 5 j j X iajs-3461 241 6 ̈ ̈ PUNCT iajs-3461 241 7 2 2 NUM iajs-3461 241 8 βˆ’ βˆ’ NOUN iajs-3461 241 9 Ξ»2wi Ξ»2wi NOUN iajs-3461 241 10 ) ) PUNCT iajs-3461 241 11 Γ§ Γ§ NOUN iajs-3461 242 1 [ [ X iajs-3461 242 2 2 2 NUM iajs-3461 242 3 ] ] X iajs-3461 242 4 = = SYM iajs-3461 242 5 0 0 X iajs-3461 242 6 . . PUNCT iajs-3461 242 7 by by ADP iajs-3461 242 8 4solving 4solve VERB iajs-3461 242 9 this this DET iajs-3461 242 10 4equation 4equation PROPN iajs-3461 242 11 for for ADP iajs-3461 242 12 Γ§[2 Γ§[2 NOUN iajs-3461 242 13 ] ] PUNCT iajs-3461 242 14 , , PUNCT iajs-3461 242 15 we we PRON iajs-3461 242 16 4obtain 4obtain NUM iajs-3461 242 17 Γ§[2 Γ§[2 X iajs-3461 242 18 ] ] X iajs-3461 242 19 = = SYM iajs-3461 242 20 ( ( PUNCT iajs-3461 242 21 € € SYM iajs-3461 242 22 3Γ§2 3Γ§2 NUM iajs-3461 242 23 [ [ X iajs-3461 242 24 2 2 NUM iajs-3461 242 25 ] ] PUNCT iajs-3461 242 26 , , PUNCT iajs-3461 242 27 Γ§2 Γ§2 PROPN iajs-3461 242 28 [ [ X iajs-3461 242 29 2 2 NUM iajs-3461 242 30 ] ] PUNCT iajs-3461 242 31 , , PUNCT iajs-3461 242 32 € € SYM iajs-3461 242 33 4Γ§2 4Γ§2 NUM iajs-3461 242 34 [ [ X iajs-3461 242 35 2 2 NUM iajs-3461 242 36 ] ] SYM iajs-3461 242 37 ) ) PUNCT iajs-3461 242 38 𝑇 𝑇 PROPN iajs-3461 242 39 , , PUNCT iajs-3461 242 40 where where SCONJ iajs-3461 242 41 Γ§2 Γ§2 PROPN iajs-3461 242 42 [ [ X iajs-3461 242 43 2 2 X iajs-3461 242 44 ] ] PUNCT iajs-3461 242 45 any any DET iajs-3461 242 46 real real ADJ iajs-3461 242 47 numbers number NOUN iajs-3461 242 48 that that PRON iajs-3461 242 49 are be AUX iajs-3461 242 50 not not PART iajs-3461 242 51 zero zero NUM iajs-3461 242 52 , , PUNCT iajs-3461 242 53 with with ADP iajs-3461 242 54 € € SYM iajs-3461 242 55 3 3 NUM iajs-3461 242 56 , , PUNCT iajs-3461 242 57 € € SYM iajs-3461 242 58 4 4 NUM iajs-3461 242 59 𝑀hich 𝑀hich NOUN iajs-3461 242 60 are be AUX iajs-3461 242 61 54mentioned 54mentione VERB iajs-3461 242 62 in in ADP iajs-3461 242 63 the the DET iajs-3461 242 64 state state NOUN iajs-3461 242 65 of of ADP iajs-3461 242 66 the the DET iajs-3461 242 67 theorem45 theorem45 NOUN iajs-3461 242 68 . . PUNCT iajs-3461 243 1 now now ADV iajs-3461 243 2 , , PUNCT iajs-3461 243 3 consider consider VERB iajs-3461 243 4 : : PUNCT iajs-3461 243 5 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 243 6 βˆ‚π‘Ÿ8 βˆ‚π‘Ÿ8 PROPN iajs-3461 243 7 ( ( PUNCT iajs-3461 243 8 Γ½ Γ½ PROPN iajs-3461 243 9 , , PUNCT iajs-3461 243 10 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 243 11 ) ) PUNCT iajs-3461 243 12 = = SYM iajs-3461 243 13 π‘“π‘Ÿ8(Γ½ π‘“π‘Ÿ8(Γ½ PROPN iajs-3461 243 14 , , PUNCT iajs-3461 243 15 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 243 16 ) ) PUNCT iajs-3461 243 17 = = SYM iajs-3461 243 18 ( ( PUNCT iajs-3461 243 19 βˆ‚π‘“1 βˆ‚π‘“1 PROPN iajs-3461 243 20 βˆ‚π‘Ÿ8 βˆ‚π‘Ÿ8 PROPN iajs-3461 243 21 , , PUNCT iajs-3461 243 22 βˆ‚π‘“2 βˆ‚π‘“2 PROPN iajs-3461 243 23 βˆ‚π‘Ÿ8 βˆ‚π‘Ÿ8 PROPN iajs-3461 243 24 , , PUNCT iajs-3461 243 25 βˆ‚π‘“3 βˆ‚π‘“3 PROPN iajs-3461 243 26 βˆ‚π‘Ÿ8 βˆ‚π‘Ÿ8 PROPN iajs-3461 243 27 ) ) PUNCT iajs-3461 243 28 t t PROPN iajs-3461 244 1 = = SYM iajs-3461 244 2 ( ( PUNCT iajs-3461 244 3 0 0 NUM iajs-3461 244 4 , , PUNCT iajs-3461 244 5 0 0 NUM iajs-3461 244 6 , , PUNCT iajs-3461 244 7 𝑗)𝑇 𝑗)𝑇 ADJ iajs-3461 245 1 so so ADV iajs-3461 245 2 , , PUNCT iajs-3461 245 3 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 245 4 βˆ‚π‘Ÿ8 βˆ‚π‘Ÿ8 PROPN iajs-3461 245 5 ( ( PUNCT iajs-3461 245 6 a2 a2 PROPN iajs-3461 245 7 , , PUNCT iajs-3461 245 8 π‘Ÿ8 π‘Ÿ8 ADJ iajs-3461 245 9 ) ) PUNCT iajs-3461 245 10 = = SYM iajs-3461 245 11 ( ( PUNCT iajs-3461 245 12 0 0 NUM iajs-3461 245 13 , , PUNCT iajs-3461 245 14 0 0 NUM iajs-3461 245 15 , , PUNCT iajs-3461 245 16 0 0 NUM iajs-3461 245 17 ) ) PUNCT iajs-3461 245 18 𝑇 𝑇 PROPN iajs-3461 245 19 . . PUNCT iajs-3461 246 1 that that PRON iajs-3461 246 2 's be AUX iajs-3461 246 3 why why SCONJ iajs-3461 246 4 ( ( PUNCT iajs-3461 246 5 Γ§[2 Γ§[2 NOUN iajs-3461 246 6 ] ] X iajs-3461 246 7 ) ) PUNCT iajs-3461 246 8 𝑇 𝑇 PROPN iajs-3461 246 9 π‘“π‘Ÿ8(a2 π‘“π‘Ÿ8(a2 PROPN iajs-3461 246 10 , , PUNCT iajs-3461 246 11 οΏ½ οΏ½ NOUN iajs-3461 246 12 ̈ ̈ X iajs-3461 246 13 οΏ½ οΏ½ PROPN iajs-3461 246 14 8) 8) NUM iajs-3461 246 15 = = SYM iajs-3461 246 16 0 0 NUM iajs-3461 246 17 . . PUNCT iajs-3461 246 18 ihjpas ihjpas PROPN iajs-3461 246 19 . . PUNCT iajs-3461 247 1 2025 2025 NUM iajs-3461 247 2 , , PUNCT iajs-3461 247 3 38 38 NUM iajs-3461 247 4 ( ( PUNCT iajs-3461 247 5 1 1 NUM iajs-3461 247 6 ) ) PUNCT iajs-3461 247 7 376 376 NUM iajs-3461 247 8 therefore9 therefore9 NOUN iajs-3461 247 9 , , PUNCT iajs-3461 247 10 according accord VERB iajs-3461 247 11 to to ADP iajs-3461 247 12 sotomayor9 sotomayor9 PROPN iajs-3461 247 13 ’s ’s PART iajs-3461 247 14 theorem theorem NOUN iajs-3461 247 15 there there PRON iajs-3461 247 16 can can AUX iajs-3461 247 17 be be AUX iajs-3461 247 18 no no DET iajs-3461 247 19 saddle saddle NOUN iajs-3461 247 20 - - PUNCT iajs-3461 247 21 node node NOUN iajs-3461 247 22 bifurcation bifurcation NOUN iajs-3461 247 23 . . PUNCT iajs-3461 248 1 although although SCONJ iajs-3461 248 2 the the DET iajs-3461 248 3 first first ADJ iajs-3461 248 4 need need NOUN iajs-3461 248 5 for for ADP iajs-3461 248 6 transcritical transcritical ADJ iajs-3461 248 7 bifurcation bifurcation NOUN iajs-3461 248 8 has have AUX iajs-3461 248 9 been be AUX iajs-3461 248 10 satisfied satisfied ADJ iajs-3461 248 11 . . PUNCT iajs-3461 249 1 now now ADV iajs-3461 249 2 , , PUNCT iajs-3461 249 3 since since SCONJ iajs-3461 249 4 π·π‘“π‘Ÿ8 π·π‘“π‘Ÿ8 PROPN iajs-3461 249 5 ( ( PUNCT iajs-3461 249 6 𝐴2 𝐴2 PROPN iajs-3461 249 7 , , PUNCT iajs-3461 249 8 οΏ½ οΏ½ PROPN iajs-3461 249 9 ̈ ̈ X iajs-3461 249 10 οΏ½ οΏ½ PROPN iajs-3461 249 11 8) 8) NUM iajs-3461 249 12 οΏ½ οΏ½ NOUN iajs-3461 249 13 Μ‡ Μ‡ NOUN iajs-3461 249 14 οΏ½ οΏ½ PROPN iajs-3461 250 1 [ [ X iajs-3461 250 2 2 2 NUM iajs-3461 250 3 ] ] X iajs-3461 250 4 = = PUNCT iajs-3461 250 5 ( ( PUNCT iajs-3461 250 6 0 0 NUM iajs-3461 250 7 0 0 NUM iajs-3461 250 8 0 0 NUM iajs-3461 250 9 0 0 NUM iajs-3461 250 10 0 0 NUM iajs-3461 250 11 0 0 NUM iajs-3461 250 12 0 0 NUM iajs-3461 250 13 0 0 NUM iajs-3461 250 14 βˆ’ βˆ’ NOUN iajs-3461 250 15 1 1 NUM iajs-3461 250 16 ) ) PUNCT iajs-3461 250 17 ( ( PUNCT iajs-3461 250 18 € € SYM iajs-3461 250 19 1 1 NUM iajs-3461 250 20 οΏ½ οΏ½ NOUN iajs-3461 250 21 Μ‡ Μ‡ NOUN iajs-3461 250 22 οΏ½ οΏ½ NOUN iajs-3461 250 23 2 2 NUM iajs-3461 250 24 [ [ SYM iajs-3461 250 25 2 2 NUM iajs-3461 250 26 ] ] PUNCT iajs-3461 250 27 οΏ½ οΏ½ PROPN iajs-3461 250 28 Μ‡ Μ‡ NOUN iajs-3461 250 29 οΏ½ οΏ½ PROPN iajs-3461 250 30 2 2 NUM iajs-3461 250 31 [ [ X iajs-3461 250 32 2 2 NUM iajs-3461 250 33 ] ] SYM iajs-3461 250 34 € € SYM iajs-3461 250 35 2 2 NUM iajs-3461 250 36 οΏ½ οΏ½ NOUN iajs-3461 250 37 Μ‡ Μ‡ NOUN iajs-3461 250 38 οΏ½ οΏ½ NOUN iajs-3461 250 39 2 2 NUM iajs-3461 250 40 [ [ X iajs-3461 250 41 2 2 NUM iajs-3461 250 42 ] ] PUNCT iajs-3461 250 43 ) ) PUNCT iajs-3461 251 1 = = PUNCT iajs-3461 251 2 ( ( PUNCT iajs-3461 251 3 0 0 NUM iajs-3461 251 4 0 0 NUM iajs-3461 251 5 0 0 NUM iajs-3461 251 6 βˆ’β‚¬2 βˆ’β‚¬2 PROPN iajs-3461 251 7 οΏ½ οΏ½ PROPN iajs-3461 251 8 Μ‡ Μ‡ NOUN iajs-3461 251 9 οΏ½ οΏ½ NOUN iajs-3461 251 10 2 2 NUM iajs-3461 251 11 [ [ X iajs-3461 251 12 2 2 NUM iajs-3461 251 13 ] ] PUNCT iajs-3461 251 14 ) ) PUNCT iajs-3461 251 15 ( ( PUNCT iajs-3461 251 16 Γ§[2 Γ§[2 NOUN iajs-3461 251 17 ] ] X iajs-3461 251 18 ) ) PUNCT iajs-3461 251 19 𝑇 𝑇 PROPN iajs-3461 252 1 [ [ X iajs-3461 252 2 π·π‘“π‘Ÿ8 π·π‘“π‘Ÿ8 PROPN iajs-3461 252 3 ( ( PUNCT iajs-3461 252 4 𝐴2 𝐴2 PROPN iajs-3461 252 5 , , PUNCT iajs-3461 252 6 οΏ½ οΏ½ PROPN iajs-3461 252 7 ̈ ̈ X iajs-3461 252 8 οΏ½ οΏ½ PROPN iajs-3461 252 9 8) 8) NUM iajs-3461 252 10 οΏ½ οΏ½ NOUN iajs-3461 252 11 Μ‡ Μ‡ NOUN iajs-3461 252 12 οΏ½ οΏ½ PROPN iajs-3461 253 1 [ [ X iajs-3461 253 2 2 2 NUM iajs-3461 253 3 ] ] X iajs-3461 253 4 ] ] X iajs-3461 253 5 = = SYM iajs-3461 253 6 ( ( PUNCT iajs-3461 253 7 € € SYM iajs-3461 253 8 3Γ§2 3Γ§2 NUM iajs-3461 253 9 [ [ X iajs-3461 253 10 2 2 NUM iajs-3461 253 11 ] ] PUNCT iajs-3461 253 12 , , PUNCT iajs-3461 253 13 Γ§2 Γ§2 PROPN iajs-3461 253 14 [ [ X iajs-3461 253 15 2 2 NUM iajs-3461 253 16 ] ] PUNCT iajs-3461 253 17 , , PUNCT iajs-3461 253 18 € € SYM iajs-3461 253 19 4Γ§2 4Γ§2 NUM iajs-3461 253 20 [ [ X iajs-3461 253 21 2 2 NUM iajs-3461 253 22 ] ] NUM iajs-3461 253 23 ) ) PUNCT iajs-3461 253 24 ( ( PUNCT iajs-3461 253 25 0 0 NUM iajs-3461 253 26 0 0 NUM iajs-3461 253 27 βˆ’β‚¬2 βˆ’β‚¬2 PROPN iajs-3461 253 28 οΏ½ οΏ½ PROPN iajs-3461 253 29 Μ‡ Μ‡ NOUN iajs-3461 253 30 οΏ½ οΏ½ NOUN iajs-3461 253 31 2 2 NUM iajs-3461 253 32 [ [ X iajs-3461 253 33 2 2 NUM iajs-3461 253 34 ] ] PUNCT iajs-3461 253 35 ) ) PUNCT iajs-3461 254 1 = = SYM iajs-3461 255 1 βˆ’β‚¬2€4Γ§2 βˆ’β‚¬2€4Γ§2 PROPN iajs-3461 255 2 [ [ PUNCT iajs-3461 255 3 2] 2] NUM iajs-3461 255 4 οΏ½ οΏ½ NOUN iajs-3461 255 5 Μ‡ Μ‡ NOUN iajs-3461 255 6 οΏ½ οΏ½ NOUN iajs-3461 255 7 2 2 NUM iajs-3461 255 8 [ [ X iajs-3461 255 9 2 2 NUM iajs-3461 255 10 ] ] PUNCT iajs-3461 255 11 β‰  β‰  PROPN iajs-3461 255 12 0 0 X iajs-3461 255 13 . . PUNCT iajs-3461 256 1 moreover moreover ADV iajs-3461 256 2 , , PUNCT iajs-3461 256 3 by by ADP iajs-3461 256 4 4substituting 4substituting NUM iajs-3461 256 5 οΏ½ οΏ½ PROPN iajs-3461 256 6 Μ‡ Μ‡ NOUN iajs-3461 256 7 οΏ½ οΏ½ PROPN iajs-3461 256 8 [2 [2 X iajs-3461 256 9 ] ] PUNCT iajs-3461 256 10 in in ADP iajs-3461 256 11 ( ( PUNCT iajs-3461 256 12 23 23 NUM iajs-3461 256 13 ) ) PUNCT iajs-3461 256 14 , , PUNCT iajs-3461 256 15 we we PRON iajs-3461 256 16 get get VERB iajs-3461 256 17 : : PUNCT iajs-3461 256 18 𝐷2𝑓(𝐴2 𝐷2𝑓(𝐴2 PROPN iajs-3461 256 19 , , PUNCT iajs-3461 256 20 οΏ½ οΏ½ PROPN iajs-3461 256 21 ̈ ̈ X iajs-3461 256 22 οΏ½ οΏ½ PROPN iajs-3461 256 23 8)( 8)( NUM iajs-3461 256 24 οΏ½ οΏ½ NOUN iajs-3461 256 25 Μ‡ Μ‡ NOUN iajs-3461 256 26 οΏ½ οΏ½ PROPN iajs-3461 257 1 [ [ X iajs-3461 257 2 2 2 NUM iajs-3461 257 3 ] ] PUNCT iajs-3461 257 4 , , PUNCT iajs-3461 257 5 οΏ½ οΏ½ PROPN iajs-3461 257 6 Μ‡ Μ‡ PROPN iajs-3461 257 7 οΏ½ οΏ½ PROPN iajs-3461 257 8 [2 [2 X iajs-3461 257 9 ] ] X iajs-3461 257 10 ) ) PUNCT iajs-3461 257 11 = = PUNCT iajs-3461 257 12 [ [ PUNCT iajs-3461 257 13 βˆ’2€1( βˆ’2€1( PROPN iajs-3461 257 14 οΏ½ οΏ½ PROPN iajs-3461 257 15 Μ‡ Μ‡ NOUN iajs-3461 257 16 οΏ½ οΏ½ PROPN iajs-3461 257 17 2 2 NUM iajs-3461 257 18 [ [ X iajs-3461 257 19 2])2 2])2 NUM iajs-3461 257 20 ( ( PUNCT iajs-3461 257 21 € € SYM iajs-3461 257 22 1 1 NUM iajs-3461 257 23 + + NUM iajs-3461 257 24 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 257 25 + + CCONJ iajs-3461 257 26 π‘Ÿ2€2 π‘Ÿ2€2 NUM iajs-3461 257 27 ) ) PUNCT iajs-3461 257 28 2( 2( NUM iajs-3461 257 29 οΏ½ οΏ½ PROPN iajs-3461 257 30 Μ‡ Μ‡ NOUN iajs-3461 257 31 οΏ½ οΏ½ NOUN iajs-3461 257 32 2 2 NUM iajs-3461 257 33 [ [ X iajs-3461 257 34 2])2(π‘Ÿ3€1 2])2(π‘Ÿ3€1 NUM iajs-3461 257 35 βˆ’ βˆ’ NOUN iajs-3461 257 36 π‘Ÿ4€2 π‘Ÿ4€2 NUM iajs-3461 257 37 ) ) PUNCT iajs-3461 257 38 2€2( 2€2( NOUN iajs-3461 257 39 οΏ½ οΏ½ PROPN iajs-3461 257 40 Μ‡ Μ‡ NOUN iajs-3461 257 41 οΏ½ οΏ½ NOUN iajs-3461 257 42 2 2 NUM iajs-3461 257 43 [ [ SYM iajs-3461 257 44 2])2(π‘Ÿ6€1 2])2(π‘Ÿ6€1 NUM iajs-3461 257 45 + + CCONJ iajs-3461 257 46 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 257 47 ) ) PUNCT iajs-3461 257 48 ] ] PUNCT iajs-3461 257 49 . . PUNCT iajs-3461 258 1 hence hence ADV iajs-3461 258 2 , , PUNCT iajs-3461 258 3 it it PRON iajs-3461 258 4 is be AUX iajs-3461 258 5 4obtained 4obtained NUM iajs-3461 258 6 that that SCONJ iajs-3461 258 7 : : PUNCT iajs-3461 258 8 ( ( PUNCT iajs-3461 258 9 € € SYM iajs-3461 258 10 3Γ§2 3Γ§2 NUM iajs-3461 258 11 [ [ X iajs-3461 258 12 2 2 NUM iajs-3461 258 13 ] ] PUNCT iajs-3461 258 14 , , PUNCT iajs-3461 258 15 Γ§2 Γ§2 PROPN iajs-3461 258 16 [ [ X iajs-3461 258 17 2 2 NUM iajs-3461 258 18 ] ] PUNCT iajs-3461 258 19 , , PUNCT iajs-3461 258 20 € € SYM iajs-3461 258 21 4Γ§2 4Γ§2 NUM iajs-3461 258 22 [ [ X iajs-3461 258 23 2 2 NUM iajs-3461 258 24 ] ] NUM iajs-3461 258 25 ) ) PUNCT iajs-3461 258 26 ( ( PUNCT iajs-3461 258 27 Γ§[2 Γ§[2 NOUN iajs-3461 258 28 ] ] X iajs-3461 258 29 ) ) PUNCT iajs-3461 258 30 𝑇 𝑇 PROPN iajs-3461 258 31 𝐷2𝑓(𝐴2 𝐷2𝑓(𝐴2 PROPN iajs-3461 258 32 , , PUNCT iajs-3461 258 33 οΏ½ οΏ½ PROPN iajs-3461 258 34 ̈ ̈ X iajs-3461 258 35 οΏ½ οΏ½ PROPN iajs-3461 258 36 8)( 8)( NUM iajs-3461 258 37 οΏ½ οΏ½ NOUN iajs-3461 258 38 Μ‡ Μ‡ VERB iajs-3461 258 39 οΏ½ οΏ½ PROPN iajs-3461 259 1 [ [ X iajs-3461 259 2 2 2 NUM iajs-3461 259 3 ] ] PUNCT iajs-3461 259 4 , , PUNCT iajs-3461 259 5 οΏ½ οΏ½ PROPN iajs-3461 259 6 Μ‡ Μ‡ PROPN iajs-3461 259 7 οΏ½ οΏ½ PROPN iajs-3461 259 8 [2 [2 X iajs-3461 259 9 ] ] X iajs-3461 259 10 ) ) PUNCT iajs-3461 259 11 = = PUNCT iajs-3461 260 1 2πœ—( 2πœ—( NUM iajs-3461 260 2 οΏ½ οΏ½ NOUN iajs-3461 260 3 Μ‡ Μ‡ VERB iajs-3461 260 4 οΏ½ οΏ½ NOUN iajs-3461 260 5 2 2 NUM iajs-3461 260 6 [ [ X iajs-3461 260 7 2])2Γ§2 2])2Γ§2 NUM iajs-3461 261 1 [ [ X iajs-3461 261 2 2 2 NUM iajs-3461 261 3 ] ] PUNCT iajs-3461 261 4 where where SCONJ iajs-3461 261 5 , , PUNCT iajs-3461 261 6 € € SYM iajs-3461 261 7 1 1 NUM iajs-3461 261 8 , , PUNCT iajs-3461 261 9 € € SYM iajs-3461 261 10 3 3 NUM iajs-3461 261 11 π‘Žπ‘›π‘‘ π‘Žπ‘›π‘‘ ADJ iajs-3461 261 12 € € SYM iajs-3461 261 13 4 4 NUM iajs-3461 261 14 are be AUX iajs-3461 261 15 mentioned mention VERB iajs-3461 261 16 in in ADP iajs-3461 261 17 the the DET iajs-3461 261 18 theorem theorem NOUN iajs-3461 261 19 's 's PART iajs-3461 261 20 state state NOUN iajs-3461 261 21 . . PUNCT iajs-3461 262 1 as as ADP iajs-3461 262 2 a a DET iajs-3461 262 3 result result NOUN iajs-3461 262 4 of of ADP iajs-3461 262 5 condition condition NOUN iajs-3461 262 6 ( ( PUNCT iajs-3461 262 7 26 26 NUM iajs-3461 262 8 ) ) PUNCT iajs-3461 262 9 , , PUNCT iajs-3461 262 10 we we PRON iajs-3461 262 11 get get VERB iajs-3461 262 12 that that PRON iajs-3461 262 13 : : PUNCT iajs-3461 262 14 ( ( PUNCT iajs-3461 262 15 Γ§[2 Γ§[2 NOUN iajs-3461 262 16 ] ] X iajs-3461 262 17 ) ) PUNCT iajs-3461 262 18 𝑇 𝑇 PROPN iajs-3461 262 19 𝐷2𝑓(𝐴2 𝐷2𝑓(𝐴2 PROPN iajs-3461 262 20 , , PUNCT iajs-3461 262 21 οΏ½ οΏ½ PROPN iajs-3461 262 22 ̈ ̈ X iajs-3461 262 23 οΏ½ οΏ½ PROPN iajs-3461 262 24 8)( 8)( NUM iajs-3461 262 25 οΏ½ οΏ½ NOUN iajs-3461 262 26 Μ‡ Μ‡ VERB iajs-3461 262 27 οΏ½ οΏ½ PROPN iajs-3461 263 1 [ [ X iajs-3461 263 2 2 2 NUM iajs-3461 263 3 ] ] PUNCT iajs-3461 263 4 , , PUNCT iajs-3461 263 5 οΏ½ οΏ½ PROPN iajs-3461 263 6 Μ‡ Μ‡ PROPN iajs-3461 263 7 οΏ½ οΏ½ PROPN iajs-3461 263 8 [2 [2 X iajs-3461 263 9 ] ] X iajs-3461 263 10 ) ) PUNCT iajs-3461 263 11 β‰  β‰  PROPN iajs-3461 263 12 0 0 NUM iajs-3461 263 13 . . PUNCT iajs-3461 264 1 thus thus ADV iajs-3461 264 2 , , PUNCT iajs-3461 264 3 4according 4accorde VERB iajs-3461 264 4 to to ADP iajs-3461 264 5 4sotomayor 4sotomayor NUM iajs-3461 264 6 ’s ’s PART iajs-3461 264 7 4theorem 4theorem NUM iajs-3461 264 8 system system NOUN iajs-3461 264 9 ( ( PUNCT iajs-3461 264 10 2 2 X iajs-3461 264 11 ) ) PUNCT iajs-3461 264 12 has have VERB iajs-3461 264 13 a a DET iajs-3461 264 14 4transcritical 4transcritical NUM iajs-3461 264 15 bifurcation4 bifurcation4 NOUN iajs-3461 264 16 at at ADP iajs-3461 264 17 the the DET iajs-3461 264 18 4equilibrium 4equilibrium PROPN iajs-3461 264 19 point point NOUN iajs-3461 264 20 𝐴2 𝐴2 NOUN iajs-3461 264 21 = = SYM iajs-3461 264 22 ( ( PUNCT iajs-3461 264 23 h h NOUN iajs-3461 264 24 βˆ’ βˆ’ PROPN iajs-3461 264 25 , , PUNCT iajs-3461 264 26 𝑖 𝑖 PROPN iajs-3461 264 27 βˆ’ βˆ’ PROPN iajs-3461 264 28 , , PUNCT iajs-3461 264 29 0 0 NUM iajs-3461 264 30 ) ) PUNCT iajs-3461 264 31 with with ADP iajs-3461 264 32 the the DET iajs-3461 264 33 4parameter 4parameter PROPN iajs-3461 264 34 οΏ½ οΏ½ PROPN iajs-3461 264 35 ̈ ̈ X iajs-3461 264 36 οΏ½ οΏ½ NOUN iajs-3461 264 37 8 8 NUM iajs-3461 264 38 = = SYM iajs-3461 264 39 π‘Ÿ6β„Ž π‘Ÿ6β„Ž X iajs-3461 264 40 βˆ’ βˆ’ NOUN iajs-3461 265 1 + + CCONJ iajs-3461 265 2 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 265 3 𝑖 𝑖 SYM iajs-3461 265 4 βˆ’ βˆ’ PROPN iajs-3461 265 5 . . PUNCT iajs-3461 266 1 4.3 4.3 NUM iajs-3461 266 2 local4 local4 PROPN iajs-3461 266 3 bifurcation4 bifurcation4 PROPN iajs-3461 266 4 analysis4 analysis4 NOUN iajs-3461 266 5 near near ADP iajs-3461 266 6 π‘¨πŸ‘ π‘¨πŸ‘ NOUN iajs-3461 266 7 = = SYM iajs-3461 266 8 ( ( PUNCT iajs-3461 266 9 οΏ½ οΏ½ PROPN iajs-3461 266 10 ΜΏ ΜΏ PROPN iajs-3461 266 11 οΏ½ οΏ½ PROPN iajs-3461 266 12 , , PUNCT iajs-3461 266 13 𝟎 𝟎 NUM iajs-3461 266 14 , , PUNCT iajs-3461 266 15 𝒋 𝒋 X iajs-3461 266 16 ΜΏ ΜΏ NOUN iajs-3461 266 17 ): ): PUNCT iajs-3461 266 18 theorem theorem NOUN iajs-3461 266 19 ( ( PUNCT iajs-3461 266 20 4 4 NUM iajs-3461 266 21 ): ): PUNCT iajs-3461 266 22 assume assume VERB iajs-3461 266 23 that that SCONJ iajs-3461 266 24 the the DET iajs-3461 266 25 following follow VERB iajs-3461 266 26 criteria criterion NOUN iajs-3461 266 27 are be AUX iajs-3461 266 28 fulfilled fulfil VERB iajs-3461 266 29 : : PUNCT iajs-3461 266 30 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 266 31 > > X iajs-3461 266 32 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 266 33 β„Ž β„Ž ADP iajs-3461 266 34 ΜΏ ΜΏ NOUN iajs-3461 266 35 βˆ’ βˆ’ PROPN iajs-3461 266 36 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 266 37 𝑗 𝑗 PROPN iajs-3461 266 38 ΜΏ ΜΏ X iajs-3461 266 39 , , PUNCT iajs-3461 266 40 ( ( PUNCT iajs-3461 266 41 27 27 NUM iajs-3461 266 42 ) ) PUNCT iajs-3461 266 43 νœ€2Μ€ νœ€2Μ€ NOUN iajs-3461 266 44 β‰  β‰  PROPN iajs-3461 266 45 0 0 NUM iajs-3461 266 46 , , PUNCT iajs-3461 266 47 ( ( PUNCT iajs-3461 266 48 28 28 X iajs-3461 266 49 ) ) PUNCT iajs-3461 267 1 ℝ ℝ PROPN iajs-3461 267 2 β‰  β‰  PROPN iajs-3461 267 3 0 0 NUM iajs-3461 267 4 ( ( PUNCT iajs-3461 267 5 29 29 NUM iajs-3461 267 6 ) ) PUNCT iajs-3461 267 7 where where SCONJ iajs-3461 267 8 : : PUNCT iajs-3461 267 9 νœ€1Μ€ νœ€1Μ€ NOUN iajs-3461 267 10 = = SYM iajs-3461 267 11 π‘Ÿ1π‘Ÿ6β„ŽΜΏ π‘Ÿ1π‘Ÿ6β„ŽΜΏ NUM iajs-3461 267 12 + + CCONJ iajs-3461 267 13 π‘Ÿ7𝑗(ΜΏ1 π‘Ÿ7𝑗(ΜΏ1 PROPN iajs-3461 267 14 βˆ’ βˆ’ NOUN iajs-3461 267 15 2β„ŽΜΏ 2β„ŽΜΏ NUM iajs-3461 267 16 βˆ’ βˆ’ NOUN iajs-3461 267 17 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 267 18 ) ) PUNCT iajs-3461 267 19 β„ŽΜΏ[π‘Ÿ2 β„ŽΜΏ[π‘Ÿ2 NOUN iajs-3461 267 20 + + CCONJ iajs-3461 267 21 π‘Ÿ1(π‘Ÿ6β„ŽΜΏ π‘Ÿ1(π‘Ÿ6β„ŽΜΏ X iajs-3461 267 22 βˆ’ βˆ’ ADP iajs-3461 267 23 π‘Ÿ8 π‘Ÿ8 NOUN iajs-3461 267 24 ) ) PUNCT iajs-3461 267 25 ] ] PUNCT iajs-3461 267 26 , , PUNCT iajs-3461 267 27 νœ€2Μ€ νœ€2Μ€ NOUN iajs-3461 267 28 = = NOUN iajs-3461 267 29 1 1 NUM iajs-3461 267 30 βˆ’ βˆ’ NOUN iajs-3461 267 31 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 267 32 𝑗 𝑗 X iajs-3461 267 33 ΜΏ ΜΏ X iajs-3461 267 34 βˆ’ βˆ’ X iajs-3461 267 35 ( ( PUNCT iajs-3461 267 36 2 2 NUM iajs-3461 267 37 βˆ’ βˆ’ PROPN iajs-3461 267 38 π‘Ÿ2νœ€1Μ€)β„ŽΜΏ π‘Ÿ2νœ€1Μ€)β„ŽΜΏ NUM iajs-3461 267 39 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 267 40 β„ŽΜΏ β„ŽΜΏ X iajs-3461 267 41 , , PUNCT iajs-3461 267 42 νœ€3Μ€ νœ€3Μ€ NOUN iajs-3461 267 43 = = SYM iajs-3461 267 44 π‘Ÿ7𝑗̿ π‘Ÿ7𝑗̿ X iajs-3461 267 45 π‘Ÿ1β„ŽΜΏ π‘Ÿ1β„ŽΜΏ NOUN iajs-3461 267 46 , , PUNCT iajs-3461 267 47 ℝ ℝ PROPN iajs-3461 267 48 = = SYM iajs-3461 267 49 Γ§3 Γ§3 PROPN iajs-3461 268 1 [ [ X iajs-3461 268 2 3][€2(π‘Ÿ6€1 3][€2(π‘Ÿ6€1 NUM iajs-3461 268 3 + + CCONJ iajs-3461 268 4 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 268 5 ) ) PUNCT iajs-3461 268 6 βˆ’ βˆ’ ADP iajs-3461 268 7 € € SYM iajs-3461 268 8 1€3(€1 1€3(€1 NUM iajs-3461 268 9 + + CCONJ iajs-3461 268 10 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 268 11 + + CCONJ iajs-3461 268 12 π‘Ÿ2€2 π‘Ÿ2€2 NUM iajs-3461 268 13 ) ) PUNCT iajs-3461 268 14 ] ] PUNCT iajs-3461 269 1 + + CCONJ iajs-3461 269 2 Γ§2 Γ§2 PROPN iajs-3461 269 3 [ [ X iajs-3461 269 4 3][π‘Ÿ3€1 3][π‘Ÿ3€1 NUM iajs-3461 269 5 βˆ’ βˆ’ PROPN iajs-3461 269 6 π‘Ÿ4€2 π‘Ÿ4€2 PROPN iajs-3461 269 7 ] ] PUNCT iajs-3461 269 8 . . PUNCT iajs-3461 270 1 then then ADV iajs-3461 270 2 system system NOUN iajs-3461 270 3 ( ( PUNCT iajs-3461 270 4 2 2 NUM iajs-3461 270 5 ) ) PUNCT iajs-3461 270 6 at at ADP iajs-3461 270 7 the the DET iajs-3461 270 8 4equilibrium 4equilibrium PROPN iajs-3461 270 9 point point NOUN iajs-3461 270 10 𝐴3 𝐴3 PROPN iajs-3461 270 11 = = SYM iajs-3461 270 12 ( ( PUNCT iajs-3461 270 13 β„ŽΜΏ β„ŽΜΏ ADV iajs-3461 270 14 , , PUNCT iajs-3461 270 15 0 0 NUM iajs-3461 270 16 , , PUNCT iajs-3461 270 17 𝑗 𝑗 PROPN iajs-3461 270 18 ΜΏ ΜΏ NOUN iajs-3461 270 19 ) ) PUNCT iajs-3461 270 20 with with ADP iajs-3461 270 21 the the DET iajs-3461 270 22 4parameter 4parameter NUM iajs-3461 270 23 value value NOUN iajs-3461 270 24 : : PUNCT iajs-3461 270 25 οΏ½ οΏ½ PROPN iajs-3461 270 26 ̈ ̈ X iajs-3461 270 27 οΏ½ οΏ½ NOUN iajs-3461 270 28 4 4 NUM iajs-3461 270 29 = = SYM iajs-3461 270 30 π‘Ÿ3β„ŽΜΏ π‘Ÿ3β„ŽΜΏ NOUN iajs-3461 270 31 βˆ’ βˆ’ PROPN iajs-3461 270 32 π‘Ÿ4𝑗 π‘Ÿ4𝑗 PROPN iajs-3461 270 33 ΜΏ ΜΏ PROPN iajs-3461 270 34 has have VERB iajs-3461 270 35 a a DET iajs-3461 270 36 transcritical transcritical ADJ iajs-3461 270 37 bifurcation bifurcation NOUN iajs-3461 270 38 , , PUNCT iajs-3461 270 39 but but CCONJ iajs-3461 270 40 a a DET iajs-3461 270 41 saddle4 saddle4 NOUN iajs-3461 270 42 βˆ’ βˆ’ NOUN iajs-3461 270 43 node node NOUN iajs-3461 270 44 can can AUX iajs-3461 270 45 not not PART iajs-3461 270 46 4occur 4occur NUM iajs-3461 270 47 at at ADP iajs-3461 270 48 𝐴3 𝐴3 PROPN iajs-3461 270 49 . . PUNCT iajs-3461 271 1 proof proof NOUN iajs-3461 271 2 : : PUNCT iajs-3461 271 3 the the DET iajs-3461 271 4 characteristic characteristic ADJ iajs-3461 271 5 equation equation NOUN iajs-3461 271 6 represented represent VERB iajs-3461 271 7 by by ADP iajs-3461 271 8 eq eq PROPN iajs-3461 271 9 . . PUNCT iajs-3461 272 1 ( ( PUNCT iajs-3461 272 2 3 3 X iajs-3461 272 3 . . X iajs-3461 272 4 π‘Ž π‘Ž X iajs-3461 272 5 ) ) PUNCT iajs-3461 272 6 of of ADP iajs-3461 272 7 𝑠ystem 𝑠ystem NOUN iajs-3461 272 8 ( ( PUNCT iajs-3461 272 9 2 2 NUM iajs-3461 272 10 ) ) PUNCT iajs-3461 272 11 at at ADP iajs-3461 272 12 the the DET iajs-3461 272 13 equilibrium equilibrium NOUN iajs-3461 272 14 point point NOUN iajs-3461 272 15 𝐴3 𝐴3 PROPN iajs-3461 272 16 has have VERB iajs-3461 272 17 zeroeigenvalue zeroeigenvalue X iajs-3461 272 18 ( ( PUNCT iajs-3461 272 19 say say INTJ iajs-3461 272 20 πœ†3𝑖 πœ†3𝑖 PUNCT iajs-3461 272 21 = = NOUN iajs-3461 272 22 0 0 X iajs-3461 272 23 ) ) PUNCT iajs-3461 272 24 at at ADP iajs-3461 272 25 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 272 26 = = SYM iajs-3461 272 27 οΏ½ οΏ½ PROPN iajs-3461 272 28 ̈ ̈ X iajs-3461 272 29 οΏ½ οΏ½ NOUN iajs-3461 272 30 4 4 NUM iajs-3461 272 31 and and CCONJ iajs-3461 272 32 the the DET iajs-3461 272 33 jacobian jacobian ADJ iajs-3461 272 34 matrix matrix NOUN iajs-3461 272 35 𝐽3 𝐽3 NOUN iajs-3461 272 36 with with ADP iajs-3461 272 37 4parameter 4parameter NUM iajs-3461 272 38 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 272 39 = = SYM iajs-3461 272 40 οΏ½ οΏ½ PROPN iajs-3461 272 41 ̈ ̈ X iajs-3461 272 42 οΏ½ οΏ½ NOUN iajs-3461 272 43 4 4 NUM iajs-3461 272 44 becomes become VERB iajs-3461 272 45 : : PUNCT iajs-3461 272 46 ihjpas ihjpas PROPN iajs-3461 272 47 . . PUNCT iajs-3461 273 1 2025 2025 NUM iajs-3461 273 2 , , PUNCT iajs-3461 273 3 38 38 NUM iajs-3461 273 4 ( ( PUNCT iajs-3461 273 5 1 1 NUM iajs-3461 273 6 ) ) PUNCT iajs-3461 273 7 377 377 NUM iajs-3461 273 8 𝐽3̈ 𝐽3̈ NOUN iajs-3461 273 9 = = SYM iajs-3461 273 10 𝐽3(π‘Ÿ4 𝐽3(π‘Ÿ4 PROPN iajs-3461 273 11 = = SYM iajs-3461 273 12 οΏ½ οΏ½ PROPN iajs-3461 273 13 ̈ ̈ X iajs-3461 273 14 οΏ½ οΏ½ NOUN iajs-3461 273 15 4 4 NUM iajs-3461 273 16 ) ) PUNCT iajs-3461 273 17 = = PUNCT iajs-3461 274 1 [ [ X iajs-3461 274 2 οΏ½ οΏ½ PROPN iajs-3461 274 3 Μƒ Μƒ NOUN iajs-3461 274 4 οΏ½ οΏ½ PROPN iajs-3461 274 5 𝑖𝑗 𝑖𝑗 NOUN iajs-3461 274 6 ] ] X iajs-3461 274 7 3Γ—3 3Γ—3 NUM iajs-3461 274 8 , , PUNCT iajs-3461 274 9 where where SCONJ iajs-3461 274 10 : : PUNCT iajs-3461 274 11 οΏ½ οΏ½ PROPN iajs-3461 274 12 Μƒ Μƒ NOUN iajs-3461 274 13 οΏ½ οΏ½ PROPN iajs-3461 274 14 𝑖𝑗 𝑖𝑗 NOUN iajs-3461 274 15 = = SYM iajs-3461 274 16 πœ‚π‘–π‘— πœ‚π‘–π‘— PROPN iajs-3461 274 17 for for ADP iajs-3461 274 18 all all DET iajs-3461 274 19 i i PRON iajs-3461 274 20 , , PUNCT iajs-3461 274 21 j j PROPN iajs-3461 274 22 = = SYM iajs-3461 274 23 1,2,3 1,2,3 NUM iajs-3461 274 24 except except SCONJ iajs-3461 274 25 οΏ½ οΏ½ PROPN iajs-3461 274 26 Μƒ Μƒ PROPN iajs-3461 274 27 οΏ½ οΏ½ PROPN iajs-3461 274 28 𝑖𝑗 𝑖𝑗 NOUN iajs-3461 274 29 = = PUNCT iajs-3461 274 30 π‘Ÿ3β„ŽΜΏ π‘Ÿ3β„ŽΜΏ PROPN iajs-3461 274 31 βˆ’ βˆ’ PROPN iajs-3461 274 32 οΏ½ οΏ½ PROPN iajs-3461 274 33 ̈ ̈ X iajs-3461 274 34 οΏ½ οΏ½ PROPN iajs-3461 274 35 4𝑗̿ 4𝑗̿ NUM iajs-3461 274 36 βˆ’ βˆ’ PROPN iajs-3461 274 37 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 274 38 = = SYM iajs-3461 274 39 0 0 X iajs-3461 274 40 . . PUNCT iajs-3461 275 1 let let VERB iajs-3461 275 2 οΏ½ οΏ½ PROPN iajs-3461 275 3 Μ‡ Μ‡ VERB iajs-3461 275 4 οΏ½ οΏ½ PROPN iajs-3461 275 5 [3 [3 NOUN iajs-3461 275 6 ] ] X iajs-3461 275 7 = = SYM iajs-3461 275 8 ( ( PUNCT iajs-3461 275 9 ( ( PUNCT iajs-3461 275 10 οΏ½ οΏ½ NOUN iajs-3461 275 11 Μ‡ Μ‡ NOUN iajs-3461 275 12 οΏ½ οΏ½ PROPN iajs-3461 275 13 2 2 NUM iajs-3461 276 1 [ [ X iajs-3461 276 2 3 3 NUM iajs-3461 276 3 ] ] PUNCT iajs-3461 276 4 , , PUNCT iajs-3461 276 5 οΏ½ οΏ½ PROPN iajs-3461 276 6 Μ‡ Μ‡ PROPN iajs-3461 276 7 οΏ½ οΏ½ PROPN iajs-3461 276 8 2 2 NUM iajs-3461 276 9 [ [ X iajs-3461 276 10 3 3 NUM iajs-3461 276 11 ] ] PUNCT iajs-3461 276 12 , , PUNCT iajs-3461 276 13 οΏ½ οΏ½ PROPN iajs-3461 276 14 Μ‡ Μ‡ PROPN iajs-3461 276 15 οΏ½ οΏ½ PROPN iajs-3461 276 16 3 3 NUM iajs-3461 276 17 [ [ X iajs-3461 276 18 3 3 NUM iajs-3461 276 19 ] ] PUNCT iajs-3461 276 20 ) ) PUNCT iajs-3461 277 1 𝑇 𝑇 PROPN iajs-3461 277 2 be be VERB iajs-3461 277 3 the the DET iajs-3461 277 4 eigenvector eigenvector NOUN iajs-3461 277 5 which which PRON iajs-3461 277 6 follows follow VERB iajs-3461 277 7 the the DET iajs-3461 277 8 eigenvalue eigenvalue NOUN iajs-3461 277 9 be be AUX iajs-3461 277 10 the the DET iajs-3461 277 11 eigenvector eigenvector NOUN iajs-3461 277 12 which which PRON iajs-3461 277 13 follows follow VERB iajs-3461 277 14 the the DET iajs-3461 277 15 eigenvalue eigenvalue NOUN iajs-3461 277 16 . . PUNCT iajs-3461 277 17 πœ†3𝑠 πœ†3𝑠 PROPN iajs-3461 278 1 = = PUNCT iajs-3461 278 2 0 0 PROPN iajs-3461 278 3 . . PUNCT iajs-3461 279 1 thus thus ADV iajs-3461 279 2 ( ( PUNCT iajs-3461 279 3 𝐽3̈ 𝐽3̈ PROPN iajs-3461 279 4 βˆ’ βˆ’ PROPN iajs-3461 279 5 πœ†3𝑠𝐼 πœ†3𝑠𝐼 PROPN iajs-3461 279 6 ) ) PUNCT iajs-3461 279 7 οΏ½ οΏ½ PROPN iajs-3461 279 8 Μ‡ Μ‡ VERB iajs-3461 279 9 οΏ½ οΏ½ PROPN iajs-3461 280 1 [ [ X iajs-3461 280 2 3 3 NUM iajs-3461 280 3 ] ] X iajs-3461 280 4 = = SYM iajs-3461 280 5 0 0 NUM iajs-3461 280 6 , , PUNCT iajs-3461 280 7 which which PRON iajs-3461 280 8 gives give VERB iajs-3461 280 9 : : PUNCT iajs-3461 280 10 οΏ½ οΏ½ PROPN iajs-3461 280 11 Μ‡ Μ‡ PROPN iajs-3461 280 12 οΏ½ οΏ½ PROPN iajs-3461 280 13 [3 [3 NOUN iajs-3461 280 14 ] ] X iajs-3461 280 15 = = SYM iajs-3461 280 16 ( ( PUNCT iajs-3461 280 17 οΏ½ οΏ½ PROPN iajs-3461 280 18 Μ‡ Μ‡ NOUN iajs-3461 280 19 οΏ½ οΏ½ PROPN iajs-3461 280 20 1 1 NUM iajs-3461 280 21 [ [ X iajs-3461 280 22 3 3 NUM iajs-3461 280 23 ] ] PUNCT iajs-3461 280 24 , , PUNCT iajs-3461 280 25 νœ€2Μ€ νœ€2Μ€ PROPN iajs-3461 280 26 οΏ½ οΏ½ PROPN iajs-3461 280 27 Μ‡ Μ‡ PROPN iajs-3461 280 28 οΏ½ οΏ½ PROPN iajs-3461 280 29 1 1 NUM iajs-3461 280 30 [ [ X iajs-3461 280 31 3 3 NUM iajs-3461 280 32 ] ] PUNCT iajs-3461 280 33 , , PUNCT iajs-3461 280 34 νœ€1Μ€ νœ€1Μ€ NOUN iajs-3461 280 35 οΏ½ οΏ½ VERB iajs-3461 280 36 Μ‡ Μ‡ VERB iajs-3461 280 37 οΏ½ οΏ½ PROPN iajs-3461 280 38 1 1 NUM iajs-3461 280 39 [ [ X iajs-3461 280 40 3 3 NUM iajs-3461 280 41 ] ] SYM iajs-3461 280 42 ) ) PUNCT iajs-3461 280 43 𝑇 𝑇 PROPN iajs-3461 280 44 , , PUNCT iajs-3461 280 45 where where SCONJ iajs-3461 280 46 οΏ½ οΏ½ PROPN iajs-3461 280 47 Μ‡ Μ‡ VERB iajs-3461 280 48 οΏ½ οΏ½ PROPN iajs-3461 280 49 1 1 NUM iajs-3461 280 50 [ [ X iajs-3461 280 51 3 3 NUM iajs-3461 280 52 ] ] PUNCT iajs-3461 280 53 any any DET iajs-3461 280 54 nonzero4 nonzero4 NOUN iajs-3461 280 55 number number NOUN iajs-3461 280 56 with with ADP iajs-3461 280 57 , , PUNCT iajs-3461 280 58 νœ€1Μ€ νœ€1Μ€ NOUN iajs-3461 280 59 which which PRON iajs-3461 280 60 are be AUX iajs-3461 280 61 mentioned4 mentioned4 X iajs-3461 280 62 in in ADP iajs-3461 280 63 the the DET iajs-3461 280 64 state4 state4 NOUN iajs-3461 280 65 of of ADP iajs-3461 280 66 the the DET iajs-3461 280 67 theorem theorem PROPN iajs-3461 280 68 . . PUNCT iajs-3461 280 69 let let VERB iajs-3461 280 70 Γ§[3 Γ§[3 PROPN iajs-3461 280 71 ] ] X iajs-3461 281 1 = = SYM iajs-3461 281 2 ( ( PUNCT iajs-3461 281 3 Γ§1 Γ§1 NOUN iajs-3461 281 4 [ [ X iajs-3461 281 5 3 3 NUM iajs-3461 281 6 ] ] PUNCT iajs-3461 281 7 , , PUNCT iajs-3461 281 8 Γ§2 Γ§2 PROPN iajs-3461 281 9 [ [ X iajs-3461 281 10 3 3 NUM iajs-3461 281 11 ] ] PUNCT iajs-3461 281 12 , , PUNCT iajs-3461 281 13 Γ§3 Γ§3 PROPN iajs-3461 281 14 [ [ X iajs-3461 281 15 3 3 NUM iajs-3461 281 16 ] ] PUNCT iajs-3461 281 17 ) ) PUNCT iajs-3461 281 18 𝑇 𝑇 NOUN iajs-3461 281 19 become become VERB iajs-3461 281 20 the the DET iajs-3461 281 21 eigenvector eigenvector NOUN iajs-3461 281 22 linked link VERB iajs-3461 281 23 to to ADP iajs-3461 281 24 the the DET iajs-3461 281 25 eigenvalue eigenvalue PROPN iajs-3461 281 26 πœ†3𝑠 πœ†3𝑠 PROPN iajs-3461 281 27 = = PROPN iajs-3461 281 28 0 0 NUM iajs-3461 281 29 of of ADP iajs-3461 281 30 the the DET iajs-3461 281 31 matrix matrix NOUN iajs-3461 281 32 𝐽3̈. 𝐽3̈. NOUN iajs-3461 282 1 then then ADV iajs-3461 282 2 we we PRON iajs-3461 282 3 have have VERB iajs-3461 282 4 ( ( PUNCT iajs-3461 282 5 𝐽3̈ 𝐽3̈ PROPN iajs-3461 282 6 𝑇 𝑇 PROPN iajs-3461 282 7 βˆ’ βˆ’ PROPN iajs-3461 282 8 πœ†3𝑖𝐼 πœ†3𝑖𝐼 NOUN iajs-3461 282 9 ) ) PUNCT iajs-3461 282 10 Γ§ Γ§ PUNCT iajs-3461 283 1 [ [ X iajs-3461 283 2 3 3 X iajs-3461 283 3 ] ] X iajs-3461 283 4 = = SYM iajs-3461 283 5 0 0 X iajs-3461 283 6 . . PUNCT iajs-3461 283 7 by by ADP iajs-3461 283 8 solving solve VERB iajs-3461 283 9 this this DET iajs-3461 283 10 equation equation NOUN iajs-3461 283 11 for for ADP iajs-3461 283 12 Γ§[3 Γ§[3 PROPN iajs-3461 283 13 ] ] PUNCT iajs-3461 283 14 , , PUNCT iajs-3461 283 15 we we PRON iajs-3461 283 16 obtain obtain VERB iajs-3461 283 17 : : PUNCT iajs-3461 283 18 Γ§[3 Γ§[3 X iajs-3461 283 19 ] ] X iajs-3461 284 1 = = SYM iajs-3461 284 2 ( ( PUNCT iajs-3461 284 3 νœ€3Μ€ νœ€3Μ€ NOUN iajs-3461 284 4 Γ§3 Γ§3 PROPN iajs-3461 285 1 [ [ X iajs-3461 285 2 3 3 NUM iajs-3461 285 3 ] ] PUNCT iajs-3461 285 4 , , PUNCT iajs-3461 285 5 Γ§2 Γ§2 PROPN iajs-3461 285 6 [ [ X iajs-3461 285 7 3 3 NUM iajs-3461 285 8 ] ] PUNCT iajs-3461 285 9 , , PUNCT iajs-3461 285 10 Γ§3 Γ§3 PROPN iajs-3461 285 11 [ [ X iajs-3461 285 12 3 3 NUM iajs-3461 285 13 ] ] PUNCT iajs-3461 285 14 ) ) PUNCT iajs-3461 285 15 𝑇 𝑇 PROPN iajs-3461 285 16 where where SCONJ iajs-3461 285 17 Γ§3 Γ§3 PROPN iajs-3461 285 18 [ [ X iajs-3461 285 19 3 3 X iajs-3461 285 20 ] ] X iajs-3461 285 21 any any DET iajs-3461 285 22 nonzero nonzero PROPN iajs-3461 285 23 number9 number9 PROPN iajs-3461 285 24 with with ADP iajs-3461 285 25 νœ€3Μ€ νœ€3Μ€ PROPN iajs-3461 285 26 those those PRON iajs-3461 285 27 are be AUX iajs-3461 285 28 referred refer VERB iajs-3461 285 29 to to ADP iajs-3461 285 30 in in ADP iajs-3461 285 31 the the DET iajs-3461 285 32 theorem theorem NOUN iajs-3461 285 33 's 's PART iajs-3461 285 34 state state NOUN iajs-3461 285 35 . . PUNCT iajs-3461 286 1 now now ADV iajs-3461 286 2 , , PUNCT iajs-3461 286 3 consider consider VERB iajs-3461 286 4 : : PUNCT iajs-3461 286 5 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 286 6 βˆ‚π‘Ÿ4 βˆ‚π‘Ÿ4 PROPN iajs-3461 286 7 ( ( PUNCT iajs-3461 286 8 Γ½ Γ½ PROPN iajs-3461 286 9 , , PUNCT iajs-3461 286 10 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 286 11 ) ) PUNCT iajs-3461 286 12 = = SYM iajs-3461 286 13 π‘“π‘Ÿ4(Γ½ π‘“π‘Ÿ4(Γ½ NOUN iajs-3461 286 14 , , PUNCT iajs-3461 286 15 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 286 16 ) ) PUNCT iajs-3461 286 17 = = SYM iajs-3461 286 18 ( ( PUNCT iajs-3461 286 19 βˆ‚π‘“1 βˆ‚π‘“1 PROPN iajs-3461 286 20 βˆ‚π‘Ÿ4 βˆ‚π‘Ÿ4 PROPN iajs-3461 286 21 , , PUNCT iajs-3461 286 22 βˆ‚π‘“2 βˆ‚π‘“2 X iajs-3461 286 23 βˆ‚π‘Ÿ4 βˆ‚π‘Ÿ4 PROPN iajs-3461 286 24 , , PUNCT iajs-3461 286 25 βˆ‚π‘“3 βˆ‚π‘“3 VERB iajs-3461 286 26 βˆ‚π‘Ÿ4 βˆ‚π‘Ÿ4 PROPN iajs-3461 286 27 ) ) PUNCT iajs-3461 286 28 t t PROPN iajs-3461 287 1 = = SYM iajs-3461 287 2 ( ( PUNCT iajs-3461 287 3 0,βˆ’π‘–π‘— 0,βˆ’π‘–π‘— NUM iajs-3461 287 4 , , PUNCT iajs-3461 287 5 0)𝑇 0)𝑇 PUNCT iajs-3461 288 1 so so ADV iajs-3461 288 2 , , PUNCT iajs-3461 288 3 βˆ‚π‘“ βˆ‚π‘“ PROPN iajs-3461 288 4 βˆ‚π‘Ÿ4 βˆ‚π‘Ÿ4 PROPN iajs-3461 288 5 ( ( PUNCT iajs-3461 288 6 a3 a3 NOUN iajs-3461 288 7 , , PUNCT iajs-3461 288 8 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 288 9 ) ) PUNCT iajs-3461 288 10 = = SYM iajs-3461 288 11 ( ( PUNCT iajs-3461 288 12 0 0 NUM iajs-3461 288 13 , , PUNCT iajs-3461 288 14 0 0 NUM iajs-3461 288 15 , , PUNCT iajs-3461 288 16 0 0 NUM iajs-3461 288 17 ) ) PUNCT iajs-3461 288 18 𝑇 𝑇 PROPN iajs-3461 288 19 . . PUNCT iajs-3461 289 1 and and CCONJ iajs-3461 289 2 hence hence ADV iajs-3461 289 3 ( ( PUNCT iajs-3461 289 4 Γ§[3 Γ§[3 PROPN iajs-3461 289 5 ] ] SYM iajs-3461 289 6 ) ) PUNCT iajs-3461 289 7 𝑇 𝑇 PROPN iajs-3461 289 8 π‘“π‘Ÿ4(a3 π‘“π‘Ÿ4(a3 PROPN iajs-3461 289 9 , , PUNCT iajs-3461 289 10 οΏ½ οΏ½ PROPN iajs-3461 289 11 ̈ ̈ X iajs-3461 289 12 οΏ½ οΏ½ NOUN iajs-3461 289 13 4 4 NUM iajs-3461 289 14 ) ) PUNCT iajs-3461 289 15 = = SYM iajs-3461 290 1 0 0 X iajs-3461 290 2 . . PUNCT iajs-3461 291 1 therefore therefore ADV iajs-3461 291 2 , , PUNCT iajs-3461 291 3 the the DET iajs-3461 291 4 saddle saddle NOUN iajs-3461 291 5 - - PUNCT iajs-3461 291 6 node node NOUN iajs-3461 291 7 bifurcation bifurcation NOUN iajs-3461 291 8 is be AUX iajs-3461 291 9 ruled rule VERB iajs-3461 291 10 out out ADP iajs-3461 291 11 by by ADP iajs-3461 291 12 sotomayor sotomayor PROPN iajs-3461 291 13 's 's PART iajs-3461 291 14 theorem theorem NOUN iajs-3461 291 15 . . PUNCT iajs-3461 292 1 while while SCONJ iajs-3461 292 2 the the DET iajs-3461 292 3 first first ADJ iajs-3461 292 4 transcritical transcritical ADJ iajs-3461 292 5 bifurcation bifurcation NOUN iajs-3461 292 6 condition condition NOUN iajs-3461 292 7 has have AUX iajs-3461 292 8 been be AUX iajs-3461 292 9 satisfied satisfied ADJ iajs-3461 292 10 . . PUNCT iajs-3461 293 1 now now ADV iajs-3461 293 2 , , PUNCT iajs-3461 293 3 since since SCONJ iajs-3461 293 4 οΏ½ οΏ½ PROPN iajs-3461 293 5 Μ‡ Μ‡ VERB iajs-3461 293 6 οΏ½ οΏ½ PROPN iajs-3461 293 7 [3 [3 NOUN iajs-3461 293 8 ] ] X iajs-3461 293 9 = = SYM iajs-3461 294 1 [ [ PUNCT iajs-3461 294 2 οΏ½ οΏ½ NOUN iajs-3461 294 3 Μ‡ Μ‡ NOUN iajs-3461 294 4 οΏ½ οΏ½ PROPN iajs-3461 294 5 1 1 NUM iajs-3461 294 6 [ [ X iajs-3461 294 7 3 3 NUM iajs-3461 294 8 ] ] PUNCT iajs-3461 294 9 , , PUNCT iajs-3461 294 10 νœ€2Μ€ νœ€2Μ€ PROPN iajs-3461 294 11 οΏ½ οΏ½ PROPN iajs-3461 294 12 Μ‡ Μ‡ PROPN iajs-3461 294 13 οΏ½ οΏ½ PROPN iajs-3461 294 14 1 1 NUM iajs-3461 294 15 [ [ X iajs-3461 294 16 3 3 NUM iajs-3461 294 17 ] ] PUNCT iajs-3461 294 18 , , PUNCT iajs-3461 294 19 νœ€1Μ€ νœ€1Μ€ NOUN iajs-3461 294 20 οΏ½ οΏ½ VERB iajs-3461 294 21 Μ‡ Μ‡ VERB iajs-3461 294 22 οΏ½ οΏ½ PROPN iajs-3461 294 23 1 1 NUM iajs-3461 295 1 [ [ X iajs-3461 295 2 3 3 NUM iajs-3461 295 3 ] ] X iajs-3461 295 4 ] ] X iajs-3461 295 5 π·π‘“π‘Ÿ4 π·π‘“π‘Ÿ4 PROPN iajs-3461 295 6 ( ( PUNCT iajs-3461 295 7 𝐴3 𝐴3 PROPN iajs-3461 295 8 , , PUNCT iajs-3461 295 9 οΏ½ οΏ½ PROPN iajs-3461 295 10 ̈ ̈ X iajs-3461 295 11 οΏ½ οΏ½ PROPN iajs-3461 295 12 4) 4) NUM iajs-3461 295 13 οΏ½ οΏ½ PROPN iajs-3461 295 14 Μ‡ Μ‡ VERB iajs-3461 295 15 οΏ½ οΏ½ PROPN iajs-3461 296 1 [ [ X iajs-3461 296 2 3 3 NUM iajs-3461 296 3 ] ] X iajs-3461 296 4 = = PUNCT iajs-3461 296 5 ( ( PUNCT iajs-3461 296 6 0 0 NUM iajs-3461 296 7 0 0 NUM iajs-3461 296 8 0 0 NUM iajs-3461 296 9 0 0 NUM iajs-3461 296 10 βˆ’ βˆ’ NOUN iajs-3461 296 11 𝑗 𝑗 INTJ iajs-3461 296 12 0 0 NUM iajs-3461 296 13 0 0 NUM iajs-3461 296 14 0 0 NUM iajs-3461 296 15 0 0 NUM iajs-3461 296 16 ) ) PUNCT iajs-3461 296 17 ( ( PUNCT iajs-3461 296 18 οΏ½ οΏ½ PROPN iajs-3461 296 19 Μ‡ Μ‡ PROPN iajs-3461 296 20 οΏ½ οΏ½ PROPN iajs-3461 296 21 1 1 NUM iajs-3461 296 22 [ [ X iajs-3461 296 23 3 3 NUM iajs-3461 296 24 ] ] PUNCT iajs-3461 296 25 νœ€2Μ€ νœ€2Μ€ PROPN iajs-3461 296 26 οΏ½ οΏ½ PROPN iajs-3461 296 27 Μ‡ Μ‡ PROPN iajs-3461 296 28 οΏ½ οΏ½ PROPN iajs-3461 296 29 1 1 NUM iajs-3461 296 30 [ [ X iajs-3461 296 31 3 3 NUM iajs-3461 296 32 ] ] SYM iajs-3461 296 33 νœ€1Μ€ νœ€1Μ€ NOUN iajs-3461 296 34 οΏ½ οΏ½ PROPN iajs-3461 296 35 Μ‡ Μ‡ VERB iajs-3461 296 36 οΏ½ οΏ½ PROPN iajs-3461 296 37 1 1 NUM iajs-3461 296 38 [ [ X iajs-3461 296 39 3 3 NUM iajs-3461 296 40 ] ] PUNCT iajs-3461 296 41 ) ) PUNCT iajs-3461 297 1 = = SYM iajs-3461 297 2 ( ( PUNCT iajs-3461 297 3 0 0 NUM iajs-3461 297 4 βˆ’π‘— βˆ’π‘— PROPN iajs-3461 297 5 νœ€2Μ€ νœ€2Μ€ PROPN iajs-3461 297 6 οΏ½ οΏ½ PROPN iajs-3461 297 7 Μ‡ Μ‡ PROPN iajs-3461 297 8 οΏ½ οΏ½ PROPN iajs-3461 297 9 1 1 NUM iajs-3461 297 10 [ [ X iajs-3461 297 11 3 3 NUM iajs-3461 297 12 ] ] SYM iajs-3461 297 13 0 0 NUM iajs-3461 297 14 ) ) PUNCT iajs-3461 297 15 ( ( PUNCT iajs-3461 297 16 Γ§[2 Γ§[2 NOUN iajs-3461 297 17 ] ] X iajs-3461 297 18 ) ) PUNCT iajs-3461 297 19 𝑇 𝑇 PROPN iajs-3461 298 1 [ [ X iajs-3461 298 2 π·π‘“π‘Ÿ4 π·π‘“π‘Ÿ4 X iajs-3461 298 3 ( ( PUNCT iajs-3461 298 4 𝐴2 𝐴2 PROPN iajs-3461 298 5 , , PUNCT iajs-3461 298 6 οΏ½ οΏ½ PROPN iajs-3461 298 7 ̈ ̈ X iajs-3461 298 8 οΏ½ οΏ½ PROPN iajs-3461 298 9 4) 4) NUM iajs-3461 298 10 οΏ½ οΏ½ PROPN iajs-3461 298 11 Μ‡ Μ‡ VERB iajs-3461 298 12 οΏ½ οΏ½ PROPN iajs-3461 299 1 [ [ X iajs-3461 299 2 2 2 NUM iajs-3461 299 3 ] ] X iajs-3461 299 4 ] ] X iajs-3461 300 1 = = SYM iajs-3461 300 2 ( ( PUNCT iajs-3461 300 3 € € SYM iajs-3461 300 4 3Γ§3 3Γ§3 NUM iajs-3461 300 5 [ [ X iajs-3461 300 6 3 3 NUM iajs-3461 300 7 ] ] PUNCT iajs-3461 300 8 , , PUNCT iajs-3461 300 9 Γ§2 Γ§2 PROPN iajs-3461 300 10 [ [ X iajs-3461 300 11 3 3 NUM iajs-3461 300 12 ] ] PUNCT iajs-3461 300 13 , , PUNCT iajs-3461 300 14 Γ§3 Γ§3 PROPN iajs-3461 301 1 [ [ X iajs-3461 301 2 3 3 NUM iajs-3461 301 3 ] ] SYM iajs-3461 301 4 ) ) PUNCT iajs-3461 301 5 ( ( PUNCT iajs-3461 301 6 0 0 NUM iajs-3461 301 7 βˆ’π‘— βˆ’π‘— PROPN iajs-3461 301 8 νœ€2Μ€ νœ€2Μ€ PROPN iajs-3461 301 9 οΏ½ οΏ½ PROPN iajs-3461 301 10 Μ‡ Μ‡ PROPN iajs-3461 301 11 οΏ½ οΏ½ PROPN iajs-3461 301 12 1 1 NUM iajs-3461 301 13 [ [ X iajs-3461 301 14 3 3 NUM iajs-3461 301 15 ] ] PUNCT iajs-3461 301 16 0 0 NUM iajs-3461 301 17 ) ) PUNCT iajs-3461 302 1 = = SYM iajs-3461 302 2 βˆ’νœ€2Μ€ βˆ’νœ€2Μ€ X iajs-3461 302 3 Γ§2 Γ§2 NOUN iajs-3461 302 4 [ [ X iajs-3461 302 5 3]𝑗 3]𝑗 PROPN iajs-3461 302 6 ΜΏ ΜΏ PROPN iajs-3461 302 7 οΏ½ οΏ½ PROPN iajs-3461 302 8 Μ‡ Μ‡ PROPN iajs-3461 302 9 οΏ½ οΏ½ PROPN iajs-3461 302 10 1 1 NUM iajs-3461 302 11 [ [ X iajs-3461 302 12 3 3 NUM iajs-3461 302 13 ] ] PUNCT iajs-3461 302 14 . . PUNCT iajs-3461 303 1 so so ADV iajs-3461 303 2 , , PUNCT iajs-3461 303 3 by by ADP iajs-3461 303 4 condition condition NOUN iajs-3461 303 5 ( ( PUNCT iajs-3461 303 6 28 28 NUM iajs-3461 303 7 ) ) PUNCT iajs-3461 303 8 , , PUNCT iajs-3461 303 9 we we PRON iajs-3461 303 10 obtain obtain VERB iajs-3461 303 11 that that PRON iajs-3461 303 12 : : PUNCT iajs-3461 303 13 ( ( PUNCT iajs-3461 303 14 Γ§[2 Γ§[2 NOUN iajs-3461 303 15 ] ] X iajs-3461 303 16 ) ) PUNCT iajs-3461 303 17 𝑇 𝑇 PROPN iajs-3461 304 1 [ [ X iajs-3461 304 2 π·π‘“π‘Ÿ4 π·π‘“π‘Ÿ4 X iajs-3461 304 3 ( ( PUNCT iajs-3461 304 4 𝐴2 𝐴2 PROPN iajs-3461 304 5 , , PUNCT iajs-3461 304 6 οΏ½ οΏ½ PROPN iajs-3461 304 7 ̈ ̈ X iajs-3461 304 8 οΏ½ οΏ½ PROPN iajs-3461 304 9 4) 4) NUM iajs-3461 304 10 οΏ½ οΏ½ PROPN iajs-3461 304 11 Μ‡ Μ‡ VERB iajs-3461 304 12 οΏ½ οΏ½ PROPN iajs-3461 305 1 [ [ X iajs-3461 305 2 2 2 NUM iajs-3461 305 3 ] ] X iajs-3461 305 4 ] ] X iajs-3461 305 5 β‰  β‰  PROPN iajs-3461 305 6 0 0 PUNCT iajs-3461 306 1 moreover moreover ADV iajs-3461 306 2 , , PUNCT iajs-3461 306 3 by by ADP iajs-3461 306 4 4substituting 4substituting NUM iajs-3461 306 5 οΏ½ οΏ½ PROPN iajs-3461 306 6 Μ‡ Μ‡ NOUN iajs-3461 306 7 οΏ½ οΏ½ NOUN iajs-3461 306 8 [3 [3 NOUN iajs-3461 306 9 ] ] PUNCT iajs-3461 306 10 in in ADP iajs-3461 306 11 ( ( PUNCT iajs-3461 306 12 23 23 NUM iajs-3461 306 13 ) ) PUNCT iajs-3461 306 14 , , PUNCT iajs-3461 306 15 we we PRON iajs-3461 306 16 get get VERB iajs-3461 306 17 : : PUNCT iajs-3461 306 18 𝐷2𝑓(𝐴3 𝐷2𝑓(𝐴3 PROPN iajs-3461 306 19 , , PUNCT iajs-3461 306 20 οΏ½ οΏ½ PROPN iajs-3461 306 21 ̈ ̈ SYM iajs-3461 306 22 οΏ½ οΏ½ PROPN iajs-3461 306 23 4)( 4)( PROPN iajs-3461 306 24 οΏ½ οΏ½ PROPN iajs-3461 306 25 Μ‡ Μ‡ NOUN iajs-3461 306 26 οΏ½ οΏ½ PROPN iajs-3461 307 1 [ [ X iajs-3461 307 2 3 3 NUM iajs-3461 307 3 ] ] PUNCT iajs-3461 307 4 , , PUNCT iajs-3461 307 5 οΏ½ οΏ½ PROPN iajs-3461 307 6 Μ‡ Μ‡ PROPN iajs-3461 307 7 οΏ½ οΏ½ PROPN iajs-3461 307 8 [3 [3 NOUN iajs-3461 307 9 ] ] X iajs-3461 307 10 ) ) PUNCT iajs-3461 307 11 = = PUNCT iajs-3461 307 12 [ [ PUNCT iajs-3461 307 13 βˆ’2€1( βˆ’2€1( PROPN iajs-3461 307 14 οΏ½ οΏ½ PROPN iajs-3461 307 15 Μ‡ Μ‡ NOUN iajs-3461 307 16 οΏ½ οΏ½ PROPN iajs-3461 307 17 2 2 NUM iajs-3461 307 18 [ [ X iajs-3461 307 19 2])2 2])2 NUM iajs-3461 307 20 ( ( PUNCT iajs-3461 307 21 € € SYM iajs-3461 307 22 1 1 NUM iajs-3461 307 23 + + NUM iajs-3461 307 24 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 307 25 + + CCONJ iajs-3461 307 26 π‘Ÿ2€2 π‘Ÿ2€2 NUM iajs-3461 307 27 ) ) PUNCT iajs-3461 307 28 2( 2( NUM iajs-3461 307 29 οΏ½ οΏ½ PROPN iajs-3461 307 30 Μ‡ Μ‡ NOUN iajs-3461 307 31 οΏ½ οΏ½ NOUN iajs-3461 307 32 2 2 NUM iajs-3461 307 33 [ [ X iajs-3461 307 34 2])2(π‘Ÿ3€1 2])2(π‘Ÿ3€1 NUM iajs-3461 307 35 βˆ’ βˆ’ NOUN iajs-3461 307 36 π‘Ÿ4€2 π‘Ÿ4€2 NUM iajs-3461 307 37 ) ) PUNCT iajs-3461 307 38 2€2( 2€2( NOUN iajs-3461 307 39 οΏ½ οΏ½ PROPN iajs-3461 307 40 Μ‡ Μ‡ NOUN iajs-3461 307 41 οΏ½ οΏ½ NOUN iajs-3461 307 42 2 2 NUM iajs-3461 307 43 [ [ SYM iajs-3461 307 44 2])2(π‘Ÿ6€1 2])2(π‘Ÿ6€1 NUM iajs-3461 307 45 + + CCONJ iajs-3461 307 46 π‘Ÿ7 π‘Ÿ7 NUM iajs-3461 307 47 ) ) PUNCT iajs-3461 307 48 ] ] PUNCT iajs-3461 307 49 . . PUNCT iajs-3461 308 1 hence hence ADV iajs-3461 308 2 , , PUNCT iajs-3461 308 3 it it PRON iajs-3461 308 4 is be AUX iajs-3461 308 5 4obtained 4obtained NUM iajs-3461 308 6 that that SCONJ iajs-3461 308 7 : : PUNCT iajs-3461 308 8 ( ( PUNCT iajs-3461 308 9 € € SYM iajs-3461 308 10 3Γ§3 3Γ§3 NUM iajs-3461 308 11 [ [ X iajs-3461 308 12 3 3 NUM iajs-3461 308 13 ] ] PUNCT iajs-3461 308 14 , , PUNCT iajs-3461 308 15 Γ§2 Γ§2 PROPN iajs-3461 308 16 [ [ X iajs-3461 308 17 3 3 NUM iajs-3461 308 18 ] ] PUNCT iajs-3461 308 19 , , PUNCT iajs-3461 308 20 Γ§3 Γ§3 PROPN iajs-3461 308 21 [ [ X iajs-3461 308 22 3 3 NUM iajs-3461 308 23 ] ] PUNCT iajs-3461 308 24 ) ) PUNCT iajs-3461 308 25 ihjpas ihjpas PROPN iajs-3461 308 26 . . PUNCT iajs-3461 309 1 2025 2025 NUM iajs-3461 309 2 , , PUNCT iajs-3461 309 3 38 38 NUM iajs-3461 309 4 ( ( PUNCT iajs-3461 309 5 1 1 NUM iajs-3461 309 6 ) ) PUNCT iajs-3461 309 7 378 378 NUM iajs-3461 309 8 ( ( PUNCT iajs-3461 309 9 Γ§[2 Γ§[2 NOUN iajs-3461 309 10 ] ] X iajs-3461 309 11 ) ) PUNCT iajs-3461 309 12 𝑇 𝑇 PROPN iajs-3461 309 13 𝐷2𝑓(𝐴3 𝐷2𝑓(𝐴3 PROPN iajs-3461 309 14 , , PUNCT iajs-3461 309 15 οΏ½ οΏ½ PROPN iajs-3461 309 16 ̈ ̈ SYM iajs-3461 309 17 οΏ½ οΏ½ PROPN iajs-3461 309 18 4)( 4)( PROPN iajs-3461 309 19 οΏ½ οΏ½ PROPN iajs-3461 309 20 Μ‡ Μ‡ NOUN iajs-3461 309 21 οΏ½ οΏ½ PROPN iajs-3461 310 1 [ [ X iajs-3461 310 2 3 3 NUM iajs-3461 310 3 ] ] PUNCT iajs-3461 310 4 , , PUNCT iajs-3461 310 5 οΏ½ οΏ½ PROPN iajs-3461 310 6 Μ‡ Μ‡ PROPN iajs-3461 310 7 οΏ½ οΏ½ PROPN iajs-3461 310 8 [3])=2( [3])=2( PROPN iajs-3461 310 9 οΏ½ οΏ½ PROPN iajs-3461 310 10 Μ‡ Μ‡ NOUN iajs-3461 310 11 οΏ½ οΏ½ PROPN iajs-3461 310 12 2 2 NUM iajs-3461 311 1 [ [ X iajs-3461 311 2 2])2 2])2 NUM iajs-3461 311 3 ℝ ℝ NOUN iajs-3461 311 4 where where SCONJ iajs-3461 311 5 , , PUNCT iajs-3461 311 6 € € SYM iajs-3461 311 7 1 1 NUM iajs-3461 311 8 , , PUNCT iajs-3461 311 9 € € SYM iajs-3461 311 10 2 2 NUM iajs-3461 311 11 , , PUNCT iajs-3461 311 12 € € SYM iajs-3461 311 13 3 3 NUM iajs-3461 311 14 and and CCONJ iajs-3461 311 15 ℝ ℝ PROPN iajs-3461 311 16 are be AUX iajs-3461 311 17 mentioned mention VERB iajs-3461 311 18 in in ADP iajs-3461 311 19 the the DET iajs-3461 311 20 state state NOUN iajs-3461 311 21 of of ADP iajs-3461 311 22 the the DET iajs-3461 311 23 theorem theorem NOUN iajs-3461 311 24 . . PROPN iajs-3461 312 1 as as ADP iajs-3461 312 2 a a DET iajs-3461 312 3 result result NOUN iajs-3461 312 4 of of ADP iajs-3461 312 5 condition condition NOUN iajs-3461 312 6 ( ( PUNCT iajs-3461 312 7 29 29 NUM iajs-3461 312 8 ) ) PUNCT iajs-3461 312 9 , , PUNCT iajs-3461 312 10 we we PRON iajs-3461 312 11 get get VERB iajs-3461 312 12 that that PRON iajs-3461 312 13 : : PUNCT iajs-3461 312 14 ( ( PUNCT iajs-3461 312 15 Γ§[2 Γ§[2 NOUN iajs-3461 312 16 ] ] X iajs-3461 312 17 ) ) PUNCT iajs-3461 312 18 𝑇 𝑇 PROPN iajs-3461 312 19 𝐷2𝑓(𝐴2 𝐷2𝑓(𝐴2 PROPN iajs-3461 312 20 , , PUNCT iajs-3461 312 21 οΏ½ οΏ½ PROPN iajs-3461 312 22 ̈ ̈ X iajs-3461 312 23 οΏ½ οΏ½ PROPN iajs-3461 312 24 8)( 8)( NUM iajs-3461 312 25 οΏ½ οΏ½ NOUN iajs-3461 312 26 Μ‡ Μ‡ VERB iajs-3461 312 27 οΏ½ οΏ½ PROPN iajs-3461 313 1 [ [ X iajs-3461 313 2 2 2 NUM iajs-3461 313 3 ] ] PUNCT iajs-3461 313 4 , , PUNCT iajs-3461 313 5 οΏ½ οΏ½ PROPN iajs-3461 313 6 Μ‡ Μ‡ PROPN iajs-3461 313 7 οΏ½ οΏ½ PROPN iajs-3461 313 8 [2 [2 X iajs-3461 313 9 ] ] X iajs-3461 313 10 ) ) PUNCT iajs-3461 313 11 β‰  β‰  PROPN iajs-3461 313 12 0 0 NUM iajs-3461 313 13 . . PUNCT iajs-3461 314 1 thus thus ADV iajs-3461 314 2 , , PUNCT iajs-3461 314 3 4according 4accorde VERB iajs-3461 314 4 to to ADP iajs-3461 314 5 4sotomayor 4sotomayor NUM iajs-3461 314 6 ’s ’s PART iajs-3461 314 7 4theorem 4theorem NUM iajs-3461 314 8 system system NOUN iajs-3461 314 9 ( ( PUNCT iajs-3461 314 10 2 2 X iajs-3461 314 11 ) ) PUNCT iajs-3461 314 12 has have VERB iajs-3461 314 13 a a DET iajs-3461 314 14 4transcritical 4transcritical NUM iajs-3461 314 15 bifurcation4 bifurcation4 NOUN iajs-3461 314 16 at at ADP iajs-3461 314 17 the the DET iajs-3461 314 18 4equilibrium 4equilibrium PROPN iajs-3461 314 19 point point NOUN iajs-3461 314 20 𝐴3 𝐴3 PROPN iajs-3461 314 21 = = SYM iajs-3461 314 22 ( ( PUNCT iajs-3461 314 23 β„ŽΜΏ β„ŽΜΏ ADV iajs-3461 314 24 , , PUNCT iajs-3461 314 25 0 0 NUM iajs-3461 314 26 , , PUNCT iajs-3461 314 27 𝑗 𝑗 PROPN iajs-3461 314 28 ΜΏ ΜΏ NOUN iajs-3461 314 29 ) ) PUNCT iajs-3461 314 30 with with ADP iajs-3461 314 31 the the DET iajs-3461 314 32 4parameter 4parameter PROPN iajs-3461 314 33 οΏ½ οΏ½ PROPN iajs-3461 314 34 ̈ ̈ X iajs-3461 314 35 οΏ½ οΏ½ NOUN iajs-3461 314 36 4 4 NUM iajs-3461 314 37 = = SYM iajs-3461 314 38 π‘Ÿ3β„ŽΜΏ π‘Ÿ3β„ŽΜΏ NOUN iajs-3461 314 39 βˆ’ βˆ’ PROPN iajs-3461 314 40 π‘Ÿ4𝑗.ΜΏ π‘Ÿ4𝑗.ΜΏ PROPN iajs-3461 314 41 4.4 4.4 NUM iajs-3461 314 42 local local ADJ iajs-3461 314 43 bifurcation4analysis bifurcation4analysis NOUN iajs-3461 314 44 near near ADP iajs-3461 314 45 π‘¨πŸ’ π‘¨πŸ’ ADJ iajs-3461 314 46 = = SYM iajs-3461 314 47 ( ( PUNCT iajs-3461 314 48 οΏ½ οΏ½ PROPN iajs-3461 314 49 Μƒ Μƒ PROPN iajs-3461 314 50 οΏ½ οΏ½ PROPN iajs-3461 314 51 , , PUNCT iajs-3461 314 52 οΏ½ οΏ½ PROPN iajs-3461 314 53 Μƒ Μƒ PROPN iajs-3461 314 54 οΏ½ οΏ½ PROPN iajs-3461 314 55 , , PUNCT iajs-3461 314 56 𝒋̃ 𝒋̃ PROPN iajs-3461 314 57 ): ): PUNCT iajs-3461 314 58 theorem theorem NOUN iajs-3461 314 59 ( ( PUNCT iajs-3461 314 60 5 5 NUM iajs-3461 314 61 ): ): PUNCT iajs-3461 314 62 assume assume VERB iajs-3461 314 63 that that SCONJ iajs-3461 314 64 the the DET iajs-3461 314 65 following follow VERB iajs-3461 314 66 criteria criterion NOUN iajs-3461 314 67 are be AUX iajs-3461 314 68 fulfilled fulfil VERB iajs-3461 314 69 : : PUNCT iajs-3461 314 70 € € SYM iajs-3461 314 71 1= 1= NUM iajs-3461 314 72 βˆ’π‘Ÿ3 βˆ’π‘Ÿ3 NOUN iajs-3461 314 73 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 314 74 , , PUNCT iajs-3461 314 75 € € SYM iajs-3461 314 76 2= 2= NUM iajs-3461 314 77 π‘Ÿ8βˆ’π‘Ÿ6hΜƒ π‘Ÿ8βˆ’π‘Ÿ6hΜƒ NOUN iajs-3461 314 78 βˆ’(π‘Ÿ7 βˆ’(π‘Ÿ7 VERB iajs-3461 314 79 οΏ½ οΏ½ PROPN iajs-3461 314 80 Μƒ Μƒ NOUN iajs-3461 314 81 οΏ½ οΏ½ NOUN iajs-3461 314 82 +𝒋̃€1 +𝒋̃€1 NOUN iajs-3461 314 83 ) ) PUNCT iajs-3461 314 84 π‘Ÿ6 π‘Ÿ6 PROPN iajs-3461 314 85 οΏ½ οΏ½ PROPN iajs-3461 314 86 Μƒ Μƒ PROPN iajs-3461 314 87 οΏ½ οΏ½ PROPN iajs-3461 314 88 , , PUNCT iajs-3461 314 89 € € SYM iajs-3461 314 90 3 3 NUM iajs-3461 314 91 = = SYM iajs-3461 314 92 βˆ’π‘Ÿ3 βˆ’π‘Ÿ3 PROPN iajs-3461 314 93 οΏ½ οΏ½ PROPN iajs-3461 314 94 Μƒ Μƒ PROPN iajs-3461 314 95 οΏ½ οΏ½ PROPN iajs-3461 314 96 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 314 97 jΜƒ jΜƒ PROPN iajs-3461 314 98 , , PUNCT iajs-3461 314 99 € € SYM iajs-3461 314 100 4= 4= NUM iajs-3461 314 101 ( ( PUNCT iajs-3461 314 102 π‘Ÿ6hΜƒ+π‘Ÿ7 π‘Ÿ6hΜƒ+π‘Ÿ7 PROPN iajs-3461 314 103 οΏ½ οΏ½ PROPN iajs-3461 314 104 Μƒ Μƒ PROPN iajs-3461 314 105 οΏ½ οΏ½ PROPN iajs-3461 314 106 βˆ’π‘Ÿ8 βˆ’π‘Ÿ8 PROPN iajs-3461 314 107 ) ) PUNCT iajs-3461 315 1 € € SYM iajs-3461 315 2 3βˆ’ 3βˆ’ NUM iajs-3461 315 3 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 315 4 𝒋̃ 𝒋̃ PROPN iajs-3461 315 5 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 315 6 hΜƒ hΜƒ PROPN iajs-3461 315 7 β‰  β‰  PROPN iajs-3461 315 8 0 0 NUM iajs-3461 315 9 , , PUNCT iajs-3461 315 10 ( ( PUNCT iajs-3461 315 11 30 30 NUM iajs-3461 315 12 ) ) SYM iajs-3461 315 13 1 1 NUM iajs-3461 315 14 = = SYM iajs-3461 315 15 2 2 NUM iajs-3461 315 16 hΜƒ hΜƒ PROPN iajs-3461 315 17 + + NUM iajs-3461 315 18 π‘Ÿ1𝑖̃ π‘Ÿ1𝑖̃ NOUN iajs-3461 315 19 + + CCONJ iajs-3461 315 20 π‘Ÿ2 π‘Ÿ2 ADJ iajs-3461 315 21 jΜƒ jΜƒ PROPN iajs-3461 315 22 , , PUNCT iajs-3461 315 23 ( ( PUNCT iajs-3461 315 24 31 31 NUM iajs-3461 315 25 ) ) PUNCT iajs-3461 315 26 € € SYM iajs-3461 315 27 2€4 2€4 NUM iajs-3461 315 28 [ [ X iajs-3461 315 29 € € SYM iajs-3461 315 30 2 2 NUM iajs-3461 315 31 + + NUM iajs-3461 315 32 π‘Ÿ1€1 π‘Ÿ1€1 NUM iajs-3461 315 33 + + CCONJ iajs-3461 315 34 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 315 35 ] ] PUNCT iajs-3461 315 36 β‰  β‰  PROPN iajs-3461 315 37 € € SYM iajs-3461 315 38 1[π‘Ÿ3€2 1[π‘Ÿ3€2 NOUN iajs-3461 315 39 + + CCONJ iajs-3461 315 40 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 315 41 ] ] X iajs-3461 315 42 + + CCONJ iajs-3461 315 43 € € SYM iajs-3461 315 44 4[π‘Ÿ6€2 4[π‘Ÿ6€2 NOUN iajs-3461 315 45 + + NOUN iajs-3461 315 46 π‘Ÿ7€1 π‘Ÿ7€1 PROPN iajs-3461 315 47 ] ] PUNCT iajs-3461 315 48 . . PUNCT iajs-3461 316 1 ( ( PUNCT iajs-3461 316 2 32 32 NUM iajs-3461 316 3 ) ) PUNCT iajs-3461 316 4 then then ADV iajs-3461 316 5 system system NOUN iajs-3461 316 6 ( ( PUNCT iajs-3461 316 7 2 2 NUM iajs-3461 316 8 ) ) PUNCT iajs-3461 316 9 at at ADP iajs-3461 316 10 the the DET iajs-3461 316 11 4equilibrium 4equilibrium PROPN iajs-3461 316 12 point point NOUN iajs-3461 316 13 𝐴4 𝐴4 NOUN iajs-3461 316 14 = = SYM iajs-3461 316 15 ( ( PUNCT iajs-3461 316 16 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 316 17 , , PUNCT iajs-3461 316 18 𝑖̃ 𝑖̃ PROPN iajs-3461 316 19 , , PUNCT iajs-3461 316 20 𝑗̃ 𝑗̃ PROPN iajs-3461 316 21 ) ) PUNCT iajs-3461 316 22 with with ADP iajs-3461 316 23 the the DET iajs-3461 316 24 4parameter 4parameter NUM iajs-3461 316 25 value value NOUN iajs-3461 316 26 : : PUNCT iajs-3461 316 27 π‘Ÿ1̈ π‘Ÿ1̈ NOUN iajs-3461 316 28 = = SYM iajs-3461 316 29 1 1 NUM iajs-3461 316 30 βˆ’ βˆ’ NUM iajs-3461 316 31 2 2 NUM iajs-3461 316 32 hΜƒ hΜƒ PROPN iajs-3461 316 33 βˆ’ βˆ’ NOUN iajs-3461 316 34 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 316 35 jΜƒ jΜƒ PROPN iajs-3461 316 36 𝑖̃ 𝑖̃ PROPN iajs-3461 316 37 , , PUNCT iajs-3461 316 38 has have VERB iajs-3461 316 39 a a DET iajs-3461 316 40 saddle saddle NOUN iajs-3461 316 41 βˆ’ βˆ’ NOUN iajs-3461 316 42 node node ADJ iajs-3461 316 43 bifurcation bifurcation NOUN iajs-3461 316 44 , , PUNCT iajs-3461 316 45 but but CCONJ iajs-3461 316 46 neither neither CCONJ iajs-3461 316 47 a a DET iajs-3461 316 48 transcritical4nor transcritical4nor PUNCT iajs-3461 316 49 a a DET iajs-3461 316 50 pitchfork pitchfork NOUN iajs-3461 316 51 bifurcation4 bifurcation4 NOUN iajs-3461 316 52 at at ADP iajs-3461 316 53 𝐴4 𝐴4 PROPN iajs-3461 316 54 . . PUNCT iajs-3461 317 1 proof proof NOUN iajs-3461 317 2 : : PUNCT iajs-3461 317 3 the the DET iajs-3461 317 4 characteristic characteristic ADJ iajs-3461 317 5 equation equation NOUN iajs-3461 317 6 given give VERB iajs-3461 317 7 by by ADP iajs-3461 317 8 eq eq PROPN iajs-3461 317 9 . . PUNCT iajs-3461 318 1 ( ( PUNCT iajs-3461 318 2 4.a 4.a NUM iajs-3461 318 3 ) ) PUNCT iajs-3461 318 4 if if SCONJ iajs-3461 318 5 𝐻4 𝐻4 NOUN iajs-3461 318 6 = = SYM iajs-3461 318 7 0 0 PUNCT iajs-3461 318 8 and and CCONJ iajs-3461 318 9 𝐴4 𝐴4 PROPN iajs-3461 318 10 becomes become VERB iajs-3461 318 11 a a DET iajs-3461 318 12 non non ADJ iajs-3461 318 13 hyperbolic hyperbolic ADJ iajs-3461 318 14 equilibrium equilibrium NOUN iajs-3461 318 15 point point NOUN iajs-3461 318 16 , , PUNCT iajs-3461 318 17 of of ADP iajs-3461 318 18 system system NOUN iajs-3461 318 19 ( ( PUNCT iajs-3461 318 20 2 2 X iajs-3461 318 21 ) ) PUNCT iajs-3461 318 22 having have VERB iajs-3461 318 23 zero zero NUM iajs-3461 318 24 eigenvalue eigenvalue X iajs-3461 318 25 ( ( PUNCT iajs-3461 318 26 say say INTJ iajs-3461 318 27 πœ†4β„Ž πœ†4β„Ž ADV iajs-3461 318 28 = = SYM iajs-3461 318 29 0 0 NUM iajs-3461 318 30 ) ) PUNCT iajs-3461 318 31 . . PUNCT iajs-3461 319 1 the the DET iajs-3461 319 2 jacobian jacobian ADJ iajs-3461 319 3 matrix matrix NOUN iajs-3461 319 4 for for ADP iajs-3461 319 5 system system NOUN iajs-3461 319 6 ( ( PUNCT iajs-3461 319 7 2 2 NUM iajs-3461 319 8 ) ) PUNCT iajs-3461 319 9 at at ADP iajs-3461 319 10 equilibrium equilibrium NOUN iajs-3461 319 11 point point NOUN iajs-3461 319 12 𝐸4 𝐸4 NOUN iajs-3461 319 13 with with ADP iajs-3461 319 14 parameter parameter NOUN iajs-3461 319 15 ( ( PUNCT iajs-3461 319 16 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 319 17 = = SYM iajs-3461 319 18 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 319 19 ) ) PUNCT iajs-3461 319 20 clearlybecomes clearlybecome VERB iajs-3461 319 21 : : PUNCT iajs-3461 319 22 𝐽4̈ 𝐽4̈ PROPN iajs-3461 319 23 = = SYM iajs-3461 319 24 𝐽4 𝐽4 PROPN iajs-3461 319 25 ( ( PUNCT iajs-3461 319 26 π‘Ÿ1 π‘Ÿ1 NOUN iajs-3461 319 27 = = SYM iajs-3461 319 28 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 319 29 ) ) PUNCT iajs-3461 319 30 = = PUNCT iajs-3461 320 1 [ [ PUNCT iajs-3461 320 2 οΏ½ οΏ½ PROPN iajs-3461 320 3 ΜΏ ΜΏ NOUN iajs-3461 320 4 οΏ½ οΏ½ PROPN iajs-3461 320 5 𝑖𝑗 𝑖𝑗 X iajs-3461 320 6 ] ] X iajs-3461 320 7 3Γ—3 3Γ—3 NUM iajs-3461 320 8 , , PUNCT iajs-3461 320 9 where where SCONJ iajs-3461 320 10 , , PUNCT iajs-3461 320 11 οΏ½ οΏ½ PROPN iajs-3461 320 12 ΜΏ ΜΏ NOUN iajs-3461 320 13 οΏ½ οΏ½ PROPN iajs-3461 320 14 𝑖𝑗 𝑖𝑗 NOUN iajs-3461 320 15 = = NOUN iajs-3461 320 16 Γ±11 Γ±11 NOUN iajs-3461 320 17 for for ADP iajs-3461 320 18 all all DET iajs-3461 320 19 i i PROPN iajs-3461 320 20 , , PUNCT iajs-3461 320 21 j j PROPN iajs-3461 320 22 = = PROPN iajs-3461 320 23 1 1 NUM iajs-3461 320 24 , , PUNCT iajs-3461 320 25 2 2 NUM iajs-3461 320 26 , , PUNCT iajs-3461 320 27 3 3 NUM iajs-3461 320 28 except except SCONJ iajs-3461 320 29 Γ±11 Γ±11 NOUN iajs-3461 320 30 which which PRON iajs-3461 320 31 is be AUX iajs-3461 320 32 given give VERB iajs-3461 320 33 by by ADP iajs-3461 320 34 : : PUNCT iajs-3461 320 35 Γ±11 Γ±11 NOUN iajs-3461 320 36 = = SYM iajs-3461 321 1 1 1 NUM iajs-3461 321 2 βˆ’ βˆ’ PROPN iajs-3461 321 3 2β„ŽΜƒ 2β„ŽΜƒ NOUN iajs-3461 321 4 βˆ’ βˆ’ PROPN iajs-3461 321 5 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 321 6 𝑖̃ 𝑖̃ PROPN iajs-3461 321 7 βˆ’ βˆ’ NOUN iajs-3461 321 8 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 321 9 jΜƒ jΜƒ PROPN iajs-3461 321 10 let let VERB iajs-3461 321 11 οΏ½ οΏ½ PART iajs-3461 321 12 Μ‡ Μ‡ VERB iajs-3461 321 13 οΏ½ οΏ½ PROPN iajs-3461 321 14 [4 [4 X iajs-3461 321 15 ] ] X iajs-3461 321 16 = = SYM iajs-3461 321 17 ( ( PUNCT iajs-3461 321 18 οΏ½ οΏ½ PROPN iajs-3461 321 19 Μ‡ Μ‡ NOUN iajs-3461 321 20 οΏ½ οΏ½ PROPN iajs-3461 321 21 1 1 NUM iajs-3461 321 22 [ [ X iajs-3461 321 23 4 4 NUM iajs-3461 321 24 ] ] PUNCT iajs-3461 321 25 , , PUNCT iajs-3461 321 26 οΏ½ οΏ½ PROPN iajs-3461 321 27 Μ‡ Μ‡ PROPN iajs-3461 321 28 οΏ½ οΏ½ PROPN iajs-3461 321 29 2 2 NUM iajs-3461 321 30 [ [ SYM iajs-3461 321 31 4 4 NUM iajs-3461 321 32 ] ] PUNCT iajs-3461 321 33 , , PUNCT iajs-3461 321 34 οΏ½ οΏ½ PROPN iajs-3461 321 35 Μ‡ Μ‡ PROPN iajs-3461 321 36 οΏ½ οΏ½ PROPN iajs-3461 321 37 3 3 NUM iajs-3461 321 38 [ [ X iajs-3461 321 39 4 4 NUM iajs-3461 321 40 ] ] PUNCT iajs-3461 321 41 ) ) PUNCT iajs-3461 321 42 𝑇 𝑇 PROPN iajs-3461 321 43 be be VERB iajs-3461 321 44 the the DET iajs-3461 321 45 eigenvector9 eigenvector9 NOUN iajs-3461 321 46 that that PRON iajs-3461 321 47 follows follow VERB iajs-3461 321 48 the the DET iajs-3461 321 49 9eigenvalue 9eigenvalue NUM iajs-3461 321 50 πœ†4β„Ž πœ†4β„Ž X iajs-3461 321 51 = = SYM iajs-3461 321 52 0 0 NUM iajs-3461 321 53 . . PUNCT iajs-3461 322 1 thus thus ADV iajs-3461 322 2 ( ( PUNCT iajs-3461 322 3 𝐽4̈ 𝐽4̈ ADV iajs-3461 322 4 βˆ’ βˆ’ PROPN iajs-3461 322 5 πœ†4β„ŽπΌ πœ†4β„ŽπΌ PROPN iajs-3461 322 6 ) ) PUNCT iajs-3461 322 7 οΏ½ οΏ½ AUX iajs-3461 322 8 Μ‡ Μ‡ VERB iajs-3461 322 9 οΏ½ οΏ½ PROPN iajs-3461 323 1 [ [ X iajs-3461 323 2 4 4 NUM iajs-3461 323 3 ] ] X iajs-3461 323 4 = = SYM iajs-3461 323 5 0 0 NUM iajs-3461 323 6 , , PUNCT iajs-3461 323 7 which which PRON iajs-3461 323 8 gives give VERB iajs-3461 323 9 : : PUNCT iajs-3461 323 10 οΏ½ οΏ½ PROPN iajs-3461 323 11 Μ‡ Μ‡ PROPN iajs-3461 323 12 οΏ½ οΏ½ PROPN iajs-3461 323 13 [4 [4 X iajs-3461 323 14 ] ] X iajs-3461 323 15 = = SYM iajs-3461 323 16 ( ( PUNCT iajs-3461 323 17 € € SYM iajs-3461 323 18 2 2 NUM iajs-3461 323 19 οΏ½ οΏ½ NOUN iajs-3461 323 20 Μ‡ Μ‡ NOUN iajs-3461 323 21 οΏ½ οΏ½ NOUN iajs-3461 323 22 3 3 NUM iajs-3461 323 23 [ [ X iajs-3461 323 24 4 4 NUM iajs-3461 323 25 ] ] PUNCT iajs-3461 323 26 , , PUNCT iajs-3461 323 27 € € SYM iajs-3461 323 28 1 1 NUM iajs-3461 323 29 οΏ½ οΏ½ NOUN iajs-3461 323 30 Μ‡ Μ‡ NOUN iajs-3461 323 31 οΏ½ οΏ½ NOUN iajs-3461 323 32 3 3 NUM iajs-3461 323 33 [ [ X iajs-3461 323 34 4 4 NUM iajs-3461 323 35 ] ] PUNCT iajs-3461 323 36 , , PUNCT iajs-3461 323 37 οΏ½ οΏ½ PROPN iajs-3461 323 38 Μ‡ Μ‡ PROPN iajs-3461 323 39 οΏ½ οΏ½ PROPN iajs-3461 323 40 3 3 NUM iajs-3461 323 41 [ [ X iajs-3461 323 42 4 4 NUM iajs-3461 323 43 ] ] SYM iajs-3461 323 44 ) ) PUNCT iajs-3461 323 45 𝑇 𝑇 PROPN iajs-3461 323 46 , , PUNCT iajs-3461 323 47 where where SCONJ iajs-3461 323 48 οΏ½ οΏ½ PROPN iajs-3461 323 49 Μ‡ Μ‡ VERB iajs-3461 323 50 οΏ½ οΏ½ PROPN iajs-3461 323 51 3 3 NUM iajs-3461 323 52 [ [ X iajs-3461 323 53 4 4 NUM iajs-3461 323 54 ] ] PUNCT iajs-3461 323 55 any any DET iajs-3461 323 56 non non ADJ iajs-3461 323 57 - - ADJ iajs-3461 323 58 zero zero NUM iajs-3461 323 59 value value NOUN iajs-3461 323 60 that that PRON iajs-3461 323 61 has have VERB iajs-3461 323 62 the the DET iajs-3461 323 63 € € SYM iajs-3461 323 64 1 1 NUM iajs-3461 323 65 and and CCONJ iajs-3461 323 66 € € SYM iajs-3461 323 67 2 2 NUM iajs-3461 323 68 conditions condition NOUN iajs-3461 323 69 given give VERB iajs-3461 323 70 in in ADP iajs-3461 323 71 the the DET iajs-3461 323 72 theorem theorem NOUN iajs-3461 323 73 . . PUNCT iajs-3461 324 1 let let VERB iajs-3461 324 2 Γ§[4 Γ§[4 PRON iajs-3461 324 3 ] ] X iajs-3461 324 4 = = SYM iajs-3461 324 5 ( ( PUNCT iajs-3461 324 6 Γ§[4 Γ§[4 PROPN iajs-3461 324 7 ] ] PUNCT iajs-3461 324 8 , , PUNCT iajs-3461 324 9 Γ§[4 Γ§[4 PROPN iajs-3461 324 10 ] ] PUNCT iajs-3461 324 11 , , PUNCT iajs-3461 324 12 Γ§[4 Γ§[4 PROPN iajs-3461 324 13 ] ] PUNCT iajs-3461 324 14 ) ) PUNCT iajs-3461 324 15 𝑇 𝑇 PROPN iajs-3461 324 16 be be AUX iajs-3461 324 17 the the DET iajs-3461 324 18 eigenvector4 eigenvector4 NOUN iajs-3461 324 19 associated4 associated4 PROPN iajs-3461 324 20 with with ADP iajs-3461 324 21 the the DET iajs-3461 324 22 4eigenvalue 4eigenvalue NUM iajs-3461 324 23 πœ†4𝑖 πœ†4𝑖 PUNCT iajs-3461 324 24 = = PUNCT iajs-3461 324 25 0 0 NUM iajs-3461 324 26 of of ADP iajs-3461 324 27 the the DET iajs-3461 324 28 matrix matrix NOUN iajs-3461 324 29 𝐽4̈. 𝐽4̈. NOUN iajs-3461 324 30 next next ADV iajs-3461 324 31 , , PUNCT iajs-3461 324 32 we we PRON iajs-3461 324 33 have have VERB iajs-3461 324 34 ( ( PUNCT iajs-3461 324 35 𝐽4̈ 𝐽4̈ PROPN iajs-3461 324 36 𝑇 𝑇 PROPN iajs-3461 324 37 βˆ’ βˆ’ PROPN iajs-3461 324 38 πœ†4𝑖𝐼 πœ†4𝑖𝐼 PROPN iajs-3461 324 39 ) ) PUNCT iajs-3461 324 40 Γ§ Γ§ X iajs-3461 325 1 [ [ X iajs-3461 325 2 4 4 NUM iajs-3461 325 3 ] ] X iajs-3461 325 4 = = SYM iajs-3461 325 5 0 0 X iajs-3461 325 6 . . PUNCT iajs-3461 325 7 by by ADP iajs-3461 325 8 solving solve VERB iajs-3461 325 9 this this DET iajs-3461 325 10 4equation 4equation PROPN iajs-3461 325 11 for for ADP iajs-3461 325 12 βˆ…[4 βˆ…[4 NOUN iajs-3461 325 13 ] ] PUNCT iajs-3461 325 14 , , PUNCT iajs-3461 325 15 we we PRON iajs-3461 325 16 obtain obtain VERB iajs-3461 325 17 : : PUNCT iajs-3461 325 18 Γ§[4 Γ§[4 PROPN iajs-3461 325 19 ] ] X iajs-3461 325 20 = = PUNCT iajs-3461 325 21 ( ( PUNCT iajs-3461 325 22 € € SYM iajs-3461 325 23 4Γ§2 4Γ§2 NUM iajs-3461 325 24 [ [ X iajs-3461 325 25 4 4 NUM iajs-3461 325 26 ] ] PUNCT iajs-3461 325 27 , , PUNCT iajs-3461 325 28 Γ§2 Γ§2 PROPN iajs-3461 325 29 [ [ X iajs-3461 325 30 4 4 NUM iajs-3461 325 31 ] ] PUNCT iajs-3461 325 32 , , PUNCT iajs-3461 325 33 € € SYM iajs-3461 325 34 3Γ§2 3Γ§2 NUM iajs-3461 325 35 [ [ X iajs-3461 325 36 4 4 NUM iajs-3461 325 37 ] ] PUNCT iajs-3461 325 38 ) ) PUNCT iajs-3461 325 39 𝑇 𝑇 PROPN iajs-3461 325 40 where where SCONJ iajs-3461 325 41 Γ§2 Γ§2 PROPN iajs-3461 325 42 [ [ X iajs-3461 325 43 4 4 X iajs-3461 325 44 ] ] PUNCT iajs-3461 325 45 any any DET iajs-3461 325 46 4nonzero 4nonzero NUM iajs-3461 325 47 4number 4number NUM iajs-3461 325 48 with with ADP iajs-3461 325 49 € € SYM iajs-3461 325 50 3 3 NUM iajs-3461 325 51 and and CCONJ iajs-3461 325 52 € € SYM iajs-3461 325 53 4 4 NUM iajs-3461 325 54 which which PRON iajs-3461 325 55 are be AUX iajs-3461 325 56 4mentioned 4mentioned NUM iajs-3461 325 57 in in ADP iajs-3461 325 58 the the DET iajs-3461 325 59 state4 state4 NOUN iajs-3461 325 60 of of ADP iajs-3461 325 61 the the DET iajs-3461 325 62 theorem theorem NOUN iajs-3461 325 63 . . PUNCT iajs-3461 326 1 now now ADV iajs-3461 326 2 , , PUNCT iajs-3461 326 3 ihjpas ihjpas PROPN iajs-3461 326 4 . . PUNCT iajs-3461 327 1 2025 2025 NUM iajs-3461 327 2 , , PUNCT iajs-3461 327 3 38 38 NUM iajs-3461 327 4 ( ( PUNCT iajs-3461 327 5 1 1 NUM iajs-3461 327 6 ) ) PUNCT iajs-3461 327 7 379 379 NUM iajs-3461 327 8 πœ•π‘“ πœ•π‘“ ADJ iajs-3461 327 9 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3461 327 10 = = SYM iajs-3461 327 11 π‘“π‘Ÿ1(Γ½ π‘“π‘Ÿ1(Γ½ PROPN iajs-3461 327 12 , , PUNCT iajs-3461 327 13 π‘Ÿ1 π‘Ÿ1 PROPN iajs-3461 327 14 ) ) PUNCT iajs-3461 327 15 = = SYM iajs-3461 327 16 ( ( PUNCT iajs-3461 327 17 πœ•π‘“1 πœ•π‘“1 NOUN iajs-3461 327 18 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3461 327 19 , , PUNCT iajs-3461 327 20 πœ•π‘“2 πœ•π‘“2 PROPN iajs-3461 327 21 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3461 327 22 , , PUNCT iajs-3461 327 23 πœ•π‘“3 πœ•π‘“3 NOUN iajs-3461 327 24 πœ•π‘Ÿ1 πœ•π‘Ÿ1 NOUN iajs-3461 327 25 , , PUNCT iajs-3461 327 26 ) ) PUNCT iajs-3461 327 27 𝑇 𝑇 PROPN iajs-3461 327 28 = = SYM iajs-3461 327 29 ( ( PUNCT iajs-3461 327 30 βˆ’β„Žπ‘– βˆ’β„Žπ‘– PROPN iajs-3461 327 31 , , PUNCT iajs-3461 327 32 0 0 NUM iajs-3461 327 33 , , PUNCT iajs-3461 327 34 0)𝑇 0)𝑇 NOUN iajs-3461 327 35 , , PUNCT iajs-3461 327 36 so so ADV iajs-3461 327 37 , , PUNCT iajs-3461 327 38 π‘“π‘Ž1(𝐴4 π‘“π‘Ž1(𝐴4 PROPN iajs-3461 327 39 , , PUNCT iajs-3461 327 40 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 327 41 ) ) PUNCT iajs-3461 327 42 = = SYM iajs-3461 327 43 ( ( PUNCT iajs-3461 327 44 βˆ’β„ŽΜƒ βˆ’β„ŽΜƒ PUNCT iajs-3461 327 45 𝑖̃ 𝑖̃ PROPN iajs-3461 327 46 , , PUNCT iajs-3461 327 47 0 0 NUM iajs-3461 327 48 , , PUNCT iajs-3461 327 49 0 0 NUM iajs-3461 327 50 ) ) PUNCT iajs-3461 327 51 𝑇 𝑇 NOUN iajs-3461 327 52 , , PUNCT iajs-3461 327 53 and and CCONJ iajs-3461 327 54 hence hence ADV iajs-3461 327 55 , , PUNCT iajs-3461 327 56 it it PRON iajs-3461 327 57 is be AUX iajs-3461 327 58 4obtained 4obtained NUM iajs-3461 327 59 that that SCONJ iajs-3461 327 60 : : PUNCT iajs-3461 327 61 ( ( PUNCT iajs-3461 327 62 Γ§[4 Γ§[4 PROPN iajs-3461 327 63 ] ] PUNCT iajs-3461 327 64 ) ) PUNCT iajs-3461 327 65 𝑇 𝑇 PROPN iajs-3461 327 66 π‘“π‘Ÿ1(𝐴4 π‘“π‘Ÿ1(𝐴4 PROPN iajs-3461 327 67 , , PUNCT iajs-3461 327 68 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 327 69 ) ) PUNCT iajs-3461 327 70 = = SYM iajs-3461 327 71 βˆ’β„ŽΜƒ βˆ’β„ŽΜƒ PUNCT iajs-3461 327 72 𝑖̃ 𝑖̃ PROPN iajs-3461 327 73 € € SYM iajs-3461 327 74 4Γ§2 4Γ§2 NUM iajs-3461 327 75 [ [ X iajs-3461 327 76 4 4 NUM iajs-3461 327 77 ] ] PUNCT iajs-3461 327 78 . . PUNCT iajs-3461 328 1 in in ADP iajs-3461 328 2 the the DET iajs-3461 328 3 context context NOUN iajs-3461 328 4 of of ADP iajs-3461 328 5 condition condition NOUN iajs-3461 328 6 ( ( PUNCT iajs-3461 328 7 30 30 NUM iajs-3461 328 8 ) ) PUNCT iajs-3461 328 9 , , PUNCT iajs-3461 328 10 we we PRON iajs-3461 328 11 thus thus ADV iajs-3461 328 12 have have VERB iajs-3461 328 13 that that PRON iajs-3461 328 14 : : PUNCT iajs-3461 328 15 ( ( PUNCT iajs-3461 328 16 Γ§[4 Γ§[4 PROPN iajs-3461 328 17 ] ] PUNCT iajs-3461 328 18 ) ) PUNCT iajs-3461 328 19 𝑇 𝑇 PROPN iajs-3461 328 20 π‘“π‘Ÿ1(𝐴4 π‘“π‘Ÿ1(𝐴4 PROPN iajs-3461 328 21 , , PUNCT iajs-3461 328 22 π‘Ÿ1̈ π‘Ÿ1̈ PROPN iajs-3461 328 23 ) ) PUNCT iajs-3461 328 24 β‰  β‰  PROPN iajs-3461 328 25 0 0 NUM iajs-3461 328 26 . . PUNCT iajs-3461 329 1 𝐷2𝑓(𝐴4 𝐷2𝑓(𝐴4 PROPN iajs-3461 329 2 , , PUNCT iajs-3461 329 3 π‘Ÿ1̈)( π‘Ÿ1̈)( PROPN iajs-3461 329 4 οΏ½ οΏ½ PROPN iajs-3461 329 5 Μ‡ Μ‡ NOUN iajs-3461 329 6 οΏ½ οΏ½ PROPN iajs-3461 329 7 [ [ X iajs-3461 329 8 4 4 NUM iajs-3461 329 9 ] ] PUNCT iajs-3461 329 10 , , PUNCT iajs-3461 329 11 οΏ½ οΏ½ PROPN iajs-3461 329 12 Μ‡ Μ‡ PROPN iajs-3461 329 13 οΏ½ οΏ½ PROPN iajs-3461 329 14 [4 [4 NOUN iajs-3461 329 15 ] ] X iajs-3461 329 16 ) ) PUNCT iajs-3461 329 17 = = PUNCT iajs-3461 330 1 [ [ PUNCT iajs-3461 330 2 βˆ’2 βˆ’2 NOUN iajs-3461 330 3 € € SYM iajs-3461 330 4 2 2 NUM iajs-3461 330 5 ( ( PUNCT iajs-3461 330 6 οΏ½ οΏ½ PROPN iajs-3461 330 7 Μ‡ Μ‡ NOUN iajs-3461 330 8 οΏ½ οΏ½ X iajs-3461 330 9 3 3 NUM iajs-3461 331 1 [ [ X iajs-3461 331 2 4])2[€2 4])2[€2 NOUN iajs-3461 331 3 + + NUM iajs-3461 331 4 π‘Ÿ1€1 π‘Ÿ1€1 NUM iajs-3461 331 5 + + CCONJ iajs-3461 331 6 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 331 7 ] ] SYM iajs-3461 331 8 2 2 NUM iajs-3461 331 9 € € SYM iajs-3461 331 10 1 1 NUM iajs-3461 331 11 ( ( PUNCT iajs-3461 331 12 οΏ½ οΏ½ PROPN iajs-3461 331 13 Μ‡ Μ‡ NOUN iajs-3461 331 14 οΏ½ οΏ½ X iajs-3461 331 15 3 3 NUM iajs-3461 332 1 [ [ X iajs-3461 332 2 4])2[π‘Ÿ3€2 4])2[π‘Ÿ3€2 NOUN iajs-3461 332 3 + + CCONJ iajs-3461 332 4 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 332 5 ] ] X iajs-3461 332 6 2 2 NUM iajs-3461 332 7 ( ( PUNCT iajs-3461 332 8 οΏ½ οΏ½ PROPN iajs-3461 332 9 Μ‡ Μ‡ NOUN iajs-3461 332 10 οΏ½ οΏ½ PROPN iajs-3461 332 11 3 3 NUM iajs-3461 332 12 [ [ X iajs-3461 332 13 4])2[π‘Ÿ6€2 4])2[π‘Ÿ6€2 NOUN iajs-3461 332 14 + + CCONJ iajs-3461 332 15 π‘Ÿ7€1 π‘Ÿ7€1 PROPN iajs-3461 332 16 ] ] PUNCT iajs-3461 332 17 ] ] PUNCT iajs-3461 332 18 . . PUNCT iajs-3461 333 1 hence hence ADV iajs-3461 333 2 , , PUNCT iajs-3461 333 3 it it PRON iajs-3461 333 4 is be AUX iajs-3461 333 5 4obtained 4obtained NUM iajs-3461 333 6 that that SCONJ iajs-3461 333 7 : : PUNCT iajs-3461 333 8 ( ( PUNCT iajs-3461 333 9 Γ§[4 Γ§[4 PROPN iajs-3461 333 10 ] ] PUNCT iajs-3461 333 11 ) ) PUNCT iajs-3461 333 12 𝑇 𝑇 PROPN iajs-3461 333 13 𝐷2𝑓(𝐴4 𝐷2𝑓(𝐴4 PROPN iajs-3461 333 14 , , PUNCT iajs-3461 333 15 π‘Ÿ1̈)( π‘Ÿ1̈)( PROPN iajs-3461 333 16 οΏ½ οΏ½ PROPN iajs-3461 333 17 Μ‡ Μ‡ NOUN iajs-3461 333 18 οΏ½ οΏ½ PROPN iajs-3461 334 1 [ [ X iajs-3461 334 2 4 4 NUM iajs-3461 334 3 ] ] PUNCT iajs-3461 334 4 , , PUNCT iajs-3461 334 5 οΏ½ οΏ½ PROPN iajs-3461 334 6 Μ‡ Μ‡ PROPN iajs-3461 334 7 οΏ½ οΏ½ PROPN iajs-3461 334 8 [4 [4 NOUN iajs-3461 334 9 ] ] X iajs-3461 334 10 ) ) PUNCT iajs-3461 334 11 = = SYM iajs-3461 334 12 βˆ’2 βˆ’2 NOUN iajs-3461 334 13 ( ( PUNCT iajs-3461 334 14 οΏ½ οΏ½ PROPN iajs-3461 334 15 Μ‡ Μ‡ NOUN iajs-3461 334 16 οΏ½ οΏ½ X iajs-3461 334 17 3 3 NUM iajs-3461 335 1 [ [ X iajs-3461 335 2 4])2Γ§2 4])2Γ§2 NOUN iajs-3461 336 1 [ [ X iajs-3461 336 2 4 4 NUM iajs-3461 336 3 ] ] PUNCT iajs-3461 336 4 ( ( PUNCT iajs-3461 336 5 € € SYM iajs-3461 336 6 2€4 2€4 NUM iajs-3461 336 7 [ [ X iajs-3461 336 8 € € SYM iajs-3461 336 9 2 2 NUM iajs-3461 336 10 + + NUM iajs-3461 336 11 π‘Ÿ1€1 π‘Ÿ1€1 NUM iajs-3461 337 1 + + CCONJ iajs-3461 337 2 π‘Ÿ2 π‘Ÿ2 NOUN iajs-3461 337 3 ] ] PUNCT iajs-3461 337 4 βˆ’ βˆ’ PROPN iajs-3461 337 5 € € SYM iajs-3461 337 6 1[π‘Ÿ3€2 1[π‘Ÿ3€2 NOUN iajs-3461 337 7 + + CCONJ iajs-3461 337 8 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 337 9 ] ] X iajs-3461 337 10 βˆ’ βˆ’ NOUN iajs-3461 337 11 € € SYM iajs-3461 337 12 4[π‘Ÿ6€2 4[π‘Ÿ6€2 NOUN iajs-3461 337 13 + + NOUN iajs-3461 337 14 π‘Ÿ7€1 π‘Ÿ7€1 PROPN iajs-3461 337 15 ] ] PUNCT iajs-3461 337 16 ) ) PUNCT iajs-3461 337 17 . . PUNCT iajs-3461 338 1 therefore therefore ADV iajs-3461 338 2 , , PUNCT iajs-3461 338 3 in in ADP iajs-3461 338 4 accordance accordance NOUN iajs-3461 338 5 with with ADP iajs-3461 338 6 condition condition NOUN iajs-3461 338 7 ( ( PUNCT iajs-3461 338 8 32 32 NUM iajs-3461 338 9 ) ) PUNCT iajs-3461 338 10 we we PRON iajs-3461 338 11 get get VERB iajs-3461 338 12 that that PRON iajs-3461 338 13 : : PUNCT iajs-3461 338 14 ( ( PUNCT iajs-3461 338 15 Γ§[4 Γ§[4 PROPN iajs-3461 338 16 ] ] PUNCT iajs-3461 338 17 ) ) PUNCT iajs-3461 338 18 𝑇 𝑇 PROPN iajs-3461 338 19 𝐷2𝑓(𝐴4 𝐷2𝑓(𝐴4 PROPN iajs-3461 338 20 , , PUNCT iajs-3461 338 21 π‘Ÿ1̈)( π‘Ÿ1̈)( PROPN iajs-3461 338 22 οΏ½ οΏ½ PROPN iajs-3461 338 23 Μ‡ Μ‡ NOUN iajs-3461 338 24 οΏ½ οΏ½ PROPN iajs-3461 339 1 [ [ X iajs-3461 339 2 4 4 NUM iajs-3461 339 3 ] ] PUNCT iajs-3461 339 4 , , PUNCT iajs-3461 339 5 οΏ½ οΏ½ PROPN iajs-3461 339 6 Μ‡ Μ‡ PROPN iajs-3461 339 7 οΏ½ οΏ½ PROPN iajs-3461 339 8 [4 [4 NOUN iajs-3461 339 9 ] ] X iajs-3461 339 10 ) ) PUNCT iajs-3461 339 11 β‰  β‰  PROPN iajs-3461 339 12 0 0 NUM iajs-3461 339 13 sotomayor sotomayor PROPN iajs-3461 339 14 's 's PART iajs-3461 339 15 theorem theorem NOUN iajs-3461 339 16 is be AUX iajs-3461 339 17 used use VERB iajs-3461 339 18 to to PART iajs-3461 339 19 show show VERB iajs-3461 339 20 that that DET iajs-3461 339 21 system system NOUN iajs-3461 339 22 ( ( PUNCT iajs-3461 339 23 2 2 X iajs-3461 339 24 ) ) PUNCT iajs-3461 339 25 has have VERB iajs-3461 339 26 a a DET iajs-3461 339 27 saddle saddle NOUN iajs-3461 339 28 - - PUNCT iajs-3461 339 29 node node NOUN iajs-3461 339 30 bifurcation bifurcation NOUN iajs-3461 339 31 𝐴4 𝐴4 NOUN iajs-3461 339 32 = = SYM iajs-3461 339 33 ( ( PUNCT iajs-3461 339 34 β„ŽΜƒ β„ŽΜƒ PROPN iajs-3461 339 35 , , PUNCT iajs-3461 339 36 𝑖̃ 𝑖̃ PROPN iajs-3461 339 37 , , PUNCT iajs-3461 339 38 j j PROPN iajs-3461 339 39 Μƒ Μƒ PROPN iajs-3461 339 40 ) ) PUNCT iajs-3461 339 41 at at ADP iajs-3461 339 42 π‘Ÿ1̈. π‘Ÿ1̈. NOUN iajs-3461 339 43 5 5 NUM iajs-3461 339 44 . . PUNCT iajs-3461 339 45 numerical numerical PROPN iajs-3461 339 46 simulation simulation PROPN iajs-3461 340 1 this this DET iajs-3461 340 2 section section NOUN iajs-3461 340 3 numerically numerically ADV iajs-3461 340 4 analyzes analyze VERB iajs-3461 340 5 system system NOUN iajs-3461 340 6 ( ( PUNCT iajs-3461 340 7 2 2 NUM iajs-3461 340 8 ) ) PUNCT iajs-3461 340 9 's 's PART iajs-3461 340 10 dynamical dynamical ADJ iajs-3461 340 11 behavior behavior NOUN iajs-3461 340 12 for for ADP iajs-3461 340 13 a a DET iajs-3461 340 14 given give VERB iajs-3461 340 15 set set NOUN iajs-3461 340 16 of of ADP iajs-3461 340 17 parameters parameter NOUN iajs-3461 340 18 and and CCONJ iajs-3461 340 19 various various ADJ iajs-3461 340 20 initial initial ADJ iajs-3461 340 21 point point NOUN iajs-3461 340 22 sets set NOUN iajs-3461 340 23 . . PUNCT iajs-3461 341 1 these these PRON iajs-3461 341 2 are be AUX iajs-3461 341 3 the the DET iajs-3461 341 4 study study NOUN iajs-3461 341 5 's 's PART iajs-3461 341 6 objectives objective NOUN iajs-3461 341 7 , , PUNCT iajs-3461 341 8 including include VERB iajs-3461 341 9 : : PUNCT iajs-3461 341 10 1 1 X iajs-3461 341 11 . . X iajs-3461 341 12 analyze analyze VERB iajs-3461 341 13 the the DET iajs-3461 341 14 impact impact NOUN iajs-3461 341 15 of of ADP iajs-3461 341 16 changing change VERB iajs-3461 341 17 the the DET iajs-3461 341 18 value value NOUN iajs-3461 341 19 of of ADP iajs-3461 341 20 each each DET iajs-3461 341 21 parameter parameter NOUN iajs-3461 341 22 on on ADP iajs-3461 341 23 the the DET iajs-3461 341 24 system system NOUN iajs-3461 341 25 's 's PART iajs-3461 341 26 dynamic dynamic ADJ iajs-3461 341 27 behavior behavior NOUN iajs-3461 341 28 ( ( PUNCT iajs-3461 341 29 2 2 NUM iajs-3461 341 30 ) ) PUNCT iajs-3461 341 31 . . PUNCT iajs-3461 342 1 2 2 X iajs-3461 342 2 . . X iajs-3461 342 3 verify verify VERB iajs-3461 342 4 the the DET iajs-3461 342 5 analytical analytical ADJ iajs-3461 342 6 results result NOUN iajs-3461 342 7 that that PRON iajs-3461 342 8 were be AUX iajs-3461 342 9 found find VERB iajs-3461 342 10 . . PUNCT iajs-3461 343 1 the the DET iajs-3461 343 2 following follow VERB iajs-3461 343 3 hypothetical hypothetical ADJ iajs-3461 343 4 collection collection NOUN iajs-3461 343 5 of of ADP iajs-3461 343 6 parameters parameter NOUN iajs-3461 343 7 is be AUX iajs-3461 343 8 found find VERB iajs-3461 343 9 to to PART iajs-3461 343 10 satisfy satisfy VERB iajs-3461 343 11 the the DET iajs-3461 343 12 stability stability NOUN iajs-3461 343 13 requirements requirement NOUN iajs-3461 343 14 of of ADP iajs-3461 343 15 the the DET iajs-3461 343 16 positive positive ADJ iajs-3461 343 17 equilibrium equilibrium NOUN iajs-3461 343 18 point point NOUN iajs-3461 343 19 , , PUNCT iajs-3461 343 20 system system NOUN iajs-3461 343 21 ( ( PUNCT iajs-3461 343 22 2 2 NUM iajs-3461 343 23 ) ) PUNCT iajs-3461 343 24 has have VERB iajs-3461 343 25 a a DET iajs-3461 343 26 globally globally ADV iajs-3461 343 27 asymptotically9 asymptotically9 NOUN iajs-3461 343 28 stable stable ADJ iajs-3461 343 29 positive positive ADJ iajs-3461 343 30 equilibrium equilibrium NOUN iajs-3461 343 31 point point NOUN iajs-3461 343 32 as as SCONJ iajs-3461 343 33 shown show VERB iajs-3461 343 34 in in ADP iajs-3461 343 35 figure figure NOUN iajs-3461 343 36 πŸ“. πŸ“. X iajs-3461 343 37 𝟏 𝟏 NUM iajs-3461 343 38 ( ( PUNCT iajs-3461 343 39 𝒂 𝒂 NOUN iajs-3461 343 40 , , PUNCT iajs-3461 343 41 𝒃 𝒃 NOUN iajs-3461 343 42 , , PUNCT iajs-3461 343 43 𝒄 𝒄 NOUN iajs-3461 343 44 ) ) PUNCT iajs-3461 343 45 , , PUNCT iajs-3461 343 46 ( ( PUNCT iajs-3461 343 47 5.1 5.1 NUM iajs-3461 343 48 ) ) PUNCT iajs-3461 343 49 π’“πŸ π’“πŸ NOUN iajs-3461 343 50 = = SYM iajs-3461 344 1 𝟎. 𝟎. X iajs-3461 344 2 πŸŽπŸ“ πŸŽπŸ“ ADJ iajs-3461 344 3 , , PUNCT iajs-3461 344 4 π’“πŸ π’“πŸ PROPN iajs-3461 344 5 = = SYM iajs-3461 344 6 𝟎. 𝟎. NUM iajs-3461 344 7 𝟏 𝟏 NUM iajs-3461 344 8 , , PUNCT iajs-3461 344 9 π’“πŸ‘ π’“πŸ‘ PROPN iajs-3461 344 10 = = SYM iajs-3461 344 11 𝟎. 𝟎. X iajs-3461 344 12 πŸŽπŸ‘ πŸŽπŸ‘ NUM iajs-3461 344 13 , , PUNCT iajs-3461 344 14 π’“πŸ’= π’“πŸ’= PROPN iajs-3461 344 15 𝟎. 𝟎. X iajs-3461 344 16 𝟏 𝟏 NUM iajs-3461 344 17 , , PUNCT iajs-3461 344 18 π’“πŸ“= π’“πŸ“= PROPN iajs-3461 344 19 𝟎. 𝟎. PUNCT iajs-3461 344 20 𝟎𝟏 𝟎𝟏 NUM iajs-3461 344 21 , , PUNCT iajs-3461 344 22 π’“πŸ” π’“πŸ” PROPN iajs-3461 344 23 = = SYM iajs-3461 344 24 𝟎. 𝟎. X iajs-3461 344 25 πŸŽπŸ” πŸŽπŸ” NUM iajs-3461 344 26 , , PUNCT iajs-3461 344 27 π’“πŸ• π’“πŸ• PROPN iajs-3461 344 28 = = SYM iajs-3461 344 29 𝟎. 𝟎. NUM iajs-3461 344 30 πŸŽπŸ• πŸŽπŸ• NUM iajs-3461 344 31 , , PUNCT iajs-3461 344 32 π’“πŸ– π’“πŸ– ADV iajs-3461 344 33 = = SYM iajs-3461 344 34 𝟎. 𝟎. NUM iajs-3461 344 35 𝟏 𝟏 NUM iajs-3461 344 36 ihjpas ihjpa NOUN iajs-3461 344 37 . . PUNCT iajs-3461 345 1 2025 2025 NUM iajs-3461 345 2 , , PUNCT iajs-3461 345 3 38 38 NUM iajs-3461 345 4 ( ( PUNCT iajs-3461 345 5 1 1 NUM iajs-3461 345 6 ) ) PUNCT iajs-3461 345 7 380 380 NUM iajs-3461 345 8 figure1 figure1 NUM iajs-3461 345 9 : : PUNCT iajs-3461 345 10 the the DET iajs-3461 345 11 time time NOUN iajs-3461 345 12 series series NOUN iajs-3461 345 13 of of ADP iajs-3461 345 14 the the DET iajs-3461 345 15 solution solution NOUN iajs-3461 345 16 of of ADP iajs-3461 345 17 system system NOUN iajs-3461 345 18 ( ( PUNCT iajs-3461 345 19 2 2 X iajs-3461 345 20 ) ) PUNCT iajs-3461 345 21 started start VERB iajs-3461 345 22 from from ADP iajs-3461 345 23 the the DET iajs-3461 345 24 three three NUM iajs-3461 345 25 different different ADJ iajs-3461 345 26 initial initial ADJ iajs-3461 345 27 point point NOUN iajs-3461 345 28 ( ( PUNCT iajs-3461 345 29 0.8 0.8 NUM iajs-3461 345 30 , , PUNCT iajs-3461 345 31 0.7 0.7 NUM iajs-3461 345 32 , , PUNCT iajs-3461 345 33 1.64 1.64 NUM iajs-3461 345 34 ) ) PUNCT iajs-3461 345 35 , , PUNCT iajs-3461 345 36 ( ( PUNCT iajs-3461 345 37 1.2 1.2 NUM iajs-3461 345 38 , , PUNCT iajs-3461 345 39 0.7 0.7 NUM iajs-3461 345 40 , , PUNCT iajs-3461 345 41 1.64 1.64 NUM iajs-3461 345 42 ) ) PUNCT iajs-3461 345 43 and and CCONJ iajs-3461 345 44 ( ( PUNCT iajs-3461 345 45 0.2 0.2 NUM iajs-3461 345 46 , , PUNCT iajs-3461 345 47 0.4 0.4 NUM iajs-3461 345 48 , , PUNCT iajs-3461 345 49 0.64 0.64 NUM iajs-3461 345 50 ) ) PUNCT iajs-3461 345 51 for for ADP iajs-3461 345 52 the the DET iajs-3461 345 53 data datum NOUN iajs-3461 345 54 given give VERB iajs-3461 345 55 by by ADP iajs-3461 345 56 ( ( PUNCT iajs-3461 345 57 5.1 5.1 NUM iajs-3461 345 58 ) ) PUNCT iajs-3461 345 59 , , PUNCT iajs-3461 345 60 ( ( PUNCT iajs-3461 345 61 a a X iajs-3461 345 62 ) ) PUNCT iajs-3461 345 63 the the DET iajs-3461 345 64 trajectories trajectory NOUN iajs-3461 345 65 of of ADP iajs-3461 345 66 h h NOUN iajs-3461 345 67 as as ADP iajs-3461 345 68 a a DET iajs-3461 345 69 function function NOUN iajs-3461 345 70 of of ADP iajs-3461 345 71 time time NOUN iajs-3461 345 72 , , PUNCT iajs-3461 345 73 ( ( PUNCT iajs-3461 345 74 b b X iajs-3461 345 75 ) ) PUNCT iajs-3461 345 76 the the DET iajs-3461 345 77 trajectories trajectory NOUN iajs-3461 345 78 of of ADP iajs-3461 345 79 i i PRON iajs-3461 345 80 as as ADP iajs-3461 345 81 a a DET iajs-3461 345 82 function function NOUN iajs-3461 345 83 of of ADP iajs-3461 345 84 time time NOUN iajs-3461 345 85 , , PUNCT iajs-3461 345 86 ( ( PUNCT iajs-3461 345 87 c c X iajs-3461 345 88 ) ) PUNCT iajs-3461 345 89 the the DET iajs-3461 345 90 trajectories trajectory NOUN iajs-3461 345 91 of of ADP iajs-3461 345 92 j j PROPN iajs-3461 345 93 as as ADP iajs-3461 345 94 a a DET iajs-3461 345 95 function function NOUN iajs-3461 345 96 of of ADP iajs-3461 345 97 time time NOUN iajs-3461 345 98 . . PUNCT iajs-3461 346 1 as as SCONJ iajs-3461 346 2 the the DET iajs-3461 346 3 solution solution NOUN iajs-3461 346 4 of of ADP iajs-3461 346 5 system system NOUN iajs-3461 346 6 ( ( PUNCT iajs-3461 346 7 2 2 X iajs-3461 346 8 ) ) PUNCT iajs-3461 346 9 approaches approach VERB iajs-3461 346 10 asymptotically asymptotically ADV iajs-3461 346 11 to to ADP iajs-3461 346 12 the the DET iajs-3461 346 13 positive positive ADJ iajs-3461 346 14 equilibrium equilibrium NOUN iajs-3461 346 15 point point NOUN iajs-3461 346 16 a4= a4= PROPN iajs-3461 346 17 ( ( PUNCT iajs-3461 346 18 0.4544 0.4544 PROPN iajs-3461 346 19 , , PUNCT iajs-3461 346 20 0.6567 0.6567 NUM iajs-3461 346 21 , , PUNCT iajs-3461 346 22 2 2 NUM iajs-3461 346 23 . . NUM iajs-3461 346 24 1722 1722 NUM iajs-3461 346 25 ) ) PUNCT iajs-3461 346 26 beginning begin VERB iajs-3461 346 27 with with ADP iajs-3461 346 28 three three NUM iajs-3461 346 29 distinct distinct ADJ iajs-3461 346 30 starting starting NOUN iajs-3461 346 31 points point NOUN iajs-3461 346 32 , , PUNCT iajs-3461 346 33 figure1 figure1 PROPN iajs-3461 346 34 . . PUNCT iajs-3461 347 1 clearly clearly ADV iajs-3461 347 2 demonstrates demonstrate VERB iajs-3461 347 3 that that DET iajs-3461 347 4 system system NOUN iajs-3461 347 5 ( ( PUNCT iajs-3461 347 6 2 2 X iajs-3461 347 7 ) ) PUNCT iajs-3461 347 8 has have VERB iajs-3461 347 9 a a DET iajs-3461 347 10 globally globally ADV iajs-3461 347 11 asymptotically asymptotically ADV iajs-3461 347 12 stable stable ADJ iajs-3461 347 13 , , PUNCT iajs-3461 347 14 and and CCONJ iajs-3461 347 15 this this PRON iajs-3461 347 16 is be AUX iajs-3461 347 17 supporting support VERB iajs-3461 347 18 our our PRON iajs-3461 347 19 obtained obtain VERB iajs-3461 347 20 analytical analytical ADJ iajs-3461 347 21 results result NOUN iajs-3461 347 22 . . PUNCT iajs-3461 348 1 we we PRON iajs-3461 348 2 will will AUX iajs-3461 348 3 now now ADV iajs-3461 348 4 talk talk VERB iajs-3461 348 5 about about ADP iajs-3461 348 6 how how SCONJ iajs-3461 348 7 system system NOUN iajs-3461 348 8 's 's PART iajs-3461 348 9 ( ( PUNCT iajs-3461 348 10 2 2 NUM iajs-3461 348 11 ) ) PUNCT iajs-3461 348 12 parameter parameter NOUN iajs-3461 348 13 settings setting NOUN iajs-3461 348 14 affect affect VERB iajs-3461 348 15 the the DET iajs-3461 348 16 system system NOUN iajs-3461 348 17 's 's PART iajs-3461 348 18 dynamical dynamical ADJ iajs-3461 348 19 behavior behavior NOUN iajs-3461 348 20 . . PUNCT iajs-3461 349 1 the the DET iajs-3461 349 2 system system NOUN iajs-3461 349 3 is be AUX iajs-3461 349 4 numerically numerically ADV iajs-3461 349 5 solved solve VERB iajs-3461 349 6 for for ADP iajs-3461 349 7 the the DET iajs-3461 349 8 data datum NOUN iajs-3461 349 9 in in ADP iajs-3461 349 10 ( ( PUNCT iajs-3461 349 11 5.1 5.1 NUM iajs-3461 349 12 ) ) PUNCT iajs-3461 349 13 by by ADP iajs-3461 349 14 changing change VERB iajs-3461 349 15 one one NUM iajs-3461 349 16 parameter parameter NOUN iajs-3461 349 17 at at ADP iajs-3461 349 18 a a DET iajs-3461 349 19 time time NOUN iajs-3461 349 20 , , PUNCT iajs-3461 349 21 sometimes sometimes ADV iajs-3461 349 22 even even ADV iajs-3461 349 23 two two NUM iajs-3461 349 24 , , PUNCT iajs-3461 349 25 and and CCONJ iajs-3461 349 26 the the DET iajs-3461 349 27 results result NOUN iajs-3461 349 28 are be AUX iajs-3461 349 29 shown show VERB iajs-3461 349 30 below below ADP iajs-3461 349 31 . . PUNCT iajs-3461 350 1 the the DET iajs-3461 350 2 system system NOUN iajs-3461 350 3 will will AUX iajs-3461 350 4 get get VERB iajs-3461 350 5 closer close ADJ iajs-3461 350 6 to to ADP iajs-3461 350 7 the the DET iajs-3461 350 8 point point NOUN iajs-3461 350 9 of of ADP iajs-3461 350 10 positive positive ADJ iajs-3461 350 11 equilibrium equilibrium NOUN iajs-3461 350 12 a4 a4 NOUN iajs-3461 350 13 in in ADP iajs-3461 350 14 the the DET iajs-3461 350 15 interior interior NOUN iajs-3461 350 16 of of ADP iajs-3461 350 17 the the DET iajs-3461 350 18 positive positive ADJ iajs-3461 350 19 quadrant quadrant NOUN iajs-3461 350 20 of of ADP iajs-3461 350 21 the the DET iajs-3461 350 22 hij hij NOUN iajs-3461 350 23 - - PUNCT iajs-3461 350 24 plane plane NOUN iajs-3461 350 25 as as SCONJ iajs-3461 350 26 shown show VERB iajs-3461 350 27 in in ADP iajs-3461 350 28 figure2 figure2 ADJ iajs-3461 350 29 . . PUNCT iajs-3461 351 1 a1 a1 NOUN iajs-3461 351 2 when when SCONJ iajs-3461 351 3 the the DET iajs-3461 351 4 predation predation NOUN iajs-3461 351 5 rate rate NOUN iajs-3461 351 6 on on ADP iajs-3461 351 7 a a DET iajs-3461 351 8 prey prey NOUN iajs-3461 351 9 is be AUX iajs-3461 351 10 varied varied ADJ iajs-3461 351 11 from from ADP iajs-3461 351 12 the the DET iajs-3461 351 13 specialist specialist ADJ iajs-3461 351 14 predator predator NOUN iajs-3461 351 15 in in ADP iajs-3461 351 16 the the DET iajs-3461 351 17 range range NOUN iajs-3461 351 18 0.0001 0.0001 NUM iajs-3461 351 19 < < X iajs-3461 351 20 r1 r1 PROPN iajs-3461 351 21 ≀ ≀ NOUN iajs-3461 351 22 0.595 0.595 NUM iajs-3461 351 23 while while SCONJ iajs-3461 351 24 maintaining maintain VERB iajs-3461 351 25 other other ADJ iajs-3461 351 26 parameters parameter NOUN iajs-3461 351 27 as as ADP iajs-3461 351 28 data datum NOUN iajs-3461 351 29 given give VERB iajs-3461 351 30 in in ADP iajs-3461 351 31 ( ( PUNCT iajs-3461 351 32 5.1 5.1 NUM iajs-3461 351 33 ) ) PUNCT iajs-3461 351 34 , , PUNCT iajs-3461 351 35 r1 r1 PROPN iajs-3461 351 36 has have VERB iajs-3461 351 37 a a DET iajs-3461 351 38 usual usual ADJ iajs-3461 351 39 value value NOUN iajs-3461 351 40 of of ADP iajs-3461 351 41 0.15 0.15 NUM iajs-3461 351 42 . . PUNCT iajs-3461 352 1 as as SCONJ iajs-3461 352 2 shown show VERB iajs-3461 352 3 in in ADP iajs-3461 352 4 figure2.a2 figure2.a2 NOUN iajs-3461 352 5 for for ADP iajs-3461 352 6 average average ADJ iajs-3461 352 7 value value NOUN iajs-3461 352 8 r1=0.9 r1=0.9 PROPN iajs-3461 352 9 , , PUNCT iajs-3461 352 10 it it PRON iajs-3461 352 11 is be AUX iajs-3461 352 12 seen see VERB iajs-3461 352 13 that that SCONJ iajs-3461 352 14 the the DET iajs-3461 352 15 solution solution NOUN iajs-3461 352 16 of of ADP iajs-3461 352 17 system system NOUN iajs-3461 352 18 ( ( PUNCT iajs-3461 352 19 2 2 X iajs-3461 352 20 ) ) PUNCT iajs-3461 352 21 approaches approach VERB iajs-3461 352 22 asymptotically asymptotically ADV iajs-3461 352 23 to to ADP iajs-3461 352 24 the the DET iajs-3461 352 25 equilibrium equilibrium NOUN iajs-3461 352 26 point point NOUN iajs-3461 352 27 a2 a2 PROPN iajs-3461 352 28 in in ADP iajs-3461 352 29 the the DET iajs-3461 352 30 range range NOUN iajs-3461 352 31 0.595 0.595 NUM iajs-3461 352 32 < < X iajs-3461 352 33 r1 r1 PROPN iajs-3461 352 34 < < X iajs-3461 352 35 2 2 NUM iajs-3461 352 36 figure2.a1 figure2.a1 NOUN iajs-3461 352 37 : : PUNCT iajs-3461 352 38 the the DET iajs-3461 352 39 time time NOUN iajs-3461 352 40 series series NOUN iajs-3461 352 41 of of ADP iajs-3461 352 42 the the DET iajs-3461 352 43 solution solution NOUN iajs-3461 352 44 of of ADP iajs-3461 352 45 system system NOUN iajs-3461 352 46 ( ( PUNCT iajs-3461 352 47 2 2 X iajs-3461 352 48 ) ) PUNCT iajs-3461 352 49 approaches approach VERB iajs-3461 352 50 asymptotically asymptotically ADV iajs-3461 352 51 to to ADP iajs-3461 352 52 the the DET iajs-3461 352 53 positive positive ADJ iajs-3461 352 54 equilibrium equilibrium NOUN iajs-3461 352 55 point point NOUN iajs-3461 352 56 a4= a4= PROPN iajs-3461 352 57 ( ( PUNCT iajs-3461 352 58 0.8182 0.8182 NOUN iajs-3461 352 59 , , PUNCT iajs-3461 352 60 0.7275 0.7275 NUM iajs-3461 352 61 , , PUNCT iajs-3461 352 62 0.7271 0.7271 NUM iajs-3461 352 63 ) ) PUNCT iajs-3461 352 64 in in ADP iajs-3461 352 65 the the DET iajs-3461 352 66 interior interior NOUN iajs-3461 352 67 of of ADP iajs-3461 352 68 𝑅+ 𝑅+ PROPN iajs-3461 352 69 3 3 NUM iajs-3461 352 70 . . PUNCT iajs-3461 353 1 for for ADP iajs-3461 353 2 the the DET iajs-3461 353 3 data datum NOUN iajs-3461 353 4 in in ADP iajs-3461 353 5 ( ( PUNCT iajs-3461 353 6 5.1 5.1 NUM iajs-3461 353 7 ) ) PUNCT iajs-3461 353 8 with with ADP iajs-3461 353 9 r1 r1 PROPN iajs-3461 353 10 = = PROPN iajs-3461 353 11 0.15 0.15 NUM iajs-3461 353 12 . . PUNCT iajs-3461 354 1 figure2.a2 figure2.a2 NOUN iajs-3461 354 2 : : PUNCT iajs-3461 354 3 the the DET iajs-3461 354 4 time time NOUN iajs-3461 354 5 series series NOUN iajs-3461 354 6 of of ADP iajs-3461 354 7 the the DET iajs-3461 354 8 solution solution NOUN iajs-3461 354 9 of of ADP iajs-3461 354 10 system system NOUN iajs-3461 354 11 ( ( PUNCT iajs-3461 354 12 2 2 X iajs-3461 354 13 ) ) PUNCT iajs-3461 354 14 approaches approach VERB iajs-3461 354 15 asymptotically asymptotically ADV iajs-3461 354 16 to to ADP iajs-3461 354 17 the the DET iajs-3461 354 18 positive positive ADJ iajs-3461 354 19 equilibrium equilibrium NOUN iajs-3461 354 20 point point NOUN iajs-3461 354 21 a2= a2= PROPN iajs-3461 354 22 ( ( PUNCT iajs-3461 354 23 0.3333 0.3333 NUM iajs-3461 354 24 , , PUNCT iajs-3461 354 25 0.7407 0.7407 NUM iajs-3461 354 26 , , PUNCT iajs-3461 354 27 0 0 NUM iajs-3461 354 28 ) ) PUNCT iajs-3461 354 29 in in ADP iajs-3461 354 30 the the DET iajs-3461 354 31 interior interior NOUN iajs-3461 354 32 of of ADP iajs-3461 354 33 𝑅+ 𝑅+ PROPN iajs-3461 354 34 3 3 NUM iajs-3461 354 35 . . PUNCT iajs-3461 355 1 for for ADP iajs-3461 355 2 the the DET iajs-3461 355 3 data datum NOUN iajs-3461 355 4 in in ADP iajs-3461 355 5 ( ( PUNCT iajs-3461 355 6 5.1 5.1 NUM iajs-3461 355 7 ) ) PUNCT iajs-3461 355 8 with with ADP iajs-3461 355 9 r1 r1 PROPN iajs-3461 355 10 = = SYM iajs-3461 355 11 0.9 0.9 NUM iajs-3461 355 12 . . PUNCT iajs-3461 356 1 additionally additionally ADV iajs-3461 356 2 , , PUNCT iajs-3461 356 3 changing change VERB iajs-3461 356 4 the the DET iajs-3461 356 5 specialist specialist ADJ iajs-3461 356 6 predator predator NOUN iajs-3461 356 7 's 's PART iajs-3461 356 8 mortality mortality NOUN iajs-3461 356 9 death death NOUN iajs-3461 356 10 rate rate NOUN iajs-3461 356 11 between between ADP iajs-3461 356 12 0.0001 0.0001 NUM iajs-3461 356 13 and and CCONJ iajs-3461 356 14 0.03 0.03 NUM iajs-3461 356 15 while while SCONJ iajs-3461 356 16 maintaining maintain VERB iajs-3461 356 17 the the DET iajs-3461 356 18 other other ADJ iajs-3461 356 19 parameters parameter NOUN iajs-3461 356 20 according accord VERB iajs-3461 356 21 to to ADP iajs-3461 356 22 the the DET iajs-3461 356 23 data datum NOUN iajs-3461 356 24 in in ADP iajs-3461 356 25 ( ( PUNCT iajs-3461 356 26 5.1 5.1 NUM iajs-3461 356 27 ) ) PUNCT iajs-3461 356 28 results result NOUN iajs-3461 356 29 in in ADP iajs-3461 356 30 the the DET iajs-3461 356 31 specialized specialized ADJ iajs-3461 356 32 predator predator NOUN iajs-3461 356 33 going go VERB iajs-3461 356 34 extinct extinct ADJ iajs-3461 356 35 , , PUNCT iajs-3461 356 36 and and CCONJ iajs-3461 356 37 figure3.e1 figure3.e1 NOUN iajs-3461 356 38 illustrates illustrate VERB iajs-3461 356 39 how how SCONJ iajs-3461 356 40 system system NOUN iajs-3461 356 41 ( ( PUNCT iajs-3461 356 42 2 2 X iajs-3461 356 43 ) ) PUNCT iajs-3461 356 44 approaches approach VERB iajs-3461 356 45 asymptotically asymptotically ADV iajs-3461 356 46 to to ADP iajs-3461 356 47 the the DET iajs-3461 356 48 positive positive ADJ iajs-3461 356 49 equilibrium equilibrium NOUN iajs-3461 356 50 pointa4 pointa4 NOUN iajs-3461 356 51 with with ADP iajs-3461 356 52 a a DET iajs-3461 356 53 typical typical ADJ iajs-3461 356 54 value value NOUN iajs-3461 356 55 of of ADP iajs-3461 356 56 r5=0.01 r5=0.01 NOUN iajs-3461 356 57 , , PUNCT iajs-3461 356 58 however however ADV iajs-3461 356 59 , , PUNCT iajs-3461 356 60 for for ADP iajs-3461 356 61 a a DET iajs-3461 356 62 typical typical ADJ iajs-3461 356 63 value value NOUN iajs-3461 356 64 of of ADP iajs-3461 356 65 0 0 NUM iajs-3461 356 66 1000 1000 NUM iajs-3461 356 67 2000 2000 NUM iajs-3461 356 68 3000 3000 NUM iajs-3461 356 69 4000 4000 NUM iajs-3461 356 70 5000 5000 NUM iajs-3461 356 71 6000 6000 NUM iajs-3461 356 72 7000 7000 NUM iajs-3461 356 73 8000 8000 NUM iajs-3461 356 74 9000 9000 NUM iajs-3461 356 75 10000 10000 NUM iajs-3461 356 76 0 0 NUM iajs-3461 356 77 0.2 0.2 NUM iajs-3461 356 78 0.4 0.4 NUM iajs-3461 356 79 0.6 0.6 NUM iajs-3461 356 80 0.8 0.8 NUM iajs-3461 356 81 1 1 NUM iajs-3461 356 82 1.2 1.2 NUM iajs-3461 356 83 1.4 1.4 NUM iajs-3461 356 84 time time NOUN iajs-3461 356 85 po po PROPN iajs-3461 356 86 pu pu PROPN iajs-3461 356 87 la la PROPN iajs-3461 356 88 tio tio PROPN iajs-3461 356 89 n n PROPN iajs-3461 356 90 ( ( PUNCT iajs-3461 356 91 1.1 1.1 NUM iajs-3461 356 92 ) ) PUNCT iajs-3461 357 1 h h NOUN iajs-3461 358 1 i i PRON iajs-3461 358 2 j j PROPN iajs-3461 358 3 0 0 NUM iajs-3461 358 4 1000 1000 NUM iajs-3461 358 5 2000 2000 NUM iajs-3461 358 6 3000 3000 NUM iajs-3461 358 7 4000 4000 NUM iajs-3461 358 8 5000 5000 NUM iajs-3461 358 9 6000 6000 NUM iajs-3461 358 10 7000 7000 NUM iajs-3461 358 11 8000 8000 NUM iajs-3461 358 12 9000 9000 NUM iajs-3461 358 13 10000 10000 NUM iajs-3461 358 14 0 0 NUM iajs-3461 358 15 0.2 0.2 NUM iajs-3461 358 16 0.4 0.4 NUM iajs-3461 358 17 0.6 0.6 NUM iajs-3461 358 18 0.8 0.8 NUM iajs-3461 358 19 1 1 NUM iajs-3461 358 20 1.2 1.2 NUM iajs-3461 358 21 1.4 1.4 NUM iajs-3461 358 22 time time NOUN iajs-3461 358 23 po po PROPN iajs-3461 358 24 pu pu PROPN iajs-3461 358 25 la la PROPN iajs-3461 358 26 tio tio PROPN iajs-3461 358 27 n n PROPN iajs-3461 358 28 ( ( PUNCT iajs-3461 358 29 1.2 1.2 NUM iajs-3461 358 30 ) ) PUNCT iajs-3461 358 31 h h NOUN iajs-3461 359 1 i i PRON iajs-3461 359 2 j j PROPN iajs-3461 359 3 0 0 NUM iajs-3461 359 4 0.2 0.2 NUM iajs-3461 359 5 0.4 0.4 NUM iajs-3461 359 6 0.6 0.6 NUM iajs-3461 359 7 0.8 0.8 NUM iajs-3461 359 8 1 1 NUM iajs-3461 359 9 1.2 1.2 NUM iajs-3461 359 10 1.4 1.4 NUM iajs-3461 359 11 1.6 1.6 NUM iajs-3461 359 12 1.8 1.8 NUM iajs-3461 359 13 2 2 NUM iajs-3461 359 14 x x SYM iajs-3461 359 15 10 10 NUM iajs-3461 359 16 4 4 NUM iajs-3461 359 17 0.1 0.1 NUM iajs-3461 359 18 0.2 0.2 NUM iajs-3461 359 19 0.3 0.3 NUM iajs-3461 359 20 0.4 0.4 NUM iajs-3461 359 21 0.5 0.5 NUM iajs-3461 359 22 0.6 0.6 NUM iajs-3461 359 23 0.7 0.7 NUM iajs-3461 359 24 0.8 0.8 NUM iajs-3461 359 25 0.9 0.9 NUM iajs-3461 359 26 1 1 NUM iajs-3461 359 27 1.1 1.1 NUM iajs-3461 359 28 time time NOUN iajs-3461 359 29 po po PROPN iajs-3461 359 30 pu pu PROPN iajs-3461 359 31 lat lat PROPN iajs-3461 359 32 ion ion NOUN iajs-3461 359 33 ( ( PUNCT iajs-3461 360 1 1.3 1.3 NUM iajs-3461 360 2 ) ) PUNCT iajs-3461 360 3 h h NOUN iajs-3461 361 1 i i PRON iajs-3461 361 2 j j PROPN iajs-3461 361 3 0 0 NUM iajs-3461 361 4 0.2 0.2 NUM iajs-3461 361 5 0.4 0.4 NUM iajs-3461 361 6 0.6 0.6 NUM iajs-3461 361 7 0.8 0.8 NUM iajs-3461 361 8 1 1 NUM iajs-3461 361 9 1.2 1.2 NUM iajs-3461 361 10 1.4 1.4 NUM iajs-3461 361 11 1.6 1.6 NUM iajs-3461 361 12 1.8 1.8 NUM iajs-3461 361 13 2 2 NUM iajs-3461 361 14 x x SYM iajs-3461 361 15 10 10 NUM iajs-3461 361 16 4 4 NUM iajs-3461 361 17 0.1 0.1 NUM iajs-3461 361 18 0.2 0.2 NUM iajs-3461 361 19 0.3 0.3 NUM iajs-3461 361 20 0.4 0.4 NUM iajs-3461 361 21 0.5 0.5 NUM iajs-3461 361 22 0.6 0.6 NUM iajs-3461 361 23 0.7 0.7 NUM iajs-3461 361 24 0.8 0.8 NUM iajs-3461 361 25 0.9 0.9 NUM iajs-3461 361 26 1 1 NUM iajs-3461 361 27 1.1 1.1 NUM iajs-3461 361 28 time time NOUN iajs-3461 361 29 po po PROPN iajs-3461 361 30 pu pu PROPN iajs-3461 361 31 lat lat PROPN iajs-3461 361 32 ion ion NOUN iajs-3461 361 33 ( ( PUNCT iajs-3461 362 1 1.3 1.3 NUM iajs-3461 362 2 ) ) PUNCT iajs-3461 362 3 h h NOUN iajs-3461 363 1 i i PRON iajs-3461 363 2 j j NOUN iajs-3461 363 3 0 0 NUM iajs-3461 363 4 1 1 NUM iajs-3461 363 5 2 2 NUM iajs-3461 363 6 3 3 NUM iajs-3461 363 7 4 4 NUM iajs-3461 363 8 5 5 NUM iajs-3461 363 9 6 6 NUM iajs-3461 363 10 7 7 NUM iajs-3461 363 11 8 8 NUM iajs-3461 363 12 9 9 NUM iajs-3461 363 13 10 10 NUM iajs-3461 363 14 x x SYM iajs-3461 363 15 10 10 NUM iajs-3461 363 16 4 4 NUM iajs-3461 363 17 0 0 NUM iajs-3461 363 18 0.2 0.2 NUM iajs-3461 363 19 0.4 0.4 NUM iajs-3461 363 20 0.6 0.6 NUM iajs-3461 363 21 0.8 0.8 NUM iajs-3461 363 22 1 1 NUM iajs-3461 363 23 1.2 1.2 NUM iajs-3461 363 24 1.4 1.4 NUM iajs-3461 363 25 time time NOUN iajs-3461 363 26 po po PROPN iajs-3461 363 27 pu pu PROPN iajs-3461 363 28 la la PROPN iajs-3461 363 29 tio tio PROPN iajs-3461 363 30 n n PROPN iajs-3461 363 31 ( ( PUNCT iajs-3461 363 32 8.1 8.1 NUM iajs-3461 363 33 ) ) PUNCT iajs-3461 363 34 h h NOUN iajs-3461 364 1 i i PRON iajs-3461 364 2 j j PROPN iajs-3461 364 3 ihjpas ihjpa VERB iajs-3461 364 4 . . PUNCT iajs-3461 365 1 2025 2025 NUM iajs-3461 365 2 , , PUNCT iajs-3461 365 3 38 38 NUM iajs-3461 365 4 ( ( PUNCT iajs-3461 365 5 1 1 NUM iajs-3461 365 6 ) ) PUNCT iajs-3461 365 7 381 381 NUM iajs-3461 365 8 r5=0.1 r5=0.1 NOUN iajs-3461 365 9 , , PUNCT iajs-3461 365 10 as as SCONJ iajs-3461 365 11 shown show VERB iajs-3461 365 12 in in ADP iajs-3461 365 13 figure3.e2 figure3.e2 PROPN iajs-3461 365 14 , , PUNCT iajs-3461 365 15 at at ADP iajs-3461 365 16 0.03 0.03 NUM iajs-3461 365 17 < < X iajs-3461 365 18 r5 r5 PROPN iajs-3461 365 19 < < NOUN iajs-3461 365 20 0.15 0.15 NUM iajs-3461 365 21 approaches approach VERB iajs-3461 365 22 asymptotically asymptotically ADV iajs-3461 365 23 to to ADP iajs-3461 365 24 the the DET iajs-3461 365 25 axial axial ADJ iajs-3461 365 26 equilibrium equilibrium NOUN iajs-3461 365 27 point point NOUN iajs-3461 365 28 a1 a1 NOUN iajs-3461 365 29 , , PUNCT iajs-3461 365 30 additionally additionally ADV iajs-3461 365 31 , , PUNCT iajs-3461 365 32 for for ADP iajs-3461 365 33 a a DET iajs-3461 365 34 typical typical ADJ iajs-3461 365 35 value value NOUN iajs-3461 365 36 ofr5=0.5 ofr5=0.5 VERB iajs-3461 365 37 , , PUNCT iajs-3461 365 38 figure3.e3 figure3.e3 PROPN iajs-3461 365 39 shows show VERB iajs-3461 365 40 that that SCONJ iajs-3461 365 41 for for ADP iajs-3461 365 42 0.15≀ 0.15≀ ADJ iajs-3461 365 43 r5 r5 PROPN iajs-3461 365 44 < < X iajs-3461 365 45 0.6 0.6 NUM iajs-3461 365 46 approaches approach VERB iajs-3461 365 47 asymptotically asymptotically ADV iajs-3461 365 48 to to ADP iajs-3461 365 49 the the DET iajs-3461 365 50 equilibrium equilibrium NOUN iajs-3461 365 51 point point NOUN iajs-3461 365 52 a3 a3 NOUN iajs-3461 365 53 . . PUNCT iajs-3461 366 1 figure3.e1 figure3.e1 PROPN iajs-3461 366 2 : : PUNCT iajs-3461 366 3 the the DET iajs-3461 366 4 time time NOUN iajs-3461 366 5 series series NOUN iajs-3461 366 6 of of ADP iajs-3461 366 7 the the DET iajs-3461 366 8 solution solution NOUN iajs-3461 366 9 of of ADP iajs-3461 366 10 system system NOUN iajs-3461 366 11 ( ( PUNCT iajs-3461 366 12 2 2 X iajs-3461 366 13 ) ) PUNCT iajs-3461 366 14 approaches approach VERB iajs-3461 366 15 asymptotically asymptotically ADV iajs-3461 366 16 to to ADP iajs-3461 366 17 the the DET iajs-3461 366 18 positive positive ADJ iajs-3461 366 19 equilibrium equilibrium NOUN iajs-3461 366 20 point point NOUN iajs-3461 366 21 a4= a4= PROPN iajs-3461 366 22 ( ( PUNCT iajs-3461 366 23 0.99043 0.99043 NUM iajs-3461 366 24 , , PUNCT iajs-3461 366 25 0.6535 0.6535 NUM iajs-3461 366 26 , , PUNCT iajs-3461 366 27 0.6304 0.6304 NUM iajs-3461 366 28 ) ) PUNCT iajs-3461 366 29 in in ADP iajs-3461 366 30 the the DET iajs-3461 366 31 interior interior NOUN iajs-3461 366 32 of of ADP iajs-3461 366 33 𝑅+ 𝑅+ PROPN iajs-3461 366 34 3 3 NUM iajs-3461 366 35 . . PUNCT iajs-3461 367 1 for for ADP iajs-3461 367 2 the the DET iajs-3461 367 3 data datum NOUN iajs-3461 367 4 in in ADP iajs-3461 367 5 ( ( PUNCT iajs-3461 367 6 5.1 5.1 NUM iajs-3461 367 7 ) ) PUNCT iajs-3461 367 8 with with ADP iajs-3461 367 9 r5 r5 PROPN iajs-3461 367 10 = = NOUN iajs-3461 367 11 0.01 0.01 NUM iajs-3461 367 12 . . PUNCT iajs-3461 368 1 figure3.e2 figure3.e2 PROPN iajs-3461 368 2 : : PUNCT iajs-3461 368 3 the the DET iajs-3461 368 4 time time NOUN iajs-3461 368 5 series series NOUN iajs-3461 368 6 of of ADP iajs-3461 368 7 the the DET iajs-3461 368 8 solution solution NOUN iajs-3461 368 9 of of ADP iajs-3461 368 10 system system NOUN iajs-3461 368 11 ( ( PUNCT iajs-3461 368 12 2 2 X iajs-3461 368 13 ) ) PUNCT iajs-3461 368 14 approaches approach VERB iajs-3461 368 15 asymptotically asymptotically ADV iajs-3461 368 16 to to ADP iajs-3461 368 17 the the DET iajs-3461 368 18 positive positive ADJ iajs-3461 368 19 equilibrium equilibrium NOUN iajs-3461 368 20 point point NOUN iajs-3461 368 21 a1= a1= PROPN iajs-3461 368 22 ( ( PUNCT iajs-3461 368 23 1 1 NUM iajs-3461 368 24 , , PUNCT iajs-3461 368 25 0 0 NUM iajs-3461 368 26 , , PUNCT iajs-3461 368 27 0 0 NUM iajs-3461 368 28 ) ) PUNCT iajs-3461 368 29 in in ADP iajs-3461 368 30 the the DET iajs-3461 368 31 interior interior NOUN iajs-3461 368 32 of of ADP iajs-3461 368 33 𝑅+ 𝑅+ PROPN iajs-3461 368 34 3 3 NUM iajs-3461 368 35 . . PUNCT iajs-3461 369 1 for for ADP iajs-3461 369 2 the the DET iajs-3461 369 3 data datum NOUN iajs-3461 369 4 in in ADP iajs-3461 369 5 ( ( PUNCT iajs-3461 369 6 5.1 5.1 NUM iajs-3461 369 7 ) ) PUNCT iajs-3461 369 8 with with ADP iajs-3461 369 9 r5 r5 PROPN iajs-3461 369 10 = = NOUN iajs-3461 369 11 0.1 0.1 NUM iajs-3461 369 12 . . PUNCT iajs-3461 370 1 figure3.e3 figure3.e3 ADJ iajs-3461 370 2 time time NOUN iajs-3461 370 3 series series NOUN iajs-3461 370 4 of of ADP iajs-3461 370 5 the the DET iajs-3461 370 6 solution solution NOUN iajs-3461 370 7 of of ADP iajs-3461 370 8 system system NOUN iajs-3461 370 9 ( ( PUNCT iajs-3461 370 10 2 2 X iajs-3461 370 11 ) ) PUNCT iajs-3461 370 12 approaches approach VERB iajs-3461 370 13 asymptotically asymptotically ADV iajs-3461 370 14 to to PART iajs-3461 370 15 point point VERB iajs-3461 370 16 a3= a3= PROPN iajs-3461 370 17 ( ( PUNCT iajs-3461 370 18 0.6667 0.6667 NUM iajs-3461 370 19 , , PUNCT iajs-3461 370 20 0 0 NUM iajs-3461 370 21 , , PUNCT iajs-3461 370 22 3.3333 3.3333 NUM iajs-3461 370 23 ) ) PUNCT iajs-3461 370 24 . . PUNCT iajs-3461 371 1 the the DET iajs-3461 371 2 system system NOUN iajs-3461 371 3 ( ( PUNCT iajs-3461 371 4 2 2 X iajs-3461 371 5 ) ) PUNCT iajs-3461 371 6 still still ADV iajs-3461 371 7 approaches approach VERB iajs-3461 371 8 asymptotically asymptotically ADV iajs-3461 371 9 to to ADP iajs-3461 371 10 the the DET iajs-3461 371 11 equilibrium equilibrium NOUN iajs-3461 371 12 point point NOUN iajs-3461 371 13 a2 a2 PROPN iajs-3461 371 14 despite despite SCONJ iajs-3461 371 15 changing change VERB iajs-3461 371 16 the the DET iajs-3461 371 17 parameter parameter NOUN iajs-3461 371 18 r6 r6 NOUN iajs-3461 371 19 which which PRON iajs-3461 371 20 represents represent VERB iajs-3461 371 21 the the DET iajs-3461 371 22 conversion conversion NOUN iajs-3461 371 23 rate rate NOUN iajs-3461 371 24 from from ADP iajs-3461 371 25 the the DET iajs-3461 371 26 prey prey NOUN iajs-3461 371 27 to to ADP iajs-3461 371 28 the the DET iajs-3461 371 29 generalist generalist ADJ iajs-3461 371 30 predator predator NOUN iajs-3461 371 31 in in ADP iajs-3461 371 32 the the DET iajs-3461 371 33 range range NOUN iajs-3461 371 34 0.0001 0.0001 NUM iajs-3461 371 35 ≀ ≀ NOUN iajs-3461 371 36 r6 r6 NOUN iajs-3461 371 37 < < X iajs-3461 371 38 0.015 0.015 NUM iajs-3461 371 39 this this PRON iajs-3461 371 40 causes cause VERB iajs-3461 371 41 extinction extinction NOUN iajs-3461 371 42 in in ADP iajs-3461 371 43 the the DET iajs-3461 371 44 prey prey NOUN iajs-3461 371 45 , , PUNCT iajs-3461 371 46 however however ADV iajs-3461 371 47 in in ADP iajs-3461 371 48 additional additional ADJ iajs-3461 371 49 for for ADP iajs-3461 371 50 0.015≀ 0.015≀ ADJ iajs-3461 371 51 r6<0.1 r6<0.1 NOUN iajs-3461 371 52 approaches approach VERB iajs-3461 371 53 asymptotically asymptotically ADV iajs-3461 371 54 to to ADP iajs-3461 371 55 the the DET iajs-3461 371 56 positive positive ADJ iajs-3461 371 57 equilibrium equilibrium NOUN iajs-3461 371 58 point point NOUN iajs-3461 371 59 a4 a4 NOUN iajs-3461 371 60 , , PUNCT iajs-3461 371 61 as as SCONJ iajs-3461 371 62 shown show VERB iajs-3461 371 63 in in ADP iajs-3461 371 64 figure4 figure4 PROPN iajs-3461 371 65 . . PUNCT iajs-3461 372 1 g2 g2 PROPN iajs-3461 372 2 , , PUNCT iajs-3461 372 3 for for ADP iajs-3461 372 4 typical typical ADJ iajs-3461 372 5 value value NOUN iajs-3461 372 6 r6=0.08 r6=0.08 NOUN iajs-3461 372 7 . . PUNCT iajs-3461 373 1 figure4.g1 figure4.g1 PROPN iajs-3461 373 2 : : PUNCT iajs-3461 373 3 the the DET iajs-3461 373 4 time time NOUN iajs-3461 373 5 series series NOUN iajs-3461 373 6 of of ADP iajs-3461 373 7 the the DET iajs-3461 373 8 solution solution NOUN iajs-3461 373 9 of of ADP iajs-3461 373 10 system system NOUN iajs-3461 373 11 ( ( PUNCT iajs-3461 373 12 2 2 X iajs-3461 373 13 ) ) PUNCT iajs-3461 373 14 approaches approach VERB iajs-3461 373 15 asymptotically asymptotically ADV iajs-3461 373 16 to to ADP iajs-3461 373 17 the the DET iajs-3461 373 18 positive positive ADJ iajs-3461 373 19 equilibrium equilibrium NOUN iajs-3461 373 20 point point NOUN iajs-3461 373 21 a2= a2= PROPN iajs-3461 373 22 ( ( PUNCT iajs-3461 373 23 0.3392 0.3392 NUM iajs-3461 373 24 , , PUNCT iajs-3461 373 25 13.2187 13.2187 NUM iajs-3461 373 26 , , PUNCT iajs-3461 373 27 0 0 NUM iajs-3461 373 28 ) ) PUNCT iajs-3461 373 29 in in ADP iajs-3461 373 30 the the DET iajs-3461 373 31 interior interior NOUN iajs-3461 373 32 of of ADP iajs-3461 373 33 𝑅+ 𝑅+ PROPN iajs-3461 373 34 3 3 NUM iajs-3461 373 35 . . PUNCT iajs-3461 374 1 for for ADP iajs-3461 374 2 the the DET iajs-3461 374 3 data datum NOUN iajs-3461 374 4 in in ADP iajs-3461 374 5 ( ( PUNCT iajs-3461 374 6 5.1 5.1 NUM iajs-3461 374 7 ) ) PUNCT iajs-3461 374 8 with with ADP iajs-3461 374 9 r6 r6 NOUN iajs-3461 374 10 = = NOUN iajs-3461 374 11 0.013 0.013 NUM iajs-3461 374 12 . . PUNCT iajs-3461 375 1 figure4.g2 figure4.g2 NOUN iajs-3461 375 2 : : PUNCT iajs-3461 375 3 the the DET iajs-3461 375 4 time time NOUN iajs-3461 375 5 series series NOUN iajs-3461 375 6 of of ADP iajs-3461 375 7 the the DET iajs-3461 375 8 solution solution NOUN iajs-3461 375 9 of of ADP iajs-3461 375 10 system system NOUN iajs-3461 375 11 ( ( PUNCT iajs-3461 375 12 2 2 X iajs-3461 375 13 ) ) PUNCT iajs-3461 375 14 approaches approach VERB iajs-3461 375 15 asymptotically asymptotically ADV iajs-3461 375 16 to to ADP iajs-3461 375 17 the the DET iajs-3461 375 18 positive positive ADJ iajs-3461 375 19 equilibrium equilibrium NOUN iajs-3461 375 20 point point NOUN iajs-3461 375 21 a4= a4= PROPN iajs-3461 375 22 ( ( PUNCT iajs-3461 375 23 0.7311 0.7311 NUM iajs-3461 375 24 , , PUNCT iajs-3461 375 25 4.1682 4.1682 NUM iajs-3461 375 26 , , PUNCT iajs-3461 375 27 0.6044 0.6044 NUM iajs-3461 375 28 ) ) PUNCT iajs-3461 375 29 in in ADP iajs-3461 375 30 the the DET iajs-3461 375 31 interior interior NOUN iajs-3461 375 32 of of ADP iajs-3461 375 33 𝑅+ 𝑅+ PROPN iajs-3461 375 34 3 3 NUM iajs-3461 375 35 . . PUNCT iajs-3461 376 1 for for ADP iajs-3461 376 2 the the DET iajs-3461 376 3 data datum NOUN iajs-3461 376 4 in in ADP iajs-3461 376 5 ( ( PUNCT iajs-3461 376 6 5.1 5.1 NUM iajs-3461 376 7 ) ) PUNCT iajs-3461 376 8 with with ADP iajs-3461 376 9 r6 r6 NOUN iajs-3461 376 10 = = NOUN iajs-3461 376 11 0.08 0.08 NUM iajs-3461 376 12 6 6 NUM iajs-3461 376 13 . . PUNCT iajs-3461 376 14 discussion discussion NOUN iajs-3461 376 15 starting start VERB iajs-3461 376 16 with with ADP iajs-3461 376 17 the the DET iajs-3461 376 18 hypothetical hypothetical ADJ iajs-3461 376 19 set set NOUN iajs-3461 376 20 of of ADP iajs-3461 376 21 data datum NOUN iajs-3461 376 22 provided provide VERB iajs-3461 376 23 by by ADP iajs-3461 376 24 eq eq PROPN iajs-3461 376 25 . . PUNCT iajs-3461 377 1 ( ( PUNCT iajs-3461 377 2 5.1 5.1 NUM iajs-3461 377 3 ) ) PUNCT iajs-3461 377 4 , , PUNCT iajs-3461 377 5 system system NOUN iajs-3461 377 6 ( ( PUNCT iajs-3461 377 7 2 2 X iajs-3461 377 8 ) ) PUNCT iajs-3461 377 9 has have AUX iajs-3461 377 10 been be AUX iajs-3461 377 11 numerically numerically ADV iajs-3461 377 12 solved solve VERB iajs-3461 377 13 for for ADP iajs-3461 377 14 several several ADJ iajs-3461 377 15 sets set NOUN iajs-3461 377 16 of of ADP iajs-3461 377 17 initial initial ADJ iajs-3461 377 18 points point NOUN iajs-3461 377 19 and and CCONJ iajs-3461 377 20 various various ADJ iajs-3461 377 21 sets set NOUN iajs-3461 377 22 of of ADP iajs-3461 377 23 parameters parameter NOUN iajs-3461 377 24 , , PUNCT iajs-3461 377 25 and and CCONJ iajs-3461 377 26 the the DET iajs-3461 377 27 following follow VERB iajs-3461 377 28 observations observation NOUN iajs-3461 377 29 are be AUX iajs-3461 377 30 obtained obtain VERB iajs-3461 377 31 : : PUNCT iajs-3461 377 32 1 1 X iajs-3461 377 33 . . X iajs-3461 377 34 when when SCONJ iajs-3461 377 35 approaching approach VERB iajs-3461 377 36 globally globally ADV iajs-3461 377 37 stable stable ADJ iajs-3461 377 38 locations location NOUN iajs-3461 377 39 via via ADP iajs-3461 377 40 int int NOUN iajs-3461 377 41 . . PUNCT iajs-3461 378 1 𝑅+ 𝑅+ PROPN iajs-3461 378 2 3 3 NUM iajs-3461 378 3 techniques technique NOUN iajs-3461 378 4 system system NOUN iajs-3461 378 5 ( ( PUNCT iajs-3461 378 6 2 2 NUM iajs-3461 378 7 ) ) PUNCT iajs-3461 378 8 only only ADV iajs-3461 378 9 has have VERB iajs-3461 378 10 two two NUM iajs-3461 378 11 types type NOUN iajs-3461 378 12 of of ADP iajs-3461 378 13 attractors attractor NOUN iajs-3461 378 14 . . PUNCT iajs-3461 379 1 the the DET iajs-3461 379 2 system system NOUN iajs-3461 379 3 ( ( PUNCT iajs-3461 379 4 2 2 X iajs-3461 379 5 ) ) PUNCT iajs-3461 379 6 approaches approach VERB iajs-3461 379 7 asymptotically asymptotically ADV iajs-3461 379 8 to to ADP iajs-3461 379 9 the the DET iajs-3461 379 10 globally globally ADV iajs-3461 379 11 stable stable ADJ iajs-3461 379 12 positive positive ADJ iajs-3461 379 13 point𝐴4 point𝐴4 NOUN iajs-3461 379 14 = = SYM iajs-3461 379 15 ( ( PUNCT iajs-3461 379 16 0.4544 0.4544 NUM iajs-3461 379 17 , , PUNCT iajs-3461 379 18 0.6567 0.6567 NUM iajs-3461 379 19 , , PUNCT iajs-3461 379 20 2.1722 2.1722 NUM iajs-3461 379 21 ) ) PUNCT iajs-3461 379 22 for for ADP iajs-3461 379 23 the the DET iajs-3461 379 24 set set VERB iajs-3461 379 25 parameter parameter NOUN iajs-3461 379 26 value value NOUN iajs-3461 379 27 specified specify VERB iajs-3461 379 28 in in ADP iajs-3461 379 29 ( ( PUNCT iajs-3461 379 30 5.1 5.1 NUM iajs-3461 379 31 ) ) PUNCT iajs-3461 379 32 . . PUNCT iajs-3461 380 1 2 2 X iajs-3461 380 2 . . X iajs-3461 380 3 the the DET iajs-3461 380 4 positive positive ADJ iajs-3461 380 5 equilibrium equilibrium NOUN iajs-3461 380 6 point point NOUN iajs-3461 380 7 𝐴4 𝐴4 PROPN iajs-3461 380 8 being be AUX iajs-3461 380 9 approached approach VERB iajs-3461 380 10 by by ADP iajs-3461 380 11 the the DET iajs-3461 380 12 solution solution NOUN iajs-3461 380 13 of of ADP iajs-3461 380 14 system system NOUN iajs-3461 380 15 ( ( PUNCT iajs-3461 380 16 2 2 NUM iajs-3461 380 17 ) ) PUNCT iajs-3461 380 18 as as ADP iajs-3461 380 19 the the DET iajs-3461 380 20 assault assault NOUN iajs-3461 380 21 rate rate NOUN iajs-3461 380 22 on on ADP iajs-3461 380 23 a a DET iajs-3461 380 24 victim victim NOUN iajs-3461 380 25 from from ADP iajs-3461 380 26 the the DET iajs-3461 380 27 specialized specialized ADJ iajs-3461 380 28 predator predator NOUN iajs-3461 380 29 r1 r1 NOUN iajs-3461 380 30 increases increase VERB iajs-3461 380 31 to to ADP iajs-3461 380 32 0.15 0.15 NUM iajs-3461 380 33 while while SCONJ iajs-3461 380 34 maintaining maintain VERB iajs-3461 380 35 the the DET iajs-3461 380 36 0 0 NUM iajs-3461 380 37 1 1 NUM iajs-3461 380 38 2 2 NUM iajs-3461 380 39 3 3 NUM iajs-3461 380 40 4 4 NUM iajs-3461 380 41 5 5 NUM iajs-3461 380 42 6 6 NUM iajs-3461 380 43 7 7 NUM iajs-3461 380 44 8 8 NUM iajs-3461 380 45 9 9 NUM iajs-3461 380 46 10 10 NUM iajs-3461 380 47 x x SYM iajs-3461 380 48 10 10 NUM iajs-3461 380 49 4 4 NUM iajs-3461 380 50 0.1 0.1 NUM iajs-3461 380 51 0.2 0.2 NUM iajs-3461 380 52 0.3 0.3 NUM iajs-3461 380 53 0.4 0.4 NUM iajs-3461 380 54 0.5 0.5 NUM iajs-3461 380 55 0.6 0.6 NUM iajs-3461 380 56 0.7 0.7 NUM iajs-3461 380 57 0.8 0.8 NUM iajs-3461 380 58 0.9 0.9 NUM iajs-3461 380 59 1 1 NUM iajs-3461 380 60 1.1 1.1 NUM iajs-3461 380 61 time time NOUN iajs-3461 380 62 po po PROPN iajs-3461 380 63 pu pu PROPN iajs-3461 380 64 lat lat PROPN iajs-3461 380 65 ion ion NOUN iajs-3461 380 66 ( ( PUNCT iajs-3461 380 67 7.1 7.1 NUM iajs-3461 380 68 ) ) PUNCT iajs-3461 380 69 h h NOUN iajs-3461 381 1 i i PRON iajs-3461 381 2 j j NOUN iajs-3461 381 3 0 0 NUM iajs-3461 381 4 1 1 NUM iajs-3461 381 5 2 2 NUM iajs-3461 381 6 3 3 NUM iajs-3461 381 7 4 4 NUM iajs-3461 381 8 5 5 NUM iajs-3461 381 9 6 6 NUM iajs-3461 381 10 x x SYM iajs-3461 381 11 10 10 NUM iajs-3461 381 12 5 5 NUM iajs-3461 381 13 0 0 NUM iajs-3461 381 14 0.2 0.2 NUM iajs-3461 381 15 0.4 0.4 NUM iajs-3461 381 16 0.6 0.6 NUM iajs-3461 381 17 0.8 0.8 NUM iajs-3461 381 18 1 1 NUM iajs-3461 381 19 1.2 1.2 NUM iajs-3461 381 20 1.4 1.4 NUM iajs-3461 381 21 time time NOUN iajs-3461 381 22 po po PROPN iajs-3461 381 23 pu pu PROPN iajs-3461 381 24 lat lat PROPN iajs-3461 381 25 ion ion NOUN iajs-3461 381 26 ( ( PUNCT iajs-3461 381 27 14 14 NUM iajs-3461 381 28 ) ) PUNCT iajs-3461 381 29 h h NOUN iajs-3461 382 1 i i PRON iajs-3461 382 2 j j NOUN iajs-3461 382 3 0 0 NUM iajs-3461 382 4 0.5 0.5 NUM iajs-3461 382 5 1 1 NUM iajs-3461 382 6 1.5 1.5 NUM iajs-3461 382 7 2 2 NUM iajs-3461 382 8 2.5 2.5 NUM iajs-3461 382 9 3 3 NUM iajs-3461 382 10 3.5 3.5 NUM iajs-3461 382 11 4 4 NUM iajs-3461 382 12 4.5 4.5 NUM iajs-3461 382 13 5 5 NUM iajs-3461 382 14 x x SYM iajs-3461 382 15 10 10 NUM iajs-3461 382 16 5 5 NUM iajs-3461 382 17 0 0 NUM iajs-3461 382 18 0.5 0.5 NUM iajs-3461 382 19 1 1 NUM iajs-3461 382 20 1.5 1.5 NUM iajs-3461 382 21 2 2 NUM iajs-3461 382 22 2.5 2.5 NUM iajs-3461 382 23 3 3 NUM iajs-3461 382 24 time time NOUN iajs-3461 382 25 po po PROPN iajs-3461 382 26 pu pu PROPN iajs-3461 382 27 lat lat PROPN iajs-3461 383 1 ion ion NOUN iajs-3461 384 1 ( ( PUNCT iajs-3461 385 1 10.1 10.1 NUM iajs-3461 385 2 ) ) PUNCT iajs-3461 385 3 h h NOUN iajs-3461 386 1 i i PRON iajs-3461 386 2 j j NOUN iajs-3461 386 3 0 0 NUM iajs-3461 386 4 0.5 0.5 NUM iajs-3461 386 5 1 1 NUM iajs-3461 386 6 1.5 1.5 NUM iajs-3461 386 7 2 2 NUM iajs-3461 386 8 2.5 2.5 NUM iajs-3461 386 9 3 3 NUM iajs-3461 386 10 x x SYM iajs-3461 386 11 10 10 NUM iajs-3461 386 12 5 5 NUM iajs-3461 386 13 0 0 NUM iajs-3461 386 14 0.2 0.2 NUM iajs-3461 386 15 0.4 0.4 NUM iajs-3461 386 16 0.6 0.6 NUM iajs-3461 386 17 0.8 0.8 NUM iajs-3461 386 18 1 1 NUM iajs-3461 386 19 1.2 1.2 NUM iajs-3461 386 20 1.4 1.4 NUM iajs-3461 386 21 time time NOUN iajs-3461 386 22 po po PROPN iajs-3461 386 23 pu pu PROPN iajs-3461 386 24 lat lat PROPN iajs-3461 386 25 ion ion NOUN iajs-3461 386 26 ( ( PUNCT iajs-3461 386 27 8.2 8.2 NUM iajs-3461 386 28 ) ) PUNCT iajs-3461 386 29 h h NOUN iajs-3461 387 1 i i PRON iajs-3461 387 2 j j NOUN iajs-3461 387 3 0 0 NUM iajs-3461 387 4 1 1 NUM iajs-3461 387 5 2 2 NUM iajs-3461 387 6 3 3 NUM iajs-3461 387 7 4 4 NUM iajs-3461 387 8 5 5 NUM iajs-3461 387 9 6 6 NUM iajs-3461 387 10 7 7 NUM iajs-3461 387 11 8 8 NUM iajs-3461 387 12 9 9 NUM iajs-3461 387 13 10 10 NUM iajs-3461 387 14 x x SYM iajs-3461 387 15 10 10 NUM iajs-3461 387 16 4 4 NUM iajs-3461 387 17 0 0 NUM iajs-3461 387 18 0.2 0.2 NUM iajs-3461 387 19 0.4 0.4 NUM iajs-3461 387 20 0.6 0.6 NUM iajs-3461 387 21 0.8 0.8 NUM iajs-3461 387 22 1 1 NUM iajs-3461 387 23 1.2 1.2 NUM iajs-3461 387 24 1.4 1.4 NUM iajs-3461 387 25 time time NOUN iajs-3461 387 26 po po PROPN iajs-3461 387 27 pu pu PROPN iajs-3461 387 28 lat lat PROPN iajs-3461 387 29 ion ion NOUN iajs-3461 387 30 ( ( PUNCT iajs-3461 387 31 8.1 8.1 NUM iajs-3461 387 32 ) ) PUNCT iajs-3461 387 33 h h NOUN iajs-3461 388 1 i i PRON iajs-3461 388 2 j j PROPN iajs-3461 388 3 ihjpas ihjpa VERB iajs-3461 388 4 . . PUNCT iajs-3461 389 1 2025 2025 NUM iajs-3461 389 2 , , PUNCT iajs-3461 389 3 38 38 NUM iajs-3461 389 4 ( ( PUNCT iajs-3461 389 5 1 1 NUM iajs-3461 389 6 ) ) PUNCT iajs-3461 389 7 382 382 NUM iajs-3461 389 8 other other ADJ iajs-3461 389 9 parameters parameter NOUN iajs-3461 389 10 as as ADP iajs-3461 389 11 in in ADP iajs-3461 389 12 eq eq ADP iajs-3461 389 13 . . PUNCT iajs-3461 390 1 ( ( PUNCT iajs-3461 390 2 5.1 5.1 NUM iajs-3461 390 3 ) ) PUNCT iajs-3461 390 4 , , PUNCT iajs-3461 390 5 but but CCONJ iajs-3461 390 6 if if SCONJ iajs-3461 390 7 r1= r1= NOUN iajs-3461 390 8 0.59 0.59 NUM iajs-3461 390 9 it it PRON iajs-3461 390 10 can can AUX iajs-3461 390 11 be be AUX iajs-3461 390 12 seen see VERB iajs-3461 390 13 that that SCONJ iajs-3461 390 14 the the DET iajs-3461 390 15 solution solution NOUN iajs-3461 390 16 to to ADP iajs-3461 390 17 system system NOUN iajs-3461 390 18 ( ( PUNCT iajs-3461 390 19 2 2 X iajs-3461 390 20 ) ) PUNCT iajs-3461 390 21 approaches approach VERB iajs-3461 390 22 the the DET iajs-3461 390 23 equilibrium equilibrium NOUN iajs-3461 390 24 point point NOUN iajs-3461 390 25 a2 a2 PROPN iajs-3461 390 26 asymptotically asymptotically ADV iajs-3461 390 27 ; ; PUNCT iajs-3461 390 28 r1= r1= NOUN iajs-3461 390 29 0.595 0.595 NUM iajs-3461 390 30 is be AUX iajs-3461 390 31 a a DET iajs-3461 390 32 bifurcation bifurcation NOUN iajs-3461 390 33 point point NOUN iajs-3461 390 34 . . PUNCT iajs-3461 391 1 3 3 X iajs-3461 391 2 . . X iajs-3461 391 3 the the DET iajs-3461 391 4 trajectory trajectory NOUN iajs-3461 391 5 changed change VERB iajs-3461 391 6 from from ADP iajs-3461 391 7 the the DET iajs-3461 391 8 axial axial ADJ iajs-3461 391 9 point point NOUN iajs-3461 391 10 a1 a1 NOUN iajs-3461 391 11 to to ADP iajs-3461 391 12 the the DET iajs-3461 391 13 equilibrium equilibrium NOUN iajs-3461 391 14 point point NOUN iajs-3461 391 15 a2 a2 PROPN iajs-3461 391 16 in in ADP iajs-3461 391 17 the the DET iajs-3461 391 18 range range NOUN iajs-3461 391 19 0.0001 0.0001 NUM iajs-3461 391 20 ≀ ≀ NUM iajs-3461 391 21 r3 r3 PROPN iajs-3461 391 22 < < X iajs-3461 391 23 0.006 0.006 NUM iajs-3461 391 24 , , PUNCT iajs-3461 391 25 and and CCONJ iajs-3461 391 26 from from ADP iajs-3461 391 27 the the DET iajs-3461 391 28 equilibrium equilibrium NOUN iajs-3461 391 29 point point NOUN iajs-3461 391 30 a2 a2 PROPN iajs-3461 391 31 to to ADP iajs-3461 391 32 the the DET iajs-3461 391 33 equilibrium equilibrium NOUN iajs-3461 391 34 point point NOUN iajs-3461 391 35 a4 a4 NOUN iajs-3461 391 36 in in ADP iajs-3461 391 37 the the DET iajs-3461 391 38 range range NOUN iajs-3461 391 39 0.006 0.006 NUM iajs-3461 391 40 ≀ ≀ NUM iajs-3461 391 41 r3 r3 PROPN iajs-3461 391 42 < < X iajs-3461 391 43 0.0125 0.0125 NUM iajs-3461 391 44 consequently consequently ADV iajs-3461 391 45 the the DET iajs-3461 391 46 bifurcation bifurcation NOUN iajs-3461 391 47 points point VERB iajs-3461 391 48 for for ADP iajs-3461 391 49 the the DET iajs-3461 391 50 parameter parameter PROPN iajs-3461 391 51 r3 r3 PROPN iajs-3461 391 52 are be AUX iajs-3461 391 53 at at ADP iajs-3461 391 54 π‘Ÿ3= π‘Ÿ3= PROPN iajs-3461 391 55 0.006 0.006 NUM iajs-3461 391 56 and and CCONJ iajs-3461 391 57 π‘Ÿ3 π‘Ÿ3 NOUN iajs-3461 391 58 = = NOUN iajs-3461 391 59 0.01 0.01 NUM iajs-3461 391 60 respectively respectively ADV iajs-3461 391 61 . . PUNCT iajs-3461 392 1 4 4 X iajs-3461 392 2 . . X iajs-3461 392 3 as as ADP iajs-3461 392 4 the the DET iajs-3461 392 5 attack attack NOUN iajs-3461 392 6 rate rate NOUN iajs-3461 392 7 of of ADP iajs-3461 392 8 generalist generalist ADJ iajs-3461 392 9 predator predator NOUN iajs-3461 392 10 on on ADP iajs-3461 392 11 specialist specialist ADJ iajs-3461 392 12 predator predator NOUN iajs-3461 392 13 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 392 14 keeping keep VERB iajs-3461 392 15 the the DET iajs-3461 392 16 rest rest NOUN iajs-3461 392 17 of of ADP iajs-3461 392 18 parameters parameter NOUN iajs-3461 392 19 as as ADP iajs-3461 392 20 in in ADP iajs-3461 392 21 eq.(5.1 eq.(5.1 NOUN iajs-3461 392 22 ) ) PUNCT iajs-3461 392 23 , , PUNCT iajs-3461 392 24 the the DET iajs-3461 392 25 solution solution NOUN iajs-3461 392 26 of of ADP iajs-3461 392 27 system system NOUN iajs-3461 392 28 ( ( PUNCT iajs-3461 392 29 2 2 X iajs-3461 392 30 ) ) PUNCT iajs-3461 392 31 approaches approach NOUN iajs-3461 392 32 to to ADP iajs-3461 392 33 the the DET iajs-3461 392 34 positive positive ADJ iajs-3461 392 35 point point NOUN iajs-3461 392 36 𝐴4 𝐴4 NOUN iajs-3461 392 37 , , PUNCT iajs-3461 392 38 if if SCONJ iajs-3461 392 39 π‘Ÿ4 π‘Ÿ4 PROPN iajs-3461 392 40 = = SYM iajs-3461 392 41 1.5 1.5 NUM iajs-3461 392 42 , , PUNCT iajs-3461 392 43 it it PRON iajs-3461 392 44 is be AUX iajs-3461 392 45 shown show VERB iajs-3461 392 46 that that SCONJ iajs-3461 392 47 the the DET iajs-3461 392 48 solution solution NOUN iajs-3461 392 49 of of ADP iajs-3461 392 50 system system NOUN iajs-3461 392 51 ( ( PUNCT iajs-3461 392 52 2 2 X iajs-3461 392 53 ) ) PUNCT iajs-3461 392 54 approaches approach VERB iajs-3461 392 55 asymptotically asymptotically ADV iajs-3461 392 56 to to ADP iajs-3461 392 57 the the DET iajs-3461 392 58 equilibrium equilibrium NOUN iajs-3461 392 59 point point NOUN iajs-3461 392 60 𝐴3 𝐴3 PROPN iajs-3461 392 61 , , PUNCT iajs-3461 392 62 indicating indicate VERB iajs-3461 392 63 that that SCONJ iajs-3461 392 64 π‘Ÿ4=1.5 π‘Ÿ4=1.5 PROPN iajs-3461 392 65 is be AUX iajs-3461 392 66 a a DET iajs-3461 392 67 bifurcation bifurcation NOUN iajs-3461 392 68 point point NOUN iajs-3461 392 69 . . PUNCT iajs-3461 393 1 5 5 X iajs-3461 393 2 . . X iajs-3461 393 3 the the DET iajs-3461 393 4 natural natural ADJ iajs-3461 393 5 death death NOUN iajs-3461 393 6 rate rate NOUN iajs-3461 393 7 of of ADP iajs-3461 393 8 specialist specialist ADJ iajs-3461 393 9 predator predator NOUN iajs-3461 393 10 r5 r5 PROPN iajs-3461 393 11 the the DET iajs-3461 393 12 solution solution NOUN iajs-3461 393 13 of of ADP iajs-3461 393 14 system system NOUN iajs-3461 393 15 ( ( PUNCT iajs-3461 393 16 2 2 X iajs-3461 393 17 ) ) PUNCT iajs-3461 393 18 advances advance NOUN iajs-3461 393 19 from from ADP iajs-3461 393 20 the the DET iajs-3461 393 21 positive positive ADJ iajs-3461 393 22 equilibrium equilibrium NOUN iajs-3461 393 23 point point NOUN iajs-3461 393 24 a4 a4 NUM iajs-3461 393 25 to to ADP iajs-3461 393 26 the the DET iajs-3461 393 27 axial axial ADJ iajs-3461 393 28 equilibrium equilibrium NOUN iajs-3461 393 29 point point NOUN iajs-3461 393 30 a1with a1with ADP iajs-3461 393 31 0.03 0.03 NUM iajs-3461 393 32 < < X iajs-3461 393 33 r5 r5 PROPN iajs-3461 393 34 < < X iajs-3461 393 35 0.15 0.15 NUM iajs-3461 393 36 keeping keep VERB iajs-3461 393 37 the the DET iajs-3461 393 38 other other ADJ iajs-3461 393 39 parameters parameter NOUN iajs-3461 393 40 as as ADP iajs-3461 393 41 in in ADP iajs-3461 393 42 eq.(5.1 eq.(5.1 NOUN iajs-3461 393 43 ) ) PUNCT iajs-3461 393 44 , , PUNCT iajs-3461 393 45 and and CCONJ iajs-3461 393 46 from from ADP iajs-3461 393 47 the the DET iajs-3461 393 48 axial axial ADJ iajs-3461 393 49 equilibrim equilibrim NOUN iajs-3461 393 50 point point NOUN iajs-3461 393 51 a1 a1 NOUN iajs-3461 393 52 to to ADP iajs-3461 393 53 the the DET iajs-3461 393 54 equilibrium equilibrium NOUN iajs-3461 393 55 point point NOUN iajs-3461 393 56 a3 a3 NOUN iajs-3461 393 57 in in ADP iajs-3461 393 58 the the DET iajs-3461 393 59 ring ring NOUN iajs-3461 393 60 0.15≀ 0.15≀ PROPN iajs-3461 393 61 r5 r5 PROPN iajs-3461 393 62 < < X iajs-3461 393 63 0.6 0.6 NUM iajs-3461 393 64 , , PUNCT iajs-3461 393 65 the the DET iajs-3461 393 66 parameter parameter NOUN iajs-3461 393 67 π‘Ÿ5 π‘Ÿ5 PROPN iajs-3461 393 68 = = PRON iajs-3461 393 69 0.045 0.045 NUM iajs-3461 393 70 is be AUX iajs-3461 393 71 bifurcation bifurcation NOUN iajs-3461 393 72 point point NOUN iajs-3461 393 73 . . PUNCT iajs-3461 394 1 6 6 X iajs-3461 394 2 . . X iajs-3461 395 1 as as SCONJ iajs-3461 395 2 the the DET iajs-3461 395 3 conversion9 conversion9 NOUN iajs-3461 395 4 rate rate NOUN iajs-3461 395 5 of of ADP iajs-3461 395 6 prey prey NOUN iajs-3461 395 7 to to ADP iajs-3461 395 8 the the DET iajs-3461 395 9 generalist generalist ADJ iajs-3461 395 10 predator predator NOUN iajs-3461 395 11 π‘Ÿ6 π‘Ÿ6 NOUN iajs-3461 395 12 decreasing decrease VERB iajs-3461 395 13 , , PUNCT iajs-3461 395 14 and and CCONJ iajs-3461 395 15 keeping keep VERB iajs-3461 395 16 the the DET iajs-3461 395 17 rest rest NOUN iajs-3461 395 18 parameters parameter NOUN iajs-3461 395 19 values value NOUN iajs-3461 395 20 as as ADP iajs-3461 395 21 in in ADP iajs-3461 395 22 eq eq NOUN iajs-3461 395 23 . . PUNCT iajs-3461 396 1 ( ( PUNCT iajs-3461 396 2 5.1 5.1 NUM iajs-3461 396 3 ) ) PUNCT iajs-3461 396 4 the the DET iajs-3461 396 5 solution solution NOUN iajs-3461 396 6 of of ADP iajs-3461 396 7 system system NOUN iajs-3461 396 8 ( ( PUNCT iajs-3461 396 9 2 2 X iajs-3461 396 10 ) ) PUNCT iajs-3461 396 11 approaches approach VERB iajs-3461 396 12 the the DET iajs-3461 396 13 equilibrium equilibrium NOUN iajs-3461 396 14 point point NOUN iajs-3461 396 15 a4 a4 NOUN iajs-3461 396 16 , , PUNCT iajs-3461 396 17 while while SCONJ iajs-3461 396 18 for for ADP iajs-3461 396 19 0.0001 0.0001 NUM iajs-3461 396 20 ≀ ≀ NOUN iajs-3461 396 21 r6 r6 NOUN iajs-3461 396 22 < < X iajs-3461 396 23 0.015 0.015 NUM iajs-3461 396 24 , , PUNCT iajs-3461 396 25 the the DET iajs-3461 396 26 generalist generalist ADJ iajs-3461 396 27 predator predator NOUN iajs-3461 396 28 population population NOUN iajs-3461 396 29 revives revive VERB iajs-3461 396 30 and and CCONJ iajs-3461 396 31 then then ADV iajs-3461 396 32 the the DET iajs-3461 396 33 trajectory trajectory NOUN iajs-3461 396 34 changed change VERB iajs-3461 396 35 from from ADP iajs-3461 396 36 the the DET iajs-3461 396 37 point point NOUN iajs-3461 396 38 a2 a2 PROPN iajs-3461 396 39 to to ADP iajs-3461 396 40 the the DET iajs-3461 396 41 positive positive ADJ iajs-3461 396 42 equilibrium equilibrium NOUN iajs-3461 396 43 point point NOUN iajs-3461 396 44 a4 a4 NOUN iajs-3461 396 45 , , PUNCT iajs-3461 396 46 while while SCONJ iajs-3461 396 47 for for ADP iajs-3461 396 48 0.015 0.015 NUM iajs-3461 396 49 ≀ ≀ NOUN iajs-3461 397 1 r6 r6 NOUN iajs-3461 397 2 < < X iajs-3461 397 3 0.1 0.1 NUM iajs-3461 397 4 thus thus ADV iajs-3461 397 5 , , PUNCT iajs-3461 397 6 the the DET iajs-3461 397 7 parameter parameter NOUN iajs-3461 397 8 π‘Ÿ7 π‘Ÿ7 PROPN iajs-3461 397 9 = = SYM iajs-3461 397 10 0.015 0.015 NUM iajs-3461 397 11 is be AUX iajs-3461 397 12 a a DET iajs-3461 397 13 bifurcation bifurcation NOUN iajs-3461 397 14 point point NOUN iajs-3461 397 15 . . PUNCT iajs-3461 398 1 7 7 X iajs-3461 398 2 . . X iajs-3461 398 3 in in ADP iajs-3461 398 4 light light NOUN iajs-3461 398 5 of of ADP iajs-3461 398 6 the the DET iajs-3461 398 7 conversion conversion NOUN iajs-3461 398 8 rate rate NOUN iajs-3461 398 9 of of ADP iajs-3461 398 10 specialist specialist ADJ iajs-3461 398 11 predator predator NOUN iajs-3461 398 12 to to ADP iajs-3461 398 13 the the DET iajs-3461 398 14 generalist generalist ADJ iajs-3461 398 15 predator predator NOUN iajs-3461 398 16 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 398 17 decreasing decrease VERB iajs-3461 398 18 to to ADP iajs-3461 398 19 0.0002 0.0002 NUM iajs-3461 398 20 and and CCONJ iajs-3461 398 21 keeping keep VERB iajs-3461 398 22 the the DET iajs-3461 398 23 rest rest NOUN iajs-3461 398 24 parameters parameter NOUN iajs-3461 398 25 values value NOUN iajs-3461 398 26 as as ADP iajs-3461 398 27 in in ADP iajs-3461 398 28 eq eq NOUN iajs-3461 398 29 . . PUNCT iajs-3461 398 30 ( ( PUNCT iajs-3461 398 31 5.1 5.1 NUM iajs-3461 398 32 ) ) PUNCT iajs-3461 398 33 the the DET iajs-3461 398 34 solution solution NOUN iajs-3461 398 35 of of ADP iajs-3461 398 36 system system NOUN iajs-3461 398 37 ( ( PUNCT iajs-3461 398 38 2 2 X iajs-3461 398 39 ) ) PUNCT iajs-3461 398 40 approaches approach VERB iajs-3461 398 41 the the DET iajs-3461 398 42 equilibrium equilibrium NOUN iajs-3461 398 43 point point NOUN iajs-3461 398 44 a2 a2 PROPN iajs-3461 398 45 , , PUNCT iajs-3461 398 46 while while SCONJ iajs-3461 398 47 for for ADP iajs-3461 398 48 0.00001 0.00001 NUM iajs-3461 398 49 ≀ ≀ NUM iajs-3461 398 50 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 398 51 < < X iajs-3461 398 52 0.005 0.005 NUM iajs-3461 398 53 the the DET iajs-3461 398 54 generalist generalist ADJ iajs-3461 398 55 predator predator NOUN iajs-3461 398 56 population population NOUN iajs-3461 398 57 revives revive VERB iajs-3461 398 58 and and CCONJ iajs-3461 398 59 when when SCONJ iajs-3461 398 60 0.005≀ 0.005≀ VERB iajs-3461 398 61 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 398 62 < < X iajs-3461 398 63 0.2 0.2 NUM iajs-3461 398 64 , , PUNCT iajs-3461 398 65 the the DET iajs-3461 398 66 trajectory trajectory NOUN iajs-3461 398 67 changed change VERB iajs-3461 398 68 from from ADP iajs-3461 398 69 the the DET iajs-3461 398 70 point point NOUN iajs-3461 398 71 a2 a2 PROPN iajs-3461 398 72 to to ADP iajs-3461 398 73 the the DET iajs-3461 398 74 positive positive ADJ iajs-3461 398 75 equilibrium equilibrium NOUN iajs-3461 398 76 point point NOUN iajs-3461 398 77 a4 a4 PROPN iajs-3461 398 78 , , PUNCT iajs-3461 398 79 the the DET iajs-3461 398 80 parameter parameter NOUN iajs-3461 398 81 π‘Ÿ7 π‘Ÿ7 NOUN iajs-3461 398 82 = = NOUN iajs-3461 399 1 0.005 0.005 NUM iajs-3461 399 2 is be AUX iajs-3461 399 3 a a DET iajs-3461 399 4 bifurcation bifurcation NOUN iajs-3461 399 5 point point NOUN iajs-3461 399 6 . . PUNCT iajs-3461 400 1 7 7 X iajs-3461 400 2 . . X iajs-3461 400 3 conclusion conclusion NOUN iajs-3461 400 4 this this DET iajs-3461 400 5 study study NOUN iajs-3461 400 6 used use VERB iajs-3461 400 7 an an DET iajs-3461 400 8 ecological ecological ADJ iajs-3461 400 9 mathematical mathematical ADJ iajs-3461 400 10 model model NOUN iajs-3461 400 11 that that PRON iajs-3461 400 12 includes include VERB iajs-3461 400 13 a a DET iajs-3461 400 14 predator predator NOUN iajs-3461 400 15 - - PUNCT iajs-3461 400 16 prey prey NOUN iajs-3461 400 17 model model NOUN iajs-3461 400 18 and and CCONJ iajs-3461 400 19 a a DET iajs-3461 400 20 food food NOUN iajs-3461 400 21 web web NOUN iajs-3461 400 22 , , PUNCT iajs-3461 400 23 as as ADV iajs-3461 400 24 well well ADV iajs-3461 400 25 as as ADP iajs-3461 400 26 a a DET iajs-3461 400 27 population population NOUN iajs-3461 400 28 of of ADP iajs-3461 400 29 prey prey NOUN iajs-3461 400 30 and and CCONJ iajs-3461 400 31 a a DET iajs-3461 400 32 population population NOUN iajs-3461 400 33 of of ADP iajs-3461 400 34 specialized specialized ADJ iajs-3461 400 35 predators predator NOUN iajs-3461 400 36 as as ADP iajs-3461 400 37 refuges refuge NOUN iajs-3461 400 38 . . PUNCT iajs-3461 401 1 additionally additionally ADV iajs-3461 401 2 , , PUNCT iajs-3461 401 3 this this DET iajs-3461 401 4 model model NOUN iajs-3461 401 5 includes include VERB iajs-3461 401 6 linear linear ADJ iajs-3461 401 7 types type NOUN iajs-3461 401 8 of of ADP iajs-3461 401 9 functional functional ADJ iajs-3461 401 10 reactions reaction NOUN iajs-3461 401 11 for for ADP iajs-3461 401 12 the the DET iajs-3461 401 13 predation predation NOUN iajs-3461 401 14 of of ADP iajs-3461 401 15 creatures creature NOUN iajs-3461 401 16 that that PRON iajs-3461 401 17 were be AUX iajs-3461 401 18 not not PART iajs-3461 401 19 protected protect VERB iajs-3461 401 20 . . PUNCT iajs-3461 402 1 acknowledgment acknowledgment NOUN iajs-3461 402 2 our our PRON iajs-3461 402 3 researcher researcher NOUN iajs-3461 402 4 extends extend VERB iajs-3461 402 5 his his PRON iajs-3461 402 6 sincere sincere ADJ iajs-3461 402 7 thanks thank NOUN iajs-3461 402 8 to to ADP iajs-3461 402 9 the the DET iajs-3461 402 10 editor editor NOUN iajs-3461 402 11 and and CCONJ iajs-3461 402 12 members member NOUN iajs-3461 402 13 of of ADP iajs-3461 402 14 the the DET iajs-3461 402 15 preparatory preparatory PROPN iajs-3461 402 16 committee committee NOUN iajs-3461 402 17 of of ADP iajs-3461 402 18 the the DET iajs-3461 402 19 ibn ibn PROPN iajs-3461 402 20 al al PROPN iajs-3461 402 21 - - PUNCT iajs-3461 402 22 haitham haitham PROPN iajs-3461 402 23 journal journal PROPN iajs-3461 402 24 of of ADP iajs-3461 402 25 pure pure ADJ iajs-3461 402 26 and and CCONJ iajs-3461 402 27 applied applied ADJ iajs-3461 402 28 sciences science NOUN iajs-3461 402 29 . . PUNCT iajs-3461 403 1 conflict conflict NOUN iajs-3461 403 2 of of ADP iajs-3461 403 3 interest interest NOUN iajs-3461 403 4 there there PRON iajs-3461 403 5 are be VERB iajs-3461 403 6 no no DET iajs-3461 403 7 conflicts conflict NOUN iajs-3461 403 8 of of ADP iajs-3461 403 9 interest interest NOUN iajs-3461 403 10 . . PUNCT iajs-3461 404 1 funding funding NOUN iajs-3461 404 2 there there PRON iajs-3461 404 3 is be VERB iajs-3461 404 4 no no DET iajs-3461 404 5 funding funding NOUN iajs-3461 404 6 for for ADP iajs-3461 404 7 the the DET iajs-3461 404 8 article article NOUN iajs-3461 404 9 . . PUNCT iajs-3461 405 1 references reference NOUN iajs-3461 405 2 1 1 NUM iajs-3461 405 3 . . PUNCT iajs-3461 406 1 mondal mondal PROPN iajs-3461 406 2 n n PROPN iajs-3461 406 3 , , PUNCT iajs-3461 406 4 barman barman PROPN iajs-3461 406 5 d d PROPN iajs-3461 406 6 , , PUNCT iajs-3461 406 7 alam alam PROPN iajs-3461 406 8 s.a.m s.a.m PROPN iajs-3461 406 9 . . PUNCT iajs-3461 406 10 impact impact NOUN iajs-3461 406 11 of of ADP iajs-3461 406 12 adult adult NOUN iajs-3461 406 13 predator predator NOUN iajs-3461 406 14 incited incite VERB iajs-3461 406 15 fear fear NOUN iajs-3461 406 16 in in ADP iajs-3461 406 17 a a DET iajs-3461 406 18 stage stage NOUN iajs-3461 406 19 structured structure VERB iajs-3461 406 20 prey prey NOUN iajs-3461 406 21 – – PUNCT iajs-3461 406 22 predator predator NOUN iajs-3461 406 23 model model NOUN iajs-3461 406 24 . . PUNCT iajs-3461 407 1 environ environ PROPN iajs-3461 407 2 dev dev PROPN iajs-3461 407 3 sustain sustain NOUN iajs-3461 407 4 . . PUNCT iajs-3461 408 1 2021;23(6):9280 2021;23(6):9280 NUM iajs-3461 408 2 - - SYM iajs-3461 408 3 9307 9307 NUM iajs-3461 408 4 . . PUNCT iajs-3461 409 1 ihjpas ihjpas PROPN iajs-3461 409 2 . . PUNCT iajs-3461 410 1 2025 2025 NUM iajs-3461 410 2 , , PUNCT iajs-3461 410 3 38 38 NUM iajs-3461 410 4 ( ( PUNCT iajs-3461 410 5 1 1 NUM iajs-3461 410 6 ) ) PUNCT iajs-3461 410 7 383 383 NUM iajs-3461 410 8 2 2 NUM iajs-3461 410 9 . . PUNCT iajs-3461 411 1 xie xie PROPN iajs-3461 411 2 y y PROPN iajs-3461 411 3 , , PUNCT iajs-3461 411 4 lu lu PROPN iajs-3461 411 5 j j PROPN iajs-3461 411 6 , , PUNCT iajs-3461 411 7 li li PROPN iajs-3461 411 8 y. y. PROPN iajs-3461 411 9 stability stability PROPN iajs-3461 411 10 and and CCONJ iajs-3461 411 11 bifurcation bifurcation NOUN iajs-3461 411 12 of of ADP iajs-3461 411 13 a a DET iajs-3461 411 14 delayed delay VERB iajs-3461 411 15 generalized generalized ADJ iajs-3461 411 16 fractional fractional ADJ iajs-3461 411 17 - - PUNCT iajs-3461 411 18 order order NOUN iajs-3461 411 19 prey prey NOUN iajs-3461 411 20 – – PUNCT iajs-3461 411 21 predator predator NOUN iajs-3461 411 22 model model NOUN iajs-3461 411 23 with with ADP iajs-3461 411 24 interspecific interspecific ADJ iajs-3461 411 25 competition competition NOUN iajs-3461 411 26 . . PUNCT iajs-3461 412 1 appl appl PROPN iajs-3461 412 2 math math PROPN iajs-3461 412 3 comput comput NOUN iajs-3461 412 4 . . PUNCT iajs-3461 413 1 2019 2019 NUM iajs-3461 413 2 . . PUNCT iajs-3461 414 1 3 3 X iajs-3461 414 2 . . X iajs-3461 414 3 wang wang PROPN iajs-3461 414 4 z z PROPN iajs-3461 414 5 , , PUNCT iajs-3461 414 6 et et PROPN iajs-3461 414 7 al al PROPN iajs-3461 414 8 . . PROPN iajs-3461 414 9 stability stability NOUN iajs-3461 414 10 and and CCONJ iajs-3461 414 11 bifurcation bifurcation NOUN iajs-3461 414 12 of of ADP iajs-3461 414 13 a a DET iajs-3461 414 14 delayed delay VERB iajs-3461 414 15 generalized generalized ADJ iajs-3461 414 16 fractional fractional ADJ iajs-3461 414 17 - - PUNCT iajs-3461 414 18 order order NOUN iajs-3461 414 19 prey prey NOUN iajs-3461 414 20 – – PUNCT iajs-3461 414 21 predator predator NOUN iajs-3461 414 22 model model NOUN iajs-3461 414 23 with with ADP iajs-3461 414 24 interspecific interspecific ADJ iajs-3461 414 25 competition competition NOUN iajs-3461 414 26 . . PUNCT iajs-3461 415 1 appl appl PROPN iajs-3461 415 2 math math PROPN iajs-3461 415 3 comput comput PROPN iajs-3461 415 4 . . PUNCT iajs-3461 416 1 2019;347 2019;347 NUM iajs-3461 416 2 . . NOUN iajs-3461 417 1 4 4 X iajs-3461 417 2 . . X iajs-3461 417 3 naji naji PROPN iajs-3461 417 4 rk rk PROPN iajs-3461 417 5 , , PUNCT iajs-3461 417 6 majeed majeed PROPN iajs-3461 417 7 sj sj PROPN iajs-3461 417 8 . . PUNCT iajs-3461 418 1 a a DET iajs-3461 418 2 prey prey NOUN iajs-3461 418 3 – – PUNCT iajs-3461 418 4 predator predator NOUN iajs-3461 418 5 model model NOUN iajs-3461 418 6 with with ADP iajs-3461 418 7 a a DET iajs-3461 418 8 refuge refuge NOUN iajs-3461 418 9 – – PUNCT iajs-3461 418 10 stage stage NOUN iajs-3461 418 11 structure structure NOUN iajs-3461 418 12 prey prey PROPN iajs-3461 418 13 population population NOUN iajs-3461 418 14 . . PUNCT iajs-3461 419 1 int int PROPN iajs-3461 419 2 j j PROPN iajs-3461 419 3 differ differ VERB iajs-3461 419 4 equ equ PROPN iajs-3461 419 5 . . PROPN iajs-3461 419 6 2016 2016 NUM iajs-3461 419 7 ; ; PUNCT iajs-3461 419 8 2016:2010464:10 2016:2010464:10 NUM iajs-3461 419 9 - - SYM iajs-3461 419 10 25 25 NUM iajs-3461 419 11 . . NOUN iajs-3461 420 1 5 5 NUM iajs-3461 420 2 . . X iajs-3461 420 3 guckenheimer guckenheimer PROPN iajs-3461 420 4 j. j. PROPN iajs-3461 420 5 dynamical dynamical PROPN iajs-3461 420 6 systems system NOUN iajs-3461 420 7 and and CCONJ iajs-3461 420 8 bifurcations bifurcation NOUN iajs-3461 420 9 of of ADP iajs-3461 420 10 vector vector NOUN iajs-3461 420 11 fields field NOUN iajs-3461 420 12 . . PUNCT iajs-3461 421 1 appl appl PROPN iajs-3461 421 2 math math PROPN iajs-3461 421 3 sci sci PROPN iajs-3461 421 4 . . PROPN iajs-3461 421 5 1986 1986 NUM iajs-3461 421 6 . . PUNCT iajs-3461 422 1 6 6 NUM iajs-3461 422 2 . . X iajs-3461 422 3 kadhim kadhim PROPN iajs-3461 422 4 zj zj PROPN iajs-3461 422 5 , , PUNCT iajs-3461 422 6 majeed majeed PROPN iajs-3461 422 7 aa aa PROPN iajs-3461 422 8 , , PUNCT iajs-3461 422 9 naji naji PROPN iajs-3461 422 10 rk rk PROPN iajs-3461 422 11 . . PUNCT iajs-3461 423 1 the the DET iajs-3461 423 2 bifurcation bifurcation NOUN iajs-3461 423 3 analysis analysis NOUN iajs-3461 423 4 of of ADP iajs-3461 423 5 a a DET iajs-3461 423 6 stage stage NOUN iajs-3461 423 7 - - PUNCT iajs-3461 423 8 structured structure VERB iajs-3461 423 9 prey prey NOUN iajs-3461 423 10 food food PROPN iajs-3461 423 11 web web NOUN iajs-3461 423 12 model model NOUN iajs-3461 423 13 with with ADP iajs-3461 423 14 refuge refuge NOUN iajs-3461 423 15 . . PUNCT iajs-3461 424 1 iraq iraq PROPN iajs-3461 424 2 j j PROPN iajs-3461 424 3 sci sci PROPN iajs-3461 424 4 . . PUNCT iajs-3461 425 1 2016;special 2016;special NUM iajs-3461 425 2 issue issue NOUN iajs-3461 425 3 , , PUNCT iajs-3461 425 4 part part NOUN iajs-3461 425 5 a:139 a:139 NOUN iajs-3461 425 6 - - PUNCT iajs-3461 425 7 155 155 NUM iajs-3461 425 8 . . PUNCT iajs-3461 426 1 7 7 X iajs-3461 426 2 . . X iajs-3461 426 3 alabacy alabacy PROPN iajs-3461 426 4 zkh zkh PROPN iajs-3461 426 5 , , PUNCT iajs-3461 426 6 majeed majeed PROPN iajs-3461 426 7 aa aa PROPN iajs-3461 426 8 . . PUNCT iajs-3461 427 1 the the DET iajs-3461 427 2 local local ADJ iajs-3461 427 3 bifurcation bifurcation NOUN iajs-3461 427 4 analysis analysis NOUN iajs-3461 427 5 of of ADP iajs-3461 427 6 two two NUM iajs-3461 427 7 preys prey NOUN iajs-3461 427 8 stage stage NOUN iajs-3461 427 9 structured structure VERB iajs-3461 427 10 predator predator NOUN iajs-3461 427 11 model model NOUN iajs-3461 427 12 with with ADP iajs-3461 427 13 anti anti ADJ iajs-3461 427 14 - - ADJ iajs-3461 427 15 predator predator ADJ iajs-3461 427 16 behavior behavior NOUN iajs-3461 427 17 . . PUNCT iajs-3461 428 1 j j PROPN iajs-3461 428 2 phys phys PROPN iajs-3461 428 3 conf conf NOUN iajs-3461 428 4 ser ser PROPN iajs-3461 428 5 . . PROPN iajs-3461 428 6 2022;012061 2022;012061 NUM iajs-3461 428 7 . . NOUN iajs-3461 429 1 8 8 X iajs-3461 429 2 . . PUNCT iajs-3461 430 1 kafi kafi PROPN iajs-3461 430 2 em em PRON iajs-3461 430 3 , , PUNCT iajs-3461 430 4 majeed majeed PROPN iajs-3461 430 5 aa aa PROPN iajs-3461 430 6 . . PUNCT iajs-3461 431 1 the the DET iajs-3461 431 2 local local ADJ iajs-3461 431 3 bifurcation bifurcation NOUN iajs-3461 431 4 of of ADP iajs-3461 431 5 an an DET iajs-3461 431 6 eco eco NOUN iajs-3461 431 7 - - PUNCT iajs-3461 431 8 epidemiological epidemiological ADJ iajs-3461 431 9 model model NOUN iajs-3461 431 10 in in ADP iajs-3461 431 11 the the DET iajs-3461 431 12 presence presence NOUN iajs-3461 431 13 of of ADP iajs-3461 431 14 stage stage NOUN iajs-3461 431 15 - - PUNCT iajs-3461 431 16 structured structure VERB iajs-3461 431 17 with with ADP iajs-3461 431 18 refuge refuge NOUN iajs-3461 431 19 . . PUNCT iajs-3461 432 1 iraqi iraqi ADJ iajs-3461 432 2 j j PROPN iajs-3461 432 3 sci sci PROPN iajs-3461 432 4 . . PROPN iajs-3461 432 5 2020:2087 2020:2087 NUM iajs-3461 432 6 - - SYM iajs-3461 432 7 2105 2105 NUM iajs-3461 432 8 . . PUNCT iajs-3461 433 1 9 9 X iajs-3461 433 2 . . X iajs-3461 433 3 wiggins wiggins PROPN iajs-3461 433 4 s. s. PROPN iajs-3461 433 5 introduction introduction NOUN iajs-3461 433 6 to to AUX iajs-3461 433 7 applied apply VERB iajs-3461 433 8 nonlinear nonlinear ADJ iajs-3461 433 9 dynamical dynamical ADJ iajs-3461 433 10 systems system NOUN iajs-3461 433 11 and and CCONJ iajs-3461 433 12 chaos chaos NOUN iajs-3461 433 13 . . PUNCT iajs-3461 434 1 springer springer NOUN iajs-3461 434 2 - - PUNCT iajs-3461 434 3 verlag verlag PROPN iajs-3461 434 4 , , PUNCT iajs-3461 434 5 new new PROPN iajs-3461 434 6 york york PROPN iajs-3461 434 7 . . PUNCT iajs-3461 434 8 1990 1990 NUM iajs-3461 434 9 . . PUNCT iajs-3461 435 1 10 10 NUM iajs-3461 435 2 . . PUNCT iajs-3461 436 1 mortoja mortoja PROPN iajs-3461 436 2 sg sg PROPN iajs-3461 436 3 , , PUNCT iajs-3461 436 4 panja panja PROPN iajs-3461 436 5 p p X iajs-3461 436 6 , , PUNCT iajs-3461 436 7 mondal mondal PROPN iajs-3461 436 8 sk sk PROPN iajs-3461 436 9 . . PROPN iajs-3461 436 10 dynamics dynamic NOUN iajs-3461 436 11 of of ADP iajs-3461 436 12 a a DET iajs-3461 436 13 predator predator NOUN iajs-3461 436 14 - - PUNCT iajs-3461 436 15 prey prey NOUN iajs-3461 436 16 model model NOUN iajs-3461 436 17 with with ADP iajs-3461 436 18 stage stage NOUN iajs-3461 436 19 - - PUNCT iajs-3461 436 20 structure structure NOUN iajs-3461 436 21 on on ADP iajs-3461 436 22 both both DET iajs-3461 436 23 species specie NOUN iajs-3461 436 24 and and CCONJ iajs-3461 436 25 anti anti ADJ iajs-3461 436 26 - - ADJ iajs-3461 436 27 predator predator ADJ iajs-3461 436 28 behavior behavior NOUN iajs-3461 436 29 . . PUNCT iajs-3461 437 1 inform inform VERB iajs-3461 437 2 med me VERB iajs-3461 437 3 unlocked unlocked ADJ iajs-3461 437 4 . . PUNCT iajs-3461 438 1 2018;10:50 2018;10:50 NUM iajs-3461 438 2 - - SYM iajs-3461 438 3 57 57 NUM iajs-3461 438 4 . . PUNCT iajs-3461 439 1 11 11 NUM iajs-3461 439 2 . . PUNCT iajs-3461 439 3 perko perko PROPN iajs-3461 439 4 l. l. PROPN iajs-3461 439 5 differential differential PROPN iajs-3461 439 6 equations equation NOUN iajs-3461 439 7 and and CCONJ iajs-3461 439 8 dynamical dynamical ADJ iajs-3461 439 9 systems system NOUN iajs-3461 439 10 . . PUNCT iajs-3461 440 1 springer springer NOUN iajs-3461 440 2 science science PROPN iajs-3461 440 3 & & CCONJ iajs-3461 440 4 business business NOUN iajs-3461 440 5 media medium NOUN iajs-3461 440 6 . . PUNCT iajs-3461 441 1 2013 2013 NUM iajs-3461 441 2 . . PUNCT iajs-3461 442 1 12 12 NUM iajs-3461 442 2 . . PUNCT iajs-3461 443 1 carr carr PROPN iajs-3461 443 2 jcw jcw PROPN iajs-3461 443 3 , , PUNCT iajs-3461 443 4 hale hale PROPN iajs-3461 443 5 j. j. PROPN iajs-3461 443 6 abelian abelian PROPN iajs-3461 443 7 integrals integral NOUN iajs-3461 443 8 and and CCONJ iajs-3461 443 9 bifurcation bifurcation NOUN iajs-3461 443 10 theory theory NOUN iajs-3461 443 11 . . PUNCT iajs-3461 444 1 j j PROPN iajs-3461 444 2 differ differ VERB iajs-3461 444 3 equations equation NOUN iajs-3461 444 4 . . PUNCT iajs-3461 445 1 1985;59:413 1985;59:413 X iajs-3461 445 2 - - SYM iajs-3461 445 3 436 436 NUM iajs-3461 445 4 . . NOUN iajs-3461 446 1 13 13 NUM iajs-3461 446 2 . . PUNCT iajs-3461 447 1 hallam hallam PROPN iajs-3461 447 2 tg tg PROPN iajs-3461 447 3 , , PUNCT iajs-3461 447 4 de de PROPN iajs-3461 447 5 luna luna PROPN iajs-3461 447 6 jt jt PROPN iajs-3461 447 7 . . PUNCT iajs-3461 448 1 effects effect NOUN iajs-3461 448 2 of of ADP iajs-3461 448 3 toxicants toxicant NOUN iajs-3461 448 4 on on ADP iajs-3461 448 5 populations population NOUN iajs-3461 448 6 : : PUNCT iajs-3461 448 7 a a DET iajs-3461 448 8 qualitative qualitative ADJ iajs-3461 448 9 approach approach NOUN iajs-3461 448 10 iii iii PROPN iajs-3461 448 11 . . PUNCT iajs-3461 449 1 environmental environmental ADJ iajs-3461 449 2 and and CCONJ iajs-3461 449 3 food food NOUN iajs-3461 449 4 chain chain NOUN iajs-3461 449 5 pathways pathway NOUN iajs-3461 449 6 . . PUNCT iajs-3461 450 1 academic academic PROPN iajs-3461 450 2 press press PROPN iajs-3461 450 3 inc inc PROPN iajs-3461 450 4 ( ( PUNCT iajs-3461 450 5 london london PROPN iajs-3461 450 6 ) ) PUNCT iajs-3461 450 7 ltd ltd PROPN iajs-3461 450 8 . . PROPN iajs-3461 450 9 1984 1984 NUM iajs-3461 450 10 . . PUNCT iajs-3461 451 1 14 14 NUM iajs-3461 451 2 . . PUNCT iajs-3461 452 1 hastings hastings PROPN iajs-3461 452 2 a a PROPN iajs-3461 452 3 , , PUNCT iajs-3461 452 4 powell powell PROPN iajs-3461 452 5 t. t. NOUN iajs-3461 452 6 chaos chaos NOUN iajs-3461 452 7 in in ADP iajs-3461 452 8 a a DET iajs-3461 452 9 three three NUM iajs-3461 452 10 - - PUNCT iajs-3461 452 11 species species NOUN iajs-3461 452 12 food food NOUN iajs-3461 452 13 chain chain NOUN iajs-3461 452 14 . . PUNCT iajs-3461 453 1 ecology ecology NOUN iajs-3461 453 2 . . PUNCT iajs-3461 454 1 1991;72(3):896 1991;72(3):896 NUM iajs-3461 454 2 - - PUNCT iajs-3461 454 3 903 903 NUM iajs-3461 454 4 . . PUNCT iajs-3461 455 1 15 15 NUM iajs-3461 455 2 . . PUNCT iajs-3461 455 3 molla molla PROPN iajs-3461 455 4 h h PROPN iajs-3461 455 5 , , PUNCT iajs-3461 455 6 sarwardi sarwardi PROPN iajs-3461 455 7 s s PROPN iajs-3461 455 8 , , PUNCT iajs-3461 455 9 haque haque PROPN iajs-3461 455 10 m. m. 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PROPN iajs-3461 458 4 mutual mutual ADJ iajs-3461 458 5 interference interference NOUN iajs-3461 458 6 between between ADP iajs-3461 458 7 parasites parasite NOUN iajs-3461 458 8 or or CCONJ iajs-3461 458 9 predators predator NOUN iajs-3461 458 10 and and CCONJ iajs-3461 458 11 its its PRON iajs-3461 458 12 effect effect NOUN iajs-3461 458 13 on on ADP iajs-3461 458 14 searching search VERB iajs-3461 458 15 efficiency efficiency NOUN iajs-3461 458 16 . . PUNCT iajs-3461 459 1 j j PROPN iajs-3461 459 2 anim anim PROPN iajs-3461 459 3 ecol ecol NOUN iajs-3461 459 4 . . PUNCT iajs-3461 460 1 1975;44:331 1975;44:331 NUM iajs-3461 460 2 - - SYM iajs-3461 460 3 340 340 NUM iajs-3461 460 4 . . PUNCT iajs-3461 461 1 17 17 NUM iajs-3461 461 2 . . PUNCT iajs-3461 462 1 chakraborty chakraborty PROPN iajs-3461 462 2 k k PROPN iajs-3461 462 3 , , PUNCT iajs-3461 462 4 jana jana PROPN iajs-3461 462 5 s s PROPN iajs-3461 462 6 , , PUNCT iajs-3461 462 7 kar kar PROPN iajs-3461 462 8 tk tk PROPN iajs-3461 462 9 . . PROPN iajs-3461 462 10 global global ADJ iajs-3461 462 11 dynamics dynamic NOUN iajs-3461 462 12 and and CCONJ iajs-3461 462 13 bifurcation bifurcation NOUN iajs-3461 462 14 in in ADP iajs-3461 462 15 a a DET iajs-3461 462 16 stage stage NOUN iajs-3461 462 17 structured structure VERB iajs-3461 462 18 prey prey NOUN iajs-3461 462 19 - - PUNCT iajs-3461 462 20 predator predator NOUN iajs-3461 462 21 fishery fishery NOUN iajs-3461 462 22 model model NOUN iajs-3461 462 23 with with ADP iajs-3461 462 24 harvesting harvesting NOUN iajs-3461 462 25 . . PUNCT iajs-3461 463 1 appl appl PROPN iajs-3461 463 2 math math PROPN iajs-3461 463 3 comput comput NOUN iajs-3461 463 4 . . 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