id sid tid token lemma pos iajs-3505 1 1 370 370 NUM iajs-3505 1 2 Β© Β© ADP iajs-3505 1 3 2024 2024 NUM iajs-3505 1 4 the the DET iajs-3505 1 5 author(s author(s NOUN iajs-3505 1 6 ) ) PUNCT iajs-3505 1 7 . . PUNCT iajs-3505 2 1 published publish VERB iajs-3505 2 2 by by ADP iajs-3505 2 3 college college NOUN iajs-3505 2 4 of of ADP iajs-3505 2 5 education education NOUN iajs-3505 2 6 for for ADP iajs-3505 2 7 pure pure ADJ iajs-3505 2 8 science science NOUN iajs-3505 2 9 ( ( PUNCT iajs-3505 2 10 ibn ibn PROPN iajs-3505 2 11 al al PROPN iajs-3505 2 12 - - PUNCT iajs-3505 2 13 haitham haitham PROPN iajs-3505 2 14 ) ) PUNCT iajs-3505 2 15 , , PUNCT iajs-3505 2 16 university university NOUN iajs-3505 2 17 of of ADP iajs-3505 2 18 baghdad baghdad PROPN iajs-3505 2 19 . . PUNCT iajs-3505 3 1 this this PRON iajs-3505 3 2 is be AUX iajs-3505 3 3 an an DET iajs-3505 3 4 open open ADJ iajs-3505 3 5 - - PUNCT iajs-3505 3 6 access access NOUN iajs-3505 3 7 article article NOUN iajs-3505 3 8 distributed distribute VERB iajs-3505 3 9 under under ADP iajs-3505 3 10 the the DET iajs-3505 3 11 terms term NOUN iajs-3505 3 12 of of ADP iajs-3505 3 13 the the DET iajs-3505 3 14 creative creative ADJ iajs-3505 3 15 commons common NOUN iajs-3505 3 16 attribution attribution NOUN iajs-3505 3 17 4.0 4.0 NUM iajs-3505 3 18 international international ADJ iajs-3505 3 19 license license NOUN iajs-3505 3 20 ibn ibn PROPN iajs-3505 3 21 al al PROPN iajs-3505 3 22 - - PUNCT iajs-3505 3 23 haitham haitham PROPN iajs-3505 3 24 journal journal PROPN iajs-3505 3 25 for for ADP iajs-3505 3 26 pure pure ADJ iajs-3505 3 27 and and CCONJ iajs-3505 3 28 applied applied ADJ iajs-3505 3 29 sciences sciences PROPN iajs-3505 3 30 journal journal PROPN iajs-3505 3 31 homepage homepage NOUN iajs-3505 3 32 : : PUNCT iajs-3505 3 33 jih.uobaghdad.edu.iq jih.uobaghdad.edu.iq NOUN iajs-3505 3 34 pissn pissn ADJ iajs-3505 3 35 : : PUNCT iajs-3505 3 36 1609 1609 NUM iajs-3505 3 37 - - SYM iajs-3505 3 38 4042 4042 NUM iajs-3505 3 39 , , PUNCT iajs-3505 3 40 eissn eissn NOUN iajs-3505 3 41 : : PUNCT iajs-3505 3 42 2521 2521 NUM iajs-3505 3 43 - - SYM iajs-3505 3 44 3407 3407 NUM iajs-3505 3 45 ihjpas ihjpa NOUN iajs-3505 3 46 . . PUNCT iajs-3505 4 1 2024 2024 NUM iajs-3505 4 2 , , PUNCT iajs-3505 4 3 37(4 37(4 X iajs-3505 4 4 ) ) PUNCT iajs-3505 4 5 stability stability NOUN iajs-3505 4 6 and and CCONJ iajs-3505 4 7 bifurcation bifurcation NOUN iajs-3505 4 8 analysis analysis NOUN iajs-3505 4 9 of of ADP iajs-3505 4 10 the the DET iajs-3505 4 11 impact impact NOUN iajs-3505 4 12 of of ADP iajs-3505 4 13 refuge refuge NOUN iajs-3505 4 14 on on ADP iajs-3505 4 15 the the DET iajs-3505 4 16 dissolved dissolve VERB iajs-3505 4 17 oxygen oxygen NOUN iajs-3505 4 18 - - PUNCT iajs-3505 4 19 plankton plankton NOUN iajs-3505 4 20 interaction interaction NOUN iajs-3505 4 21 ahmed ahmed PROPN iajs-3505 4 22 ali1 ali1 PROPN iajs-3505 4 23 , , PUNCT iajs-3505 4 24 * * PUNCT iajs-3505 4 25 , , PUNCT iajs-3505 4 26 shireen shireen PROPN iajs-3505 4 27 jawad2 jawad2 PROPN iajs-3505 4 28 and and CCONJ iajs-3505 4 29 fengde fengde PROPN iajs-3505 4 30 chen3 chen3 NOUN iajs-3505 4 31 1,2department 1,2department NUM iajs-3505 4 32 of of ADP iajs-3505 4 33 mathematics mathematic NOUN iajs-3505 4 34 , , PUNCT iajs-3505 4 35 college college NOUN iajs-3505 4 36 of of ADP iajs-3505 4 37 science science NOUN iajs-3505 4 38 , , PUNCT iajs-3505 4 39 university university NOUN iajs-3505 4 40 of of ADP iajs-3505 4 41 baghdad baghdad PROPN iajs-3505 4 42 , , PUNCT iajs-3505 4 43 baghdad baghdad PROPN iajs-3505 4 44 , , PUNCT iajs-3505 4 45 iraq iraq PROPN iajs-3505 4 46 . . PUNCT iajs-3505 5 1 3college 3college NUM iajs-3505 5 2 of of ADP iajs-3505 5 3 mathematics mathematic NOUN iajs-3505 5 4 and and CCONJ iajs-3505 5 5 statistics statistic NOUN iajs-3505 5 6 , , PUNCT iajs-3505 5 7 fuzhou fuzhou PROPN iajs-3505 5 8 university university PROPN iajs-3505 5 9 , , PUNCT iajs-3505 5 10 fuzhou fuzhou PROPN iajs-3505 5 11 350108 350108 NUM iajs-3505 5 12 , , PUNCT iajs-3505 5 13 p.r p.r PROPN iajs-3505 5 14 . . PROPN iajs-3505 5 15 china china PROPN iajs-3505 5 16 . . PUNCT iajs-3505 6 1 * * PUNCT iajs-3505 6 2 corresponding correspond VERB iajs-3505 6 3 author author NOUN iajs-3505 6 4 . . PUNCT iajs-3505 7 1 received receive VERB iajs-3505 7 2 : : PUNCT iajs-3505 7 3 21 21 NUM iajs-3505 7 4 may may PROPN iajs-3505 7 5 2023 2023 NUM iajs-3505 7 6 accepted accept VERB iajs-3505 7 7 : : PUNCT iajs-3505 7 8 9 9 NUM iajs-3505 7 9 july july PROPN iajs-3505 7 10 2023 2023 NUM iajs-3505 7 11 published publish VERB iajs-3505 7 12 : : PUNCT iajs-3505 7 13 20 20 NUM iajs-3505 7 14 october october NOUN iajs-3505 7 15 2024 2024 NUM iajs-3505 7 16 doi.org/10.30526/37.4.3505 doi.org/10.30526/37.4.3505 NOUN iajs-3505 7 17 abstract abstract ADV iajs-3505 7 18 the the DET iajs-3505 7 19 suggested suggest VERB iajs-3505 7 20 mathematical mathematical ADJ iajs-3505 7 21 model model NOUN iajs-3505 7 22 for for ADP iajs-3505 7 23 studying study VERB iajs-3505 7 24 the the DET iajs-3505 7 25 effect effect NOUN iajs-3505 7 26 of of ADP iajs-3505 7 27 refuge refuge NOUN iajs-3505 7 28 on on ADP iajs-3505 7 29 the the DET iajs-3505 7 30 dissolved dissolve VERB iajs-3505 7 31 oxygen oxygen NOUN iajs-3505 7 32 in in ADP iajs-3505 7 33 the the DET iajs-3505 7 34 plankton plankton NOUN iajs-3505 7 35 ecosystem ecosystem NOUN iajs-3505 7 36 is be AUX iajs-3505 7 37 based base VERB iajs-3505 7 38 on on ADP iajs-3505 7 39 measurements measurement NOUN iajs-3505 7 40 of of ADP iajs-3505 7 41 dissolved dissolved ADJ iajs-3505 7 42 oxygen oxygen NOUN iajs-3505 7 43 , , PUNCT iajs-3505 7 44 phytoplankton phytoplankton NOUN iajs-3505 7 45 , , PUNCT iajs-3505 7 46 and and CCONJ iajs-3505 7 47 zooplankton zooplankton NOUN iajs-3505 7 48 populations population NOUN iajs-3505 7 49 . . PUNCT iajs-3505 8 1 the the DET iajs-3505 8 2 aim aim NOUN iajs-3505 8 3 of of ADP iajs-3505 8 4 this this DET iajs-3505 8 5 work work NOUN iajs-3505 8 6 is be AUX iajs-3505 8 7 to to PART iajs-3505 8 8 find find VERB iajs-3505 8 9 out out ADP iajs-3505 8 10 the the DET iajs-3505 8 11 potential potential ADJ iajs-3505 8 12 equilibrium equilibrium NOUN iajs-3505 8 13 and and CCONJ iajs-3505 8 14 to to PART iajs-3505 8 15 investigate investigate VERB iajs-3505 8 16 their their PRON iajs-3505 8 17 behaviour behaviour NOUN iajs-3505 8 18 . . PUNCT iajs-3505 9 1 the the DET iajs-3505 9 2 study study NOUN iajs-3505 9 3 shows show VERB iajs-3505 9 4 that that SCONJ iajs-3505 9 5 there there PRON iajs-3505 9 6 are be VERB iajs-3505 9 7 three three NUM iajs-3505 9 8 points point NOUN iajs-3505 9 9 of of ADP iajs-3505 9 10 equilibrium equilibrium NOUN iajs-3505 9 11 . . PUNCT iajs-3505 10 1 the the DET iajs-3505 10 2 feasibility feasibility NOUN iajs-3505 10 3 requirements requirement NOUN iajs-3505 10 4 and and CCONJ iajs-3505 10 5 stability stability NOUN iajs-3505 10 6 conditions condition NOUN iajs-3505 10 7 for for ADP iajs-3505 10 8 all all DET iajs-3505 10 9 steady steady ADJ iajs-3505 10 10 states state NOUN iajs-3505 10 11 are be AUX iajs-3505 10 12 determined determine VERB iajs-3505 10 13 . . PUNCT iajs-3505 11 1 using use VERB iajs-3505 11 2 the the DET iajs-3505 11 3 consumption consumption NOUN iajs-3505 11 4 of of ADP iajs-3505 11 5 oxygen oxygen NOUN iajs-3505 11 6 by by ADP iajs-3505 11 7 zooplankton zooplankton NOUN iajs-3505 11 8 as as ADP iajs-3505 11 9 a a DET iajs-3505 11 10 bifurcation bifurcation NOUN iajs-3505 11 11 parameter parameter NOUN iajs-3505 11 12 , , PUNCT iajs-3505 11 13 we we PRON iajs-3505 11 14 test test VERB iajs-3505 11 15 for for ADP iajs-3505 11 16 the the DET iajs-3505 11 17 presence presence NOUN iajs-3505 11 18 of of ADP iajs-3505 11 19 hopfbifurcation hopfbifurcation NOUN iajs-3505 11 20 for for ADP iajs-3505 11 21 the the DET iajs-3505 11 22 interior interior ADJ iajs-3505 11 23 equilibrium equilibrium NOUN iajs-3505 11 24 . . PUNCT iajs-3505 12 1 it it PRON iajs-3505 12 2 is be AUX iajs-3505 12 3 shown show VERB iajs-3505 12 4 what what DET iajs-3505 12 5 conditions condition NOUN iajs-3505 12 6 must must AUX iajs-3505 12 7 be be AUX iajs-3505 12 8 met meet VERB iajs-3505 12 9 for for ADP iajs-3505 12 10 stable stable ADJ iajs-3505 12 11 limit limit NOUN iajs-3505 12 12 cycles cycle NOUN iajs-3505 12 13 . . PUNCT iajs-3505 13 1 finally finally ADV iajs-3505 13 2 , , PUNCT iajs-3505 13 3 a a DET iajs-3505 13 4 numerical numerical ADJ iajs-3505 13 5 simulation simulation NOUN iajs-3505 13 6 is be AUX iajs-3505 13 7 conducted conduct VERB iajs-3505 13 8 to to PART iajs-3505 13 9 back back VERB iajs-3505 13 10 up up ADP iajs-3505 13 11 the the DET iajs-3505 13 12 analytical analytical ADJ iajs-3505 13 13 findings finding NOUN iajs-3505 13 14 . . PUNCT iajs-3505 14 1 it it PRON iajs-3505 14 2 shows show VERB iajs-3505 14 3 when when SCONJ iajs-3505 14 4 the the DET iajs-3505 14 5 stability stability NOUN iajs-3505 14 6 criteria criterion NOUN iajs-3505 14 7 are be AUX iajs-3505 14 8 met meet VERB iajs-3505 14 9 , , PUNCT iajs-3505 14 10 the the DET iajs-3505 14 11 solution solution NOUN iajs-3505 14 12 of of ADP iajs-3505 14 13 the the DET iajs-3505 14 14 proposed propose VERB iajs-3505 14 15 system system NOUN iajs-3505 14 16 constantly constantly ADV iajs-3505 14 17 oscillates oscillate VERB iajs-3505 14 18 around around ADP iajs-3505 14 19 the the DET iajs-3505 14 20 positive positive ADJ iajs-3505 14 21 stable stable ADJ iajs-3505 14 22 state state NOUN iajs-3505 14 23 . . PUNCT iajs-3505 15 1 in in ADP iajs-3505 15 2 addition addition NOUN iajs-3505 15 3 , , PUNCT iajs-3505 15 4 the the DET iajs-3505 15 5 solution solution NOUN iajs-3505 15 6 exhibits exhibit VERB iajs-3505 15 7 limit limit VERB iajs-3505 15 8 cycle cycle NOUN iajs-3505 15 9 behaviour behaviour NOUN iajs-3505 15 10 for for ADP iajs-3505 15 11 small small ADJ iajs-3505 15 12 changes change NOUN iajs-3505 15 13 in in ADP iajs-3505 15 14 certain certain ADJ iajs-3505 15 15 parameters parameter NOUN iajs-3505 15 16 . . PUNCT iajs-3505 16 1 keywords keyword NOUN iajs-3505 16 2 : : PUNCT iajs-3505 16 3 plankton plankton NOUN iajs-3505 16 4 interaction interaction NOUN iajs-3505 16 5 , , PUNCT iajs-3505 16 6 stability stability NOUN iajs-3505 16 7 analysis analysis NOUN iajs-3505 16 8 , , PUNCT iajs-3505 16 9 bifurcation bifurcation NOUN iajs-3505 16 10 , , PUNCT iajs-3505 16 11 dissolved dissolve VERB iajs-3505 16 12 oxygen oxygen NOUN iajs-3505 16 13 model model NOUN iajs-3505 16 14 . . PUNCT iajs-3505 17 1 1 1 X iajs-3505 17 2 . . X iajs-3505 17 3 introduction introduction NOUN iajs-3505 17 4 much much ADJ iajs-3505 17 5 effort effort NOUN iajs-3505 17 6 has have AUX iajs-3505 17 7 been be AUX iajs-3505 17 8 put put VERB iajs-3505 17 9 into into ADP iajs-3505 17 10 understanding understanding NOUN iajs-3505 17 11 dissolved dissolve VERB iajs-3505 17 12 oxygen oxygen NOUN iajs-3505 17 13 dynamics dynamic NOUN iajs-3505 17 14 better well ADV iajs-3505 17 15 since since SCONJ iajs-3505 17 16 they they PRON iajs-3505 17 17 are be AUX iajs-3505 17 18 such such DET iajs-3505 17 19 a a DET iajs-3505 17 20 crucial crucial ADJ iajs-3505 17 21 indicator indicator NOUN iajs-3505 17 22 of of ADP iajs-3505 17 23 the the DET iajs-3505 17 24 health health NOUN iajs-3505 17 25 of of ADP iajs-3505 17 26 marine marine ADJ iajs-3505 17 27 ecosystems ecosystem NOUN iajs-3505 18 1 [ [ X iajs-3505 18 2 1 1 NUM iajs-3505 18 3 - - SYM iajs-3505 18 4 3 3 NUM iajs-3505 18 5 ] ] PUNCT iajs-3505 18 6 . . PUNCT iajs-3505 19 1 most most ADJ iajs-3505 19 2 of of ADP iajs-3505 19 3 the the DET iajs-3505 19 4 oxygen oxygen NOUN iajs-3505 19 5 in in ADP iajs-3505 19 6 the the DET iajs-3505 19 7 oceans ocean NOUN iajs-3505 19 8 comes come VERB iajs-3505 19 9 from from ADP iajs-3505 19 10 phytoplankton phytoplankton NOUN iajs-3505 19 11 which which PRON iajs-3505 19 12 also also ADV iajs-3505 19 13 serves serve VERB iajs-3505 19 14 as as ADP iajs-3505 19 15 the the DET iajs-3505 19 16 foundation foundation NOUN iajs-3505 19 17 of of ADP iajs-3505 19 18 the the DET iajs-3505 19 19 marine marine ADJ iajs-3505 19 20 food food NOUN iajs-3505 19 21 chain chain NOUN iajs-3505 19 22 through through ADP iajs-3505 19 23 their their PRON iajs-3505 19 24 photosynthesis photosynthesis NOUN iajs-3505 19 25 [ [ X iajs-3505 19 26 4 4 NUM iajs-3505 19 27 ] ] PUNCT iajs-3505 19 28 . . PUNCT iajs-3505 20 1 it it PRON iajs-3505 20 2 ’s ’ VERB iajs-3505 20 3 well well INTJ iajs-3505 20 4 knowledge knowledge NOUN iajs-3505 20 5 that that SCONJ iajs-3505 20 6 salinity salinity NOUN iajs-3505 20 7 , , PUNCT iajs-3505 20 8 temperature temperature NOUN iajs-3505 20 9 , , PUNCT iajs-3505 20 10 and and CCONJ iajs-3505 20 11 nutrient nutrient NOUN iajs-3505 20 12 availability availability NOUN iajs-3505 20 13 all all PRON iajs-3505 20 14 play play VERB iajs-3505 20 15 significant significant ADJ iajs-3505 20 16 roles role NOUN iajs-3505 20 17 in in ADP iajs-3505 20 18 determining determine VERB iajs-3505 20 19 how how SCONJ iajs-3505 20 20 much much ADJ iajs-3505 20 21 oxygen oxygen NOUN iajs-3505 20 22 phytoplankton phytoplankton NOUN iajs-3505 20 23 can can AUX iajs-3505 20 24 create create VERB iajs-3505 20 25 . . PUNCT iajs-3505 21 1 additionally additionally ADV iajs-3505 21 2 , , PUNCT iajs-3505 21 3 there there PRON iajs-3505 21 4 is be VERB iajs-3505 21 5 a a DET iajs-3505 21 6 considerable considerable ADJ iajs-3505 21 7 diurnal diurnal ADJ iajs-3505 21 8 variation variation NOUN iajs-3505 21 9 in in ADP iajs-3505 21 10 oxygen oxygen NOUN iajs-3505 21 11 generation generation NOUN iajs-3505 21 12 by by ADP iajs-3505 21 13 phytoplankton phytoplankton NOUN iajs-3505 21 14 . . PUNCT iajs-3505 22 1 therefore therefore ADV iajs-3505 22 2 , , PUNCT iajs-3505 22 3 the the DET iajs-3505 22 4 link link NOUN iajs-3505 22 5 between between ADP iajs-3505 22 6 phytoplankton phytoplankton NOUN iajs-3505 22 7 and and CCONJ iajs-3505 22 8 dissolved dissolve VERB iajs-3505 22 9 oxygen oxygen NOUN iajs-3505 22 10 is be AUX iajs-3505 22 11 crucial crucial ADJ iajs-3505 22 12 to to ADP iajs-3505 22 13 the the DET iajs-3505 22 14 existence existence NOUN iajs-3505 22 15 of of ADP iajs-3505 22 16 organisms organism NOUN iajs-3505 22 17 . . PUNCT iajs-3505 23 1 oxygen oxygen NOUN iajs-3505 23 2 production production NOUN iajs-3505 23 3 fluctuations fluctuation NOUN iajs-3505 23 4 can can AUX iajs-3505 23 5 have have VERB iajs-3505 23 6 severe severe ADJ iajs-3505 23 7 consequences consequence NOUN iajs-3505 23 8 for for ADP iajs-3505 23 9 marine marine ADJ iajs-3505 23 10 life life NOUN iajs-3505 23 11 . . PUNCT iajs-3505 24 1 [ [ X iajs-3505 24 2 5 5 NUM iajs-3505 24 3 ] ] PUNCT iajs-3505 24 4 . . PUNCT iajs-3505 25 1 since since SCONJ iajs-3505 25 2 oxygen oxygen NOUN iajs-3505 25 3 is be AUX iajs-3505 25 4 used use VERB iajs-3505 25 5 by by ADP iajs-3505 25 6 living live VERB iajs-3505 25 7 things thing NOUN iajs-3505 25 8 for for ADP iajs-3505 25 9 photosynthesis photosynthesis NOUN iajs-3505 25 10 ( ( PUNCT iajs-3505 25 11 during during ADP iajs-3505 25 12 the the DET iajs-3505 25 13 day day NOUN iajs-3505 25 14 ) ) PUNCT iajs-3505 25 15 and and CCONJ iajs-3505 25 16 respiration respiration NOUN iajs-3505 25 17 ( ( PUNCT iajs-3505 25 18 at at ADP iajs-3505 25 19 night night NOUN iajs-3505 25 20 ) ) PUNCT iajs-3505 25 21 , , PUNCT iajs-3505 25 22 the the DET iajs-3505 25 23 amount amount NOUN iajs-3505 25 24 of of ADP iajs-3505 25 25 oxygen oxygen NOUN iajs-3505 25 26 in in ADP iajs-3505 25 27 the the DET iajs-3505 25 28 water water NOUN iajs-3505 25 29 changes change VERB iajs-3505 25 30 day day NOUN iajs-3505 25 31 and and CCONJ iajs-3505 25 32 night night NOUN iajs-3505 25 33 . . PUNCT iajs-3505 26 1 as as ADP iajs-3505 26 2 a a DET iajs-3505 26 3 result result NOUN iajs-3505 26 4 , , PUNCT iajs-3505 26 5 phytoplankton phytoplankton NOUN iajs-3505 26 6 colonies colony NOUN iajs-3505 26 7 can can AUX iajs-3505 26 8 serve serve VERB iajs-3505 26 9 as as ADP iajs-3505 26 10 reliable reliable ADJ iajs-3505 26 11 markers marker NOUN iajs-3505 26 12 of of ADP iajs-3505 26 13 ecological ecological ADJ iajs-3505 26 14 status status NOUN iajs-3505 26 15 . . PUNCT iajs-3505 27 1 [ [ X iajs-3505 27 2 6 6 NUM iajs-3505 27 3 ] ] PUNCT iajs-3505 27 4 , , PUNCT iajs-3505 27 5 [ [ X iajs-3505 27 6 7 7 NUM iajs-3505 27 7 ] ] PUNCT iajs-3505 27 8 . . PUNCT iajs-3505 28 1 mondal mondal PROPN iajs-3505 28 2 , , PUNCT iajs-3505 28 3 and and CCONJ iajs-3505 28 4 his his PRON iajs-3505 28 5 colleague colleague NOUN iajs-3505 28 6 , , PUNCT iajs-3505 28 7 for for ADP iajs-3505 28 8 instance instance NOUN iajs-3505 28 9 , , PUNCT iajs-3505 28 10 have have AUX iajs-3505 28 11 studied study VERB iajs-3505 28 12 how how SCONJ iajs-3505 28 13 the the DET iajs-3505 28 14 coupled couple VERB iajs-3505 28 15 plankton plankton NOUN iajs-3505 28 16 - - PUNCT iajs-3505 28 17 oxygen oxygen NOUN iajs-3505 28 18 dynamics dynamic NOUN iajs-3505 28 19 in in ADP iajs-3505 28 20 the the DET iajs-3505 28 21 ocean ocean NOUN iajs-3505 28 22 are be AUX iajs-3505 28 23 affected affect VERB iajs-3505 28 24 by by ADP iajs-3505 28 25 a a DET iajs-3505 28 26 low low ADJ iajs-3505 28 27 oxygen oxygen NOUN iajs-3505 28 28 production production NOUN iajs-3505 28 29 rate rate NOUN iajs-3505 28 30 , , PUNCT iajs-3505 28 31 which which PRON iajs-3505 28 32 can can AUX iajs-3505 28 33 result result VERB iajs-3505 28 34 in in ADP iajs-3505 28 35 oxygen oxygen NOUN iajs-3505 28 36 depletion depletion NOUN iajs-3505 28 37 and and CCONJ iajs-3505 28 38 species species NOUN iajs-3505 28 39 extinction extinction NOUN iajs-3505 28 40 [ [ X iajs-3505 28 41 8 8 NUM iajs-3505 28 42 ] ] PUNCT iajs-3505 28 43 . . PUNCT iajs-3505 29 1 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3505 29 2 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3505 29 3 https://orcid.org/0009-0004-7408-6238 https://orcid.org/0009-0004-7408-6238 NOUN iajs-3505 29 4 mailto:ahmedali.970q6@gmail.com mailto:ahmedali.970q6@gmail.com PROPN iajs-3505 29 5 https://orcid.org/0000-0002-3090-8357 https://orcid.org/0000-0002-3090-8357 NOUN iajs-3505 29 6 mailto:shireen.jawad@sc.uobaghdad.edu.iq mailto:shireen.jawad@sc.uobaghdad.edu.iq PROPN iajs-3505 30 1 https://orcid.org/0000-0003-3617-5550 https://orcid.org/0000-0003-3617-5550 PROPN iajs-3505 30 2 mailto:fdchen@263.ne mailto:fdchen@263.ne PROPN iajs-3505 30 3 ihjpas ihjpas PROPN iajs-3505 30 4 . . PUNCT iajs-3505 31 1 2024 2024 NUM iajs-3505 31 2 , , PUNCT iajs-3505 31 3 37(4 37(4 NOUN iajs-3505 31 4 ) ) PUNCT iajs-3505 31 5 371 371 NUM iajs-3505 31 6 furthermore furthermore ADV iajs-3505 31 7 , , PUNCT iajs-3505 31 8 the the DET iajs-3505 31 9 primary primary ADJ iajs-3505 31 10 objective objective NOUN iajs-3505 31 11 of of ADP iajs-3505 31 12 the the DET iajs-3505 31 13 study study NOUN iajs-3505 31 14 of of ADP iajs-3505 31 15 theoretical theoretical ADJ iajs-3505 31 16 ecology ecology NOUN iajs-3505 31 17 is be AUX iajs-3505 31 18 to to PART iajs-3505 31 19 identify identify VERB iajs-3505 31 20 the the DET iajs-3505 31 21 various various ADJ iajs-3505 31 22 dynamical dynamical ADJ iajs-3505 31 23 mechanisms mechanism NOUN iajs-3505 31 24 underlying underlie VERB iajs-3505 31 25 interactions interaction NOUN iajs-3505 31 26 between between ADP iajs-3505 31 27 prey prey NOUN iajs-3505 31 28 and and CCONJ iajs-3505 31 29 predator predator NOUN iajs-3505 32 1 [ [ PUNCT iajs-3505 32 2 9 9 NUM iajs-3505 32 3 - - SYM iajs-3505 32 4 11 11 NUM iajs-3505 32 5 ] ] PUNCT iajs-3505 32 6 . . PUNCT iajs-3505 33 1 the the DET iajs-3505 33 2 relationship relationship NOUN iajs-3505 33 3 between between ADP iajs-3505 33 4 phytoplankton phytoplankton NOUN iajs-3505 33 5 and and CCONJ iajs-3505 33 6 zooplankton zooplankton NOUN iajs-3505 33 7 is be AUX iajs-3505 33 8 an an DET iajs-3505 33 9 example example NOUN iajs-3505 33 10 of of ADP iajs-3505 33 11 a a DET iajs-3505 33 12 predator predator NOUN iajs-3505 33 13 - - PUNCT iajs-3505 33 14 prey prey NOUN iajs-3505 33 15 interaction interaction NOUN iajs-3505 33 16 that that PRON iajs-3505 33 17 reveals reveal VERB iajs-3505 33 18 numerous numerous ADJ iajs-3505 33 19 aspects aspect NOUN iajs-3505 33 20 of of ADP iajs-3505 33 21 marine marine ADJ iajs-3505 33 22 ecology ecology NOUN iajs-3505 33 23 . . PUNCT iajs-3505 34 1 phytoplankton phytoplankton NOUN iajs-3505 34 2 contributes contribute VERB iajs-3505 34 3 substantially substantially ADV iajs-3505 34 4 to to ADP iajs-3505 34 5 aquatic aquatic ADJ iajs-3505 34 6 ecosystems ecosystem NOUN iajs-3505 34 7 , , PUNCT iajs-3505 34 8 including include VERB iajs-3505 34 9 producing produce VERB iajs-3505 34 10 vast vast ADJ iajs-3505 34 11 quantities quantity NOUN iajs-3505 34 12 of of ADP iajs-3505 34 13 oxygen oxygen NOUN iajs-3505 34 14 , , PUNCT iajs-3505 34 15 managing manage VERB iajs-3505 34 16 natural natural ADJ iajs-3505 34 17 resources resource NOUN iajs-3505 34 18 and and CCONJ iajs-3505 34 19 water water NOUN iajs-3505 34 20 quality quality NOUN iajs-3505 34 21 , , PUNCT iajs-3505 34 22 and and CCONJ iajs-3505 34 23 establishing establish VERB iajs-3505 34 24 multiple multiple ADJ iajs-3505 34 25 food food NOUN iajs-3505 34 26 webs webs NOUN iajs-3505 34 27 [ [ X iajs-3505 34 28 12 12 NUM iajs-3505 34 29 - - SYM iajs-3505 34 30 14 14 NUM iajs-3505 34 31 ] ] PUNCT iajs-3505 34 32 . . PUNCT iajs-3505 35 1 plankton plankton PROPN iajs-3505 35 2 dynamics dynamic NOUN iajs-3505 35 3 research research NOUN iajs-3505 35 4 is be AUX iajs-3505 35 5 a a DET iajs-3505 35 6 fascinating fascinating ADJ iajs-3505 35 7 field field NOUN iajs-3505 35 8 of of ADP iajs-3505 35 9 study study NOUN iajs-3505 35 10 . . PUNCT iajs-3505 36 1 plankton plankton PROPN iajs-3505 36 2 constitutes constitute VERB iajs-3505 36 3 the the DET iajs-3505 36 4 building building NOUN iajs-3505 36 5 elements element NOUN iajs-3505 36 6 of of ADP iajs-3505 36 7 all all DET iajs-3505 36 8 aquatic aquatic ADJ iajs-3505 36 9 food food NOUN iajs-3505 36 10 chains chain NOUN iajs-3505 36 11 , , PUNCT iajs-3505 36 12 with with ADP iajs-3505 36 13 phytoplankton phytoplankton NOUN iajs-3505 36 14 occupying occupy VERB iajs-3505 36 15 the the DET iajs-3505 36 16 first first ADJ iajs-3505 36 17 trophic trophic ADJ iajs-3505 36 18 level level NOUN iajs-3505 36 19 [ [ X iajs-3505 36 20 15 15 NUM iajs-3505 36 21 ] ] PUNCT iajs-3505 36 22 . . PUNCT iajs-3505 37 1 bagheli bagheli PROPN iajs-3505 37 2 and and CCONJ iajs-3505 37 3 dhar dhar PROPN iajs-3505 37 4 examine examine VERB iajs-3505 37 5 the the DET iajs-3505 37 6 effect effect NOUN iajs-3505 37 7 of of ADP iajs-3505 37 8 dissolved dissolve VERB iajs-3505 37 9 oxygen oxygen NOUN iajs-3505 37 10 on on ADP iajs-3505 37 11 the the DET iajs-3505 37 12 presence presence NOUN iajs-3505 37 13 of of ADP iajs-3505 37 14 a a DET iajs-3505 37 15 planktonic planktonic ADJ iajs-3505 37 16 population population NOUN iajs-3505 37 17 that that PRON iajs-3505 37 18 interacts interact VERB iajs-3505 37 19 . . PUNCT iajs-3505 38 1 they they PRON iajs-3505 38 2 conclude conclude VERB iajs-3505 38 3 that that SCONJ iajs-3505 38 4 hopf hopf NOUN iajs-3505 38 5 - - PUNCT iajs-3505 38 6 bifurcation bifurcation NOUN iajs-3505 38 7 in in ADP iajs-3505 38 8 the the DET iajs-3505 38 9 interior interior ADJ iajs-3505 38 10 equilibrium equilibrium NOUN iajs-3505 38 11 is be AUX iajs-3505 38 12 possible possible ADJ iajs-3505 38 13 if if SCONJ iajs-3505 38 14 the the DET iajs-3505 38 15 phytoplankton phytoplankton NOUN iajs-3505 38 16 growth growth NOUN iajs-3505 38 17 rate rate NOUN iajs-3505 38 18 is be AUX iajs-3505 38 19 selected select VERB iajs-3505 38 20 as as ADP iajs-3505 38 21 the the DET iajs-3505 38 22 bifurcation bifurcation NOUN iajs-3505 38 23 parameter parameter NOUN iajs-3505 39 1 [ [ X iajs-3505 39 2 16 16 NUM iajs-3505 39 3 ] ] PUNCT iajs-3505 39 4 . . PUNCT iajs-3505 40 1 the the DET iajs-3505 40 2 purpose purpose NOUN iajs-3505 40 3 of of ADP iajs-3505 40 4 this this DET iajs-3505 40 5 research research NOUN iajs-3505 40 6 is be AUX iajs-3505 40 7 to to PART iajs-3505 40 8 examine examine VERB iajs-3505 40 9 how how SCONJ iajs-3505 40 10 the the DET iajs-3505 40 11 phytoplankton phytoplankton NOUN iajs-3505 40 12 refuge refuge NOUN iajs-3505 40 13 affects affect VERB iajs-3505 40 14 the the DET iajs-3505 40 15 dynamics dynamic NOUN iajs-3505 40 16 of of ADP iajs-3505 40 17 the the DET iajs-3505 40 18 oxygen oxygen NOUN iajs-3505 40 19 - - PUNCT iajs-3505 40 20 plankton plankton NOUN iajs-3505 40 21 model model NOUN iajs-3505 40 22 . . PUNCT iajs-3505 41 1 some some DET iajs-3505 41 2 phytoplankton phytoplankton NOUN iajs-3505 41 3 may may AUX iajs-3505 41 4 evade evade VERB iajs-3505 41 5 their their PRON iajs-3505 41 6 zooplankton zooplankton NOUN iajs-3505 41 7 prey prey NOUN iajs-3505 41 8 by by ADP iajs-3505 41 9 relocating relocate VERB iajs-3505 41 10 to to ADP iajs-3505 41 11 deeper deep ADJ iajs-3505 41 12 water water NOUN iajs-3505 41 13 column column NOUN iajs-3505 41 14 layers layer NOUN iajs-3505 41 15 . . PUNCT iajs-3505 42 1 these these DET iajs-3505 42 2 sediments sediment NOUN iajs-3505 42 3 offer offer VERB iajs-3505 42 4 shelter shelter NOUN iajs-3505 42 5 from from ADP iajs-3505 42 6 predators predator NOUN iajs-3505 42 7 to to ADP iajs-3505 42 8 the the DET iajs-3505 42 9 prey prey NOUN iajs-3505 42 10 . . PUNCT iajs-3505 43 1 here here ADV iajs-3505 43 2 is be AUX iajs-3505 43 3 how how SCONJ iajs-3505 43 4 the the DET iajs-3505 43 5 article article NOUN iajs-3505 43 6 is be AUX iajs-3505 43 7 laid lay VERB iajs-3505 43 8 out out ADP iajs-3505 43 9 : : PUNCT iajs-3505 43 10 in in ADP iajs-3505 43 11 sec sec PROPN iajs-3505 43 12 . . PROPN iajs-3505 43 13 2 2 NUM iajs-3505 43 14 , , PUNCT iajs-3505 43 15 we we PRON iajs-3505 43 16 built build VERB iajs-3505 43 17 the the DET iajs-3505 43 18 structure structure NOUN iajs-3505 43 19 of of ADP iajs-3505 43 20 the the DET iajs-3505 43 21 proposed propose VERB iajs-3505 43 22 model model NOUN iajs-3505 43 23 . . PUNCT iajs-3505 44 1 sec sec PROPN iajs-3505 44 2 . . PROPN iajs-3505 44 3 3 3 NUM iajs-3505 44 4 explains explain VERB iajs-3505 44 5 the the DET iajs-3505 44 6 feasibility feasibility NOUN iajs-3505 44 7 requirements requirement NOUN iajs-3505 44 8 and and CCONJ iajs-3505 44 9 stability stability NOUN iajs-3505 44 10 conditions condition NOUN iajs-3505 44 11 for for ADP iajs-3505 44 12 all all DET iajs-3505 44 13 steady steady ADJ iajs-3505 44 14 states state NOUN iajs-3505 44 15 . . PUNCT iajs-3505 45 1 the the DET iajs-3505 45 2 prevalence prevalence NOUN iajs-3505 45 3 of of ADP iajs-3505 45 4 hopf hopf ADJ iajs-3505 45 5 bifurcations bifurcation NOUN iajs-3505 45 6 is be AUX iajs-3505 45 7 also also ADV iajs-3505 45 8 illustrated illustrate VERB iajs-3505 45 9 in in ADP iajs-3505 45 10 sec sec PROPN iajs-3505 45 11 . . PROPN iajs-3505 46 1 4 4 NUM iajs-3505 46 2 . . X iajs-3505 46 3 in in ADP iajs-3505 46 4 sec sec PROPN iajs-3505 46 5 . . PROPN iajs-3505 47 1 5 5 NUM iajs-3505 47 2 . . X iajs-3505 47 3 we we PRON iajs-3505 47 4 undertake undertake VERB iajs-3505 47 5 matlab matlab PROPN iajs-3505 47 6 - - PUNCT iajs-3505 47 7 based base VERB iajs-3505 47 8 numerical numerical ADJ iajs-3505 47 9 simulations simulation NOUN iajs-3505 47 10 to to PART iajs-3505 47 11 validate validate VERB iajs-3505 47 12 the the DET iajs-3505 47 13 analytical analytical ADJ iajs-3505 47 14 results result NOUN iajs-3505 47 15 . . PUNCT iajs-3505 48 1 2 2 X iajs-3505 48 2 . . X iajs-3505 48 3 construction construction NOUN iajs-3505 48 4 of of ADP iajs-3505 48 5 the the DET iajs-3505 48 6 model model NOUN iajs-3505 48 7 include include VERB iajs-3505 48 8 let let VERB iajs-3505 48 9 𝑒(𝑑 𝑒(𝑑 NOUN iajs-3505 48 10 ) ) PUNCT iajs-3505 48 11 and and CCONJ iajs-3505 48 12 𝑣(𝑑 𝑣(𝑑 NOUN iajs-3505 48 13 ) ) PUNCT iajs-3505 48 14 indicate indicate VERB iajs-3505 48 15 the the DET iajs-3505 48 16 phytoplankton phytoplankton NOUN iajs-3505 48 17 and and CCONJ iajs-3505 48 18 zooplankton zooplankton NOUN iajs-3505 48 19 populations population NOUN iajs-3505 48 20 at at ADP iajs-3505 48 21 the the DET iajs-3505 48 22 time time NOUN iajs-3505 48 23 𝑑 𝑑 NOUN iajs-3505 48 24 , , PUNCT iajs-3505 48 25 respectively respectively ADV iajs-3505 48 26 . . PUNCT iajs-3505 49 1 we we PRON iajs-3505 49 2 assume assume VERB iajs-3505 49 3 some some DET iajs-3505 49 4 phytoplankton phytoplankton NOUN iajs-3505 49 5 populations population NOUN iajs-3505 49 6 are be AUX iajs-3505 49 7 safe safe ADJ iajs-3505 49 8 from from ADP iajs-3505 49 9 zooplankton zooplankton NOUN iajs-3505 49 10 predation predation NOUN iajs-3505 49 11 because because SCONJ iajs-3505 49 12 they they PRON iajs-3505 49 13 can can AUX iajs-3505 49 14 conceal conceal VERB iajs-3505 49 15 themselves themselves PRON iajs-3505 49 16 in in ADP iajs-3505 49 17 the the DET iajs-3505 49 18 ocean ocean NOUN iajs-3505 49 19 ’s ’s PART iajs-3505 49 20 floor floor NOUN iajs-3505 49 21 - - PUNCT iajs-3505 49 22 diverse diverse ADJ iajs-3505 49 23 sediments sediment NOUN iajs-3505 49 24 . . PUNCT iajs-3505 50 1 these these DET iajs-3505 50 2 sediments sediment NOUN iajs-3505 50 3 offer offer VERB iajs-3505 50 4 refuge refuge NOUN iajs-3505 50 5 from from ADP iajs-3505 50 6 predators predator NOUN iajs-3505 50 7 to to ADP iajs-3505 50 8 the the DET iajs-3505 50 9 hunted hunted NOUN iajs-3505 50 10 . . PUNCT iajs-3505 51 1 𝑀(𝑑 𝑀(𝑑 NOUN iajs-3505 51 2 ) ) PUNCT iajs-3505 51 3 represents represent VERB iajs-3505 51 4 the the DET iajs-3505 51 5 dissolved dissolve VERB iajs-3505 51 6 oxygen oxygen NOUN iajs-3505 51 7 concentration concentration NOUN iajs-3505 51 8 in in ADP iajs-3505 51 9 the the DET iajs-3505 51 10 marine marine NOUN iajs-3505 51 11 . . PUNCT iajs-3505 52 1 since since SCONJ iajs-3505 52 2 phytoplankton phytoplankton NOUN iajs-3505 52 3 do do VERB iajs-3505 52 4 photosynthesis photosynthesis NOUN iajs-3505 52 5 throughout throughout ADP iajs-3505 52 6 the the DET iajs-3505 52 7 day day NOUN iajs-3505 52 8 , , PUNCT iajs-3505 52 9 they they PRON iajs-3505 52 10 also also ADV iajs-3505 52 11 contribute contribute VERB iajs-3505 52 12 to to ADP iajs-3505 52 13 atmospheric atmospheric ADJ iajs-3505 52 14 oxygen oxygen NOUN iajs-3505 52 15 production production NOUN iajs-3505 52 16 . . PUNCT iajs-3505 53 1 several several ADJ iajs-3505 53 2 additional additional ADJ iajs-3505 53 3 factors factor NOUN iajs-3505 53 4 , , PUNCT iajs-3505 53 5 such such ADJ iajs-3505 53 6 as as ADP iajs-3505 53 7 the the DET iajs-3505 53 8 respiration respiration NOUN iajs-3505 53 9 of of ADP iajs-3505 53 10 marine marine ADJ iajs-3505 53 11 organisms organism NOUN iajs-3505 53 12 , , PUNCT iajs-3505 53 13 the the DET iajs-3505 53 14 consumption consumption NOUN iajs-3505 53 15 of of ADP iajs-3505 53 16 oxygen oxygen NOUN iajs-3505 53 17 by by ADP iajs-3505 53 18 phytoplankton phytoplankton NOUN iajs-3505 53 19 at at ADP iajs-3505 53 20 night night NOUN iajs-3505 53 21 , , PUNCT iajs-3505 53 22 and and CCONJ iajs-3505 53 23 the the DET iajs-3505 53 24 gradual gradual ADJ iajs-3505 53 25 drop drop NOUN iajs-3505 53 26 in in ADP iajs-3505 53 27 oxygen oxygen NOUN iajs-3505 53 28 concentration concentration NOUN iajs-3505 53 29 brought bring VERB iajs-3505 53 30 about about ADP iajs-3505 53 31 by by ADP iajs-3505 53 32 chemical chemical ADJ iajs-3505 53 33 reactions reaction NOUN iajs-3505 53 34 in in ADP iajs-3505 53 35 the the DET iajs-3505 53 36 water water NOUN iajs-3505 53 37 , , PUNCT iajs-3505 53 38 all all PRON iajs-3505 53 39 affect affect VERB iajs-3505 53 40 the the DET iajs-3505 53 41 rate rate NOUN iajs-3505 53 42 at at ADP iajs-3505 53 43 which which PRON iajs-3505 53 44 oxygen oxygen NOUN iajs-3505 53 45 is be AUX iajs-3505 53 46 depleted deplete VERB iajs-3505 53 47 . . PUNCT iajs-3505 54 1 the the DET iajs-3505 54 2 following follow VERB iajs-3505 54 3 set set NOUN iajs-3505 54 4 of of ADP iajs-3505 54 5 ordinary ordinary ADJ iajs-3505 54 6 differential differential ADJ iajs-3505 54 7 equations equation NOUN iajs-3505 54 8 serves serve VERB iajs-3505 54 9 as as ADP iajs-3505 54 10 the the DET iajs-3505 54 11 governing govern VERB iajs-3505 54 12 structure structure NOUN iajs-3505 54 13 for for ADP iajs-3505 54 14 the the DET iajs-3505 54 15 dynamical dynamical ADJ iajs-3505 54 16 system system NOUN iajs-3505 54 17 of of ADP iajs-3505 54 18 the the DET iajs-3505 54 19 system system NOUN iajs-3505 54 20 ( ( PUNCT iajs-3505 54 21 1 1 NUM iajs-3505 54 22 ): ): PUNCT iajs-3505 54 23 𝑑𝑒 𝑑𝑒 PROPN iajs-3505 54 24 𝑑𝑑 𝑑𝑑 ADP iajs-3505 54 25 = = PUNCT iajs-3505 54 26 π‘Ÿπ‘’ π‘Ÿπ‘’ PROPN iajs-3505 54 27 ( ( PUNCT iajs-3505 54 28 π‘Ž1 π‘Ž1 NOUN iajs-3505 54 29 + + NUM iajs-3505 54 30 𝑀0 𝑀0 NOUN iajs-3505 54 31 βˆ’π‘€ βˆ’π‘€ NOUN iajs-3505 54 32 ) ) PUNCT iajs-3505 55 1 βˆ’ βˆ’ PROPN iajs-3505 55 2 𝛼1𝑒(1 𝛼1𝑒(1 PROPN iajs-3505 55 3 βˆ’ βˆ’ PROPN iajs-3505 55 4 π‘š)𝑣 π‘š)𝑣 PUNCT iajs-3505 55 5 βˆ’ βˆ’ PROPN iajs-3505 56 1 𝛿1𝑒 𝛿1𝑒 PROPN iajs-3505 56 2 , , PUNCT iajs-3505 56 3 𝑑𝑣 𝑑𝑣 X iajs-3505 56 4 𝑑𝑑 𝑑𝑑 ADP iajs-3505 56 5 = = PUNCT iajs-3505 56 6 𝛼2𝑒(1 𝛼2𝑒(1 PROPN iajs-3505 56 7 βˆ’π‘š)𝑣 βˆ’π‘š)𝑣 NOUN iajs-3505 56 8 ( ( PUNCT iajs-3505 56 9 π‘Ž2 π‘Ž2 PROPN iajs-3505 56 10 + + NUM iajs-3505 56 11 𝑀0 𝑀0 PROPN iajs-3505 56 12 βˆ’ βˆ’ PROPN iajs-3505 56 13 𝑀 𝑀 PROPN iajs-3505 56 14 ) ) PUNCT iajs-3505 56 15 βˆ’ βˆ’ NOUN iajs-3505 56 16 𝛿2𝑣 𝛿2𝑣 NOUN iajs-3505 56 17 , , PUNCT iajs-3505 56 18 𝑑𝑀 𝑑𝑀 PROPN iajs-3505 56 19 𝑑𝑑 𝑑𝑑 ADP iajs-3505 56 20 = = PUNCT iajs-3505 56 21 𝑠(𝑀0 𝑠(𝑀0 NOUN iajs-3505 56 22 βˆ’ βˆ’ PROPN iajs-3505 56 23 𝑀 𝑀 ADP iajs-3505 56 24 ) ) PUNCT iajs-3505 56 25 + + CCONJ iajs-3505 57 1 𝑑𝑒 𝑑𝑒 ADP iajs-3505 57 2 βˆ’ βˆ’ PROPN iajs-3505 57 3 𝛾𝑀 𝛾𝑀 NOUN iajs-3505 57 4 βˆ’ βˆ’ ADP iajs-3505 57 5 𝛾1𝑒𝑀 𝛾1𝑒𝑀 NUM iajs-3505 57 6 βˆ’ βˆ’ PROPN iajs-3505 57 7 𝛾2𝑣𝑀 𝛾2𝑣𝑀 PROPN iajs-3505 57 8 , , PUNCT iajs-3505 57 9 ( ( PUNCT iajs-3505 57 10 1 1 X iajs-3505 57 11 ) ) PUNCT iajs-3505 57 12 with with ADP iajs-3505 57 13 the the DET iajs-3505 57 14 initial initial ADJ iajs-3505 57 15 conditions condition NOUN iajs-3505 57 16 𝑒(0 𝑒(0 PROPN iajs-3505 57 17 ) ) PUNCT iajs-3505 57 18 β‰₯ β‰₯ NOUN iajs-3505 57 19 0 0 NUM iajs-3505 57 20 , , PUNCT iajs-3505 57 21 𝑣(0 𝑣(0 NUM iajs-3505 57 22 ) ) PUNCT iajs-3505 57 23 β‰₯ β‰₯ NOUN iajs-3505 57 24 0 0 NUM iajs-3505 57 25 and and CCONJ iajs-3505 57 26 𝑀(0 𝑀(0 NOUN iajs-3505 57 27 ) ) PUNCT iajs-3505 57 28 β‰₯ β‰₯ NOUN iajs-3505 57 29 0 0 NUM iajs-3505 57 30 . . PUNCT iajs-3505 58 1 in in ADP iajs-3505 58 2 the the DET iajs-3505 58 3 first first ADJ iajs-3505 58 4 equation equation NOUN iajs-3505 58 5 of of ADP iajs-3505 58 6 the the DET iajs-3505 58 7 system system NOUN iajs-3505 58 8 ( ( PUNCT iajs-3505 58 9 1 1 NUM iajs-3505 58 10 ) ) PUNCT iajs-3505 58 11 , , PUNCT iajs-3505 58 12 π‘Ÿπ‘’ π‘Ÿπ‘’ PROPN iajs-3505 58 13 ( ( PUNCT iajs-3505 58 14 π‘Ž1+𝑀0βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘€ PROPN iajs-3505 58 15 ) ) PUNCT iajs-3505 58 16 represents represent VERB iajs-3505 58 17 the the DET iajs-3505 58 18 absorption absorption NOUN iajs-3505 58 19 of of ADP iajs-3505 58 20 dissolved dissolved ADJ iajs-3505 58 21 oxygen oxygen NOUN iajs-3505 58 22 from from ADP iajs-3505 58 23 phytoplankton phytoplankton NOUN iajs-3505 58 24 with with ADP iajs-3505 58 25 the the DET iajs-3505 58 26 growth growth NOUN iajs-3505 58 27 rate rate NOUN iajs-3505 58 28 π‘Ÿ. π‘Ÿ. NOUN iajs-3505 59 1 the the DET iajs-3505 59 2 maximum maximum ADJ iajs-3505 59 3 growth growth NOUN iajs-3505 59 4 rate rate NOUN iajs-3505 59 5 of of ADP iajs-3505 59 6 the the DET iajs-3505 59 7 phytoplankton phytoplankton NOUN iajs-3505 59 8 population population NOUN iajs-3505 59 9 is be AUX iajs-3505 59 10 π‘Ÿ/π‘Ž1 π‘Ÿ/π‘Ž1 NOUN iajs-3505 59 11 at at ADP iajs-3505 59 12 𝑀 𝑀 PRON iajs-3505 59 13 = = SYM iajs-3505 59 14 𝑀0 𝑀0 PROPN iajs-3505 59 15 . . PUNCT iajs-3505 60 1 𝛼1 𝛼1 NOUN iajs-3505 60 2 is be AUX iajs-3505 60 3 the the DET iajs-3505 60 4 phytoplankton phytoplankton NOUN iajs-3505 60 5 ’s ’s PART iajs-3505 60 6 capture capture NOUN iajs-3505 60 7 rate rate NOUN iajs-3505 60 8 by by ADP iajs-3505 60 9 zooplankton zooplankton NOUN iajs-3505 60 10 . . PUNCT iajs-3505 61 1 π‘š π‘š X iajs-3505 61 2 ∈ ∈ PROPN iajs-3505 61 3 ( ( PUNCT iajs-3505 61 4 0,1 0,1 NUM iajs-3505 61 5 ) ) PUNCT iajs-3505 61 6 is be AUX iajs-3505 61 7 the the DET iajs-3505 61 8 proportion proportion NOUN iajs-3505 61 9 of of ADP iajs-3505 61 10 protected protect VERB iajs-3505 61 11 phytoplankton phytoplankton NOUN iajs-3505 61 12 . . PUNCT iajs-3505 62 1 ( ( PUNCT iajs-3505 62 2 1 1 NUM iajs-3505 62 3 βˆ’ βˆ’ NOUN iajs-3505 62 4 π‘š π‘š NOUN iajs-3505 62 5 ) ) PUNCT iajs-3505 62 6 is be AUX iajs-3505 62 7 the the DET iajs-3505 62 8 ratio ratio NOUN iajs-3505 62 9 of of ADP iajs-3505 62 10 unprotected unprotected ADJ iajs-3505 62 11 phytoplankton phytoplankton NOUN iajs-3505 62 12 devoured devour VERB iajs-3505 62 13 by by ADP iajs-3505 62 14 different different ADJ iajs-3505 62 15 zooplankton zooplankton NOUN iajs-3505 62 16 groups group NOUN iajs-3505 62 17 . . PUNCT iajs-3505 63 1 𝛼2 𝛼2 PROPN iajs-3505 63 2 is be AUX iajs-3505 63 3 the the DET iajs-3505 63 4 conversion conversion NOUN iajs-3505 63 5 rate rate NOUN iajs-3505 63 6 from from ADP iajs-3505 63 7 phytoplankton phytoplankton NOUN iajs-3505 63 8 to to ADP iajs-3505 63 9 zooplankton zooplankton NOUN iajs-3505 63 10 . . PUNCT iajs-3505 64 1 𝛿1 𝛿1 NOUN iajs-3505 64 2 and and CCONJ iajs-3505 64 3 𝛿2 𝛿2 NOUN iajs-3505 64 4 are be AUX iajs-3505 64 5 the the DET iajs-3505 64 6 phytoplankton phytoplankton NOUN iajs-3505 64 7 and and CCONJ iajs-3505 64 8 zooplankton zooplankton NOUN iajs-3505 64 9 ’s ’s PART iajs-3505 64 10 natural natural ADJ iajs-3505 64 11 death death NOUN iajs-3505 64 12 rates rate NOUN iajs-3505 64 13 . . PUNCT iajs-3505 65 1 π‘Ž1 π‘Ž1 PROPN iajs-3505 65 2 is be AUX iajs-3505 65 3 the the DET iajs-3505 65 4 phytoplankton phytoplankton NOUN iajs-3505 65 5 saturation saturation NOUN iajs-3505 65 6 constant constant ADJ iajs-3505 65 7 . . PUNCT iajs-3505 66 1 π‘Ž2 π‘Ž2 NOUN iajs-3505 66 2 is be AUX iajs-3505 66 3 the the DET iajs-3505 66 4 zooplankton zooplankton NOUN iajs-3505 66 5 saturation saturation NOUN iajs-3505 66 6 constant constant ADJ iajs-3505 66 7 . . PUNCT iajs-3505 67 1 𝑀0 𝑀0 NOUN iajs-3505 67 2 is be AUX iajs-3505 67 3 the the DET iajs-3505 67 4 constant constant ADJ iajs-3505 67 5 concentration concentration NOUN iajs-3505 67 6 of of ADP iajs-3505 67 7 dissolved dissolve VERB iajs-3505 67 8 ihjpas ihjpa NOUN iajs-3505 67 9 . . PUNCT iajs-3505 68 1 2024 2024 NUM iajs-3505 68 2 , , PUNCT iajs-3505 68 3 37(4 37(4 NOUN iajs-3505 68 4 ) ) PUNCT iajs-3505 68 5 372 372 NUM iajs-3505 68 6 oxygen oxygen NOUN iajs-3505 68 7 that that PRON iajs-3505 68 8 comes come VERB iajs-3505 68 9 from from ADP iajs-3505 68 10 external external ADJ iajs-3505 68 11 sources source NOUN iajs-3505 68 12 . . PUNCT iajs-3505 69 1 𝑠 𝑠 PROPN iajs-3505 69 2 is be AUX iajs-3505 69 3 the the DET iajs-3505 69 4 replenishment replenishment NOUN iajs-3505 69 5 rate rate NOUN iajs-3505 69 6 of of ADP iajs-3505 69 7 oxygen oxygen NOUN iajs-3505 69 8 in in ADP iajs-3505 69 9 marine marine PROPN iajs-3505 69 10 . . PUNCT iajs-3505 70 1 𝑑 𝑑 PROPN iajs-3505 70 2 is be AUX iajs-3505 70 3 the the DET iajs-3505 70 4 amount amount NOUN iajs-3505 70 5 of of ADP iajs-3505 70 6 oxygen oxygen NOUN iajs-3505 70 7 produced produce VERB iajs-3505 70 8 as as ADP iajs-3505 70 9 a a DET iajs-3505 70 10 result result NOUN iajs-3505 70 11 of of ADP iajs-3505 70 12 the the DET iajs-3505 70 13 process process NOUN iajs-3505 70 14 of of ADP iajs-3505 70 15 photosynthesis photosynthesis NOUN iajs-3505 70 16 carried carry VERB iajs-3505 70 17 out out ADP iajs-3505 70 18 by by ADP iajs-3505 70 19 phytoplankton phytoplankton NOUN iajs-3505 70 20 . . PUNCT iajs-3505 71 1 𝛾 𝛾 PROPN iajs-3505 71 2 is be AUX iajs-3505 71 3 the the DET iajs-3505 71 4 natural natural ADJ iajs-3505 71 5 depletion depletion NOUN iajs-3505 71 6 rate rate NOUN iajs-3505 71 7 of of ADP iajs-3505 71 8 oxygen oxygen NOUN iajs-3505 71 9 . . PUNCT iajs-3505 72 1 𝛾1 𝛾1 PROPN iajs-3505 72 2 is be AUX iajs-3505 72 3 the the DET iajs-3505 72 4 consumption consumption NOUN iajs-3505 72 5 of of ADP iajs-3505 72 6 oxygen oxygen NOUN iajs-3505 72 7 by by ADP iajs-3505 72 8 phytoplankton phytoplankton NOUN iajs-3505 72 9 during during ADP iajs-3505 72 10 the the DET iajs-3505 72 11 night night NOUN iajs-3505 72 12 . . PUNCT iajs-3505 73 1 𝛾2 𝛾2 PROPN iajs-3505 73 2 is be AUX iajs-3505 73 3 the the DET iajs-3505 73 4 consumption consumption NOUN iajs-3505 73 5 of of ADP iajs-3505 73 6 oxygen oxygen NOUN iajs-3505 73 7 by by ADP iajs-3505 73 8 zooplankton zooplankton NOUN iajs-3505 73 9 . . PUNCT iajs-3505 74 1 the the DET iajs-3505 74 2 equations equation NOUN iajs-3505 74 3 of of ADP iajs-3505 74 4 the the DET iajs-3505 74 5 system system NOUN iajs-3505 74 6 ( ( PUNCT iajs-3505 74 7 1 1 X iajs-3505 74 8 ) ) PUNCT iajs-3505 74 9 are be AUX iajs-3505 74 10 𝐢1(𝑅+ 𝐢1(𝑅+ PROPN iajs-3505 74 11 3 3 NUM iajs-3505 74 12 ) ) PUNCT iajs-3505 74 13 , , PUNCT iajs-3505 74 14 where where SCONJ iajs-3505 74 15 𝑅+ 𝑅+ PROPN iajs-3505 74 16 3 3 NUM iajs-3505 74 17 = = SYM iajs-3505 74 18 { { PUNCT iajs-3505 74 19 ( ( PUNCT iajs-3505 74 20 𝑒 𝑒 X iajs-3505 74 21 , , PUNCT iajs-3505 74 22 𝑣 𝑣 NOUN iajs-3505 74 23 , , PUNCT iajs-3505 74 24 𝑀 𝑀 X iajs-3505 74 25 ) ) PUNCT iajs-3505 74 26 , , PUNCT iajs-3505 74 27 𝑒 𝑒 X iajs-3505 74 28 β‰₯ β‰₯ NOUN iajs-3505 74 29 0 0 NUM iajs-3505 74 30 , , PUNCT iajs-3505 74 31 𝑣 𝑣 DET iajs-3505 74 32 β‰₯ β‰₯ NOUN iajs-3505 74 33 0 0 NUM iajs-3505 74 34 , , PUNCT iajs-3505 74 35 𝑀 𝑀 ADP iajs-3505 74 36 β‰₯ β‰₯ NOUN iajs-3505 74 37 0 0 NUM iajs-3505 74 38 } } PUNCT iajs-3505 74 39 . . PUNCT iajs-3505 75 1 consequently consequently ADV iajs-3505 75 2 , , PUNCT iajs-3505 75 3 they they PRON iajs-3505 75 4 are be AUX iajs-3505 75 5 lipschitz lipschitz VERB iajs-3505 75 6 ian ian PROPN iajs-3505 75 7 [ [ X iajs-3505 75 8 17 17 NUM iajs-3505 75 9 ] ] PUNCT iajs-3505 75 10 . . PUNCT iajs-3505 76 1 therefore therefore ADV iajs-3505 76 2 , , PUNCT iajs-3505 76 3 the the DET iajs-3505 76 4 system system NOUN iajs-3505 76 5 ’s ’s PART iajs-3505 76 6 ( ( PUNCT iajs-3505 76 7 1 1 NUM iajs-3505 76 8 ) ) PUNCT iajs-3505 76 9 solution solution NOUN iajs-3505 76 10 exists exist VERB iajs-3505 76 11 and and CCONJ iajs-3505 76 12 is be AUX iajs-3505 76 13 unique unique ADJ iajs-3505 76 14 . . PUNCT iajs-3505 77 1 2 2 X iajs-3505 77 2 . . X iajs-3505 77 3 existence existence NOUN iajs-3505 77 4 of of ADP iajs-3505 77 5 equilibria equilibrium NOUN iajs-3505 77 6 system system NOUN iajs-3505 77 7 ( ( PUNCT iajs-3505 77 8 1 1 X iajs-3505 77 9 ) ) PUNCT iajs-3505 77 10 has have VERB iajs-3505 77 11 three three NUM iajs-3505 77 12 non non ADJ iajs-3505 77 13 - - ADJ iajs-3505 77 14 negative negative ADJ iajs-3505 77 15 steady steady ADJ iajs-3505 77 16 states state NOUN iajs-3505 77 17 , , PUNCT iajs-3505 77 18 namely namely ADV iajs-3505 77 19 1 1 NUM iajs-3505 77 20 . . PUNCT iajs-3505 78 1 𝑧1 𝑧1 NOUN iajs-3505 78 2 = = SYM iajs-3505 78 3 ( ( PUNCT iajs-3505 78 4 0,0 0,0 NOUN iajs-3505 78 5 , , PUNCT iajs-3505 78 6 οΏ½ οΏ½ NOUN iajs-3505 78 7 Μ‚ Μ‚ NOUN iajs-3505 78 8 οΏ½ οΏ½ NOUN iajs-3505 78 9 ) ) PUNCT iajs-3505 78 10 , , PUNCT iajs-3505 78 11 where where SCONJ iajs-3505 78 12 οΏ½ οΏ½ PROPN iajs-3505 78 13 Μ‚ Μ‚ VERB iajs-3505 78 14 οΏ½ οΏ½ NOUN iajs-3505 78 15 = = SYM iajs-3505 78 16 𝑠𝑀0 𝑠𝑀0 NOUN iajs-3505 78 17 𝑠+𝛾 𝑠+𝛾 NUM iajs-3505 78 18 . . PUNCT iajs-3505 79 1 2 2 X iajs-3505 79 2 . . X iajs-3505 79 3 𝑧2 𝑧2 NOUN iajs-3505 79 4 = = SYM iajs-3505 79 5 ( ( PUNCT iajs-3505 79 6 οΏ½ οΏ½ NOUN iajs-3505 79 7 Μ… Μ… NOUN iajs-3505 79 8 οΏ½ οΏ½ PROPN iajs-3505 79 9 , , PUNCT iajs-3505 79 10 0 0 NUM iajs-3505 79 11 , , PUNCT iajs-3505 79 12 οΏ½ οΏ½ NOUN iajs-3505 79 13 Μ… Μ… NOUN iajs-3505 79 14 οΏ½ οΏ½ NOUN iajs-3505 79 15 ) ) PUNCT iajs-3505 79 16 , , PUNCT iajs-3505 79 17 where where SCONJ iajs-3505 79 18 οΏ½ οΏ½ NOUN iajs-3505 79 19 Μ… Μ… NOUN iajs-3505 79 20 οΏ½ οΏ½ NOUN iajs-3505 79 21 = = SYM iajs-3505 79 22 ( ( PUNCT iajs-3505 79 23 𝑠+𝛾)(π‘Ž1𝛿1βˆ’π‘Ÿ 𝑠+𝛾)(π‘Ž1𝛿1βˆ’π‘Ÿ NOUN iajs-3505 79 24 ) ) PUNCT iajs-3505 79 25 𝑑𝛿1βˆ’π›Ύ1(𝛿1(π‘Ž1+𝑀0)βˆ’π‘Ÿ 𝑑𝛿1βˆ’π›Ύ1(𝛿1(π‘Ž1+𝑀0)βˆ’π‘Ÿ PROPN iajs-3505 79 26 ) ) PUNCT iajs-3505 79 27 and and CCONJ iajs-3505 79 28 οΏ½ οΏ½ PROPN iajs-3505 79 29 Μ… Μ… NOUN iajs-3505 79 30 οΏ½ οΏ½ NOUN iajs-3505 79 31 = = SYM iajs-3505 79 32 π‘Ž1 π‘Ž1 PROPN iajs-3505 79 33 + + CCONJ iajs-3505 79 34 𝑀0 𝑀0 NOUN iajs-3505 79 35 βˆ’ βˆ’ NOUN iajs-3505 79 36 π‘Ÿ π‘Ÿ X iajs-3505 79 37 𝛿1 𝛿1 NOUN iajs-3505 79 38 , , PUNCT iajs-3505 79 39 and and CCONJ iajs-3505 79 40 . . PUNCT iajs-3505 79 41 clearly clearly ADV iajs-3505 79 42 , , PUNCT iajs-3505 79 43 οΏ½ οΏ½ PROPN iajs-3505 79 44 Μ… Μ… NOUN iajs-3505 79 45 οΏ½ οΏ½ PROPN iajs-3505 79 46 and and CCONJ iajs-3505 79 47 οΏ½ οΏ½ PROPN iajs-3505 79 48 Μ… Μ… NOUN iajs-3505 79 49 οΏ½ οΏ½ NOUN iajs-3505 79 50 are be AUX iajs-3505 79 51 positive positive ADJ iajs-3505 79 52 if if SCONJ iajs-3505 79 53 the the DET iajs-3505 79 54 following follow VERB iajs-3505 79 55 condition condition NOUN iajs-3505 79 56 is be AUX iajs-3505 79 57 satisfied satisfied ADJ iajs-3505 79 58 : : PUNCT iajs-3505 79 59 π‘Ÿ π‘Ÿ X iajs-3505 79 60 < < X iajs-3505 79 61 π‘šπ‘–π‘›. π‘šπ‘–π‘›. X iajs-3505 79 62 { { PUNCT iajs-3505 79 63 π‘Ž1𝛿1 π‘Ž1𝛿1 PROPN iajs-3505 79 64 , , PUNCT iajs-3505 79 65 𝛿1(𝑑 𝛿1(𝑑 PROPN iajs-3505 79 66 βˆ’ βˆ’ PROPN iajs-3505 79 67 𝛾1(π‘Ž1 𝛾1(π‘Ž1 PROPN iajs-3505 79 68 + + CCONJ iajs-3505 79 69 𝑀0 𝑀0 NOUN iajs-3505 79 70 ) ) PUNCT iajs-3505 79 71 ) ) PUNCT iajs-3505 79 72 } } PUNCT iajs-3505 79 73 . . PUNCT iajs-3505 80 1 ( ( PUNCT iajs-3505 80 2 2 2 NUM iajs-3505 80 3 ) ) PUNCT iajs-3505 80 4 3 3 NUM iajs-3505 80 5 . . PUNCT iajs-3505 81 1 𝑧3 𝑧3 PROPN iajs-3505 81 2 = = PUNCT iajs-3505 81 3 ( ( PUNCT iajs-3505 81 4 𝑒 𝑒 NOUN iajs-3505 81 5 βˆ— βˆ— NOUN iajs-3505 81 6 , , PUNCT iajs-3505 81 7 π‘£βˆ— π‘£βˆ— ADJ iajs-3505 81 8 , , PUNCT iajs-3505 81 9 π‘€βˆ— π‘€βˆ— PROPN iajs-3505 81 10 ) ) PUNCT iajs-3505 81 11 , , PUNCT iajs-3505 81 12 where where SCONJ iajs-3505 81 13 𝑒 𝑒 X iajs-3505 81 14 = = SYM iajs-3505 81 15 𝛿2(π‘Ž2+𝑀0βˆ’π‘€ 𝛿2(π‘Ž2+𝑀0βˆ’π‘€ X iajs-3505 81 16 ) ) PUNCT iajs-3505 81 17 ( ( PUNCT iajs-3505 81 18 1βˆ’π‘š)[𝛼2βˆ’π‘Ž(π‘Ž2+𝑀0βˆ’π‘€ 1βˆ’π‘š)[𝛼2βˆ’π‘Ž(π‘Ž2+𝑀0βˆ’π‘€ NOUN iajs-3505 81 19 ) ) PUNCT iajs-3505 81 20 ] ] PUNCT iajs-3505 81 21 , , PUNCT iajs-3505 81 22 𝑣 𝑣 X iajs-3505 81 23 = = PUNCT iajs-3505 81 24 π‘Ÿ π‘Ÿ DET iajs-3505 81 25 𝛼1(1βˆ’π‘š)(π‘Ž1+𝑀0βˆ’π‘€ 𝛼1(1βˆ’π‘š)(π‘Ž1+𝑀0βˆ’π‘€ PROPN iajs-3505 81 26 ) ) PUNCT iajs-3505 81 27 βˆ’ βˆ’ PROPN iajs-3505 81 28 𝛿1 𝛿1 PROPN iajs-3505 81 29 𝛼1(1βˆ’π‘š 𝛼1(1βˆ’π‘š PROPN iajs-3505 81 30 ) ) PUNCT iajs-3505 81 31 , , PUNCT iajs-3505 81 32 and and CCONJ iajs-3505 81 33 𝑀 𝑀 X iajs-3505 81 34 is be AUX iajs-3505 81 35 the the DET iajs-3505 81 36 root root NOUN iajs-3505 81 37 of of ADP iajs-3505 81 38 the the DET iajs-3505 81 39 following follow VERB iajs-3505 81 40 equation equation NOUN iajs-3505 81 41 : : PUNCT iajs-3505 81 42 𝐡0𝑀 𝐡0𝑀 PROPN iajs-3505 81 43 3 3 NUM iajs-3505 81 44 + + CCONJ iajs-3505 81 45 𝐡1𝑀 𝐡1𝑀 PROPN iajs-3505 81 46 2 2 NUM iajs-3505 81 47 + + CCONJ iajs-3505 81 48 𝐡2𝑀 𝐡2𝑀 PROPN iajs-3505 81 49 + + CCONJ iajs-3505 81 50 𝐡3 𝐡3 NOUN iajs-3505 81 51 = = SYM iajs-3505 81 52 0 0 NUM iajs-3505 81 53 , , PUNCT iajs-3505 81 54 ( ( PUNCT iajs-3505 81 55 3 3 X iajs-3505 81 56 ) ) PUNCT iajs-3505 81 57 where where SCONJ iajs-3505 81 58 , , PUNCT iajs-3505 81 59 𝐡0 𝐡0 NOUN iajs-3505 81 60 = = PUNCT iajs-3505 81 61 π‘Žπ›Ό1(1 π‘Žπ›Ό1(1 ADJ iajs-3505 81 62 βˆ’ βˆ’ NUM iajs-3505 81 63 π‘š)(𝑠 π‘š)(𝑠 PROPN iajs-3505 81 64 + + CCONJ iajs-3505 81 65 𝛾 𝛾 X iajs-3505 81 66 ) ) PUNCT iajs-3505 81 67 > > X iajs-3505 81 68 0 0 NUM iajs-3505 81 69 , , PUNCT iajs-3505 81 70 𝐡1 𝐡1 NOUN iajs-3505 81 71 = = PUNCT iajs-3505 81 72 𝛼1π‘Žπ‘ π‘€0[2 𝛼1π‘Žπ‘ π‘€0[2 NUM iajs-3505 81 73 βˆ’ βˆ’ NOUN iajs-3505 81 74 ( ( PUNCT iajs-3505 81 75 1 1 NUM iajs-3505 81 76 βˆ’ βˆ’ PROPN iajs-3505 81 77 π‘š π‘š NOUN iajs-3505 81 78 ) ) PUNCT iajs-3505 81 79 ] ] PUNCT iajs-3505 82 1 + + PUNCT iajs-3505 82 2 𝑑𝛿2𝛼1 𝑑𝛿2𝛼1 NOUN iajs-3505 82 3 βˆ’ βˆ’ PROPN iajs-3505 82 4 𝛼1(𝑠 𝛼1(𝑠 PROPN iajs-3505 82 5 + + NOUN iajs-3505 82 6 𝛾)(𝛼2 𝛾)(𝛼2 PROPN iajs-3505 82 7 βˆ’ βˆ’ PROPN iajs-3505 82 8 π‘Žπ‘Ž2 π‘Žπ‘Ž2 NOUN iajs-3505 82 9 ) ) PUNCT iajs-3505 82 10 βˆ’ βˆ’ NOUN iajs-3505 82 11 𝛼1π‘Žπ›Ύ[π‘Ž1(1 𝛼1π‘Žπ›Ύ[π‘Ž1(1 NUM iajs-3505 82 12 βˆ’ βˆ’ PROPN iajs-3505 82 13 π‘š π‘š NOUN iajs-3505 82 14 ) ) PUNCT iajs-3505 82 15 βˆ’ βˆ’ PROPN iajs-3505 83 1 2𝑀0 2𝑀0 NUM iajs-3505 83 2 ] ] X iajs-3505 83 3 𝐡2 𝐡2 NOUN iajs-3505 83 4 = = SYM iajs-3505 84 1 𝛼1π‘Žπ‘ π‘€0(1 𝛼1π‘Žπ‘ π‘€0(1 ADJ iajs-3505 84 2 βˆ’ βˆ’ PROPN iajs-3505 84 3 π‘š)(2π‘Ž1 π‘š)(2π‘Ž1 PROPN iajs-3505 84 4 + + CCONJ iajs-3505 84 5 3𝑀0 3𝑀0 NUM iajs-3505 84 6 ) ) PUNCT iajs-3505 84 7 βˆ’ βˆ’ PROPN iajs-3505 84 8 𝛼1𝑀0(1 𝛼1𝑀0(1 NUM iajs-3505 84 9 βˆ’ βˆ’ NOUN iajs-3505 84 10 π‘š)(𝑠 π‘š)(𝑠 PROPN iajs-3505 84 11 + + CCONJ iajs-3505 84 12 𝛾)[𝛼2 𝛾)[𝛼2 PUNCT iajs-3505 84 13 βˆ’ βˆ’ PROPN iajs-3505 85 1 π‘Žπ‘Ž2 π‘Žπ‘Ž2 X iajs-3505 85 2 ] ] X iajs-3505 85 3 βˆ’ βˆ’ PROPN iajs-3505 85 4 𝑑𝛿2𝛼1π‘Ž1 𝑑𝛿2𝛼1π‘Ž1 PROPN iajs-3505 86 1 + + CCONJ iajs-3505 86 2 𝛼1(1 𝛼1(1 ADJ iajs-3505 86 3 βˆ’ βˆ’ NUM iajs-3505 86 4 π‘š)(𝑠 π‘š)(𝑠 NOUN iajs-3505 86 5 + + PUNCT iajs-3505 86 6 𝛾)[π‘Žπ‘Ž1π‘Ž2 𝛾)[π‘Žπ‘Ž1π‘Ž2 ADJ iajs-3505 86 7 βˆ’ βˆ’ PROPN iajs-3505 86 8 𝛼2π‘Ž1 𝛼2π‘Ž1 X iajs-3505 86 9 + + CCONJ iajs-3505 86 10 π‘Žπ‘€0 π‘Žπ‘€0 X iajs-3505 86 11 2 2 NUM iajs-3505 86 12 ] ] PUNCT iajs-3505 86 13 + + NUM iajs-3505 86 14 𝛼1π‘Žπ›Ύπ‘€0(1 𝛼1π‘Žπ›Ύπ‘€0(1 PROPN iajs-3505 86 15 βˆ’ βˆ’ NUM iajs-3505 86 16 π‘š)(π‘Ž1 π‘š)(π‘Ž1 PUNCT iajs-3505 86 17 + + NUM iajs-3505 86 18 𝑀0 𝑀0 NOUN iajs-3505 86 19 ) ) PUNCT iajs-3505 86 20 𝐡3 𝐡3 NOUN iajs-3505 87 1 = = PUNCT iajs-3505 87 2 𝛼1𝑠𝑀0(1 𝛼1𝑠𝑀0(1 ADJ iajs-3505 87 3 βˆ’ βˆ’ PROPN iajs-3505 87 4 π‘š)[𝛼2π‘Ž1 π‘š)[𝛼2π‘Ž1 NOUN iajs-3505 87 5 βˆ’ βˆ’ PROPN iajs-3505 88 1 π‘Žπ‘Ž1π‘Ž2 π‘Žπ‘Ž1π‘Ž2 PUNCT iajs-3505 88 2 βˆ’ βˆ’ PROPN iajs-3505 88 3 π‘Žπ‘Ž1𝑀0 π‘Žπ‘Ž1𝑀0 PROPN iajs-3505 88 4 + + NUM iajs-3505 88 5 𝛼2𝑀0 𝛼2𝑀0 PROPN iajs-3505 88 6 βˆ’ βˆ’ PROPN iajs-3505 88 7 π‘Žπ›Ό2𝑀0 π‘Žπ›Ό2𝑀0 PROPN iajs-3505 88 8 βˆ’ βˆ’ PROPN iajs-3505 88 9 π‘Žπ‘€0 π‘Žπ‘€0 NOUN iajs-3505 88 10 2 2 NUM iajs-3505 88 11 ] ] X iajs-3505 88 12 + + CCONJ iajs-3505 88 13 𝑑𝛿2𝛼1[π‘Ž1π‘Ž2 𝑑𝛿2𝛼1[π‘Ž1π‘Ž2 ADJ iajs-3505 88 14 + + CCONJ iajs-3505 88 15 π‘Ž1𝑀0 π‘Ž1𝑀0 PROPN iajs-3505 88 16 + + ADJ iajs-3505 88 17 π‘Ž2𝑀0 π‘Ž2𝑀0 PROPN iajs-3505 88 18 + + NUM iajs-3505 88 19 𝑀0 𝑀0 NOUN iajs-3505 88 20 2 2 NUM iajs-3505 88 21 ] ] PUNCT iajs-3505 88 22 . . PUNCT iajs-3505 89 1 using use VERB iajs-3505 89 2 descartes descartes PROPN iajs-3505 89 3 ’s ’s PART iajs-3505 89 4 rule rule NOUN iajs-3505 89 5 of of ADP iajs-3505 89 6 sign sign NOUN iajs-3505 89 7 [ [ X iajs-3505 89 8 15 15 NUM iajs-3505 89 9 ] ] PUNCT iajs-3505 89 10 , , PUNCT iajs-3505 89 11 equation equation NOUN iajs-3505 89 12 ( ( PUNCT iajs-3505 89 13 3 3 X iajs-3505 89 14 ) ) PUNCT iajs-3505 89 15 has have VERB iajs-3505 89 16 a a DET iajs-3505 89 17 unique unique ADJ iajs-3505 89 18 positive positive ADJ iajs-3505 89 19 root root NOUN iajs-3505 89 20 , , PUNCT iajs-3505 89 21 say say VERB iajs-3505 89 22 𝑀 𝑀 ADP iajs-3505 89 23 = = PUNCT iajs-3505 89 24 π‘€βˆ— π‘€βˆ— PROPN iajs-3505 89 25 , , PUNCT iajs-3505 89 26 if if SCONJ iajs-3505 89 27 one one NUM iajs-3505 89 28 of of ADP iajs-3505 89 29 the the DET iajs-3505 89 30 following follow VERB iajs-3505 89 31 sets set VERB iajs-3505 89 32 conditions condition NOUN iajs-3505 89 33 hold hold VERB iajs-3505 89 34 : : PUNCT iajs-3505 89 35 𝐡1 𝐡1 NOUN iajs-3505 89 36 > > X iajs-3505 89 37 0 0 PUNCT iajs-3505 89 38 and and CCONJ iajs-3505 89 39 𝐡3 𝐡3 NOUN iajs-3505 89 40 < < X iajs-3505 89 41 0 0 PROPN iajs-3505 89 42 , , PUNCT iajs-3505 89 43 𝐡2 𝐡2 NOUN iajs-3505 89 44 < < X iajs-3505 89 45 0 0 PROPN iajs-3505 89 46 and and CCONJ iajs-3505 89 47 𝐡3 𝐡3 NOUN iajs-3505 89 48 < < X iajs-3505 89 49 0 0 X iajs-3505 89 50 . . PUNCT iajs-3505 90 1 ( ( PUNCT iajs-3505 90 2 4 4 NUM iajs-3505 90 3 ) ) PUNCT iajs-3505 90 4 for for ADP iajs-3505 90 5 π‘’βˆ—and π‘’βˆ—and PROPN iajs-3505 90 6 π‘£βˆ—to π‘£βˆ—to NOUN iajs-3505 90 7 be be AUX iajs-3505 90 8 positive positive ADJ iajs-3505 90 9 , , PUNCT iajs-3505 90 10 the the DET iajs-3505 90 11 following follow VERB iajs-3505 90 12 two two NUM iajs-3505 90 13 conditions condition NOUN iajs-3505 90 14 must must AUX iajs-3505 90 15 be be AUX iajs-3505 90 16 satisfied satisfied ADJ iajs-3505 90 17 : : PUNCT iajs-3505 90 18 𝛼2 𝛼2 VERB iajs-3505 90 19 > > X iajs-3505 90 20 π‘Ž(π‘Ž2 π‘Ž(π‘Ž2 NOUN iajs-3505 90 21 + + PROPN iajs-3505 90 22 𝑀0 𝑀0 NOUN iajs-3505 90 23 βˆ’ βˆ’ NOUN iajs-3505 90 24 𝑀 𝑀 PROPN iajs-3505 90 25 ) ) PUNCT iajs-3505 90 26 , , PUNCT iajs-3505 90 27 π‘Ÿ π‘Ÿ X iajs-3505 90 28 > > X iajs-3505 90 29 𝛿1(π‘Ž1 𝛿1(π‘Ž1 PROPN iajs-3505 90 30 + + PROPN iajs-3505 90 31 𝑀0 𝑀0 NOUN iajs-3505 90 32 βˆ’ βˆ’ NOUN iajs-3505 90 33 𝑀 𝑀 ADP iajs-3505 90 34 ) ) PUNCT iajs-3505 90 35 . . PUNCT iajs-3505 91 1 ( ( PUNCT iajs-3505 91 2 5 5 NUM iajs-3505 91 3 ) ) PUNCT iajs-3505 91 4 3 3 NUM iajs-3505 91 5 . . X iajs-3505 91 6 stability stability NOUN iajs-3505 91 7 analysis analysis NOUN iajs-3505 91 8 the the DET iajs-3505 91 9 feature feature NOUN iajs-3505 91 10 of of ADP iajs-3505 91 11 the the DET iajs-3505 91 12 eigenvalues eigenvalue NOUN iajs-3505 91 13 of of ADP iajs-3505 91 14 the the DET iajs-3505 91 15 jacobian jacobian ADJ iajs-3505 91 16 matrix matrix NOUN iajs-3505 91 17 𝐽(𝑒 𝐽(𝑒 PRON iajs-3505 91 18 , , PUNCT iajs-3505 91 19 𝑣 𝑣 NOUN iajs-3505 91 20 , , PUNCT iajs-3505 91 21 𝑀 𝑀 ADP iajs-3505 91 22 ) ) PUNCT iajs-3505 91 23 at at ADP iajs-3505 91 24 an an DET iajs-3505 91 25 equilibrium equilibrium NOUN iajs-3505 91 26 point point NOUN iajs-3505 91 27 is be AUX iajs-3505 91 28 directly directly ADV iajs-3505 91 29 related relate VERB iajs-3505 91 30 to to ADP iajs-3505 91 31 the the DET iajs-3505 91 32 behaviour behaviour NOUN iajs-3505 91 33 of of ADP iajs-3505 91 34 the the DET iajs-3505 91 35 system system NOUN iajs-3505 91 36 ( ( PUNCT iajs-3505 91 37 1 1 NUM iajs-3505 91 38 ) ) PUNCT iajs-3505 91 39 near near ADP iajs-3505 91 40 an an DET iajs-3505 91 41 equilibrium equilibrium NOUN iajs-3505 91 42 . . PUNCT iajs-3505 92 1 the the DET iajs-3505 92 2 𝐽(𝑒 𝐽(𝑒 PROPN iajs-3505 92 3 , , PUNCT iajs-3505 92 4 𝑣 𝑣 NOUN iajs-3505 92 5 , , PUNCT iajs-3505 92 6 𝑀 𝑀 ADP iajs-3505 92 7 ) ) PUNCT iajs-3505 92 8 at at ADP iajs-3505 92 9 any any DET iajs-3505 92 10 point point NOUN iajs-3505 92 11 , , PUNCT iajs-3505 92 12 say say VERB iajs-3505 92 13 ( ( PUNCT iajs-3505 92 14 𝑒 𝑒 X iajs-3505 92 15 , , PUNCT iajs-3505 92 16 𝑣 𝑣 NOUN iajs-3505 92 17 , , PUNCT iajs-3505 92 18 𝑀 𝑀 ADP iajs-3505 92 19 ) ) PUNCT iajs-3505 92 20 , , PUNCT iajs-3505 92 21 can can AUX iajs-3505 92 22 be be AUX iajs-3505 92 23 written write VERB iajs-3505 92 24 as as ADP iajs-3505 92 25 : : PUNCT iajs-3505 92 26 𝐽 𝐽 PROPN iajs-3505 92 27 = = PUNCT iajs-3505 93 1 [ [ PUNCT iajs-3505 93 2 π‘Ÿ π‘Ÿ NOUN iajs-3505 93 3 ( ( PUNCT iajs-3505 93 4 π‘Ž1 π‘Ž1 PROPN iajs-3505 93 5 + + PROPN iajs-3505 93 6 𝑀0 𝑀0 NOUN iajs-3505 93 7 βˆ’ βˆ’ NOUN iajs-3505 93 8 𝑀 𝑀 SYM iajs-3505 93 9 ) ) PUNCT iajs-3505 93 10 βˆ’ βˆ’ PROPN iajs-3505 94 1 𝛼1𝑣(1 𝛼1𝑣(1 PROPN iajs-3505 94 2 βˆ’π‘š βˆ’π‘š PROPN iajs-3505 94 3 ) ) PUNCT iajs-3505 95 1 βˆ’ βˆ’ PROPN iajs-3505 95 2 𝛿1 𝛿1 NOUN iajs-3505 95 3 βˆ’π›Ό1𝑒(1 βˆ’π›Ό1𝑒(1 ADP iajs-3505 95 4 βˆ’ βˆ’ NOUN iajs-3505 95 5 π‘š π‘š NOUN iajs-3505 95 6 ) ) PUNCT iajs-3505 95 7 π‘Ÿπ‘’ π‘Ÿπ‘’ NOUN iajs-3505 95 8 ( ( PUNCT iajs-3505 95 9 π‘Ž1 π‘Ž1 NOUN iajs-3505 95 10 + + NUM iajs-3505 95 11 𝑀0 𝑀0 NOUN iajs-3505 95 12 βˆ’ βˆ’ NOUN iajs-3505 95 13 𝑀 𝑀 SYM iajs-3505 95 14 ) ) PUNCT iajs-3505 95 15 2 2 NUM iajs-3505 95 16 𝛼2𝑣(1 𝛼2𝑣(1 PROPN iajs-3505 95 17 βˆ’ βˆ’ ADP iajs-3505 95 18 π‘š π‘š NOUN iajs-3505 95 19 ) ) PUNCT iajs-3505 95 20 ( ( PUNCT iajs-3505 95 21 π‘Ž2 π‘Ž2 NOUN iajs-3505 95 22 + + NUM iajs-3505 95 23 𝑀0 𝑀0 PROPN iajs-3505 95 24 βˆ’ βˆ’ NOUN iajs-3505 95 25 𝑀 𝑀 NOUN iajs-3505 95 26 ) ) PUNCT iajs-3505 95 27 𝛼2𝑒(1 𝛼2𝑒(1 PROPN iajs-3505 95 28 βˆ’ βˆ’ ADP iajs-3505 95 29 π‘š π‘š NOUN iajs-3505 95 30 ) ) PUNCT iajs-3505 95 31 ( ( PUNCT iajs-3505 95 32 π‘Ž2 π‘Ž2 NOUN iajs-3505 95 33 + + NUM iajs-3505 95 34 𝑀0 𝑀0 PROPN iajs-3505 95 35 βˆ’ βˆ’ PROPN iajs-3505 95 36 𝑀 𝑀 PROPN iajs-3505 95 37 ) ) PUNCT iajs-3505 95 38 βˆ’ βˆ’ PROPN iajs-3505 96 1 𝛿2 𝛿2 PROPN iajs-3505 96 2 𝛼2𝑒(1 𝛼2𝑒(1 PROPN iajs-3505 96 3 βˆ’ βˆ’ PROPN iajs-3505 96 4 π‘š)𝑣 π‘š)𝑣 PUNCT iajs-3505 97 1 ( ( PUNCT iajs-3505 97 2 π‘Ž2 π‘Ž2 PROPN iajs-3505 97 3 + + PROPN iajs-3505 97 4 𝑀0 𝑀0 PROPN iajs-3505 97 5 βˆ’ βˆ’ NOUN iajs-3505 97 6 𝑀 𝑀 SYM iajs-3505 97 7 ) ) PUNCT iajs-3505 97 8 2 2 NUM iajs-3505 97 9 𝑑 𝑑 PROPN iajs-3505 97 10 βˆ’ βˆ’ PROPN iajs-3505 97 11 𝛾1𝑀 𝛾1𝑀 PROPN iajs-3505 97 12 βˆ’π›Ύ2𝑀 βˆ’π›Ύ2𝑀 NUM iajs-3505 97 13 βˆ’(𝑠 βˆ’(𝑠 NOUN iajs-3505 97 14 + + CCONJ iajs-3505 97 15 𝛾 𝛾 ADP iajs-3505 97 16 + + CCONJ iajs-3505 97 17 𝛾1𝑒 𝛾1𝑒 PROPN iajs-3505 97 18 + + CCONJ iajs-3505 97 19 𝛾2𝑣 𝛾2𝑣 NOUN iajs-3505 97 20 ) ) PUNCT iajs-3505 97 21 ] ] PUNCT iajs-3505 98 1 the the DET iajs-3505 98 2 local local ADJ iajs-3505 98 3 stability stability NOUN iajs-3505 98 4 of of ADP iajs-3505 98 5 system system NOUN iajs-3505 98 6 ( ( PUNCT iajs-3505 98 7 1 1 NUM iajs-3505 98 8 ) ) PUNCT iajs-3505 98 9 around around ADP iajs-3505 98 10 each each DET iajs-3505 98 11 equilibrium equilibrium NOUN iajs-3505 98 12 is be AUX iajs-3505 98 13 : : PUNCT iajs-3505 98 14 ihjpas ihjpas PROPN iajs-3505 98 15 . . PUNCT iajs-3505 99 1 2024 2024 NUM iajs-3505 99 2 , , PUNCT iajs-3505 99 3 37(4 37(4 NOUN iajs-3505 99 4 ) ) PUNCT iajs-3505 99 5 373 373 NUM iajs-3505 99 6 1 1 NUM iajs-3505 99 7 . . PUNCT iajs-3505 100 1 the the DET iajs-3505 100 2 jacobian jacobian ADJ iajs-3505 100 3 matrix matrix NOUN iajs-3505 100 4 at at ADP iajs-3505 100 5 𝑧1 𝑧1 NOUN iajs-3505 100 6 = = SYM iajs-3505 100 7 ( ( PUNCT iajs-3505 100 8 0,0 0,0 NOUN iajs-3505 100 9 , , PUNCT iajs-3505 100 10 οΏ½ οΏ½ NOUN iajs-3505 100 11 Μ‚ Μ‚ NOUN iajs-3505 100 12 οΏ½ οΏ½ NOUN iajs-3505 100 13 ) ) PUNCT iajs-3505 100 14 is be AUX iajs-3505 100 15 given give VERB iajs-3505 100 16 as as ADP iajs-3505 100 17 : : PUNCT iajs-3505 100 18 𝐽(𝑧1 𝐽(𝑧1 NOUN iajs-3505 100 19 ) ) PUNCT iajs-3505 100 20 = = NOUN iajs-3505 101 1 [ [ PUNCT iajs-3505 101 2 π‘Ÿ π‘Ÿ X iajs-3505 101 3 ( ( PUNCT iajs-3505 101 4 π‘Ž1 π‘Ž1 NOUN iajs-3505 101 5 + + NUM iajs-3505 101 6 𝑀0 𝑀0 PROPN iajs-3505 101 7 βˆ’ βˆ’ PROPN iajs-3505 101 8 οΏ½ οΏ½ PROPN iajs-3505 101 9 Μ‚ Μ‚ SYM iajs-3505 101 10 οΏ½ οΏ½ NOUN iajs-3505 101 11 ) ) PUNCT iajs-3505 102 1 βˆ’ βˆ’ PROPN iajs-3505 102 2 𝛿1 𝛿1 NOUN iajs-3505 102 3 0 0 NUM iajs-3505 102 4 0 0 NUM iajs-3505 102 5 0 0 NUM iajs-3505 103 1 βˆ’π›Ώ2 βˆ’π›Ώ2 NOUN iajs-3505 103 2 0 0 PUNCT iajs-3505 104 1 𝑑 𝑑 PROPN iajs-3505 104 2 βˆ’ βˆ’ PROPN iajs-3505 104 3 𝛾1 𝛾1 PROPN iajs-3505 104 4 οΏ½ οΏ½ PROPN iajs-3505 104 5 Μ‚ Μ‚ PROPN iajs-3505 104 6 οΏ½ οΏ½ PROPN iajs-3505 104 7 βˆ’π›Ύ2 βˆ’π›Ύ2 PROPN iajs-3505 104 8 οΏ½ οΏ½ PROPN iajs-3505 104 9 Μ‚ Μ‚ PROPN iajs-3505 104 10 οΏ½ οΏ½ PROPN iajs-3505 104 11 βˆ’π‘  βˆ’π‘  ADV iajs-3505 104 12 βˆ’ βˆ’ PROPN iajs-3505 104 13 𝛾 𝛾 NOUN iajs-3505 104 14 ] ] PUNCT iajs-3505 104 15 then then ADV iajs-3505 104 16 , , PUNCT iajs-3505 104 17 𝐽(𝑧1 𝐽(𝑧1 PROPN iajs-3505 104 18 ) ) PUNCT iajs-3505 104 19 has have VERB iajs-3505 104 20 the the DET iajs-3505 104 21 eigenvalues eigenvalue NOUN iajs-3505 104 22 πœ†11 πœ†11 NOUN iajs-3505 104 23 = = SYM iajs-3505 104 24 π‘Ÿ π‘Ÿ NOUN iajs-3505 104 25 ( ( PUNCT iajs-3505 104 26 π‘Ž1+𝑀0βˆ’ π‘Ž1+𝑀0βˆ’ PROPN iajs-3505 104 27 οΏ½ οΏ½ PROPN iajs-3505 104 28 Μ‚ Μ‚ PROPN iajs-3505 104 29 οΏ½ οΏ½ PROPN iajs-3505 104 30 ) ) PUNCT iajs-3505 104 31 βˆ’ βˆ’ PROPN iajs-3505 104 32 𝛿1 𝛿1 NOUN iajs-3505 104 33 , , PUNCT iajs-3505 104 34 πœ†12 πœ†12 NOUN iajs-3505 104 35 = = PUNCT iajs-3505 104 36 βˆ’π›Ώ2 βˆ’π›Ώ2 PROPN iajs-3505 104 37 < < X iajs-3505 104 38 0 0 PROPN iajs-3505 104 39 , , PUNCT iajs-3505 104 40 and and CCONJ iajs-3505 104 41 πœ†13 πœ†13 NOUN iajs-3505 104 42 = = SYM iajs-3505 104 43 βˆ’π‘  βˆ’π‘  ADV iajs-3505 104 44 βˆ’ βˆ’ X iajs-3505 104 45 𝛾. 𝛾. NOUN iajs-3505 104 46 𝑧1is 𝑧1is X iajs-3505 104 47 a a DET iajs-3505 104 48 locally locally ADV iajs-3505 104 49 asymptotically asymptotically ADV iajs-3505 104 50 stable stable ADJ iajs-3505 104 51 point point NOUN iajs-3505 104 52 if if SCONJ iajs-3505 104 53 and and CCONJ iajs-3505 104 54 only only ADV iajs-3505 104 55 if if SCONJ iajs-3505 104 56 π‘Ÿ π‘Ÿ PRON iajs-3505 104 57 < < X iajs-3505 104 58 𝛿1(π‘Ž1 𝛿1(π‘Ž1 PROPN iajs-3505 104 59 + + PROPN iajs-3505 104 60 𝑀0 𝑀0 PROPN iajs-3505 104 61 βˆ’ βˆ’ PROPN iajs-3505 104 62 οΏ½ οΏ½ PROPN iajs-3505 104 63 Μ‚ Μ‚ NOUN iajs-3505 104 64 οΏ½ οΏ½ NOUN iajs-3505 104 65 ) ) PUNCT iajs-3505 104 66 ( ( PUNCT iajs-3505 104 67 6 6 NUM iajs-3505 104 68 ) ) PUNCT iajs-3505 104 69 2 2 NUM iajs-3505 104 70 . . PUNCT iajs-3505 105 1 the the DET iajs-3505 105 2 jacobian jacobian ADJ iajs-3505 105 3 matrix matrix NOUN iajs-3505 105 4 at at ADP iajs-3505 105 5 𝑧2 𝑧2 PROPN iajs-3505 105 6 = = SYM iajs-3505 105 7 ( ( PUNCT iajs-3505 105 8 οΏ½ οΏ½ NOUN iajs-3505 105 9 Μ… Μ… NOUN iajs-3505 105 10 οΏ½ οΏ½ PROPN iajs-3505 105 11 , , PUNCT iajs-3505 105 12 0 0 NUM iajs-3505 105 13 , , PUNCT iajs-3505 105 14 οΏ½ οΏ½ NOUN iajs-3505 105 15 Μ… Μ… NOUN iajs-3505 105 16 οΏ½ οΏ½ NOUN iajs-3505 105 17 ) ) PUNCT iajs-3505 105 18 is be AUX iajs-3505 105 19 given give VERB iajs-3505 105 20 as as ADP iajs-3505 105 21 : : PUNCT iajs-3505 105 22 𝐽(𝑧2 𝐽(𝑧2 ADJ iajs-3505 105 23 ) ) PUNCT iajs-3505 105 24 = = NOUN iajs-3505 106 1 [ [ PUNCT iajs-3505 106 2 π‘Ÿ π‘Ÿ X iajs-3505 106 3 ( ( PUNCT iajs-3505 106 4 π‘Ž1 π‘Ž1 PROPN iajs-3505 106 5 + + PROPN iajs-3505 106 6 𝑀0 𝑀0 PROPN iajs-3505 106 7 βˆ’ βˆ’ PROPN iajs-3505 106 8 οΏ½ οΏ½ PROPN iajs-3505 106 9 Μ… Μ… NOUN iajs-3505 106 10 οΏ½ οΏ½ NOUN iajs-3505 106 11 ) ) PUNCT iajs-3505 106 12 βˆ’ βˆ’ PROPN iajs-3505 106 13 𝛿1 𝛿1 PROPN iajs-3505 106 14 βˆ’π›Ό1 βˆ’π›Ό1 PROPN iajs-3505 106 15 οΏ½ οΏ½ NOUN iajs-3505 106 16 Μ… Μ… NOUN iajs-3505 106 17 οΏ½ οΏ½ NOUN iajs-3505 106 18 (1 (1 PUNCT iajs-3505 106 19 βˆ’ βˆ’ NOUN iajs-3505 106 20 π‘š π‘š X iajs-3505 106 21 ) ) PUNCT iajs-3505 106 22 π‘Ÿ π‘Ÿ NOUN iajs-3505 106 23 οΏ½ οΏ½ NOUN iajs-3505 106 24 Μ… Μ… VERB iajs-3505 106 25 οΏ½ οΏ½ NOUN iajs-3505 106 26 ( ( PUNCT iajs-3505 106 27 π‘Ž1 π‘Ž1 PROPN iajs-3505 106 28 + + NUM iajs-3505 106 29 𝑀0 𝑀0 PROPN iajs-3505 106 30 βˆ’ βˆ’ PROPN iajs-3505 106 31 οΏ½ οΏ½ PROPN iajs-3505 106 32 Μ… Μ… NOUN iajs-3505 106 33 οΏ½ οΏ½ NOUN iajs-3505 106 34 ) ) PUNCT iajs-3505 106 35 2 2 NUM iajs-3505 106 36 0 0 NUM iajs-3505 106 37 𝛼2 𝛼2 PROPN iajs-3505 106 38 οΏ½ οΏ½ PROPN iajs-3505 106 39 Μ… Μ… NOUN iajs-3505 106 40 οΏ½ οΏ½ NOUN iajs-3505 106 41 (1 (1 PUNCT iajs-3505 106 42 βˆ’ βˆ’ NOUN iajs-3505 106 43 π‘š π‘š NOUN iajs-3505 106 44 ) ) PUNCT iajs-3505 106 45 ( ( PUNCT iajs-3505 106 46 π‘Ž2 π‘Ž2 PROPN iajs-3505 106 47 + + PROPN iajs-3505 106 48 𝑀0 𝑀0 PROPN iajs-3505 106 49 βˆ’ βˆ’ PROPN iajs-3505 106 50 οΏ½ οΏ½ PROPN iajs-3505 106 51 Μ… Μ… NOUN iajs-3505 106 52 οΏ½ οΏ½ NOUN iajs-3505 106 53 ) ) PUNCT iajs-3505 106 54 βˆ’ βˆ’ PROPN iajs-3505 107 1 𝛿2 𝛿2 NOUN iajs-3505 107 2 0 0 NUM iajs-3505 107 3 𝑑 𝑑 PROPN iajs-3505 107 4 βˆ’ βˆ’ PROPN iajs-3505 107 5 𝛾1 𝛾1 PROPN iajs-3505 107 6 οΏ½ οΏ½ PROPN iajs-3505 107 7 Μ… Μ… NOUN iajs-3505 107 8 οΏ½ οΏ½ PROPN iajs-3505 107 9 βˆ’π›Ύ2 βˆ’π›Ύ2 PROPN iajs-3505 107 10 οΏ½ οΏ½ PROPN iajs-3505 107 11 Μ… Μ… NOUN iajs-3505 107 12 οΏ½ οΏ½ NOUN iajs-3505 107 13 βˆ’π‘  βˆ’π‘  ADV iajs-3505 107 14 βˆ’ βˆ’ PROPN iajs-3505 107 15 𝛾 𝛾 ADP iajs-3505 107 16 βˆ’ βˆ’ PROPN iajs-3505 107 17 𝛾1 𝛾1 PROPN iajs-3505 107 18 οΏ½ οΏ½ PROPN iajs-3505 107 19 Μ… Μ… NOUN iajs-3505 107 20 οΏ½ οΏ½ PROPN iajs-3505 107 21 ] ] PUNCT iajs-3505 107 22 then then ADV iajs-3505 107 23 , , PUNCT iajs-3505 107 24 |𝐽(𝑧2 |𝐽(𝑧2 PROPN iajs-3505 107 25 ) ) PUNCT iajs-3505 107 26 βˆ’ βˆ’ ADP iajs-3505 107 27 πΌπœ†| πΌπœ†| ADV iajs-3505 107 28 = = SYM iajs-3505 107 29 0 0 NUM iajs-3505 107 30 gives give VERB iajs-3505 107 31 : : PUNCT iajs-3505 107 32 ( ( PUNCT iajs-3505 107 33 𝛼2 𝛼2 PROPN iajs-3505 107 34 οΏ½ οΏ½ PROPN iajs-3505 107 35 Μ… Μ… NOUN iajs-3505 107 36 οΏ½ οΏ½ NOUN iajs-3505 107 37 (1 (1 PUNCT iajs-3505 107 38 βˆ’ βˆ’ NOUN iajs-3505 107 39 π‘š π‘š NOUN iajs-3505 107 40 ) ) PUNCT iajs-3505 107 41 ( ( PUNCT iajs-3505 107 42 π‘Ž2 π‘Ž2 NOUN iajs-3505 107 43 + + NUM iajs-3505 107 44 𝑀0 𝑀0 PROPN iajs-3505 107 45 βˆ’ βˆ’ PROPN iajs-3505 107 46 οΏ½ οΏ½ PROPN iajs-3505 107 47 Μ… Μ… NOUN iajs-3505 107 48 οΏ½ οΏ½ NOUN iajs-3505 107 49 ) ) PUNCT iajs-3505 107 50 βˆ’ βˆ’ PROPN iajs-3505 108 1 𝛿2 𝛿2 NOUN iajs-3505 108 2 βˆ’ βˆ’ NOUN iajs-3505 108 3 πœ† πœ† NOUN iajs-3505 108 4 ) ) PUNCT iajs-3505 109 1 [ [ X iajs-3505 109 2 πœ† πœ† ADP iajs-3505 109 3 2 2 NUM iajs-3505 109 4 βˆ’ βˆ’ NOUN iajs-3505 109 5 π‘‡π‘Ÿ(𝐽(𝑧2))πœ† π‘‡π‘Ÿ(𝐽(𝑧2))πœ† ADJ iajs-3505 109 6 + + CCONJ iajs-3505 109 7 𝐷𝑒𝑑(𝐽(𝑧2 𝐷𝑒𝑑(𝐽(𝑧2 X iajs-3505 109 8 ) ) PUNCT iajs-3505 109 9 ) ) PUNCT iajs-3505 109 10 ] ] PUNCT iajs-3505 110 1 the the DET iajs-3505 110 2 eigenvalues eigenvalue NOUN iajs-3505 110 3 of of ADP iajs-3505 110 4 the the DET iajs-3505 110 5 above above ADJ iajs-3505 110 6 equation equation NOUN iajs-3505 110 7 can can AUX iajs-3505 110 8 be be AUX iajs-3505 110 9 written write VERB iajs-3505 110 10 as as SCONJ iajs-3505 110 11 follows follow VERB iajs-3505 110 12 πœ†21 πœ†21 NOUN iajs-3505 110 13 = = SYM iajs-3505 110 14 𝛼2 𝛼2 PROPN iajs-3505 110 15 οΏ½ οΏ½ PROPN iajs-3505 110 16 Μ… Μ… NOUN iajs-3505 110 17 οΏ½ οΏ½ NOUN iajs-3505 110 18 (1βˆ’π‘š (1βˆ’π‘š NOUN iajs-3505 110 19 ) ) PUNCT iajs-3505 110 20 ( ( PUNCT iajs-3505 110 21 π‘Ž2+𝑀0βˆ’ π‘Ž2+𝑀0βˆ’ PROPN iajs-3505 110 22 οΏ½ οΏ½ PROPN iajs-3505 110 23 Μ… Μ… NOUN iajs-3505 110 24 οΏ½ οΏ½ NOUN iajs-3505 110 25 ) ) PUNCT iajs-3505 110 26 βˆ’ βˆ’ PROPN iajs-3505 110 27 𝛿2 𝛿2 NOUN iajs-3505 110 28 , , PUNCT iajs-3505 110 29 π‘‡π‘Ÿ(𝐽(𝑧2 π‘‡π‘Ÿ(𝐽(𝑧2 NOUN iajs-3505 110 30 ) ) PUNCT iajs-3505 110 31 ) ) PUNCT iajs-3505 111 1 = = SYM iajs-3505 111 2 π‘Ÿ π‘Ÿ NOUN iajs-3505 111 3 ( ( PUNCT iajs-3505 111 4 π‘Ž1+𝑀0βˆ’ π‘Ž1+𝑀0βˆ’ PROPN iajs-3505 111 5 οΏ½ οΏ½ PROPN iajs-3505 111 6 Μ… Μ… NOUN iajs-3505 111 7 οΏ½ οΏ½ NOUN iajs-3505 111 8 ) ) PUNCT iajs-3505 111 9 βˆ’ βˆ’ PROPN iajs-3505 111 10 𝛿1 𝛿1 NOUN iajs-3505 111 11 βˆ’ βˆ’ PROPN iajs-3505 111 12 ( ( PUNCT iajs-3505 111 13 𝑠 𝑠 PROPN iajs-3505 111 14 + + NUM iajs-3505 111 15 𝛾 𝛾 PROPN iajs-3505 111 16 + + X iajs-3505 111 17 𝛾1 𝛾1 PROPN iajs-3505 111 18 οΏ½ οΏ½ PROPN iajs-3505 111 19 Μ… Μ… NOUN iajs-3505 111 20 οΏ½ οΏ½ NOUN iajs-3505 111 21 ) ) PUNCT iajs-3505 111 22 , , PUNCT iajs-3505 111 23 𝐷𝑒𝑑(𝐽(𝑧2 𝐷𝑒𝑑(𝐽(𝑧2 NOUN iajs-3505 111 24 ) ) PUNCT iajs-3505 111 25 ) ) PUNCT iajs-3505 112 1 = = PUNCT iajs-3505 113 1 𝛿1(𝑠 𝛿1(𝑠 PROPN iajs-3505 113 2 + + CCONJ iajs-3505 113 3 𝛾 𝛾 PROPN iajs-3505 113 4 + + CCONJ iajs-3505 113 5 𝛾1 𝛾1 PROPN iajs-3505 113 6 οΏ½ οΏ½ PROPN iajs-3505 113 7 Μ… Μ… NOUN iajs-3505 113 8 οΏ½ οΏ½ NOUN iajs-3505 113 9 ) ) PUNCT iajs-3505 114 1 + + NUM iajs-3505 114 2 π‘Ÿπ‘’ π‘Ÿπ‘’ NOUN iajs-3505 114 3 Μ… Μ… NOUN iajs-3505 114 4 ( ( PUNCT iajs-3505 114 5 𝛾1 𝛾1 PROPN iajs-3505 114 6 οΏ½ οΏ½ PROPN iajs-3505 114 7 Μ… Μ… NOUN iajs-3505 114 8 οΏ½ οΏ½ NOUN iajs-3505 114 9 βˆ’π‘‘ βˆ’π‘‘ VERB iajs-3505 114 10 ) ) PUNCT iajs-3505 114 11 ( ( PUNCT iajs-3505 114 12 π‘Ž1+𝑀0βˆ’ π‘Ž1+𝑀0βˆ’ PROPN iajs-3505 114 13 οΏ½ οΏ½ PROPN iajs-3505 114 14 Μ… Μ… NOUN iajs-3505 114 15 οΏ½ οΏ½ NOUN iajs-3505 114 16 ) ) PUNCT iajs-3505 114 17 2 2 NUM iajs-3505 114 18 βˆ’ βˆ’ NOUN iajs-3505 114 19 π‘Ÿ(𝑠+𝛾+𝛾1 π‘Ÿ(𝑠+𝛾+𝛾1 NUM iajs-3505 114 20 οΏ½ οΏ½ NOUN iajs-3505 114 21 Μ… Μ… NOUN iajs-3505 114 22 οΏ½ οΏ½ NOUN iajs-3505 114 23 ) ) PUNCT iajs-3505 114 24 ( ( PUNCT iajs-3505 114 25 π‘Ž1+𝑀0βˆ’ π‘Ž1+𝑀0βˆ’ PROPN iajs-3505 114 26 οΏ½ οΏ½ PROPN iajs-3505 114 27 Μ… Μ… NOUN iajs-3505 114 28 οΏ½ οΏ½ NOUN iajs-3505 114 29 ) ) PUNCT iajs-3505 114 30 . . PUNCT iajs-3505 115 1 clearly clearly ADV iajs-3505 115 2 , , PUNCT iajs-3505 115 3 𝑧2 𝑧2 PROPN iajs-3505 115 4 is be AUX iajs-3505 115 5 a a DET iajs-3505 115 6 locally locally ADV iajs-3505 115 7 asymptotical asymptotical ADJ iajs-3505 115 8 stable stable ADJ iajs-3505 115 9 point point NOUN iajs-3505 115 10 if if SCONJ iajs-3505 115 11 and and CCONJ iajs-3505 115 12 only only ADV iajs-3505 115 13 if if SCONJ iajs-3505 115 14 the the DET iajs-3505 115 15 following follow VERB iajs-3505 115 16 conditions condition NOUN iajs-3505 115 17 are be AUX iajs-3505 115 18 satisfied satisfied ADJ iajs-3505 115 19 : : PUNCT iajs-3505 115 20 𝛿2 𝛿2 NOUN iajs-3505 115 21 > > X iajs-3505 115 22 𝛼2 𝛼2 PROPN iajs-3505 115 23 οΏ½ οΏ½ PROPN iajs-3505 115 24 Μ… Μ… NOUN iajs-3505 115 25 οΏ½ οΏ½ NOUN iajs-3505 115 26 (1 (1 PUNCT iajs-3505 115 27 βˆ’ βˆ’ NOUN iajs-3505 115 28 π‘š π‘š NOUN iajs-3505 115 29 ) ) PUNCT iajs-3505 115 30 ( ( PUNCT iajs-3505 115 31 π‘Ž2 π‘Ž2 NOUN iajs-3505 115 32 + + NUM iajs-3505 115 33 𝑀0 𝑀0 PROPN iajs-3505 115 34 βˆ’ βˆ’ PROPN iajs-3505 115 35 οΏ½ οΏ½ PROPN iajs-3505 115 36 Μ… Μ… NOUN iajs-3505 115 37 οΏ½ οΏ½ NOUN iajs-3505 115 38 ) ) PUNCT iajs-3505 115 39 , , PUNCT iajs-3505 115 40 π‘Ÿ π‘Ÿ X iajs-3505 115 41 < < X iajs-3505 116 1 [ [ X iajs-3505 116 2 𝛿2 𝛿2 NOUN iajs-3505 116 3 + + CCONJ iajs-3505 116 4 ( ( PUNCT iajs-3505 116 5 𝑠 𝑠 PROPN iajs-3505 116 6 + + NUM iajs-3505 116 7 𝛾 𝛾 PROPN iajs-3505 116 8 + + X iajs-3505 116 9 𝛾1 𝛾1 PROPN iajs-3505 116 10 οΏ½ οΏ½ PROPN iajs-3505 116 11 Μ… Μ… NOUN iajs-3505 116 12 οΏ½ οΏ½ NOUN iajs-3505 116 13 )](π‘Ž1 )](π‘Ž1 PROPN iajs-3505 116 14 + + NUM iajs-3505 116 15 𝑀0 𝑀0 PROPN iajs-3505 116 16 βˆ’ βˆ’ PROPN iajs-3505 116 17 οΏ½ οΏ½ PROPN iajs-3505 116 18 Μ… Μ… NOUN iajs-3505 116 19 οΏ½ οΏ½ NOUN iajs-3505 116 20 ) ) PUNCT iajs-3505 116 21 , , PUNCT iajs-3505 116 22 𝛿1(𝑠 𝛿1(𝑠 PROPN iajs-3505 116 23 + + CCONJ iajs-3505 116 24 𝛾 𝛾 PROPN iajs-3505 116 25 + + CCONJ iajs-3505 116 26 𝛾1 𝛾1 PROPN iajs-3505 116 27 οΏ½ οΏ½ PROPN iajs-3505 116 28 Μ… Μ… NOUN iajs-3505 116 29 οΏ½ οΏ½ NOUN iajs-3505 116 30 ) ) PUNCT iajs-3505 116 31 + + NUM iajs-3505 116 32 π‘Ÿπ‘’ π‘Ÿπ‘’ NOUN iajs-3505 116 33 Μ… Μ… NOUN iajs-3505 116 34 ( ( PUNCT iajs-3505 116 35 𝛾1 𝛾1 PROPN iajs-3505 116 36 οΏ½ οΏ½ NOUN iajs-3505 116 37 Μ… Μ… NOUN iajs-3505 116 38 οΏ½ οΏ½ NOUN iajs-3505 116 39 βˆ’ βˆ’ NOUN iajs-3505 116 40 𝑑 𝑑 NOUN iajs-3505 116 41 ) ) PUNCT iajs-3505 116 42 ( ( PUNCT iajs-3505 116 43 π‘Ž1 π‘Ž1 PROPN iajs-3505 116 44 + + NUM iajs-3505 116 45 𝑀0 𝑀0 PROPN iajs-3505 116 46 βˆ’ βˆ’ PROPN iajs-3505 116 47 οΏ½ οΏ½ PROPN iajs-3505 116 48 Μ… Μ… NOUN iajs-3505 116 49 οΏ½ οΏ½ NOUN iajs-3505 116 50 ) ) PUNCT iajs-3505 116 51 2 2 NUM iajs-3505 116 52 > > X iajs-3505 116 53 π‘Ÿ(𝑠 π‘Ÿ(𝑠 PROPN iajs-3505 116 54 + + CCONJ iajs-3505 116 55 𝛾 𝛾 PROPN iajs-3505 116 56 + + CCONJ iajs-3505 116 57 𝛾1 𝛾1 PROPN iajs-3505 116 58 οΏ½ οΏ½ PROPN iajs-3505 116 59 Μ… Μ… NOUN iajs-3505 116 60 οΏ½ οΏ½ NOUN iajs-3505 116 61 ) ) PUNCT iajs-3505 116 62 ( ( PUNCT iajs-3505 116 63 π‘Ž1 π‘Ž1 NOUN iajs-3505 116 64 + + NUM iajs-3505 116 65 𝑀0 𝑀0 PROPN iajs-3505 116 66 βˆ’ βˆ’ PROPN iajs-3505 116 67 οΏ½ οΏ½ PROPN iajs-3505 116 68 Μ… Μ… NOUN iajs-3505 116 69 οΏ½ οΏ½ NOUN iajs-3505 116 70 ) ) PUNCT iajs-3505 116 71 . . PUNCT iajs-3505 117 1 } } PUNCT iajs-3505 117 2 ( ( PUNCT iajs-3505 117 3 7 7 X iajs-3505 117 4 ) ) PUNCT iajs-3505 117 5 3 3 NUM iajs-3505 117 6 . . PUNCT iajs-3505 118 1 the the DET iajs-3505 118 2 jacobian jacobian ADJ iajs-3505 118 3 matrix matrix NOUN iajs-3505 118 4 at at ADP iajs-3505 118 5 𝑧3 𝑧3 PROPN iajs-3505 118 6 = = PUNCT iajs-3505 118 7 ( ( PUNCT iajs-3505 118 8 𝑒 𝑒 NOUN iajs-3505 118 9 βˆ— βˆ— NOUN iajs-3505 118 10 , , PUNCT iajs-3505 118 11 π‘£βˆ— π‘£βˆ— ADJ iajs-3505 118 12 , , PUNCT iajs-3505 118 13 π‘€βˆ— π‘€βˆ— PROPN iajs-3505 118 14 ) ) PUNCT iajs-3505 118 15 is be AUX iajs-3505 118 16 given give VERB iajs-3505 118 17 as as ADP iajs-3505 118 18 : : PUNCT iajs-3505 118 19 𝐽(𝑧3 𝐽(𝑧3 X iajs-3505 118 20 ) ) PUNCT iajs-3505 118 21 = = NOUN iajs-3505 119 1 [ [ PUNCT iajs-3505 119 2 π‘Ÿ π‘Ÿ X iajs-3505 119 3 ( ( PUNCT iajs-3505 119 4 π‘Ž1+𝑀0βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘€ PROPN iajs-3505 119 5 βˆ— βˆ— NOUN iajs-3505 119 6 ) ) PUNCT iajs-3505 119 7 βˆ’ βˆ’ PROPN iajs-3505 119 8 𝛿1 𝛿1 NOUN iajs-3505 119 9 βˆ’ βˆ’ PROPN iajs-3505 119 10 𝛼1𝑣 𝛼1𝑣 PROPN iajs-3505 119 11 βˆ—(1 βˆ—(1 VERB iajs-3505 119 12 βˆ’ βˆ’ NOUN iajs-3505 119 13 π‘š π‘š NOUN iajs-3505 119 14 ) ) PUNCT iajs-3505 119 15 βˆ’π›Ό1𝑒 βˆ’π›Ό1𝑒 NOUN iajs-3505 120 1 βˆ—(1 βˆ—(1 NOUN iajs-3505 121 1 βˆ’ βˆ’ NOUN iajs-3505 121 2 π‘š π‘š NOUN iajs-3505 121 3 ) ) PUNCT iajs-3505 121 4 π‘Ÿπ‘’βˆ— π‘Ÿπ‘’βˆ— NOUN iajs-3505 121 5 ( ( PUNCT iajs-3505 121 6 π‘Ž1+𝑀0βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘€ PROPN iajs-3505 121 7 βˆ—)2 βˆ—)2 PROPN iajs-3505 121 8 𝛼2𝑣 𝛼2𝑣 NUM iajs-3505 121 9 βˆ—(1βˆ’π‘š βˆ—(1βˆ’π‘š ADJ iajs-3505 121 10 ) ) PUNCT iajs-3505 121 11 ( ( PUNCT iajs-3505 121 12 π‘Ž2+𝑀0βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘€ PROPN iajs-3505 121 13 βˆ— βˆ— NOUN iajs-3505 121 14 ) ) PUNCT iajs-3505 121 15 0 0 PUNCT iajs-3505 122 1 𝛼2𝑒 𝛼2𝑒 NOUN iajs-3505 122 2 βˆ—(1βˆ’π‘š)π‘£βˆ— βˆ—(1βˆ’π‘š)π‘£βˆ— PUNCT iajs-3505 122 3 ( ( PUNCT iajs-3505 122 4 π‘Ž2+𝑀0βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘€ PROPN iajs-3505 122 5 βˆ—)2 βˆ—)2 ADV iajs-3505 122 6 π‘‘βˆ’π›Ύ1𝑀 π‘‘βˆ’π›Ύ1𝑀 NOUN iajs-3505 122 7 βˆ— βˆ— VERB iajs-3505 122 8 βˆ’π›Ύ2𝑀 βˆ’π›Ύ2𝑀 NUM iajs-3505 122 9 βˆ— βˆ— NOUN iajs-3505 122 10 βˆ’π‘  βˆ’π‘  ADV iajs-3505 122 11 βˆ’ βˆ’ NOUN iajs-3505 122 12 𝛾 𝛾 ADP iajs-3505 122 13 βˆ’ βˆ’ PROPN iajs-3505 122 14 𝛾1𝑒 𝛾1𝑒 PROPN iajs-3505 122 15 βˆ— βˆ— NOUN iajs-3505 122 16 βˆ’ βˆ’ PROPN iajs-3505 122 17 𝛾2𝑣 𝛾2𝑣 NOUN iajs-3505 122 18 βˆ— βˆ— NOUN iajs-3505 122 19 ] ] X iajs-3505 122 20 = = SYM iajs-3505 122 21 ( ( PUNCT iajs-3505 122 22 π‘Žπ‘–π‘—)3Γ—3 π‘Žπ‘–π‘—)3Γ—3 X iajs-3505 122 23 ( ( PUNCT iajs-3505 122 24 8) 8) NUM iajs-3505 122 25 so so ADV iajs-3505 122 26 , , PUNCT iajs-3505 122 27 the the DET iajs-3505 122 28 characteristic characteristic ADJ iajs-3505 122 29 equation equation NOUN iajs-3505 122 30 of of ADP iajs-3505 122 31 𝐽(𝑧3 𝐽(𝑧3 PROPN iajs-3505 122 32 ) ) PUNCT iajs-3505 122 33 can can AUX iajs-3505 122 34 be be AUX iajs-3505 122 35 written write VERB iajs-3505 122 36 as as ADP iajs-3505 122 37 : : PUNCT iajs-3505 122 38 πœ†3 πœ†3 NOUN iajs-3505 122 39 + + CCONJ iajs-3505 123 1 𝐴1πœ† 𝐴1πœ† VERB iajs-3505 123 2 2 2 NUM iajs-3505 123 3 + + CCONJ iajs-3505 123 4 𝐴2πœ† 𝐴2πœ† PROPN iajs-3505 123 5 + + CCONJ iajs-3505 123 6 𝐴3 𝐴3 PROPN iajs-3505 123 7 = = SYM iajs-3505 123 8 0 0 NUM iajs-3505 123 9 , , PUNCT iajs-3505 123 10 ( ( PUNCT iajs-3505 123 11 9 9 NUM iajs-3505 123 12 ) ) PUNCT iajs-3505 123 13 where where SCONJ iajs-3505 123 14 , , PUNCT iajs-3505 123 15 𝐴1 𝐴1 PROPN iajs-3505 123 16 = = SYM iajs-3505 124 1 βˆ’(π‘Ž11 βˆ’(π‘Ž11 ADV iajs-3505 124 2 + + CCONJ iajs-3505 124 3 π‘Ž33 π‘Ž33 NOUN iajs-3505 124 4 ) ) PUNCT iajs-3505 124 5 , , PUNCT iajs-3505 124 6 𝐴2 𝐴2 PROPN iajs-3505 124 7 = = PUNCT iajs-3505 124 8 βˆ’(π‘Ž13π‘Ž31 βˆ’(π‘Ž13π‘Ž31 NOUN iajs-3505 124 9 + + NUM iajs-3505 124 10 π‘Ž23π‘Ž32 π‘Ž23π‘Ž32 NOUN iajs-3505 124 11 + + X iajs-3505 124 12 π‘Ž12π‘Ž21 π‘Ž12π‘Ž21 PROPN iajs-3505 124 13 βˆ’ βˆ’ PROPN iajs-3505 124 14 π‘Ž11π‘Ž33 π‘Ž11π‘Ž33 NOUN iajs-3505 124 15 ) ) PUNCT iajs-3505 124 16 , , PUNCT iajs-3505 124 17 𝐴3 𝐴3 PROPN iajs-3505 124 18 = = PUNCT iajs-3505 124 19 π‘Ž11π‘Ž23π‘Ž32 π‘Ž11π‘Ž23π‘Ž32 PROPN iajs-3505 124 20 + + CCONJ iajs-3505 124 21 π‘Ž12π‘Ž21π‘Ž33 π‘Ž12π‘Ž21π‘Ž33 PROPN iajs-3505 124 22 βˆ’ βˆ’ ADP iajs-3505 124 23 π‘Ž13π‘Ž21π‘Ž32 π‘Ž13π‘Ž21π‘Ž32 NOUN iajs-3505 124 24 βˆ’ βˆ’ NOUN iajs-3505 124 25 π‘Ž12π‘Ž23π‘Ž31 π‘Ž12π‘Ž23π‘Ž31 ADJ iajs-3505 124 26 , , PUNCT iajs-3505 124 27 βˆ†= βˆ†= ADJ iajs-3505 124 28 𝐴1𝐴2 𝐴1𝐴2 VERB iajs-3505 124 29 βˆ’ βˆ’ PROPN iajs-3505 124 30 𝐴3 𝐴3 PROPN iajs-3505 124 31 = = PRON iajs-3505 124 32 ( ( PUNCT iajs-3505 124 33 π‘Ž11 π‘Ž11 VERB iajs-3505 124 34 + + NUM iajs-3505 124 35 π‘Ž33)(π‘Ž13π‘Ž31 π‘Ž33)(π‘Ž13π‘Ž31 NOUN iajs-3505 124 36 βˆ’ βˆ’ NOUN iajs-3505 124 37 π‘Ž11π‘Ž33 π‘Ž11π‘Ž33 NOUN iajs-3505 124 38 ) ) PUNCT iajs-3505 124 39 + + CCONJ iajs-3505 125 1 π‘Ž11π‘Ž12π‘Ž21 π‘Ž11π‘Ž12π‘Ž21 PROPN iajs-3505 125 2 + + CCONJ iajs-3505 125 3 π‘Ž23π‘Ž32π‘Ž33 π‘Ž23π‘Ž32π‘Ž33 PROPN iajs-3505 125 4 + + CCONJ iajs-3505 125 5 π‘Ž12π‘Ž23π‘Ž31 π‘Ž12π‘Ž23π‘Ž31 ADJ iajs-3505 125 6 + + NUM iajs-3505 125 7 ihjpas ihjpa NOUN iajs-3505 125 8 . . PUNCT iajs-3505 126 1 2024 2024 NUM iajs-3505 126 2 , , PUNCT iajs-3505 126 3 37(4 37(4 NOUN iajs-3505 126 4 ) ) PUNCT iajs-3505 126 5 374 374 NUM iajs-3505 126 6 π‘Ž13π‘Ž21π‘Ž32 π‘Ž13π‘Ž21π‘Ž32 NOUN iajs-3505 126 7 . . PUNCT iajs-3505 126 8 now now ADV iajs-3505 126 9 , , PUNCT iajs-3505 126 10 from from ADP iajs-3505 126 11 the the DET iajs-3505 126 12 routh routh PROPN iajs-3505 126 13 - - PUNCT iajs-3505 126 14 hurwitz hurwitz PROPN iajs-3505 126 15 criteria criterion NOUN iajs-3505 126 16 [ [ X iajs-3505 126 17 18 18 NUM iajs-3505 126 18 ] ] PUNCT iajs-3505 126 19 , , PUNCT iajs-3505 126 20 𝑧3 𝑧3 PROPN iajs-3505 126 21 is be AUX iajs-3505 126 22 a a DET iajs-3505 126 23 las las NOUN iajs-3505 126 24 point point NOUN iajs-3505 126 25 , , PUNCT iajs-3505 126 26 under under ADP iajs-3505 126 27 the the DET iajs-3505 126 28 condition condition NOUN iajs-3505 126 29 that that PRON iajs-3505 126 30 𝐴1 𝐴1 PROPN iajs-3505 126 31 > > X iajs-3505 126 32 0 0 PROPN iajs-3505 126 33 , , PUNCT iajs-3505 126 34 𝐴3 𝐴3 PROPN iajs-3505 126 35 > > X iajs-3505 126 36 0 0 PUNCT iajs-3505 126 37 and and CCONJ iajs-3505 126 38 βˆ† βˆ† X iajs-3505 126 39 > > X iajs-3505 126 40 0 0 NUM iajs-3505 126 41 . . PROPN iajs-3505 127 1 4 4 NUM iajs-3505 127 2 . . X iajs-3505 127 3 hop hop NOUN iajs-3505 127 4 bifurcation bifurcation NOUN iajs-3505 127 5 from from ADP iajs-3505 127 6 theorem theorem ADJ iajs-3505 127 7 2 2 NUM iajs-3505 127 8 , , PUNCT iajs-3505 127 9 the the DET iajs-3505 127 10 steady steady ADJ iajs-3505 127 11 state state NOUN iajs-3505 127 12 𝑧3 𝑧3 ADJ iajs-3505 127 13 changes change NOUN iajs-3505 127 14 as as ADP iajs-3505 127 15 the the DET iajs-3505 127 16 parameter parameter NOUN iajs-3505 127 17 𝛾2 𝛾2 NOUN iajs-3505 127 18 crosses cross VERB iajs-3505 127 19 the the DET iajs-3505 127 20 threshold threshold NOUN iajs-3505 127 21 value value NOUN iajs-3505 127 22 𝛾2 𝛾2 VERB iajs-3505 127 23 βˆ— βˆ— NOUN iajs-3505 127 24 , , PUNCT iajs-3505 127 25 which which PRON iajs-3505 127 26 implies imply VERB iajs-3505 127 27 that that SCONJ iajs-3505 127 28 𝑧3 𝑧3 PROPN iajs-3505 127 29 may may AUX iajs-3505 127 30 become become VERB iajs-3505 127 31 unstable unstable ADJ iajs-3505 127 32 due due ADJ iajs-3505 127 33 to to ADP iajs-3505 127 34 hopf hopf ADJ iajs-3505 127 35 bifurcation bifurcation NOUN iajs-3505 127 36 when when SCONJ iajs-3505 127 37 forced force VERB iajs-3505 127 38 to to PART iajs-3505 127 39 operate operate VERB iajs-3505 127 40 within within ADP iajs-3505 127 41 particular particular ADJ iajs-3505 127 42 restrictions restriction NOUN iajs-3505 127 43 on on ADP iajs-3505 127 44 its its PRON iajs-3505 127 45 parameters parameter NOUN iajs-3505 127 46 [ [ X iajs-3505 127 47 19 19 NUM iajs-3505 127 48 - - SYM iajs-3505 127 49 25 25 NUM iajs-3505 127 50 ] ] PUNCT iajs-3505 127 51 . . PUNCT iajs-3505 128 1 in in ADP iajs-3505 128 2 the the DET iajs-3505 128 3 case case NOUN iajs-3505 128 4 where where SCONJ iajs-3505 128 5 we we PRON iajs-3505 128 6 use use VERB iajs-3505 128 7 𝛾2 𝛾2 NOUN iajs-3505 128 8 βˆ— βˆ— NOUN iajs-3505 128 9 as as ADP iajs-3505 128 10 the the DET iajs-3505 128 11 bifurcation bifurcation NOUN iajs-3505 128 12 parameter parameter NOUN iajs-3505 128 13 , , PUNCT iajs-3505 128 14 the the DET iajs-3505 128 15 hopf hopf ADJ iajs-3505 128 16 bifurcation bifurcation NOUN iajs-3505 128 17 threshold threshold NOUN iajs-3505 128 18 and and CCONJ iajs-3505 128 19 its its PRON iajs-3505 128 20 conditions condition NOUN iajs-3505 128 21 are be AUX iajs-3505 128 22 clearly clearly ADV iajs-3505 128 23 clarified clarify VERB iajs-3505 128 24 in in ADP iajs-3505 128 25 the the DET iajs-3505 128 26 following following NOUN iajs-3505 128 27 theorem theorem NOUN iajs-3505 128 28 [ [ X iajs-3505 128 29 ] ] X iajs-3505 128 30 . . PUNCT iajs-3505 129 1 theorem theorem NOUN iajs-3505 129 2 2 2 NUM iajs-3505 129 3 . . PUNCT iajs-3505 130 1 under under ADP iajs-3505 130 2 the the DET iajs-3505 130 3 following follow VERB iajs-3505 130 4 assumptions assumption NOUN iajs-3505 130 5 𝐴𝑖 𝐴𝑖 PROPN iajs-3505 130 6 > > X iajs-3505 130 7 0 0 PROPN iajs-3505 130 8 , , PUNCT iajs-3505 130 9 𝑖 𝑖 SYM iajs-3505 130 10 = = SYM iajs-3505 130 11 1,2 1,2 NUM iajs-3505 130 12 π‘€βˆ— π‘€βˆ— NOUN iajs-3505 130 13 > > SYM iajs-3505 130 14 π‘£βˆ— π‘£βˆ— ADJ iajs-3505 130 15 𝛾2 𝛾2 VERB iajs-3505 130 16 βˆ— βˆ— NOUN iajs-3505 130 17 > > X iajs-3505 130 18 0 0 PUNCT iajs-3505 131 1 ( ( PUNCT iajs-3505 131 2 11 11 NUM iajs-3505 131 3 ) ) PUNCT iajs-3505 131 4 ( ( PUNCT iajs-3505 131 5 12 12 NUM iajs-3505 131 6 ) ) PUNCT iajs-3505 131 7 ( ( PUNCT iajs-3505 131 8 13 13 NUM iajs-3505 131 9 ) ) PUNCT iajs-3505 131 10 where where SCONJ iajs-3505 131 11 ai ai VERB iajs-3505 131 12 ’s ’s ADV iajs-3505 131 13 are be AUX iajs-3505 131 14 the the DET iajs-3505 131 15 coefficients coefficient NOUN iajs-3505 131 16 of of ADP iajs-3505 131 17 the the DET iajs-3505 131 18 characteristic characteristic ADJ iajs-3505 131 19 equation equation NOUN iajs-3505 131 20 given give VERB iajs-3505 131 21 in in ADP iajs-3505 131 22 equation equation NOUN iajs-3505 131 23 ( ( PUNCT iajs-3505 131 24 9 9 NUM iajs-3505 131 25 ) ) PUNCT iajs-3505 131 26 with with ADP iajs-3505 131 27 𝛾2 𝛾2 NOUN iajs-3505 131 28 = = SYM iajs-3505 131 29 𝛾2 𝛾2 VERB iajs-3505 131 30 βˆ— βˆ— NOUN iajs-3505 131 31 and and CCONJ iajs-3505 131 32 the the DET iajs-3505 131 33 formula formula NOUN iajs-3505 131 34 for for ADP iajs-3505 131 35 𝛾2 𝛾2 NOUN iajs-3505 131 36 βˆ— βˆ— NOUN iajs-3505 131 37 is be AUX iajs-3505 131 38 shown show VERB iajs-3505 131 39 in in ADP iajs-3505 131 40 the the DET iajs-3505 131 41 following following ADJ iajs-3505 131 42 proof proof NOUN iajs-3505 131 43 . . PUNCT iajs-3505 132 1 then then ADV iajs-3505 132 2 , , PUNCT iajs-3505 132 3 there there PRON iajs-3505 132 4 exists exist VERB iajs-3505 132 5 a a DET iajs-3505 132 6 hopf hopf ADJ iajs-3505 132 7 bifurcation bifurcation NOUN iajs-3505 132 8 for for ADP iajs-3505 132 9 𝑧3 𝑧3 PROPN iajs-3505 132 10 at at ADP iajs-3505 132 11 𝛾2 𝛾2 NOUN iajs-3505 132 12 = = PUNCT iajs-3505 132 13 𝛾2 𝛾2 VERB iajs-3505 132 14 βˆ—. βˆ—. PROPN iajs-3505 132 15 proof proof NOUN iajs-3505 132 16 : : PUNCT iajs-3505 132 17 the the DET iajs-3505 132 18 value value NOUN iajs-3505 132 19 of of ADP iajs-3505 132 20 the the DET iajs-3505 132 21 bifurcation bifurcation NOUN iajs-3505 132 22 parameter parameter NOUN iajs-3505 132 23 can can AUX iajs-3505 132 24 be be AUX iajs-3505 132 25 found find VERB iajs-3505 132 26 if if SCONJ iajs-3505 132 27 we we PRON iajs-3505 132 28 set set VERB iajs-3505 132 29 𝐴1(𝛾2 𝐴1(𝛾2 PROPN iajs-3505 132 30 βˆ—)𝐴2(𝛾2 βˆ—)𝐴2(𝛾2 ADJ iajs-3505 132 31 βˆ— βˆ— NOUN iajs-3505 132 32 ) ) PUNCT iajs-3505 133 1 βˆ’ βˆ’ PROPN iajs-3505 134 1 𝐴3(𝛾2 𝐴3(𝛾2 PROPN iajs-3505 134 2 βˆ— βˆ— NOUN iajs-3505 134 3 ) ) PUNCT iajs-3505 135 1 = = SYM iajs-3505 135 2 0 0 NUM iajs-3505 136 1 in in ADP iajs-3505 136 2 equation equation NOUN iajs-3505 136 3 ( ( PUNCT iajs-3505 136 4 9 9 NUM iajs-3505 136 5 ) ) PUNCT iajs-3505 136 6 . . PUNCT iajs-3505 137 1 this this PRON iajs-3505 137 2 gives give VERB iajs-3505 137 3 : : PUNCT iajs-3505 137 4 𝛾2 𝛾2 VERB iajs-3505 137 5 βˆ— βˆ— NOUN iajs-3505 137 6 = = PUNCT iajs-3505 137 7 ( ( PUNCT iajs-3505 137 8 π‘Ž11+π‘Ž33)(π‘Ž13π‘Ž31βˆ’π‘Ž11π‘Ž33)+π‘Ž11π‘Ž12π‘Ž21+π‘Ž12π‘Ž23π‘Ž31 π‘Ž11+π‘Ž33)(π‘Ž13π‘Ž31βˆ’π‘Ž11π‘Ž33)+π‘Ž11π‘Ž12π‘Ž21+π‘Ž12π‘Ž23π‘Ž31 NOUN iajs-3505 137 9 ) ) PUNCT iajs-3505 137 10 ( ( PUNCT iajs-3505 137 11 π‘Ž23π‘Ž33+π‘Ž13π‘Ž21)𝑀 π‘Ž23π‘Ž33+π‘Ž13π‘Ž21)𝑀 X iajs-3505 137 12 βˆ— βˆ— NOUN iajs-3505 137 13 . . PUNCT iajs-3505 138 1 clearly clearly ADV iajs-3505 138 2 , , PUNCT iajs-3505 138 3 𝛾2 𝛾2 VERB iajs-3505 138 4 βˆ— βˆ— NOUN iajs-3505 138 5 > > X iajs-3505 138 6 0 0 PUNCT iajs-3505 139 1 if if SCONJ iajs-3505 139 2 condition condition NOUN iajs-3505 139 3 ( ( PUNCT iajs-3505 139 4 13 13 NUM iajs-3505 139 5 ) ) PUNCT iajs-3505 139 6 holds hold VERB iajs-3505 139 7 . . PUNCT iajs-3505 140 1 now now ADV iajs-3505 140 2 , , PUNCT iajs-3505 140 3 at at ADP iajs-3505 140 4 𝛾2 𝛾2 NOUN iajs-3505 140 5 = = PUNCT iajs-3505 140 6 𝛾2 𝛾2 VERB iajs-3505 140 7 βˆ— βˆ— NOUN iajs-3505 140 8 , , PUNCT iajs-3505 140 9 equation equation NOUN iajs-3505 140 10 ( ( PUNCT iajs-3505 140 11 9 9 NUM iajs-3505 140 12 ) ) PUNCT iajs-3505 140 13 can can AUX iajs-3505 140 14 be be AUX iajs-3505 140 15 written write VERB iajs-3505 140 16 as as ADP iajs-3505 140 17 ( ( PUNCT iajs-3505 140 18 πœ† πœ† X iajs-3505 140 19 + + X iajs-3505 140 20 𝐴1)(πœ† 𝐴1)(πœ† X iajs-3505 140 21 2 2 NUM iajs-3505 140 22 + + NUM iajs-3505 140 23 𝐴2 𝐴2 PROPN iajs-3505 140 24 ) ) PUNCT iajs-3505 141 1 = = NOUN iajs-3505 141 2 0 0 X iajs-3505 141 3 . . PUNCT iajs-3505 142 1 according accord VERB iajs-3505 142 2 to to ADP iajs-3505 142 3 condition condition NOUN iajs-3505 142 4 ( ( PUNCT iajs-3505 142 5 11 11 NUM iajs-3505 142 6 ) ) PUNCT iajs-3505 142 7 , , PUNCT iajs-3505 142 8 the the DET iajs-3505 142 9 above above ADJ iajs-3505 142 10 equation equation NOUN iajs-3505 142 11 has have VERB iajs-3505 142 12 three three NUM iajs-3505 142 13 roots root NOUN iajs-3505 142 14 , , PUNCT iajs-3505 142 15 a a DET iajs-3505 142 16 negative negative ADJ iajs-3505 142 17 root root NOUN iajs-3505 142 18 πœ†1 πœ†1 NOUN iajs-3505 142 19 = = PUNCT iajs-3505 142 20 βˆ’π΄1 βˆ’π΄1 PROPN iajs-3505 142 21 and and CCONJ iajs-3505 142 22 two two NUM iajs-3505 142 23 purely purely ADV iajs-3505 142 24 imaginary imaginary ADJ iajs-3505 142 25 roots root NOUN iajs-3505 142 26 πœ†2,3 πœ†2,3 NOUN iajs-3505 142 27 = = SYM iajs-3505 142 28 Β±π‘–βˆšπ΄2 Β±π‘–βˆšπ΄2 X iajs-3505 142 29 . . PUNCT iajs-3505 143 1 in in ADP iajs-3505 143 2 a a DET iajs-3505 143 3 neighbourhood neighbourhood NOUN iajs-3505 143 4 of of ADP iajs-3505 143 5 𝛾2 𝛾2 NOUN iajs-3505 143 6 βˆ— βˆ— NOUN iajs-3505 143 7 , , PUNCT iajs-3505 143 8 the the DET iajs-3505 143 9 roots root NOUN iajs-3505 143 10 have have VERB iajs-3505 143 11 the the DET iajs-3505 143 12 following follow VERB iajs-3505 143 13 forms form NOUN iajs-3505 143 14 : : PUNCT iajs-3505 143 15 πœ†1 πœ†1 NOUN iajs-3505 143 16 = = PUNCT iajs-3505 143 17 βˆ’π΄1 βˆ’π΄1 PROPN iajs-3505 143 18 , , PUNCT iajs-3505 143 19 πœ†2,3 πœ†2,3 PROPN iajs-3505 143 20 = = SYM iajs-3505 143 21 𝜌1(𝛾2 𝜌1(𝛾2 NUM iajs-3505 143 22 ) ) PUNCT iajs-3505 143 23 Β± Β± NOUN iajs-3505 143 24 π‘–πœŒ2(𝛾2 π‘–πœŒ2(𝛾2 NUM iajs-3505 143 25 ) ) PUNCT iajs-3505 143 26 . . PUNCT iajs-3505 144 1 clearly clearly ADV iajs-3505 144 2 , , PUNCT iajs-3505 144 3 𝑅𝑒 𝑅𝑒 PROPN iajs-3505 144 4 ( ( PUNCT iajs-3505 144 5 πœ†2,3)|𝛾2=𝛾2βˆ— πœ†2,3)|𝛾2=𝛾2βˆ— VERB iajs-3505 144 6 = = NOUN iajs-3505 144 7 𝜌1(𝛾2 𝜌1(𝛾2 NUM iajs-3505 144 8 βˆ— βˆ— NOUN iajs-3505 144 9 ) ) PUNCT iajs-3505 144 10 = = SYM iajs-3505 144 11 0 0 NUM iajs-3505 144 12 indicates indicate VERB iajs-3505 144 13 that that SCONJ iajs-3505 144 14 the the DET iajs-3505 144 15 first first ADJ iajs-3505 144 16 condition condition NOUN iajs-3505 144 17 for for ADP iajs-3505 144 18 hopf hopf ADJ iajs-3505 144 19 bifurcation bifurcation NOUN iajs-3505 144 20 has have AUX iajs-3505 144 21 been be AUX iajs-3505 144 22 met meet VERB iajs-3505 144 23 at at ADP iajs-3505 144 24 𝛾2 𝛾2 NOUN iajs-3505 144 25 = = PUNCT iajs-3505 144 26 𝛾2 𝛾2 VERB iajs-3505 144 27 βˆ—. βˆ—. PROPN iajs-3505 144 28 now now ADV iajs-3505 144 29 to to PART iajs-3505 144 30 confirm confirm VERB iajs-3505 144 31 the the DET iajs-3505 144 32 transversality transversality NOUN iajs-3505 144 33 condition condition NOUN iajs-3505 144 34 , , PUNCT iajs-3505 144 35 we we PRON iajs-3505 144 36 substitute substitute VERB iajs-3505 144 37 𝜌1(𝛾2 𝜌1(𝛾2 ADV iajs-3505 144 38 ) ) PUNCT iajs-3505 144 39 Β± Β± NOUN iajs-3505 144 40 π‘–πœŒ2(𝛾2 π‘–πœŒ2(𝛾2 VERB iajs-3505 144 41 ) ) PUNCT iajs-3505 144 42 into into ADP iajs-3505 144 43 equation equation NOUN iajs-3505 144 44 ( ( PUNCT iajs-3505 144 45 9 9 NUM iajs-3505 144 46 ) ) PUNCT iajs-3505 144 47 and and CCONJ iajs-3505 144 48 then then ADV iajs-3505 144 49 compute compute VERB iajs-3505 144 50 its its PRON iajs-3505 144 51 derivative derivative NOUN iajs-3505 144 52 with with ADP iajs-3505 144 53 respect respect NOUN iajs-3505 144 54 to to ADP iajs-3505 144 55 π‘‘βˆ— π‘‘βˆ— PROPN iajs-3505 144 56 , , PUNCT iajs-3505 144 57 𝛩(𝛾2 𝛩(𝛾2 PROPN iajs-3505 144 58 βˆ—)πœ“(𝛾2 βˆ—)πœ“(𝛾2 PROPN iajs-3505 144 59 βˆ— βˆ— NOUN iajs-3505 144 60 ) ) PUNCT iajs-3505 145 1 + + CCONJ iajs-3505 145 2 𝛀(𝛾2 𝛀(𝛾2 PROPN iajs-3505 145 3 βˆ—)πœ™(𝛾2 βˆ—)πœ™(𝛾2 NOUN iajs-3505 145 4 βˆ— βˆ— NOUN iajs-3505 145 5 ) ) PUNCT iajs-3505 145 6 β‰  β‰  PROPN iajs-3505 145 7 0 0 NUM iajs-3505 145 8 , , PUNCT iajs-3505 145 9 where where SCONJ iajs-3505 145 10 the the DET iajs-3505 145 11 form form NOUN iajs-3505 145 12 of of ADP iajs-3505 145 13 𝛩(𝛾2 𝛩(𝛾2 PROPN iajs-3505 145 14 βˆ— βˆ— NOUN iajs-3505 145 15 ) ) PUNCT iajs-3505 145 16 , , PUNCT iajs-3505 145 17 πœ“(𝛾2 πœ“(𝛾2 NOUN iajs-3505 145 18 βˆ— βˆ— NOUN iajs-3505 145 19 ) ) PUNCT iajs-3505 145 20 , , PUNCT iajs-3505 145 21 𝛀(𝛾2 𝛀(𝛾2 NOUN iajs-3505 145 22 βˆ— βˆ— NOUN iajs-3505 145 23 ) ) PUNCT iajs-3505 145 24 and and CCONJ iajs-3505 145 25 πœ™(𝛾2 πœ™(𝛾2 ADJ iajs-3505 145 26 βˆ— βˆ— NOUN iajs-3505 145 27 ) ) PUNCT iajs-3505 145 28 are be AUX iajs-3505 145 29 πœ“(𝛾2 πœ“(𝛾2 ADJ iajs-3505 145 30 ) ) PUNCT iajs-3505 145 31 = = PUNCT iajs-3505 146 1 3𝜌1 3𝜌1 NUM iajs-3505 146 2 2(𝛾2 2(𝛾2 NUM iajs-3505 146 3 ) ) PUNCT iajs-3505 147 1 + + CCONJ iajs-3505 147 2 2𝐴1(𝛾2)𝜌1(𝛾2 2𝐴1(𝛾2)𝜌1(𝛾2 NUM iajs-3505 147 3 ) ) PUNCT iajs-3505 148 1 + + NUM iajs-3505 148 2 𝐴2(𝛾2 𝐴2(𝛾2 X iajs-3505 148 3 ) ) PUNCT iajs-3505 149 1 βˆ’ βˆ’ PROPN iajs-3505 149 2 3𝜌2 3𝜌2 NUM iajs-3505 149 3 2(𝛾2 2(𝛾2 NUM iajs-3505 149 4 ) ) PUNCT iajs-3505 149 5 , , PUNCT iajs-3505 149 6 πœ™(𝛾2 πœ™(𝛾2 X iajs-3505 149 7 ) ) PUNCT iajs-3505 149 8 = = SYM iajs-3505 150 1 6𝜌1(𝛾2)𝜌2(𝛾2 6𝜌1(𝛾2)𝜌2(𝛾2 NUM iajs-3505 150 2 ) ) PUNCT iajs-3505 150 3 + + X iajs-3505 150 4 2𝐴1(𝛾2)𝜌2(𝛾2 2𝐴1(𝛾2)𝜌2(𝛾2 NUM iajs-3505 150 5 ) ) PUNCT iajs-3505 150 6 , , PUNCT iajs-3505 150 7 𝛩(𝛾2 𝛩(𝛾2 PROPN iajs-3505 150 8 ) ) PUNCT iajs-3505 150 9 = = SYM iajs-3505 150 10 𝜌1 𝜌1 PROPN iajs-3505 150 11 2(𝛾2)𝐴1 2(𝛾2)𝐴1 NOUN iajs-3505 150 12 β€²(𝛾2 β€²(𝛾2 NUM iajs-3505 150 13 ) ) PUNCT iajs-3505 151 1 + + CCONJ iajs-3505 151 2 𝐴2 𝐴2 PROPN iajs-3505 151 3 β€²(𝛾2)𝜌1(𝛾2 β€²(𝛾2)𝜌1(𝛾2 PROPN iajs-3505 151 4 ) ) PUNCT iajs-3505 152 1 + + CCONJ iajs-3505 152 2 𝐴3 𝐴3 PROPN iajs-3505 152 3 β€²(𝛾2 β€²(𝛾2 NOUN iajs-3505 152 4 ) ) PUNCT iajs-3505 153 1 βˆ’ βˆ’ PROPN iajs-3505 153 2 𝐴1 𝐴1 PROPN iajs-3505 153 3 β€²(𝛾2)𝜌2 β€²(𝛾2)𝜌2 VERB iajs-3505 153 4 2(𝛾2 2(𝛾2 NUM iajs-3505 153 5 ) ) PUNCT iajs-3505 153 6 , , PUNCT iajs-3505 153 7 𝛀(𝛾2 𝛀(𝛾2 X iajs-3505 153 8 ) ) PUNCT iajs-3505 153 9 = = SYM iajs-3505 154 1 2𝜌1(𝛾2)𝜌2(𝛾2)𝐴1 2𝜌1(𝛾2)𝜌2(𝛾2)𝐴1 NUM iajs-3505 154 2 β€² β€² NUM iajs-3505 154 3 ( ( PUNCT iajs-3505 154 4 𝛾2 𝛾2 PROPN iajs-3505 154 5 ) ) PUNCT iajs-3505 154 6 + + CCONJ iajs-3505 154 7 𝐴2 𝐴2 PROPN iajs-3505 154 8 β€² β€² NUM iajs-3505 154 9 ( ( PUNCT iajs-3505 154 10 𝛾2)𝜌2(𝛾2 𝛾2)𝜌2(𝛾2 ADJ iajs-3505 154 11 ) ) PUNCT iajs-3505 154 12 . . PUNCT iajs-3505 155 1 now now ADV iajs-3505 155 2 at𝛾2 at𝛾2 PROPN iajs-3505 155 3 = = PUNCT iajs-3505 155 4 𝛾2 𝛾2 VERB iajs-3505 155 5 βˆ— βˆ— NOUN iajs-3505 155 6 , , PUNCT iajs-3505 155 7 substitution substitution NOUN iajs-3505 155 8 𝜌1 𝜌1 NOUN iajs-3505 155 9 = = SYM iajs-3505 155 10 0 0 NUM iajs-3505 155 11 and and CCONJ iajs-3505 155 12 𝜌2 𝜌2 ADJ iajs-3505 155 13 = = SYM iajs-3505 155 14 √𝐴2 √𝐴2 PROPN iajs-3505 155 15 , , PUNCT iajs-3505 155 16 into into ADP iajs-3505 155 17 equation equation NOUN iajs-3505 155 18 ( ( PUNCT iajs-3505 155 19 9 9 NUM iajs-3505 155 20 ) ) PUNCT iajs-3505 155 21 , , PUNCT iajs-3505 155 22 the the DET iajs-3505 155 23 following follow VERB iajs-3505 155 24 is be AUX iajs-3505 155 25 obtained obtain VERB iajs-3505 155 26 : : PUNCT iajs-3505 155 27 πœ“(𝛾2 πœ“(𝛾2 ADJ iajs-3505 155 28 βˆ— βˆ— NOUN iajs-3505 155 29 ) ) PUNCT iajs-3505 155 30 = = SYM iajs-3505 156 1 βˆ’2𝐴2(𝛾2 βˆ’2𝐴2(𝛾2 NUM iajs-3505 156 2 βˆ— βˆ— NOUN iajs-3505 156 3 ) ) PUNCT iajs-3505 156 4 , , PUNCT iajs-3505 156 5 πœ™(𝛾2 πœ™(𝛾2 ADJ iajs-3505 156 6 βˆ— βˆ— NOUN iajs-3505 156 7 ) ) PUNCT iajs-3505 157 1 = = SYM iajs-3505 157 2 2𝐴1(𝛾2 2𝐴1(𝛾2 NUM iajs-3505 157 3 βˆ—)√𝐴2(𝛾2 βˆ—)√𝐴2(𝛾2 ADJ iajs-3505 157 4 βˆ— βˆ— NOUN iajs-3505 157 5 ) ) PUNCT iajs-3505 157 6 , , PUNCT iajs-3505 157 7 𝛩(𝛾2 𝛩(𝛾2 PROPN iajs-3505 157 8 βˆ— βˆ— NOUN iajs-3505 157 9 ) ) PUNCT iajs-3505 158 1 = = SYM iajs-3505 158 2 𝐴3 𝐴3 PROPN iajs-3505 158 3 β€² β€² NOUN iajs-3505 158 4 ( ( PUNCT iajs-3505 158 5 𝛾2 𝛾2 VERB iajs-3505 158 6 βˆ— βˆ— NOUN iajs-3505 158 7 ) ) PUNCT iajs-3505 158 8 βˆ’ βˆ’ NOUN iajs-3505 159 1 𝐴1 𝐴1 PROPN iajs-3505 159 2 β€² β€² NUM iajs-3505 159 3 ( ( PUNCT iajs-3505 159 4 𝛾2 𝛾2 VERB iajs-3505 159 5 βˆ—)𝐴2(𝛾2 βˆ—)𝐴2(𝛾2 ADJ iajs-3505 159 6 βˆ— βˆ— NOUN iajs-3505 159 7 ) ) PUNCT iajs-3505 159 8 , , PUNCT iajs-3505 159 9 𝛀(𝛾2 𝛀(𝛾2 NOUN iajs-3505 159 10 βˆ— βˆ— NOUN iajs-3505 159 11 ) ) PUNCT iajs-3505 160 1 = = SYM iajs-3505 160 2 𝐴2 𝐴2 PROPN iajs-3505 160 3 β€² β€² NUM iajs-3505 160 4 ( ( PUNCT iajs-3505 160 5 𝛾2 𝛾2 VERB iajs-3505 160 6 βˆ—)√𝐴2(𝛾2 βˆ—)√𝐴2(𝛾2 ADV iajs-3505 160 7 βˆ— βˆ— NOUN iajs-3505 160 8 ) ) PUNCT iajs-3505 160 9 , , PUNCT iajs-3505 160 10 where where SCONJ iajs-3505 160 11 𝐴1 𝐴1 PROPN iajs-3505 160 12 β€² β€² NUM iajs-3505 160 13 ( ( PUNCT iajs-3505 160 14 𝛾2 𝛾2 VERB iajs-3505 160 15 βˆ— βˆ— NOUN iajs-3505 160 16 ) ) PUNCT iajs-3505 160 17 = = SYM iajs-3505 160 18 π‘£βˆ— π‘£βˆ— ADJ iajs-3505 160 19 , , PUNCT iajs-3505 160 20 ihjpas ihjpa VERB iajs-3505 160 21 . . PUNCT iajs-3505 161 1 2024 2024 NUM iajs-3505 161 2 , , PUNCT iajs-3505 161 3 37(4 37(4 NOUN iajs-3505 161 4 ) ) PUNCT iajs-3505 161 5 375 375 NUM iajs-3505 161 6 𝐴2 𝐴2 NOUN iajs-3505 161 7 β€² β€² NUM iajs-3505 161 8 ( ( PUNCT iajs-3505 161 9 𝛾2 𝛾2 VERB iajs-3505 161 10 βˆ— βˆ— NOUN iajs-3505 161 11 ) ) PUNCT iajs-3505 162 1 = = SYM iajs-3505 162 2 π‘€βˆ— π‘€βˆ— PROPN iajs-3505 162 3 βˆ’ βˆ’ PROPN iajs-3505 162 4 π‘£βˆ— π‘£βˆ— PROPN iajs-3505 162 5 , , PUNCT iajs-3505 162 6 𝐴3 𝐴3 PROPN iajs-3505 162 7 β€² β€² PROPN iajs-3505 162 8 ( ( PUNCT iajs-3505 162 9 𝛾2 𝛾2 VERB iajs-3505 162 10 βˆ— βˆ— NOUN iajs-3505 162 11 ) ) PUNCT iajs-3505 162 12 = = PRON iajs-3505 163 1 βˆ’2π‘€βˆ— βˆ’2π‘€βˆ— NOUN iajs-3505 163 2 βˆ’ βˆ’ X iajs-3505 163 3 π‘£βˆ—. π‘£βˆ—. ADV iajs-3505 163 4 hence hence ADV iajs-3505 163 5 , , PUNCT iajs-3505 163 6 condition condition NOUN iajs-3505 163 7 ( ( PUNCT iajs-3505 163 8 12 12 NUM iajs-3505 163 9 ) ) PUNCT iajs-3505 163 10 gives give VERB iajs-3505 163 11 𝛩(𝛾2 𝛩(𝛾2 PROPN iajs-3505 163 12 βˆ—)πœ“(𝛾2 βˆ—)πœ“(𝛾2 PROPN iajs-3505 163 13 βˆ— βˆ— NOUN iajs-3505 163 14 ) ) PUNCT iajs-3505 164 1 + + CCONJ iajs-3505 164 2 𝛀(𝛾2 𝛀(𝛾2 PROPN iajs-3505 164 3 βˆ—)πœ™(𝛾2 βˆ—)πœ™(𝛾2 NOUN iajs-3505 164 4 βˆ— βˆ— NOUN iajs-3505 164 5 ) ) PUNCT iajs-3505 165 1 = = SYM iajs-3505 166 1 2𝐴2(𝛾2 2𝐴2(𝛾2 NUM iajs-3505 166 2 βˆ—)[π‘£βˆ— βˆ—)[π‘£βˆ— NOUN iajs-3505 167 1 + + CCONJ iajs-3505 167 2 2π‘€βˆ— 2π‘€βˆ— NUM iajs-3505 167 3 + + CCONJ iajs-3505 167 4 2π‘£βˆ—π΄2(𝛾2 2π‘£βˆ—π΄2(𝛾2 NUM iajs-3505 167 5 βˆ— βˆ— NOUN iajs-3505 167 6 ) ) PUNCT iajs-3505 168 1 + + CCONJ iajs-3505 168 2 ( ( PUNCT iajs-3505 168 3 π‘€βˆ— π‘€βˆ— ADV iajs-3505 168 4 βˆ’ βˆ’ PROPN iajs-3505 168 5 π‘£βˆ—)2𝐴1(𝛾2 π‘£βˆ—)2𝐴1(𝛾2 ADJ iajs-3505 168 6 βˆ—)𝐴2(𝛾2 βˆ—)𝐴2(𝛾2 ADJ iajs-3505 168 7 βˆ— βˆ— NOUN iajs-3505 168 8 ) ) PUNCT iajs-3505 168 9 ] ] PUNCT iajs-3505 169 1 β‰  β‰  PROPN iajs-3505 169 2 0 0 X iajs-3505 169 3 . . PUNCT iajs-3505 170 1 that that PRON iajs-3505 170 2 means mean VERB iajs-3505 170 3 the the DET iajs-3505 170 4 hop hop NOUN iajs-3505 170 5 bifurcation bifurcation NOUN iajs-3505 170 6 has have AUX iajs-3505 170 7 occurred occur VERB iajs-3505 170 8 at at ADP iajs-3505 170 9 𝛾2 𝛾2 NOUN iajs-3505 170 10 βˆ—. βˆ—. PROPN iajs-3505 170 11 from from ADP iajs-3505 170 12 theorem theorem ADJ iajs-3505 170 13 3 3 NUM iajs-3505 170 14 , , PUNCT iajs-3505 170 15 the the DET iajs-3505 170 16 stability stability NOUN iajs-3505 170 17 condition condition NOUN iajs-3505 170 18 of of ADP iajs-3505 170 19 the the DET iajs-3505 170 20 stable stable ADJ iajs-3505 170 21 limit limit NOUN iajs-3505 170 22 cycle cycle NOUN iajs-3505 170 23 in in ADP iajs-3505 170 24 𝑅(𝑒,𝑣,𝑀 𝑅(𝑒,𝑣,𝑀 NOUN iajs-3505 170 25 ) ) PUNCT iajs-3505 170 26 3 3 NUM iajs-3505 170 27 is be AUX iajs-3505 170 28 presented present VERB iajs-3505 170 29 using use VERB iajs-3505 170 30 the the DET iajs-3505 170 31 coefficient coefficient NOUN iajs-3505 170 32 of of ADP iajs-3505 170 33 curvature curvature NOUN iajs-3505 170 34 of of ADP iajs-3505 170 35 the the DET iajs-3505 170 36 limit limit NOUN iajs-3505 170 37 cycle cycle NOUN iajs-3505 170 38 . . PUNCT iajs-3505 171 1 for for ADP iajs-3505 171 2 a a DET iajs-3505 171 3 detailed detailed ADJ iajs-3505 171 4 discussion discussion NOUN iajs-3505 171 5 , , PUNCT iajs-3505 171 6 we we PRON iajs-3505 171 7 refer refer VERB iajs-3505 171 8 to to ADP iajs-3505 171 9 [ [ X iajs-3505 171 10 18 18 NUM iajs-3505 171 11 ] ] PUNCT iajs-3505 171 12 . . PUNCT iajs-3505 172 1 theorem theorem VERB iajs-3505 172 2 3 3 NUM iajs-3505 172 3 the the DET iajs-3505 172 4 system system NOUN iajs-3505 172 5 ( ( PUNCT iajs-3505 172 6 1 1 X iajs-3505 172 7 ) ) PUNCT iajs-3505 172 8 has have VERB iajs-3505 172 9 a a DET iajs-3505 172 10 stable stable ADJ iajs-3505 172 11 limit limit NOUN iajs-3505 172 12 cycle cycle NOUN iajs-3505 172 13 in in ADP iajs-3505 172 14 𝑅(𝑒,𝑣,𝑀 𝑅(𝑒,𝑣,𝑀 NOUN iajs-3505 172 15 ) ) PUNCT iajs-3505 172 16 3 3 NUM iajs-3505 172 17 , , PUNCT iajs-3505 172 18 if if SCONJ iajs-3505 172 19 the the DET iajs-3505 172 20 following follow VERB iajs-3505 172 21 conditions condition NOUN iajs-3505 172 22 are be AUX iajs-3505 172 23 true true ADJ iajs-3505 172 24 : : PUNCT iajs-3505 172 25 π‘Ÿ π‘Ÿ NOUN iajs-3505 172 26 ( ( PUNCT iajs-3505 172 27 π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ VERB iajs-3505 172 28 βˆ—)2 βˆ—)2 PRON iajs-3505 172 29 β‰  β‰  PROPN iajs-3505 172 30 𝛼2(𝑒1+𝑒 𝛼2(𝑒1+𝑒 NUM iajs-3505 172 31 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š NOUN iajs-3505 172 32 ) ) PUNCT iajs-3505 172 33 ( ( PUNCT iajs-3505 172 34 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 172 35 βˆ—)2 βˆ—)2 PROPN iajs-3505 172 36 . . PUNCT iajs-3505 173 1 ( ( PUNCT iajs-3505 173 2 14 14 NUM iajs-3505 173 3 ) ) PUNCT iajs-3505 173 4 proof proof NOUN iajs-3505 173 5 : : PUNCT iajs-3505 173 6 by by ADP iajs-3505 173 7 shifting shift VERB iajs-3505 173 8 the the DET iajs-3505 173 9 𝑧3 𝑧3 NOUN iajs-3505 173 10 = = PUNCT iajs-3505 173 11 ( ( PUNCT iajs-3505 173 12 𝑒 𝑒 NOUN iajs-3505 173 13 βˆ— βˆ— NOUN iajs-3505 173 14 , , PUNCT iajs-3505 173 15 π‘£βˆ— π‘£βˆ— ADJ iajs-3505 173 16 , , PUNCT iajs-3505 173 17 π‘€βˆ— π‘€βˆ— PROPN iajs-3505 173 18 ) ) PUNCT iajs-3505 173 19 to to ADP iajs-3505 173 20 ( ( PUNCT iajs-3505 173 21 0 0 NUM iajs-3505 173 22 , , PUNCT iajs-3505 173 23 0 0 NUM iajs-3505 173 24 , , PUNCT iajs-3505 173 25 0 0 NUM iajs-3505 173 26 ) ) PUNCT iajs-3505 173 27 by by ADP iajs-3505 173 28 using use VERB iajs-3505 173 29 the the DET iajs-3505 173 30 following follow VERB iajs-3505 173 31 transformations transformation NOUN iajs-3505 173 32 𝑒 𝑒 X iajs-3505 173 33 = = SYM iajs-3505 173 34 𝑒1 𝑒1 NOUN iajs-3505 173 35 + + CCONJ iajs-3505 173 36 π‘’βˆ— π‘’βˆ— NOUN iajs-3505 173 37 , , PUNCT iajs-3505 173 38 𝑣 𝑣 X iajs-3505 173 39 = = SYM iajs-3505 173 40 𝑒2 𝑒2 PROPN iajs-3505 173 41 + + CCONJ iajs-3505 173 42 𝑣 𝑣 ADP iajs-3505 173 43 βˆ— βˆ— NOUN iajs-3505 173 44 , , PUNCT iajs-3505 173 45 𝑀 𝑀 ADP iajs-3505 173 46 = = SYM iajs-3505 173 47 𝑒3 𝑒3 PROPN iajs-3505 173 48 + + PROPN iajs-3505 173 49 𝑀 𝑀 ADP iajs-3505 173 50 βˆ—. βˆ—. PROPN iajs-3505 173 51 then then ADV iajs-3505 173 52 the the DET iajs-3505 173 53 system system NOUN iajs-3505 173 54 ( ( PUNCT iajs-3505 173 55 1 1 X iajs-3505 173 56 ) ) PUNCT iajs-3505 173 57 becomes become VERB iajs-3505 173 58 : : PUNCT iajs-3505 173 59 𝑑𝑒1 𝑑𝑒1 NOUN iajs-3505 173 60 𝑑𝑑 𝑑𝑑 ADP iajs-3505 173 61 = = PUNCT iajs-3505 173 62 π‘Ÿ(𝑒1 π‘Ÿ(𝑒1 PROPN iajs-3505 173 63 + + CCONJ iajs-3505 173 64 𝑒 𝑒 PROPN iajs-3505 173 65 βˆ— βˆ— NOUN iajs-3505 173 66 ) ) PUNCT iajs-3505 173 67 ( ( PUNCT iajs-3505 173 68 π‘Ž1 π‘Ž1 NOUN iajs-3505 173 69 + + NUM iajs-3505 173 70 𝑀0 𝑀0 PROPN iajs-3505 173 71 βˆ’ βˆ’ PROPN iajs-3505 173 72 𝑒3 𝑒3 PROPN iajs-3505 173 73 βˆ’ βˆ’ PROPN iajs-3505 173 74 𝑀 𝑀 ADP iajs-3505 173 75 βˆ— βˆ— NOUN iajs-3505 173 76 ) ) PUNCT iajs-3505 174 1 βˆ’ βˆ’ ADP iajs-3505 174 2 𝛼1(𝑒1 𝛼1(𝑒1 NOUN iajs-3505 174 3 + + CCONJ iajs-3505 174 4 𝑒 𝑒 PROPN iajs-3505 174 5 βˆ—)(𝑒2 βˆ—)(𝑒2 PROPN iajs-3505 174 6 + + ADP iajs-3505 174 7 𝑣 𝑣 ADP iajs-3505 174 8 βˆ—)(1 βˆ—)(1 PRON iajs-3505 174 9 βˆ’ βˆ’ NOUN iajs-3505 174 10 π‘š π‘š NOUN iajs-3505 174 11 ) ) PUNCT iajs-3505 174 12 βˆ’ βˆ’ PROPN iajs-3505 174 13 ( ( PUNCT iajs-3505 174 14 𝑒1 𝑒1 PROPN iajs-3505 174 15 + + CCONJ iajs-3505 174 16 𝑒 𝑒 X iajs-3505 174 17 βˆ—)𝛿1 βˆ—)𝛿1 NOUN iajs-3505 174 18 𝑑𝑒2 𝑑𝑒2 ADJ iajs-3505 174 19 𝑑𝑑 𝑑𝑑 ADP iajs-3505 174 20 = = PROPN iajs-3505 174 21 𝛼2(𝑒1 𝛼2(𝑒1 PROPN iajs-3505 174 22 + + NUM iajs-3505 174 23 𝑒 𝑒 PROPN iajs-3505 174 24 βˆ—)(𝑒2 βˆ—)(𝑒2 PROPN iajs-3505 174 25 + + ADP iajs-3505 174 26 𝑣 𝑣 ADP iajs-3505 174 27 βˆ—)(1 βˆ—)(1 PRON iajs-3505 174 28 βˆ’ βˆ’ NOUN iajs-3505 174 29 π‘š π‘š NOUN iajs-3505 174 30 ) ) PUNCT iajs-3505 174 31 ( ( PUNCT iajs-3505 174 32 π‘Ž2 π‘Ž2 NOUN iajs-3505 174 33 + + NUM iajs-3505 174 34 𝑀0 𝑀0 PROPN iajs-3505 174 35 βˆ’ βˆ’ PROPN iajs-3505 174 36 𝑒3 𝑒3 PROPN iajs-3505 174 37 βˆ’ βˆ’ PROPN iajs-3505 174 38 𝑀 𝑀 ADP iajs-3505 174 39 βˆ— βˆ— NOUN iajs-3505 174 40 ) ) PUNCT iajs-3505 174 41 βˆ’ βˆ’ NOUN iajs-3505 175 1 𝛿2(𝑒2 𝛿2(𝑒2 NUM iajs-3505 175 2 + + CCONJ iajs-3505 175 3 𝑣 𝑣 DET iajs-3505 175 4 βˆ— βˆ— NOUN iajs-3505 175 5 ) ) PUNCT iajs-3505 175 6 𝑑𝑒3 𝑑𝑒3 NOUN iajs-3505 175 7 𝑑𝑑 𝑑𝑑 ADP iajs-3505 175 8 = = PUNCT iajs-3505 175 9 𝑠[𝑀0 𝑠[𝑀0 NOUN iajs-3505 175 10 βˆ’ βˆ’ PROPN iajs-3505 175 11 ( ( PUNCT iajs-3505 175 12 𝑒3 𝑒3 NOUN iajs-3505 175 13 + + CCONJ iajs-3505 175 14 𝑀 𝑀 ADP iajs-3505 175 15 βˆ— βˆ— NOUN iajs-3505 175 16 ) ) PUNCT iajs-3505 175 17 ] ] PUNCT iajs-3505 176 1 + + CCONJ iajs-3505 176 2 𝑑(𝑒1 𝑑(𝑒1 PRON iajs-3505 176 3 + + CCONJ iajs-3505 176 4 𝑒 𝑒 NOUN iajs-3505 176 5 βˆ— βˆ— NOUN iajs-3505 176 6 ) ) PUNCT iajs-3505 176 7 βˆ’ βˆ’ NUM iajs-3505 176 8 𝛾(𝑒3 𝛾(𝑒3 NOUN iajs-3505 176 9 + + CCONJ iajs-3505 176 10 𝑀 𝑀 X iajs-3505 176 11 βˆ— βˆ— NOUN iajs-3505 176 12 ) ) PUNCT iajs-3505 176 13 βˆ’ βˆ’ PROPN iajs-3505 176 14 𝛾1(𝑒1 𝛾1(𝑒1 NOUN iajs-3505 176 15 + + NOUN iajs-3505 176 16 𝑒 𝑒 PROPN iajs-3505 176 17 βˆ—)(𝑒3 βˆ—)(𝑒3 PROPN iajs-3505 176 18 + + CCONJ iajs-3505 176 19 𝑀 𝑀 NOUN iajs-3505 176 20 βˆ— βˆ— NOUN iajs-3505 176 21 ) ) PUNCT iajs-3505 176 22 βˆ’ βˆ’ ADP iajs-3505 176 23 𝛾2(𝑒2 𝛾2(𝑒2 NOUN iajs-3505 176 24 + + CCONJ iajs-3505 176 25 𝑣 𝑣 ADP iajs-3505 176 26 βˆ—)(𝑒3 βˆ—)(𝑒3 PROPN iajs-3505 176 27 + + ADP iajs-3505 176 28 𝑀 𝑀 ADP iajs-3505 176 29 βˆ— βˆ— NOUN iajs-3505 176 30 ) ) PUNCT iajs-3505 176 31 , , PUNCT iajs-3505 176 32 where where SCONJ iajs-3505 176 33 the the DET iajs-3505 176 34 nonlinear nonlinear ADJ iajs-3505 176 35 part part NOUN iajs-3505 176 36 of of ADP iajs-3505 176 37 the the DET iajs-3505 176 38 above above ADJ iajs-3505 176 39 system system NOUN iajs-3505 176 40 is be AUX iajs-3505 176 41 presented present VERB iajs-3505 176 42 in in ADP iajs-3505 176 43 the the DET iajs-3505 176 44 following follow VERB iajs-3505 176 45 matrix matrix NOUN iajs-3505 176 46 is be AUX iajs-3505 176 47 ℧ ℧ PROPN iajs-3505 176 48 = = PUNCT iajs-3505 176 49 ( ( PUNCT iajs-3505 176 50 ℧ ℧ PROPN iajs-3505 176 51 1 1 NUM iajs-3505 176 52 ℧ ℧ SYM iajs-3505 176 53 2 2 NUM iajs-3505 176 54 ℧ ℧ NOUN iajs-3505 176 55 3 3 NUM iajs-3505 176 56 ) ) PUNCT iajs-3505 176 57 = = PRON iajs-3505 176 58 ( ( PUNCT iajs-3505 176 59 π‘Ÿ(𝑒1 π‘Ÿ(𝑒1 NOUN iajs-3505 176 60 + + CCONJ iajs-3505 176 61 𝑒 𝑒 NOUN iajs-3505 176 62 βˆ— βˆ— NOUN iajs-3505 176 63 ) ) PUNCT iajs-3505 176 64 ( ( PUNCT iajs-3505 176 65 π‘Ž1 π‘Ž1 NOUN iajs-3505 176 66 + + NUM iajs-3505 176 67 𝑀0 𝑀0 PROPN iajs-3505 176 68 βˆ’ βˆ’ PROPN iajs-3505 176 69 𝑒3 𝑒3 PROPN iajs-3505 176 70 βˆ’ βˆ’ PROPN iajs-3505 176 71 𝑀 𝑀 ADP iajs-3505 176 72 βˆ— βˆ— NOUN iajs-3505 176 73 ) ) PUNCT iajs-3505 177 1 βˆ’ βˆ’ ADP iajs-3505 177 2 𝛼1(1 𝛼1(1 ADJ iajs-3505 177 3 βˆ’ βˆ’ PROPN iajs-3505 177 4 π‘š)𝑒1𝑒2 π‘š)𝑒1𝑒2 PUNCT iajs-3505 177 5 𝛼2(𝑒1 𝛼2(𝑒1 NOUN iajs-3505 177 6 + + CCONJ iajs-3505 177 7 𝑒 𝑒 PROPN iajs-3505 177 8 βˆ—)(𝑒2 βˆ—)(𝑒2 PROPN iajs-3505 177 9 + + ADP iajs-3505 177 10 𝑣 𝑣 ADP iajs-3505 177 11 βˆ—)(1 βˆ—)(1 PRON iajs-3505 177 12 βˆ’ βˆ’ NOUN iajs-3505 177 13 π‘š π‘š NOUN iajs-3505 177 14 ) ) PUNCT iajs-3505 177 15 ( ( PUNCT iajs-3505 177 16 π‘Ž2 π‘Ž2 NOUN iajs-3505 177 17 + + NUM iajs-3505 177 18 𝑀0 𝑀0 PROPN iajs-3505 177 19 βˆ’ βˆ’ PROPN iajs-3505 177 20 𝑒3 𝑒3 PROPN iajs-3505 177 21 βˆ’ βˆ’ PROPN iajs-3505 177 22 𝑀 𝑀 ADP iajs-3505 177 23 βˆ— βˆ— NOUN iajs-3505 177 24 ) ) PUNCT iajs-3505 178 1 βˆ’π›Ύ1𝑒1𝑒3 βˆ’π›Ύ1𝑒1𝑒3 NOUN iajs-3505 178 2 βˆ’ βˆ’ PROPN iajs-3505 179 1 𝛾2𝑒2𝑒3 𝛾2𝑒2𝑒3 PROPN iajs-3505 179 2 ) ) PUNCT iajs-3505 179 3 we we PRON iajs-3505 179 4 derive derive VERB iajs-3505 179 5 the the DET iajs-3505 179 6 following follow VERB iajs-3505 179 7 characteristic characteristic ADJ iajs-3505 179 8 quantities quantity NOUN iajs-3505 179 9 from from ADP iajs-3505 179 10 the the DET iajs-3505 179 11 nonlinear nonlinear ADJ iajs-3505 179 12 part part NOUN iajs-3505 179 13 : : PUNCT iajs-3505 179 14 𝑔20 𝑔20 NOUN iajs-3505 179 15 0 0 NUM iajs-3505 179 16 = = SYM iajs-3505 179 17 1 1 NUM iajs-3505 179 18 4 4 NUM iajs-3505 179 19 { { PUNCT iajs-3505 179 20 πœ•2 πœ•2 NOUN iajs-3505 179 21 ℧ ℧ NOUN iajs-3505 179 22 1 1 NUM iajs-3505 179 23 πœ•π‘’1 πœ•π‘’1 ADV iajs-3505 179 24 2 2 NUM iajs-3505 179 25 βˆ’ βˆ’ NOUN iajs-3505 179 26 πœ•2 πœ•2 SYM iajs-3505 179 27 ℧ ℧ NOUN iajs-3505 179 28 1 1 NUM iajs-3505 179 29 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 179 30 2 2 NUM iajs-3505 179 31 + + SYM iajs-3505 179 32 2 2 NUM iajs-3505 179 33 πœ•2 πœ•2 SYM iajs-3505 179 34 ℧ ℧ NOUN iajs-3505 179 35 2 2 NUM iajs-3505 179 36 πœ•π‘’1πœ•π‘’2 πœ•π‘’1πœ•π‘’2 NOUN iajs-3505 179 37 + + CCONJ iajs-3505 179 38 𝑖 𝑖 SYM iajs-3505 179 39 ( ( PUNCT iajs-3505 179 40 πœ•2 πœ•2 NOUN iajs-3505 179 41 ℧ ℧ SYM iajs-3505 179 42 2 2 NUM iajs-3505 179 43 πœ•π‘’1 πœ•π‘’1 ADV iajs-3505 179 44 2 2 NUM iajs-3505 179 45 βˆ’ βˆ’ NOUN iajs-3505 179 46 πœ•2 πœ•2 SYM iajs-3505 179 47 ℧ ℧ NOUN iajs-3505 179 48 2 2 NUM iajs-3505 179 49 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 179 50 2 2 NUM iajs-3505 179 51 βˆ’ βˆ’ NUM iajs-3505 179 52 2 2 NUM iajs-3505 179 53 πœ•2 πœ•2 SYM iajs-3505 179 54 ℧ ℧ NOUN iajs-3505 179 55 1 1 NUM iajs-3505 179 56 πœ•π‘’1πœ•π‘’2 πœ•π‘’1πœ•π‘’2 ADJ iajs-3505 179 57 ) ) PUNCT iajs-3505 179 58 } } PUNCT iajs-3505 179 59 = = SYM iajs-3505 179 60 1 1 NUM iajs-3505 179 61 2 2 NUM iajs-3505 179 62 { { PUNCT iajs-3505 179 63 𝛼2(1βˆ’π‘š 𝛼2(1βˆ’π‘š NUM iajs-3505 179 64 ) ) PUNCT iajs-3505 179 65 ( ( PUNCT iajs-3505 179 66 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 179 67 βˆ— βˆ— NOUN iajs-3505 179 68 ) ) PUNCT iajs-3505 179 69 βˆ’ βˆ’ ADP iajs-3505 179 70 𝛼1(1 𝛼1(1 ADJ iajs-3505 179 71 βˆ’ βˆ’ NUM iajs-3505 179 72 π‘š)𝑖 π‘š)𝑖 PUNCT iajs-3505 179 73 } } PUNCT iajs-3505 179 74 , , PUNCT iajs-3505 179 75 𝑔11 𝑔11 PROPN iajs-3505 179 76 0 0 NUM iajs-3505 180 1 = = SYM iajs-3505 180 2 1 1 NUM iajs-3505 180 3 4 4 NUM iajs-3505 180 4 { { PUNCT iajs-3505 180 5 πœ•2 πœ•2 NOUN iajs-3505 180 6 ℧ ℧ NOUN iajs-3505 180 7 1 1 NUM iajs-3505 180 8 πœ•π‘’1 πœ•π‘’1 ADV iajs-3505 180 9 2 2 NUM iajs-3505 180 10 + + SYM iajs-3505 180 11 πœ•2 πœ•2 SYM iajs-3505 180 12 ℧ ℧ SYM iajs-3505 180 13 1 1 NUM iajs-3505 180 14 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 180 15 2 2 NUM iajs-3505 180 16 + + NOUN iajs-3505 180 17 𝑖 𝑖 SYM iajs-3505 180 18 ( ( PUNCT iajs-3505 180 19 πœ•2 πœ•2 NOUN iajs-3505 180 20 ℧ ℧ SYM iajs-3505 180 21 2 2 NUM iajs-3505 180 22 πœ•π‘’1 πœ•π‘’1 ADV iajs-3505 180 23 2 2 NUM iajs-3505 180 24 + + SYM iajs-3505 180 25 πœ•2 πœ•2 SYM iajs-3505 180 26 ℧ ℧ SYM iajs-3505 180 27 2 2 NUM iajs-3505 180 28 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 180 29 2 2 NUM iajs-3505 180 30 ) ) PUNCT iajs-3505 180 31 } } PUNCT iajs-3505 180 32 = = SYM iajs-3505 180 33 0 0 NUM iajs-3505 180 34 , , PUNCT iajs-3505 180 35 𝐺110 𝐺110 NOUN iajs-3505 180 36 0 0 NUM iajs-3505 180 37 = = SYM iajs-3505 180 38 1 1 NUM iajs-3505 180 39 2 2 NUM iajs-3505 180 40 { { PUNCT iajs-3505 180 41 πœ•2 πœ•2 NOUN iajs-3505 180 42 ℧ ℧ NOUN iajs-3505 180 43 1 1 NUM iajs-3505 180 44 πœ•π‘’1πœ•π‘’3 πœ•π‘’1πœ•π‘’3 NOUN iajs-3505 180 45 + + SYM iajs-3505 180 46 πœ•2 πœ•2 SYM iajs-3505 180 47 ℧ ℧ SYM iajs-3505 180 48 2 2 NUM iajs-3505 180 49 πœ•π‘’2πœ•π‘’3 πœ•π‘’2πœ•π‘’3 NOUN iajs-3505 180 50 + + X iajs-3505 180 51 𝑖 𝑖 SYM iajs-3505 180 52 ( ( PUNCT iajs-3505 180 53 πœ•2 πœ•2 NOUN iajs-3505 180 54 ℧ ℧ SYM iajs-3505 180 55 2 2 NUM iajs-3505 180 56 πœ•π‘’1πœ•π‘’3 πœ•π‘’1πœ•π‘’3 NOUN iajs-3505 180 57 βˆ’ βˆ’ NOUN iajs-3505 180 58 πœ•2 πœ•2 NOUN iajs-3505 180 59 ℧ ℧ PROPN iajs-3505 180 60 1 1 NUM iajs-3505 180 61 πœ•π‘’2πœ•π‘’3 πœ•π‘’2πœ•π‘’3 NOUN iajs-3505 180 62 ) ) PUNCT iajs-3505 180 63 } } PUNCT iajs-3505 180 64 = = SYM iajs-3505 180 65 1 1 NUM iajs-3505 180 66 2 2 NUM iajs-3505 180 67 { { PUNCT iajs-3505 180 68 π‘Ÿ π‘Ÿ NOUN iajs-3505 180 69 ( ( PUNCT iajs-3505 180 70 π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 71 βˆ—)2 βˆ—)2 ADJ iajs-3505 180 72 + + CCONJ iajs-3505 180 73 𝛼2(𝑒1+𝑒 𝛼2(𝑒1+𝑒 NUM iajs-3505 180 74 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š NOUN iajs-3505 180 75 ) ) PUNCT iajs-3505 180 76 ( ( PUNCT iajs-3505 180 77 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 78 βˆ—)2 βˆ—)2 PROPN iajs-3505 180 79 + + CCONJ iajs-3505 180 80 𝑖 𝑖 X iajs-3505 180 81 ( ( PUNCT iajs-3505 180 82 𝛼2(𝑒2+𝑣 𝛼2(𝑒2+𝑣 X iajs-3505 180 83 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š ADV iajs-3505 180 84 ) ) PUNCT iajs-3505 180 85 ( ( PUNCT iajs-3505 180 86 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 87 βˆ—)2 βˆ—)2 ADV iajs-3505 180 88 ) ) PUNCT iajs-3505 180 89 } } PUNCT iajs-3505 180 90 , , PUNCT iajs-3505 180 91 𝐺101 𝐺101 PROPN iajs-3505 180 92 0 0 NUM iajs-3505 180 93 = = SYM iajs-3505 180 94 1 1 NUM iajs-3505 180 95 2 2 NUM iajs-3505 180 96 { { PUNCT iajs-3505 180 97 πœ•2 πœ•2 NOUN iajs-3505 180 98 ℧ ℧ SYM iajs-3505 180 99 1 1 NUM iajs-3505 180 100 πœ•π‘’1πœ•π‘’3 πœ•π‘’1πœ•π‘’3 PROPN iajs-3505 180 101 βˆ’ βˆ’ NOUN iajs-3505 180 102 πœ•2 πœ•2 NOUN iajs-3505 180 103 ℧ ℧ NOUN iajs-3505 180 104 2 2 NUM iajs-3505 180 105 πœ•π‘’2πœ•π‘’3 πœ•π‘’2πœ•π‘’3 NOUN iajs-3505 180 106 + + X iajs-3505 180 107 𝑖 𝑖 SYM iajs-3505 180 108 ( ( PUNCT iajs-3505 180 109 πœ•2 πœ•2 NOUN iajs-3505 180 110 ℧ ℧ SYM iajs-3505 180 111 2 2 NUM iajs-3505 180 112 πœ•π‘’1πœ•π‘’3 πœ•π‘’1πœ•π‘’3 NOUN iajs-3505 180 113 + + X iajs-3505 180 114 πœ•2 πœ•2 SYM iajs-3505 180 115 ℧ ℧ PROPN iajs-3505 180 116 1 1 NUM iajs-3505 180 117 πœ•π‘’2πœ•π‘’3 πœ•π‘’2πœ•π‘’3 NOUN iajs-3505 180 118 ) ) PUNCT iajs-3505 180 119 } } PUNCT iajs-3505 180 120 = = SYM iajs-3505 180 121 1 1 NUM iajs-3505 180 122 2 2 NUM iajs-3505 180 123 { { PUNCT iajs-3505 180 124 π‘Ÿ π‘Ÿ NOUN iajs-3505 180 125 ( ( PUNCT iajs-3505 180 126 π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 127 βˆ—)2 βˆ—)2 ADV iajs-3505 180 128 βˆ’ βˆ’ PROPN iajs-3505 180 129 𝛼2(𝑒1+𝑒 𝛼2(𝑒1+𝑒 NUM iajs-3505 180 130 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š NOUN iajs-3505 180 131 ) ) PUNCT iajs-3505 180 132 ( ( PUNCT iajs-3505 180 133 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 134 βˆ—)2 βˆ—)2 PROPN iajs-3505 180 135 + + CCONJ iajs-3505 180 136 𝑖 𝑖 X iajs-3505 180 137 ( ( PUNCT iajs-3505 180 138 𝛼2(𝑒2+𝑣 𝛼2(𝑒2+𝑣 X iajs-3505 180 139 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š ADV iajs-3505 180 140 ) ) PUNCT iajs-3505 180 141 ( ( PUNCT iajs-3505 180 142 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 180 143 βˆ—)2 βˆ—)2 ADV iajs-3505 180 144 ) ) PUNCT iajs-3505 180 145 } } PUNCT iajs-3505 180 146 , , PUNCT iajs-3505 180 147 π‘Š11 π‘Š11 PROPN iajs-3505 180 148 0 0 NUM iajs-3505 180 149 = = SYM iajs-3505 181 1 βˆ’ βˆ’ PROPN iajs-3505 181 2 1 1 NUM iajs-3505 181 3 4πœ†3(π‘Ž1(π‘˜ 4πœ†3(π‘Ž1(π‘˜ NUM iajs-3505 181 4 βˆ— βˆ— NOUN iajs-3505 181 5 ) ) PUNCT iajs-3505 181 6 ( ( PUNCT iajs-3505 181 7 πœ•2 πœ•2 NOUN iajs-3505 181 8 ℧ ℧ SYM iajs-3505 181 9 3 3 NUM iajs-3505 181 10 πœ•π‘’1 πœ•π‘’1 ADP iajs-3505 181 11 2 2 NUM iajs-3505 181 12 + + SYM iajs-3505 181 13 πœ•2 πœ•2 NUM iajs-3505 181 14 ℧ ℧ SYM iajs-3505 181 15 3 3 NUM iajs-3505 181 16 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 181 17 2 2 NUM iajs-3505 181 18 ) ) PUNCT iajs-3505 181 19 = = SYM iajs-3505 181 20 0 0 NUM iajs-3505 181 21 , , PUNCT iajs-3505 181 22 π‘Š20 π‘Š20 NOUN iajs-3505 181 23 0 0 NUM iajs-3505 181 24 = = SYM iajs-3505 181 25 βˆ’ βˆ’ PROPN iajs-3505 181 26 1 1 NUM iajs-3505 181 27 4πœ†3(π‘Ž1(π‘˜ 4πœ†3(π‘Ž1(π‘˜ NUM iajs-3505 181 28 βˆ— βˆ— NOUN iajs-3505 181 29 ) ) PUNCT iajs-3505 181 30 ( ( PUNCT iajs-3505 181 31 πœ•2 πœ•2 NOUN iajs-3505 181 32 ℧ ℧ SYM iajs-3505 181 33 3 3 NUM iajs-3505 181 34 πœ•π‘’1 πœ•π‘’1 ADP iajs-3505 181 35 2 2 NUM iajs-3505 181 36 + + SYM iajs-3505 181 37 πœ•2 πœ•2 NUM iajs-3505 181 38 ℧ ℧ SYM iajs-3505 181 39 3 3 NUM iajs-3505 181 40 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 181 41 2 2 NUM iajs-3505 181 42 βˆ’ βˆ’ NOUN iajs-3505 181 43 2𝑖 2𝑖 NOUN iajs-3505 181 44 πœ•2 πœ•2 SYM iajs-3505 181 45 ℧ ℧ NOUN iajs-3505 181 46 3 3 NUM iajs-3505 181 47 πœ•π‘’1πœ•π‘’2 πœ•π‘’1πœ•π‘’2 ADJ iajs-3505 181 48 ) ) PUNCT iajs-3505 182 1 = = SYM iajs-3505 182 2 0 0 NUM iajs-3505 182 3 , , PUNCT iajs-3505 182 4 ihjpas ihjpa VERB iajs-3505 182 5 . . PUNCT iajs-3505 183 1 2024 2024 NUM iajs-3505 183 2 , , PUNCT iajs-3505 183 3 37(4 37(4 NOUN iajs-3505 183 4 ) ) PUNCT iajs-3505 183 5 376 376 NUM iajs-3505 184 1 𝐺21 𝐺21 PROPN iajs-3505 184 2 0 0 NUM iajs-3505 185 1 = = SYM iajs-3505 185 2 1 1 NUM iajs-3505 185 3 8 8 NUM iajs-3505 185 4 { { PUNCT iajs-3505 185 5 πœ•3 πœ•3 NOUN iajs-3505 185 6 ℧ ℧ NOUN iajs-3505 185 7 1 1 NUM iajs-3505 185 8 πœ•π‘’1 πœ•π‘’1 ADP iajs-3505 185 9 3 3 NUM iajs-3505 185 10 + + SYM iajs-3505 185 11 πœ•3 πœ•3 NOUN iajs-3505 185 12 ℧ ℧ NOUN iajs-3505 185 13 1 1 NUM iajs-3505 185 14 πœ•π‘’1πœ•π‘’2 πœ•π‘’1πœ•π‘’2 NUM iajs-3505 185 15 2 2 NUM iajs-3505 185 16 + + NUM iajs-3505 185 17 πœ•3 πœ•3 NOUN iajs-3505 185 18 ℧ ℧ NOUN iajs-3505 185 19 2 2 NUM iajs-3505 185 20 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 185 21 3 3 NUM iajs-3505 185 22 + + SYM iajs-3505 185 23 πœ•3 πœ•3 NOUN iajs-3505 185 24 ℧ ℧ SYM iajs-3505 185 25 2 2 NUM iajs-3505 185 26 πœ•π‘’1 πœ•π‘’1 ADV iajs-3505 185 27 2πœ•π‘’2 2πœ•π‘’2 NUM iajs-3505 185 28 + + CCONJ iajs-3505 185 29 𝑖 𝑖 SYM iajs-3505 185 30 ( ( PUNCT iajs-3505 185 31 πœ•3 πœ•3 NOUN iajs-3505 185 32 ℧ ℧ NOUN iajs-3505 185 33 2 2 NUM iajs-3505 185 34 πœ•π‘’1 πœ•π‘’1 ADP iajs-3505 185 35 3 3 NUM iajs-3505 185 36 + + SYM iajs-3505 185 37 πœ•3 πœ•3 NOUN iajs-3505 185 38 ℧ ℧ NOUN iajs-3505 185 39 2 2 NUM iajs-3505 185 40 πœ•π‘’1πœ•π‘’2 πœ•π‘’1πœ•π‘’2 ADJ iajs-3505 185 41 2 2 NUM iajs-3505 185 42 βˆ’ βˆ’ NOUN iajs-3505 185 43 πœ•3 πœ•3 NOUN iajs-3505 185 44 ℧ ℧ NOUN iajs-3505 185 45 1 1 NUM iajs-3505 185 46 πœ•π‘’2 πœ•π‘’2 NOUN iajs-3505 185 47 3 3 NUM iajs-3505 185 48 βˆ’ βˆ’ NOUN iajs-3505 185 49 πœ•3 πœ•3 NOUN iajs-3505 185 50 ℧ ℧ NOUN iajs-3505 185 51 1 1 NUM iajs-3505 185 52 πœ•π‘’1 πœ•π‘’1 PROPN iajs-3505 185 53 2πœ•π‘’2 2πœ•π‘’2 NUM iajs-3505 185 54 ) ) PUNCT iajs-3505 185 55 } } PUNCT iajs-3505 185 56 = = SYM iajs-3505 185 57 0 0 NUM iajs-3505 185 58 , , PUNCT iajs-3505 185 59 thus thus ADV iajs-3505 185 60 , , PUNCT iajs-3505 185 61 the the DET iajs-3505 185 62 coefficient coefficient NOUN iajs-3505 185 63 of of ADP iajs-3505 185 64 the the DET iajs-3505 185 65 curvature curvature NOUN iajs-3505 185 66 of of ADP iajs-3505 185 67 the the DET iajs-3505 185 68 limit limit NOUN iajs-3505 185 69 cycle cycle NOUN iajs-3505 185 70 of of ADP iajs-3505 185 71 the the DET iajs-3505 185 72 dopz dopz ADJ iajs-3505 185 73 system system NOUN iajs-3505 185 74 ( ( PUNCT iajs-3505 185 75 1 1 X iajs-3505 185 76 ) ) PUNCT iajs-3505 185 77 is be AUX iajs-3505 185 78 given give VERB iajs-3505 185 79 by by ADP iajs-3505 185 80 𝜎1 𝜎1 PROPN iajs-3505 185 81 0 0 NUM iajs-3505 185 82 = = SYM iajs-3505 186 1 𝑅𝑒 𝑅𝑒 VERB iajs-3505 186 2 { { PUNCT iajs-3505 186 3 𝑔20 𝑔20 PROPN iajs-3505 186 4 0 0 NUM iajs-3505 186 5 𝑔11 𝑔11 NOUN iajs-3505 186 6 0 0 NUM iajs-3505 186 7 4 4 NUM iajs-3505 186 8 𝑖 𝑖 SYM iajs-3505 186 9 + + X iajs-3505 186 10 𝐺110 𝐺110 NOUN iajs-3505 186 11 0 0 PUNCT iajs-3505 187 1 π‘Š11 π‘Š11 NOUN iajs-3505 187 2 0 0 NUM iajs-3505 188 1 + + CCONJ iajs-3505 188 2 𝐺21 𝐺21 PROPN iajs-3505 188 3 0 0 PUNCT iajs-3505 189 1 + + NOUN iajs-3505 189 2 𝐺101 𝐺101 X iajs-3505 189 3 0 0 NUM iajs-3505 189 4 π‘Š20 π‘Š20 NUM iajs-3505 189 5 0 0 NUM iajs-3505 189 6 2 2 NUM iajs-3505 189 7 } } PUNCT iajs-3505 189 8 , , PUNCT iajs-3505 189 9 𝜎1 𝜎1 ADJ iajs-3505 189 10 0 0 PUNCT iajs-3505 190 1 = = SYM iajs-3505 191 1 𝑅𝑒 𝑅𝑒 VERB iajs-3505 191 2 1 1 NUM iajs-3505 191 3 4 4 NUM iajs-3505 191 4 { { PUNCT iajs-3505 191 5 π‘Ÿ π‘Ÿ NOUN iajs-3505 191 6 ( ( PUNCT iajs-3505 191 7 π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 191 8 βˆ—)2 βˆ—)2 ADV iajs-3505 191 9 βˆ’ βˆ’ PROPN iajs-3505 191 10 𝛼2(𝑒1+𝑒 𝛼2(𝑒1+𝑒 NUM iajs-3505 191 11 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š NOUN iajs-3505 191 12 ) ) PUNCT iajs-3505 191 13 ( ( PUNCT iajs-3505 191 14 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 191 15 βˆ—)2 βˆ—)2 PROPN iajs-3505 191 16 + + CCONJ iajs-3505 191 17 𝑖 𝑖 X iajs-3505 191 18 ( ( PUNCT iajs-3505 191 19 𝛼2(𝑒2+𝑣 𝛼2(𝑒2+𝑣 X iajs-3505 191 20 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š ADV iajs-3505 191 21 ) ) PUNCT iajs-3505 191 22 ( ( PUNCT iajs-3505 191 23 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 191 24 βˆ—)2 βˆ—)2 ADV iajs-3505 191 25 ) ) PUNCT iajs-3505 191 26 } } PUNCT iajs-3505 192 1 = = SYM iajs-3505 192 2 π‘Ÿ π‘Ÿ NOUN iajs-3505 192 3 ( ( PUNCT iajs-3505 192 4 π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž1+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 192 5 βˆ—)2 βˆ—)2 ADV iajs-3505 192 6 βˆ’ βˆ’ PROPN iajs-3505 192 7 𝛼2(𝑒1+𝑒 𝛼2(𝑒1+𝑒 NUM iajs-3505 192 8 βˆ—)(1βˆ’π‘š βˆ—)(1βˆ’π‘š NOUN iajs-3505 192 9 ) ) PUNCT iajs-3505 192 10 ( ( PUNCT iajs-3505 192 11 π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ π‘Ž2+𝑀0βˆ’π‘’3βˆ’π‘€ NOUN iajs-3505 192 12 βˆ—)2 βˆ—)2 PROPN iajs-3505 192 13 . . PUNCT iajs-3505 193 1 thus thus ADV iajs-3505 193 2 , , PUNCT iajs-3505 193 3 condition condition NOUN iajs-3505 193 4 ( ( PUNCT iajs-3505 193 5 14 14 NUM iajs-3505 193 6 ) ) PUNCT iajs-3505 193 7 guarantees guarantee VERB iajs-3505 193 8 that that SCONJ iajs-3505 193 9 system system NOUN iajs-3505 193 10 ( ( PUNCT iajs-3505 193 11 1 1 X iajs-3505 193 12 ) ) PUNCT iajs-3505 193 13 has have VERB iajs-3505 193 14 a a DET iajs-3505 193 15 stable stable ADJ iajs-3505 193 16 limit limit NOUN iajs-3505 193 17 cycle cycle NOUN iajs-3505 193 18 . . PUNCT iajs-3505 194 1 5 5 NUM iajs-3505 194 2 . . X iajs-3505 194 3 numerical numerical ADJ iajs-3505 194 4 simulations simulation NOUN iajs-3505 194 5 and and CCONJ iajs-3505 194 6 discussion discussion NOUN iajs-3505 194 7 numerical numerical PROPN iajs-3505 194 8 simulations simulation NOUN iajs-3505 194 9 support support VERB iajs-3505 194 10 our our PRON iajs-3505 194 11 theoretical theoretical ADJ iajs-3505 194 12 predictions prediction NOUN iajs-3505 194 13 and and CCONJ iajs-3505 194 14 reveal reveal VERB iajs-3505 194 15 the the DET iajs-3505 194 16 system system NOUN iajs-3505 194 17 ’s ’s PART iajs-3505 194 18 numerous numerous ADJ iajs-3505 194 19 dynamics dynamic NOUN iajs-3505 194 20 ( ( PUNCT iajs-3505 194 21 1 1 NUM iajs-3505 194 22 ) ) PUNCT iajs-3505 194 23 . . PUNCT iajs-3505 195 1 the the DET iajs-3505 195 2 ode45 ode45 NOUN iajs-3505 195 3 solver solver ADV iajs-3505 195 4 was be AUX iajs-3505 195 5 used use VERB iajs-3505 195 6 to to PART iajs-3505 195 7 find find VERB iajs-3505 195 8 the the DET iajs-3505 195 9 numerical numerical ADJ iajs-3505 195 10 solution solution NOUN iajs-3505 195 11 to to ADP iajs-3505 195 12 our our PRON iajs-3505 195 13 system system NOUN iajs-3505 195 14 , , PUNCT iajs-3505 195 15 and and CCONJ iajs-3505 195 16 all all DET iajs-3505 195 17 figures figure NOUN iajs-3505 195 18 were be AUX iajs-3505 195 19 made make VERB iajs-3505 195 20 in in ADP iajs-3505 195 21 matlab matlab PROPN iajs-3505 195 22 2019b 2019b NUM iajs-3505 195 23 . . PUNCT iajs-3505 196 1 we we PRON iajs-3505 196 2 aim aim VERB iajs-3505 196 3 to to PART iajs-3505 196 4 study study VERB iajs-3505 196 5 the the DET iajs-3505 196 6 kinetics kinetic NOUN iajs-3505 196 7 of of ADP iajs-3505 196 8 dissolved dissolve VERB iajs-3505 196 9 oxygen oxygen NOUN iajs-3505 196 10 depletion depletion NOUN iajs-3505 196 11 for for ADP iajs-3505 196 12 the the DET iajs-3505 196 13 phytoplankton phytoplankton NOUN iajs-3505 196 14 - - PUNCT iajs-3505 196 15 zooplankton zooplankton NOUN iajs-3505 196 16 interaction interaction NOUN iajs-3505 196 17 with with ADP iajs-3505 196 18 the the DET iajs-3505 196 19 following follow VERB iajs-3505 196 20 data datum NOUN iajs-3505 196 21 : : PUNCT iajs-3505 196 22 π‘Ÿ π‘Ÿ X iajs-3505 196 23 = = SYM iajs-3505 196 24 0.35 0.35 NUM iajs-3505 196 25 , , PUNCT iajs-3505 196 26 𝛼1 𝛼1 NOUN iajs-3505 196 27 = = SYM iajs-3505 196 28 0.26 0.26 NUM iajs-3505 196 29 , , PUNCT iajs-3505 196 30 𝛼2 𝛼2 PROPN iajs-3505 196 31 = = SYM iajs-3505 196 32 0.17 0.17 NUM iajs-3505 196 33 , , PUNCT iajs-3505 196 34 π‘Ž1 π‘Ž1 NOUN iajs-3505 196 35 = = SYM iajs-3505 196 36 0.21 0.21 NUM iajs-3505 196 37 , , PUNCT iajs-3505 196 38 π‘Ž2 π‘Ž2 NOUN iajs-3505 196 39 = = SYM iajs-3505 196 40 0.21 0.21 NUM iajs-3505 196 41 , , PUNCT iajs-3505 196 42 𝛿1 𝛿1 NOUN iajs-3505 196 43 = = SYM iajs-3505 196 44 0.11 0.11 NUM iajs-3505 196 45 , , PUNCT iajs-3505 196 46 𝛿2 𝛿2 NOUN iajs-3505 196 47 = = SYM iajs-3505 196 48 0.11,π‘š 0.11,π‘š NUM iajs-3505 196 49 = = SYM iajs-3505 196 50 0.25 0.25 NUM iajs-3505 196 51 , , PUNCT iajs-3505 196 52 𝛾 𝛾 NOUN iajs-3505 196 53 = = SYM iajs-3505 196 54 0.21 0.21 NUM iajs-3505 196 55 , , PUNCT iajs-3505 196 56 𝛾1 𝛾1 PROPN iajs-3505 196 57 = = SYM iajs-3505 196 58 0.19 0.19 NUM iajs-3505 196 59 , , PUNCT iajs-3505 196 60 𝛾2 𝛾2 VERB iajs-3505 196 61 = = PUNCT iajs-3505 196 62 0.41,𝑀0 0.41,𝑀0 PUNCT iajs-3505 196 63 = = SYM iajs-3505 196 64 3 3 NUM iajs-3505 196 65 , , PUNCT iajs-3505 196 66 𝑠 𝑠 PROPN iajs-3505 196 67 = = SYM iajs-3505 196 68 2.86 2.86 NUM iajs-3505 196 69 , , PUNCT iajs-3505 196 70 𝑑 𝑑 NOUN iajs-3505 196 71 = = PROPN iajs-3505 196 72 0.41 0.41 NUM iajs-3505 196 73 , , PUNCT iajs-3505 196 74 ( ( PUNCT iajs-3505 196 75 15 15 NUM iajs-3505 196 76 ) ) PUNCT iajs-3505 196 77 to to PART iajs-3505 196 78 examine examine VERB iajs-3505 196 79 the the DET iajs-3505 196 80 effect effect NOUN iajs-3505 196 81 of of ADP iajs-3505 196 82 varying vary VERB iajs-3505 196 83 𝛾2 𝛾2 NOUN iajs-3505 196 84 ( ( PUNCT iajs-3505 196 85 the the DET iajs-3505 196 86 consumption consumption NOUN iajs-3505 196 87 of of ADP iajs-3505 196 88 oxygen oxygen NOUN iajs-3505 196 89 by by ADP iajs-3505 196 90 zooplankton zooplankton NOUN iajs-3505 196 91 ) ) PUNCT iajs-3505 196 92 , , PUNCT iajs-3505 196 93 system system NOUN iajs-3505 196 94 ( ( PUNCT iajs-3505 196 95 1 1 X iajs-3505 196 96 ) ) PUNCT iajs-3505 196 97 has have AUX iajs-3505 196 98 been be AUX iajs-3505 196 99 numerically numerically ADV iajs-3505 196 100 solved solve VERB iajs-3505 196 101 for for ADP iajs-3505 196 102 the the DET iajs-3505 196 103 data datum NOUN iajs-3505 196 104 in in ADP iajs-3505 196 105 ( ( PUNCT iajs-3505 196 106 15 15 NUM iajs-3505 196 107 ) ) PUNCT iajs-3505 196 108 with with ADP iajs-3505 196 109 different different ADJ iajs-3505 196 110 values value NOUN iajs-3505 196 111 . . PUNCT iajs-3505 197 1 it it PRON iajs-3505 197 2 is be AUX iajs-3505 197 3 clear clear ADJ iajs-3505 197 4 from from ADP iajs-3505 197 5 figure figure NOUN iajs-3505 197 6 1 1 NUM iajs-3505 197 7 the the DET iajs-3505 197 8 solution solution NOUN iajs-3505 197 9 converges converge VERB iajs-3505 197 10 to to AUX iajs-3505 197 11 𝑧3 𝑧3 VERB iajs-3505 197 12 for for ADP iajs-3505 197 13 𝛾2 𝛾2 NOUN iajs-3505 197 14 > > X iajs-3505 197 15 0.22 0.22 NUM iajs-3505 197 16 . . PUNCT iajs-3505 198 1 further far ADV iajs-3505 198 2 , , PUNCT iajs-3505 198 3 the the DET iajs-3505 198 4 solution solution NOUN iajs-3505 198 5 approaches approach VERB iajs-3505 198 6 a a DET iajs-3505 198 7 periodic periodic ADJ iajs-3505 198 8 behaviour behaviour NOUN iajs-3505 198 9 for for SCONJ iajs-3505 198 10 𝛾2 𝛾2 VERB iajs-3505 198 11 ≀ ≀ ADJ iajs-3505 198 12 0.22 0.22 NUM iajs-3505 198 13 . . PUNCT iajs-3505 199 1 the the DET iajs-3505 199 2 latter latter ADJ iajs-3505 199 3 result result NOUN iajs-3505 199 4 confirms confirm VERB iajs-3505 199 5 the the DET iajs-3505 199 6 one one NOUN iajs-3505 199 7 obtained obtain VERB iajs-3505 199 8 in in ADP iajs-3505 199 9 theorems theorem NOUN iajs-3505 199 10 2 2 NUM iajs-3505 199 11 , , PUNCT iajs-3505 199 12 which which PRON iajs-3505 199 13 establishes establish VERB iajs-3505 199 14 the the DET iajs-3505 199 15 existence existence NOUN iajs-3505 199 16 of of ADP iajs-3505 199 17 hopf hopf ADJ iajs-3505 199 18 bifurcation bifurcation NOUN iajs-3505 199 19 at at ADP iajs-3505 199 20 𝛾2 𝛾2 NOUN iajs-3505 199 21 = = SYM iajs-3505 199 22 0.22 0.22 NUM iajs-3505 199 23 . . PUNCT iajs-3505 200 1 figure figure NOUN iajs-3505 200 2 1 1 NUM iajs-3505 200 3 . . PUNCT iajs-3505 201 1 dynamics dynamic NOUN iajs-3505 201 2 of of ADP iajs-3505 201 3 the the DET iajs-3505 201 4 system system NOUN iajs-3505 201 5 ( ( PUNCT iajs-3505 201 6 1 1 NUM iajs-3505 201 7 ) ) PUNCT iajs-3505 201 8 ( ( PUNCT iajs-3505 201 9 a a DET iajs-3505 201 10 ) ) PUNCT iajs-3505 201 11 time time NOUN iajs-3505 201 12 series series NOUN iajs-3505 201 13 with with ADP iajs-3505 201 14 Ξ³2 Ξ³2 PROPN iajs-3505 201 15 = = PROPN iajs-3505 201 16 0.41 0.41 NUM iajs-3505 201 17 ; ; PUNCT iajs-3505 201 18 ( ( PUNCT iajs-3505 201 19 b b NOUN iajs-3505 201 20 ) ) PUNCT iajs-3505 201 21 phase phase NOUN iajs-3505 201 22 portrait portrait NOUN iajs-3505 201 23 of of ADP iajs-3505 201 24 ( ( PUNCT iajs-3505 201 25 a a X iajs-3505 201 26 ) ) PUNCT iajs-3505 201 27 ; ; PUNCT iajs-3505 201 28 ( ( PUNCT iajs-3505 201 29 c c X iajs-3505 201 30 ) ) PUNCT iajs-3505 201 31 time time NOUN iajs-3505 201 32 series series NOUN iajs-3505 201 33 with with ADP iajs-3505 201 34 Ξ³2 Ξ³2 PROPN iajs-3505 201 35 = = SYM iajs-3505 201 36 0.22 0.22 NUM iajs-3505 201 37 ; ; PUNCT iajs-3505 201 38 ( ( PUNCT iajs-3505 201 39 d d X iajs-3505 201 40 ) ) PUNCT iajs-3505 201 41 phase phase NOUN iajs-3505 201 42 portrait portrait NOUN iajs-3505 201 43 of of ADP iajs-3505 201 44 ( ( PUNCT iajs-3505 201 45 c c NOUN iajs-3505 201 46 ) ) PUNCT iajs-3505 201 47 . . PUNCT iajs-3505 202 1 further far ADV iajs-3505 202 2 , , PUNCT iajs-3505 202 3 figure figure NOUN iajs-3505 202 4 2 2 NUM iajs-3505 202 5 investigates investigate VERB iajs-3505 202 6 the the DET iajs-3505 202 7 effect effect NOUN iajs-3505 202 8 of of ADP iajs-3505 202 9 change change NOUN iajs-3505 202 10 in in ADP iajs-3505 202 11 the the DET iajs-3505 202 12 proportion proportion NOUN iajs-3505 202 13 of of ADP iajs-3505 202 14 protected protect VERB iajs-3505 202 15 phytoplankton phytoplankton NOUN iajs-3505 202 16 ( ( PUNCT iajs-3505 202 17 π‘š π‘š NOUN iajs-3505 202 18 ) ) PUNCT iajs-3505 202 19 on on ADP iajs-3505 202 20 the the DET iajs-3505 202 21 stability stability NOUN iajs-3505 202 22 properties property NOUN iajs-3505 202 23 of of ADP iajs-3505 202 24 the the DET iajs-3505 202 25 system system NOUN iajs-3505 202 26 ( ( PUNCT iajs-3505 202 27 1 1 NUM iajs-3505 202 28 ) ) PUNCT iajs-3505 202 29 . . PUNCT iajs-3505 203 1 it it PRON iajs-3505 203 2 shows show VERB iajs-3505 203 3 for for ADP iajs-3505 203 4 π‘š π‘š X iajs-3505 203 5 < < X iajs-3505 203 6 0.65 0.65 NUM iajs-3505 203 7 , , PUNCT iajs-3505 203 8 and and CCONJ iajs-3505 203 9 the the DET iajs-3505 203 10 solution solution NOUN iajs-3505 203 11 settles settle VERB iajs-3505 203 12 to to ADP iajs-3505 203 13 the the DET iajs-3505 203 14 ihjpas ihjpa NOUN iajs-3505 203 15 . . PUNCT iajs-3505 204 1 2024 2024 NUM iajs-3505 204 2 , , PUNCT iajs-3505 204 3 37(4 37(4 PROPN iajs-3505 204 4 ) ) PUNCT iajs-3505 204 5 377 377 NUM iajs-3505 204 6 positive positive ADJ iajs-3505 204 7 equilibrium equilibrium NOUN iajs-3505 204 8 point point NOUN iajs-3505 204 9 . . PUNCT iajs-3505 205 1 while while SCONJ iajs-3505 205 2 for for ADP iajs-3505 205 3 π‘š π‘š PROPN iajs-3505 205 4 β‰₯ β‰₯ NUM iajs-3505 205 5 0.65 0.65 NUM iajs-3505 205 6 , , PUNCT iajs-3505 205 7 the the DET iajs-3505 205 8 solution solution NOUN iajs-3505 205 9 delivers deliver VERB iajs-3505 205 10 a a DET iajs-3505 205 11 periodic periodic ADJ iajs-3505 205 12 attractor attractor NOUN iajs-3505 205 13 behaviour behaviour NOUN iajs-3505 205 14 . . PUNCT iajs-3505 206 1 figure figure NOUN iajs-3505 206 2 2 2 NUM iajs-3505 206 3 . . PUNCT iajs-3505 207 1 dynamics dynamic NOUN iajs-3505 207 2 of of ADP iajs-3505 207 3 the the DET iajs-3505 207 4 system system NOUN iajs-3505 207 5 ( ( PUNCT iajs-3505 207 6 1 1 NUM iajs-3505 207 7 ) ) PUNCT iajs-3505 207 8 ( ( PUNCT iajs-3505 207 9 a a DET iajs-3505 207 10 ) ) PUNCT iajs-3505 207 11 time time NOUN iajs-3505 207 12 series series NOUN iajs-3505 207 13 with with ADP iajs-3505 207 14 π‘š π‘š PROPN iajs-3505 207 15 = = SYM iajs-3505 207 16 0.15 0.15 NUM iajs-3505 207 17 ; ; PUNCT iajs-3505 207 18 ( ( PUNCT iajs-3505 207 19 b b NOUN iajs-3505 207 20 ) ) PUNCT iajs-3505 207 21 phase phase NOUN iajs-3505 207 22 portrait portrait NOUN iajs-3505 207 23 of of ADP iajs-3505 207 24 ( ( PUNCT iajs-3505 207 25 a a X iajs-3505 207 26 ) ) PUNCT iajs-3505 207 27 ; ; PUNCT iajs-3505 207 28 ( ( PUNCT iajs-3505 207 29 c c X iajs-3505 207 30 ) ) PUNCT iajs-3505 207 31 time time NOUN iajs-3505 207 32 series series NOUN iajs-3505 207 33 with with ADP iajs-3505 207 34 m m PROPN iajs-3505 207 35 = = NOUN iajs-3505 207 36 0.65 0.65 NUM iajs-3505 207 37 ; ; PUNCT iajs-3505 207 38 ( ( PUNCT iajs-3505 207 39 d d X iajs-3505 207 40 ) ) PUNCT iajs-3505 207 41 phase phase NOUN iajs-3505 207 42 portrait portrait NOUN iajs-3505 207 43 of of ADP iajs-3505 207 44 ( ( PUNCT iajs-3505 207 45 c c NOUN iajs-3505 207 46 ) ) PUNCT iajs-3505 207 47 . . PUNCT iajs-3505 208 1 further far ADV iajs-3505 208 2 , , PUNCT iajs-3505 208 3 figure figure VERB iajs-3505 208 4 3 3 NUM iajs-3505 208 5 shows show NOUN iajs-3505 208 6 for for ADP iajs-3505 208 7 different different ADJ iajs-3505 208 8 values value NOUN iajs-3505 208 9 of of ADP iajs-3505 208 10 𝛾 𝛾 PROPN iajs-3505 208 11 ( ( PUNCT iajs-3505 208 12 the the DET iajs-3505 208 13 natural natural ADJ iajs-3505 208 14 depletion depletion NOUN iajs-3505 208 15 rate rate NOUN iajs-3505 208 16 of of ADP iajs-3505 208 17 oxygen oxygen NOUN iajs-3505 208 18 ) ) PUNCT iajs-3505 208 19 , , PUNCT iajs-3505 208 20 the the DET iajs-3505 208 21 solution solution NOUN iajs-3505 208 22 stabilizes stabilize VERB iajs-3505 208 23 at at ADP iajs-3505 208 24 𝑧3 𝑧3 PROPN iajs-3505 208 25 for for ADP iajs-3505 208 26 𝛾 𝛾 X iajs-3505 208 27 < < X iajs-3505 208 28 0.62 0.62 NUM iajs-3505 208 29 . . PUNCT iajs-3505 209 1 while while SCONJ iajs-3505 209 2 for for ADP iajs-3505 209 3 𝛾 𝛾 PROPN iajs-3505 209 4 β‰₯ β‰₯ NUM iajs-3505 209 5 0.62 0.62 NUM iajs-3505 209 6 , , PUNCT iajs-3505 209 7 the the DET iajs-3505 209 8 solution solution NOUN iajs-3505 209 9 shows show VERB iajs-3505 209 10 a a DET iajs-3505 209 11 periodic periodic ADJ iajs-3505 209 12 behaviour behaviour NOUN iajs-3505 209 13 . . PUNCT iajs-3505 210 1 figure figure VERB iajs-3505 210 2 3 3 NUM iajs-3505 210 3 . . PUNCT iajs-3505 211 1 dynamics dynamic NOUN iajs-3505 211 2 of of ADP iajs-3505 211 3 the the DET iajs-3505 211 4 system system NOUN iajs-3505 211 5 ( ( PUNCT iajs-3505 211 6 1 1 NUM iajs-3505 211 7 ) ) PUNCT iajs-3505 211 8 ( ( PUNCT iajs-3505 211 9 a a DET iajs-3505 211 10 ) ) PUNCT iajs-3505 211 11 time time NOUN iajs-3505 211 12 series series NOUN iajs-3505 211 13 with with ADP iajs-3505 211 14 Ξ³ Ξ³ X iajs-3505 211 15 = = SYM iajs-3505 211 16 0.21 0.21 NUM iajs-3505 211 17 ; ; PUNCT iajs-3505 211 18 ( ( PUNCT iajs-3505 211 19 b b NOUN iajs-3505 211 20 ) ) PUNCT iajs-3505 211 21 phase phase NOUN iajs-3505 211 22 portrait portrait NOUN iajs-3505 211 23 of of ADP iajs-3505 211 24 ( ( PUNCT iajs-3505 211 25 a a X iajs-3505 211 26 ) ) PUNCT iajs-3505 211 27 ; ; PUNCT iajs-3505 211 28 ( ( PUNCT iajs-3505 211 29 c c X iajs-3505 211 30 ) ) PUNCT iajs-3505 211 31 time time NOUN iajs-3505 211 32 series series NOUN iajs-3505 211 33 with with ADP iajs-3505 211 34 Ξ³ Ξ³ X iajs-3505 211 35 = = NOUN iajs-3505 211 36 0.62 0.62 NUM iajs-3505 211 37 ; ; PUNCT iajs-3505 211 38 ( ( PUNCT iajs-3505 211 39 d d X iajs-3505 211 40 ) ) PUNCT iajs-3505 211 41 phase phase NOUN iajs-3505 211 42 portrait portrait NOUN iajs-3505 211 43 of of ADP iajs-3505 211 44 ( ( PUNCT iajs-3505 211 45 c c NOUN iajs-3505 211 46 ) ) PUNCT iajs-3505 211 47 . . PUNCT iajs-3505 212 1 further far ADV iajs-3505 212 2 , , PUNCT iajs-3505 212 3 figure figure VERB iajs-3505 212 4 4 4 NUM iajs-3505 212 5 investigates investigate VERB iajs-3505 212 6 the the DET iajs-3505 212 7 effect effect NOUN iajs-3505 212 8 of of ADP iajs-3505 212 9 change change NOUN iajs-3505 212 10 in in ADP iajs-3505 212 11 the the DET iajs-3505 212 12 replenishment replenishment NOUN iajs-3505 212 13 rate rate NOUN iajs-3505 212 14 of of ADP iajs-3505 212 15 oxygen oxygen NOUN iajs-3505 212 16 in in ADP iajs-3505 212 17 the the DET iajs-3505 212 18 marine marine NOUN iajs-3505 212 19 ( ( PUNCT iajs-3505 212 20 𝑠 𝑠 NOUN iajs-3505 212 21 ) ) PUNCT iajs-3505 212 22 on on ADP iajs-3505 212 23 the the DET iajs-3505 212 24 stability stability NOUN iajs-3505 212 25 properties property NOUN iajs-3505 212 26 of of ADP iajs-3505 212 27 the the DET iajs-3505 212 28 system system NOUN iajs-3505 212 29 ( ( PUNCT iajs-3505 212 30 1 1 NUM iajs-3505 212 31 ) ) PUNCT iajs-3505 212 32 . . PUNCT iajs-3505 213 1 it it PRON iajs-3505 213 2 shows show VERB iajs-3505 213 3 for for ADP iajs-3505 213 4 𝑠 𝑠 PROPN iajs-3505 213 5 > > ADP iajs-3505 213 6 1.85 1.85 NUM iajs-3505 213 7 ; ; PUNCT iajs-3505 213 8 the the DET iajs-3505 213 9 solution solution NOUN iajs-3505 213 10 settles settle VERB iajs-3505 213 11 down down ADP iajs-3505 213 12 to to ADP iajs-3505 213 13 𝑧3 𝑧3 PROPN iajs-3505 213 14 . . PUNCT iajs-3505 214 1 while while SCONJ iajs-3505 214 2 for for ADP iajs-3505 214 3 𝑠 𝑠 PROPN iajs-3505 214 4 ≀ ≀ NUM iajs-3505 214 5 1.85 1.85 NUM iajs-3505 214 6 . . PUNCT iajs-3505 214 7 , , PUNCT iajs-3505 214 8 the the DET iajs-3505 214 9 solution solution NOUN iajs-3505 214 10 shows show VERB iajs-3505 214 11 a a DET iajs-3505 214 12 periodic periodic ADJ iajs-3505 214 13 behaviour behaviour NOUN iajs-3505 214 14 . . PUNCT iajs-3505 215 1 ihjpas ihjpas PROPN iajs-3505 215 2 . . PUNCT iajs-3505 216 1 2024 2024 NUM iajs-3505 216 2 , , PUNCT iajs-3505 216 3 37(4 37(4 NOUN iajs-3505 216 4 ) ) PUNCT iajs-3505 216 5 378 378 NUM iajs-3505 216 6 figure figure NOUN iajs-3505 216 7 4 4 NUM iajs-3505 216 8 . . PUNCT iajs-3505 216 9 dynamics dynamic NOUN iajs-3505 216 10 of of ADP iajs-3505 216 11 system system NOUN iajs-3505 216 12 ( ( PUNCT iajs-3505 216 13 1 1 NUM iajs-3505 216 14 ) ) PUNCT iajs-3505 216 15 ( ( PUNCT iajs-3505 216 16 a a DET iajs-3505 216 17 ) ) PUNCT iajs-3505 216 18 time time NOUN iajs-3505 216 19 series series NOUN iajs-3505 216 20 with with ADP iajs-3505 216 21 s=2.86 s=2.86 NOUN iajs-3505 216 22 ; ; PUNCT iajs-3505 216 23 ( ( PUNCT iajs-3505 216 24 b b X iajs-3505 216 25 ) ) PUNCT iajs-3505 216 26 phase phase NOUN iajs-3505 216 27 portrait portrait NOUN iajs-3505 216 28 of of ADP iajs-3505 216 29 ( ( PUNCT iajs-3505 216 30 a a X iajs-3505 216 31 ) ) PUNCT iajs-3505 216 32 ; ; PUNCT iajs-3505 216 33 ( ( PUNCT iajs-3505 216 34 c c X iajs-3505 216 35 ) ) PUNCT iajs-3505 216 36 time time NOUN iajs-3505 216 37 series series NOUN iajs-3505 216 38 with with ADP iajs-3505 216 39 s=1.85 s=1.85 PROPN iajs-3505 216 40 ; ; PUNCT iajs-3505 216 41 ( ( PUNCT iajs-3505 216 42 d d X iajs-3505 216 43 ) ) PUNCT iajs-3505 216 44 phase phase NOUN iajs-3505 216 45 portrait portrait NOUN iajs-3505 216 46 of of ADP iajs-3505 216 47 ( ( PUNCT iajs-3505 216 48 c c NOUN iajs-3505 216 49 ) ) PUNCT iajs-3505 216 50 ; ; PUNCT iajs-3505 216 51 ( ( PUNCT iajs-3505 216 52 e e NOUN iajs-3505 216 53 ) ) PUNCT iajs-3505 216 54 time time NOUN iajs-3505 216 55 series series NOUN iajs-3505 216 56 with with ADP iajs-3505 216 57 s=0.44 s=0.44 NOUN iajs-3505 216 58 ; ; PUNCT iajs-3505 216 59 ( ( PUNCT iajs-3505 216 60 f f X iajs-3505 216 61 ) ) PUNCT iajs-3505 216 62 phase phase NOUN iajs-3505 216 63 portrait portrait NOUN iajs-3505 216 64 of of ADP iajs-3505 216 65 ( ( PUNCT iajs-3505 216 66 e e NOUN iajs-3505 216 67 ) ) PUNCT iajs-3505 216 68 . . PUNCT iajs-3505 217 1 now now ADV iajs-3505 217 2 the the DET iajs-3505 217 3 effect effect NOUN iajs-3505 217 4 of of ADP iajs-3505 217 5 changing change VERB iajs-3505 217 6 the the DET iajs-3505 217 7 concentration concentration NOUN iajs-3505 217 8 of of ADP iajs-3505 217 9 dissolved dissolved ADJ iajs-3505 217 10 oxygen oxygen NOUN iajs-3505 217 11 that that PRON iajs-3505 217 12 comes come VERB iajs-3505 217 13 from from ADP iajs-3505 217 14 several several ADJ iajs-3505 217 15 sources source NOUN iajs-3505 217 16 ( ( PUNCT iajs-3505 217 17 𝑀0 𝑀0 PROPN iajs-3505 217 18 ) ) PUNCT iajs-3505 217 19 is be AUX iajs-3505 217 20 explored explore VERB iajs-3505 217 21 in in ADP iajs-3505 217 22 figure figure NOUN iajs-3505 217 23 5 5 NUM iajs-3505 217 24 . . PUNCT iajs-3505 218 1 it it PRON iajs-3505 218 2 shows show VERB iajs-3505 218 3 that that SCONJ iajs-3505 218 4 the the DET iajs-3505 218 5 solution solution NOUN iajs-3505 218 6 settles settle VERB iajs-3505 218 7 asymptotically asymptotically ADV iajs-3505 218 8 to to ADP iajs-3505 218 9 the the DET iajs-3505 218 10 , , PUNCT iajs-3505 218 11 𝑧3 𝑧3 PROPN iajs-3505 218 12 , , PUNCT iajs-3505 218 13 for for ADP iajs-3505 218 14 𝑀0 𝑀0 PROPN iajs-3505 218 15 > > X iajs-3505 218 16 2.48 2.48 NUM iajs-3505 218 17 . . PUNCT iajs-3505 219 1 further far ADV iajs-3505 219 2 , , PUNCT iajs-3505 219 3 the the DET iajs-3505 219 4 solution solution NOUN iajs-3505 219 5 approaches approach VERB iajs-3505 219 6 a a DET iajs-3505 219 7 periodic periodic ADJ iajs-3505 219 8 attractor attractor NOUN iajs-3505 219 9 for for ADP iajs-3505 219 10 𝑀0 𝑀0 NOUN iajs-3505 219 11 ≀ ≀ ADV iajs-3505 219 12 2.48 2.48 NUM iajs-3505 219 13 . . PUNCT iajs-3505 220 1 figure figure NOUN iajs-3505 220 2 5 5 NUM iajs-3505 220 3 . . PUNCT iajs-3505 221 1 dynamics dynamic NOUN iajs-3505 221 2 of of ADP iajs-3505 221 3 system system NOUN iajs-3505 221 4 ( ( PUNCT iajs-3505 221 5 1 1 NUM iajs-3505 221 6 ) ) PUNCT iajs-3505 221 7 ( ( PUNCT iajs-3505 221 8 a a DET iajs-3505 221 9 ) ) PUNCT iajs-3505 221 10 time time NOUN iajs-3505 221 11 series series NOUN iajs-3505 221 12 with with ADP iajs-3505 221 13 w0 w0 PROPN iajs-3505 221 14 =3 =3 PROPN iajs-3505 221 15 ; ; PUNCT iajs-3505 221 16 ( ( PUNCT iajs-3505 221 17 b b X iajs-3505 221 18 ) ) PUNCT iajs-3505 221 19 phase phase NOUN iajs-3505 221 20 portrait portrait NOUN iajs-3505 221 21 of of ADP iajs-3505 221 22 ( ( PUNCT iajs-3505 221 23 a a X iajs-3505 221 24 ) ) PUNCT iajs-3505 221 25 ; ; PUNCT iajs-3505 221 26 ( ( PUNCT iajs-3505 221 27 c c X iajs-3505 221 28 ) ) PUNCT iajs-3505 221 29 time time NOUN iajs-3505 221 30 series series NOUN iajs-3505 221 31 with with ADP iajs-3505 221 32 w0 w0 PROPN iajs-3505 221 33 = = PROPN iajs-3505 221 34 2.48 2.48 NUM iajs-3505 221 35 ; ; PUNCT iajs-3505 221 36 ( ( PUNCT iajs-3505 221 37 d d X iajs-3505 221 38 ) ) PUNCT iajs-3505 221 39 phase phase NOUN iajs-3505 221 40 portrait portrait NOUN iajs-3505 221 41 of of ADP iajs-3505 221 42 ( ( PUNCT iajs-3505 221 43 c c NOUN iajs-3505 221 44 ) ) PUNCT iajs-3505 221 45 . . PUNCT iajs-3505 222 1 6 6 X iajs-3505 222 2 . . X iajs-3505 222 3 conclusion conclusion NOUN iajs-3505 222 4 this this DET iajs-3505 222 5 study study NOUN iajs-3505 222 6 modified modify VERB iajs-3505 222 7 the the DET iajs-3505 222 8 dissolved dissolve VERB iajs-3505 222 9 oxygen oxygen NOUN iajs-3505 222 10 - - PUNCT iajs-3505 222 11 plankton plankton NOUN iajs-3505 222 12 model model NOUN iajs-3505 222 13 by by ADP iajs-3505 222 14 considering consider VERB iajs-3505 222 15 that that SCONJ iajs-3505 222 16 zooplankton zooplankton NOUN iajs-3505 222 17 feeds feed VERB iajs-3505 222 18 only only ADV iajs-3505 222 19 on on ADP iajs-3505 222 20 the the DET iajs-3505 222 21 available available ADJ iajs-3505 222 22 phytoplankton phytoplankton NOUN iajs-3505 222 23 . . PUNCT iajs-3505 223 1 the the DET iajs-3505 223 2 objective objective NOUN iajs-3505 223 3 is be AUX iajs-3505 223 4 to to PART iajs-3505 223 5 determine determine VERB iajs-3505 223 6 how how SCONJ iajs-3505 223 7 this this DET iajs-3505 223 8 type type NOUN iajs-3505 223 9 of of ADP iajs-3505 223 10 interaction interaction NOUN iajs-3505 223 11 ihjpas ihjpa NOUN iajs-3505 223 12 . . PUNCT iajs-3505 224 1 2024 2024 NUM iajs-3505 224 2 , , PUNCT iajs-3505 224 3 37(4 37(4 NOUN iajs-3505 224 4 ) ) PUNCT iajs-3505 224 5 379 379 NUM iajs-3505 224 6 impacts impact VERB iajs-3505 224 7 the the DET iajs-3505 224 8 dynamics dynamic NOUN iajs-3505 224 9 of of ADP iajs-3505 224 10 an an DET iajs-3505 224 11 aquatic aquatic ADJ iajs-3505 224 12 ecosystem ecosystem NOUN iajs-3505 224 13 . . PUNCT iajs-3505 225 1 the the DET iajs-3505 225 2 system system NOUN iajs-3505 225 3 was be AUX iajs-3505 225 4 analyzed analyze VERB iajs-3505 225 5 theoretically theoretically ADV iajs-3505 225 6 and and CCONJ iajs-3505 225 7 numerically numerically ADV iajs-3505 225 8 . . PUNCT iajs-3505 226 1 the the DET iajs-3505 226 2 results result NOUN iajs-3505 226 3 of of ADP iajs-3505 226 4 the the DET iajs-3505 226 5 theoretical theoretical ADJ iajs-3505 226 6 analysis analysis NOUN iajs-3505 226 7 revealed reveal VERB iajs-3505 226 8 three three NUM iajs-3505 226 9 stable stable ADJ iajs-3505 226 10 states state NOUN iajs-3505 226 11 . . PUNCT iajs-3505 227 1 depending depend VERB iajs-3505 227 2 on on ADP iajs-3505 227 3 the the DET iajs-3505 227 4 conditions condition NOUN iajs-3505 227 5 , , PUNCT iajs-3505 227 6 the the DET iajs-3505 227 7 behaviour behaviour NOUN iajs-3505 227 8 of of ADP iajs-3505 227 9 the the DET iajs-3505 227 10 three three NUM iajs-3505 227 11 constant constant ADJ iajs-3505 227 12 states state NOUN iajs-3505 227 13 was be AUX iajs-3505 227 14 either either CCONJ iajs-3505 227 15 stable stable ADJ iajs-3505 227 16 or or CCONJ iajs-3505 227 17 unstable unstable ADJ iajs-3505 227 18 . . PUNCT iajs-3505 228 1 the the DET iajs-3505 228 2 conditions condition NOUN iajs-3505 228 3 necessary necessary ADJ iajs-3505 228 4 for for SCONJ iajs-3505 228 5 a a DET iajs-3505 228 6 hopf hopf ADJ iajs-3505 228 7 bifurcation bifurcation NOUN iajs-3505 228 8 around around ADP iajs-3505 228 9 the the DET iajs-3505 228 10 positive positive ADJ iajs-3505 228 11 stable stable ADJ iajs-3505 228 12 state state NOUN iajs-3505 228 13 have have AUX iajs-3505 228 14 been be AUX iajs-3505 228 15 identified identify VERB iajs-3505 228 16 . . PUNCT iajs-3505 229 1 nonetheless nonetheless ADV iajs-3505 229 2 , , PUNCT iajs-3505 229 3 the the DET iajs-3505 229 4 numerical numerical PROPN iajs-3505 229 5 simulation simulation PROPN iajs-3505 229 6 deduced deduce VERB iajs-3505 229 7 that that SCONJ iajs-3505 229 8 system system NOUN iajs-3505 229 9 ( ( PUNCT iajs-3505 229 10 1 1 X iajs-3505 229 11 ) ) PUNCT iajs-3505 229 12 always always ADV iajs-3505 229 13 sways sway VERB iajs-3505 229 14 about about ADP iajs-3505 229 15 the the DET iajs-3505 229 16 positive positive ADJ iajs-3505 229 17 steady steady ADJ iajs-3505 229 18 state state NOUN iajs-3505 229 19 when when SCONJ iajs-3505 229 20 the the DET iajs-3505 229 21 stability stability NOUN iajs-3505 229 22 criteria criterion NOUN iajs-3505 229 23 are be AUX iajs-3505 229 24 met meet VERB iajs-3505 229 25 . . PUNCT iajs-3505 230 1 further far ADV iajs-3505 230 2 , , PUNCT iajs-3505 230 3 for for ADP iajs-3505 230 4 small small ADJ iajs-3505 230 5 changing change VERB iajs-3505 230 6 in in ADP iajs-3505 230 7 some some DET iajs-3505 230 8 parameters parameter NOUN iajs-3505 230 9 , , PUNCT iajs-3505 230 10 such such ADJ iajs-3505 230 11 that that DET iajs-3505 230 12 𝑀0 𝑀0 PROPN iajs-3505 230 13 , , PUNCT iajs-3505 230 14 𝑠 𝑠 PROPN iajs-3505 230 15 and and CCONJ iajs-3505 230 16 𝛾 𝛾 PROPN iajs-3505 230 17 , , PUNCT iajs-3505 230 18 the the DET iajs-3505 230 19 system system NOUN iajs-3505 230 20 ( ( PUNCT iajs-3505 230 21 1 1 X iajs-3505 230 22 ) ) PUNCT iajs-3505 230 23 shows shows AUX iajs-3505 230 24 limit limit VERB iajs-3505 230 25 cycle cycle NOUN iajs-3505 230 26 behaviour behaviour NOUN iajs-3505 230 27 . . PUNCT iajs-3505 231 1 for for ADP iajs-3505 231 2 future future ADJ iajs-3505 231 3 work work NOUN iajs-3505 231 4 , , PUNCT iajs-3505 231 5 we we PRON iajs-3505 231 6 suggest suggest VERB iajs-3505 231 7 considering consider VERB iajs-3505 231 8 climate climate NOUN iajs-3505 231 9 change change NOUN iajs-3505 231 10 ’s ’s PART iajs-3505 231 11 impact impact NOUN iajs-3505 231 12 on on ADP iajs-3505 231 13 the the DET iajs-3505 231 14 ocean ocean NOUN iajs-3505 231 15 ’s ’s PART iajs-3505 231 16 oxygen oxygen NOUN iajs-3505 231 17 - - PUNCT iajs-3505 231 18 plankton plankton NOUN iajs-3505 231 19 dynamics dynamic NOUN iajs-3505 231 20 . . PUNCT iajs-3505 232 1 acknowledgment acknowledgment NOUN iajs-3505 232 2 the the DET iajs-3505 232 3 authors author NOUN iajs-3505 232 4 are be AUX iajs-3505 232 5 very very ADV iajs-3505 232 6 grateful grateful ADJ iajs-3505 232 7 to to ADP iajs-3505 232 8 the the DET iajs-3505 232 9 executive executive ADJ iajs-3505 232 10 manager manager NOUN iajs-3505 232 11 and and CCONJ iajs-3505 232 12 editorial editorial PROPN iajs-3505 232 13 board board NOUN iajs-3505 232 14 members member NOUN iajs-3505 232 15 of of ADP iajs-3505 232 16 the the DET iajs-3505 232 17 ibn ibn PROPN iajs-3505 232 18 al al PROPN iajs-3505 232 19 - - PUNCT iajs-3505 232 20 haitham haitham PROPN iajs-3505 232 21 journal journal PROPN iajs-3505 232 22 of of ADP iajs-3505 232 23 pure pure ADJ iajs-3505 232 24 and and CCONJ iajs-3505 232 25 applied applied ADJ iajs-3505 232 26 sciences science NOUN iajs-3505 232 27 . . PUNCT iajs-3505 233 1 conflict conflict NOUN iajs-3505 233 2 of of ADP iajs-3505 233 3 interest interest NOUN iajs-3505 233 4 the the DET iajs-3505 233 5 authors author NOUN iajs-3505 233 6 declare declare VERB iajs-3505 233 7 that that SCONJ iajs-3505 233 8 they they PRON iajs-3505 233 9 have have VERB iajs-3505 233 10 no no DET iajs-3505 233 11 conflicts conflict NOUN iajs-3505 233 12 of of ADP iajs-3505 233 13 interest interest NOUN iajs-3505 233 14 . . PUNCT iajs-3505 234 1 funding funding NOUN iajs-3505 234 2 there there PRON iajs-3505 234 3 is be VERB iajs-3505 234 4 no no DET iajs-3505 234 5 funding funding NOUN iajs-3505 234 6 for for ADP iajs-3505 234 7 the the DET iajs-3505 234 8 article article NOUN iajs-3505 234 9 . . PUNCT iajs-3505 235 1 references reference NOUN iajs-3505 235 2 1 1 NUM iajs-3505 235 3 . . PUNCT iajs-3505 236 1 hull hull NOUN iajs-3505 236 2 , , PUNCT iajs-3505 236 3 v. v. PROPN iajs-3505 236 4 ; ; PUNCT iajs-3505 236 5 parrella parrella PROPN iajs-3505 236 6 , , PUNCT iajs-3505 236 7 l. l. PROPN iajs-3505 236 8 ; ; PUNCT iajs-3505 236 9 falcucci falcucci NOUN iajs-3505 236 10 , , PUNCT iajs-3505 236 11 m. m. NOUN iajs-3505 236 12 modelling modelling NOUN iajs-3505 236 13 dissolved dissolve VERB iajs-3505 236 14 oxygen oxygen NOUN iajs-3505 236 15 dynamics dynamic NOUN iajs-3505 236 16 in in ADP iajs-3505 236 17 coastal coastal ADJ iajs-3505 236 18 lagoons lagoon NOUN iajs-3505 236 19 . . PUNCT iajs-3505 237 1 ecological ecological ADJ iajs-3505 237 2 modelling modelling NOUN iajs-3505 237 3 , , PUNCT iajs-3505 237 4 2008 2008 NUM iajs-3505 237 5 , , PUNCT iajs-3505 237 6 211(3–4 211(3–4 NUM iajs-3505 237 7 ) ) PUNCT iajs-3505 237 8 , , PUNCT iajs-3505 237 9 468–480 468–480 NUM iajs-3505 237 10 . . PUNCT iajs-3505 238 1 https://doi.org/10.1016/j.ecolmodel.2007.09.023 https://doi.org/10.1016/j.ecolmodel.2007.09.023 PROPN iajs-3505 238 2 2 2 NUM iajs-3505 238 3 . . NUM iajs-3505 238 4 misra misra ADJ iajs-3505 238 5 , , PUNCT iajs-3505 238 6 a. a. NOUN iajs-3505 238 7 k. k. PROPN iajs-3505 239 1 modeling model VERB iajs-3505 239 2 the the DET iajs-3505 239 3 depletion depletion NOUN iajs-3505 239 4 of of ADP iajs-3505 239 5 dissolved dissolved ADJ iajs-3505 239 6 oxygen oxygen NOUN iajs-3505 239 7 in in ADP iajs-3505 239 8 a a DET iajs-3505 239 9 lake lake NOUN iajs-3505 239 10 due due ADP iajs-3505 239 11 to to ADP iajs-3505 239 12 submerged submerge VERB iajs-3505 239 13 macrophytes macrophyte NOUN iajs-3505 239 14 . . PUNCT iajs-3505 240 1 nonlinear nonlinear ADJ iajs-3505 240 2 analysis analysis NOUN iajs-3505 240 3 : : PUNCT iajs-3505 240 4 modelling modelling NOUN iajs-3505 240 5 and and CCONJ iajs-3505 240 6 control control NOUN iajs-3505 240 7 , , PUNCT iajs-3505 240 8 2010 2010 NUM iajs-3505 240 9 , , PUNCT iajs-3505 240 10 15(2 15(2 NUM iajs-3505 240 11 ) ) PUNCT iajs-3505 240 12 , , PUNCT iajs-3505 240 13 185–198 185–198 NUM iajs-3505 240 14 . . PUNCT iajs-3505 241 1 https://doi.org/10.15388/na.2010.15.2.14353 https://doi.org/10.15388/na.2010.15.2.14353 NOUN iajs-3505 241 2 3 3 NUM iajs-3505 241 3 . . NOUN iajs-3505 241 4 misra misra ADJ iajs-3505 241 5 , , PUNCT iajs-3505 241 6 a. a. PROPN iajs-3505 241 7 k. k. PROPN iajs-3505 241 8 ; ; PUNCT iajs-3505 241 9 chandra chandra PROPN iajs-3505 241 10 , , PUNCT iajs-3505 241 11 p. p. PROPN iajs-3505 241 12 ; ; PUNCT iajs-3505 241 13 raghavendra raghavendra PROPN iajs-3505 241 14 , , PUNCT iajs-3505 241 15 v. v. ADP iajs-3505 241 16 modeling model VERB iajs-3505 241 17 the the DET iajs-3505 241 18 depletion depletion NOUN iajs-3505 241 19 of of ADP iajs-3505 241 20 dissolved dissolved ADJ iajs-3505 241 21 oxygen oxygen NOUN iajs-3505 241 22 in in ADP iajs-3505 241 23 a a DET iajs-3505 241 24 lake lake NOUN iajs-3505 241 25 due due ADP iajs-3505 241 26 to to ADP iajs-3505 241 27 algal algal ADJ iajs-3505 241 28 bloom bloom NOUN iajs-3505 241 29 : : PUNCT iajs-3505 241 30 effect effect NOUN iajs-3505 241 31 of of ADP iajs-3505 241 32 time time NOUN iajs-3505 241 33 delay delay NOUN iajs-3505 241 34 . . PUNCT iajs-3505 242 1 advances advance NOUN iajs-3505 242 2 in in ADP iajs-3505 242 3 water water NOUN iajs-3505 242 4 resources resource NOUN iajs-3505 242 5 , , PUNCT iajs-3505 242 6 2011 2011 NUM iajs-3505 242 7 , , PUNCT iajs-3505 242 8 34(10 34(10 NUM iajs-3505 242 9 ) ) PUNCT iajs-3505 242 10 , , PUNCT iajs-3505 242 11 1232–1238 1232–1238 NUM iajs-3505 242 12 . . PUNCT iajs-3505 243 1 https://doi.org/10.1016/j.advwatres.2011.05.010 https://doi.org/10.1016/j.advwatres.2011.05.010 ADJ iajs-3505 243 2 4 4 NUM iajs-3505 243 3 . . PUNCT iajs-3505 243 4 gΓΆkΓ§e gΓΆkΓ§e PROPN iajs-3505 243 5 , , PUNCT iajs-3505 243 6 a. a. PROPN iajs-3505 243 7 a a DET iajs-3505 243 8 mathematical mathematical ADJ iajs-3505 243 9 study study NOUN iajs-3505 243 10 for for ADP iajs-3505 243 11 chaotic chaotic ADJ iajs-3505 243 12 dynamics dynamic NOUN iajs-3505 243 13 of of ADP iajs-3505 243 14 dissolved dissolve VERB iajs-3505 243 15 oxygen oxygen NOUN iajs-3505 243 16 - - PUNCT iajs-3505 243 17 phytoplankton phytoplankton NOUN iajs-3505 243 18 interactions interaction NOUN iajs-3505 243 19 under under ADP iajs-3505 243 20 environmental environmental ADJ iajs-3505 243 21 driving driving NOUN iajs-3505 243 22 factors factor NOUN iajs-3505 243 23 and and CCONJ iajs-3505 243 24 time time NOUN iajs-3505 243 25 lag lag PROPN iajs-3505 243 26 . . PUNCT iajs-3505 244 1 chaos chaos NOUN iajs-3505 244 2 , , PUNCT iajs-3505 244 3 solitons soliton NOUN iajs-3505 244 4 & & CCONJ iajs-3505 244 5 fractals fractal NOUN iajs-3505 244 6 , , PUNCT iajs-3505 244 7 2021 2021 NUM iajs-3505 244 8 , , PUNCT iajs-3505 244 9 151 151 NUM iajs-3505 244 10 , , PUNCT iajs-3505 244 11 111268 111268 NUM iajs-3505 244 12 . . PUNCT iajs-3505 245 1 https://doi.org/10.1016/j.chaos.2021.111268 https://doi.org/10.1016/j.chaos.2021.111268 NOUN iajs-3505 245 2 5 5 NUM iajs-3505 245 3 . . PUNCT iajs-3505 245 4 sekerci sekerci PROPN iajs-3505 245 5 , , PUNCT iajs-3505 245 6 y. y. PROPN iajs-3505 245 7 ; ; PUNCT iajs-3505 245 8 petrovskii petrovskii PROPN iajs-3505 245 9 , , PUNCT iajs-3505 245 10 s. s. PROPN iajs-3505 245 11 mathematical mathematical PROPN iajs-3505 245 12 modelling modelling NOUN iajs-3505 245 13 of of ADP iajs-3505 245 14 plankton plankton NOUN iajs-3505 245 15 – – PUNCT iajs-3505 245 16 oxygen oxygen NOUN iajs-3505 245 17 dynamics dynamic NOUN iajs-3505 245 18 under under ADP iajs-3505 245 19 the the DET iajs-3505 245 20 climate climate NOUN iajs-3505 245 21 change change NOUN iajs-3505 245 22 . . PUNCT iajs-3505 246 1 bulletin bulletin NOUN iajs-3505 246 2 of of ADP iajs-3505 246 3 mathematical mathematical ADJ iajs-3505 246 4 biology biology NOUN iajs-3505 246 5 , , PUNCT iajs-3505 246 6 2015 2015 NUM iajs-3505 246 7 , , PUNCT iajs-3505 246 8 77(12 77(12 NUM iajs-3505 246 9 ) ) PUNCT iajs-3505 246 10 , , PUNCT iajs-3505 246 11 2325–2353 2325–2353 NUM iajs-3505 246 12 . . PUNCT iajs-3505 247 1 https://doi.org/10.1007/s11538015-0126-0 https://doi.org/10.1007/s11538015-0126-0 PROPN iajs-3505 247 2 6 6 NUM iajs-3505 247 3 . . PUNCT iajs-3505 248 1 hancke hancke PROPN iajs-3505 248 2 , , PUNCT iajs-3505 248 3 k. k. PROPN iajs-3505 248 4 ; ; PUNCT iajs-3505 248 5 glud glud PROPN iajs-3505 248 6 , , PUNCT iajs-3505 248 7 r. r. PROPN iajs-3505 248 8 n. n. PROPN iajs-3505 248 9 temperature temperature NOUN iajs-3505 248 10 effects effect NOUN iajs-3505 248 11 on on ADP iajs-3505 248 12 respiration respiration NOUN iajs-3505 248 13 and and CCONJ iajs-3505 248 14 photosynthesis photosynthesis NOUN iajs-3505 248 15 in in ADP iajs-3505 248 16 three three NUM iajs-3505 248 17 diatomdominated diatomdominated ADJ iajs-3505 248 18 benthic benthic ADJ iajs-3505 248 19 communities community NOUN iajs-3505 248 20 . . PUNCT iajs-3505 249 1 aquatic aquatic ADJ iajs-3505 249 2 microbial microbial ADJ iajs-3505 249 3 ecology ecology NOUN iajs-3505 249 4 , , PUNCT iajs-3505 249 5 2004 2004 NUM iajs-3505 249 6 , , PUNCT iajs-3505 249 7 37(3 37(3 NUM iajs-3505 249 8 ) ) PUNCT iajs-3505 249 9 , , PUNCT iajs-3505 249 10 265–281 265–281 NUM iajs-3505 249 11 . . PUNCT iajs-3505 250 1 7 7 X iajs-3505 250 2 . . X iajs-3505 250 3 mandal mandal PROPN iajs-3505 250 4 , , PUNCT iajs-3505 250 5 s. s. PROPN iajs-3505 250 6 ; ; PUNCT iajs-3505 250 7 ray ray PROPN iajs-3505 250 8 , , PUNCT iajs-3505 250 9 s. s. PROPN iajs-3505 250 10 ; ; PUNCT iajs-3505 250 11 ghosh ghosh PROPN iajs-3505 250 12 , , PUNCT iajs-3505 250 13 p. p. PROPN iajs-3505 250 14 b. b. PROPN iajs-3505 251 1 modeling modeling NOUN iajs-3505 251 2 nutrient nutrient NOUN iajs-3505 251 3 ( ( PUNCT iajs-3505 251 4 dissolved dissolve VERB iajs-3505 251 5 inorganic inorganic ADJ iajs-3505 251 6 nitrogen nitrogen NOUN iajs-3505 251 7 ) ) PUNCT iajs-3505 251 8 and and CCONJ iajs-3505 251 9 plankton plankton NOUN iajs-3505 251 10 dynamics dynamic NOUN iajs-3505 251 11 at at ADP iajs-3505 251 12 sagar sagar PROPN iajs-3505 251 13 island island NOUN iajs-3505 251 14 of of ADP iajs-3505 251 15 hooghly hooghly ADV iajs-3505 251 16 – – PUNCT iajs-3505 251 17 matla matla NOUN iajs-3505 251 18 estuarine estuarine NOUN iajs-3505 251 19 system system NOUN iajs-3505 251 20 , , PUNCT iajs-3505 251 21 west west PROPN iajs-3505 251 22 bengal bengal PROPN iajs-3505 251 23 , , PUNCT iajs-3505 251 24 india india PROPN iajs-3505 251 25 . . PUNCT iajs-3505 251 26 natural natural ADJ iajs-3505 251 27 resource resource NOUN iajs-3505 251 28 modeling modeling NOUN iajs-3505 251 29 , , PUNCT iajs-3505 251 30 2012 2012 NUM iajs-3505 251 31 , , PUNCT iajs-3505 251 32 25(4 25(4 NOUN iajs-3505 251 33 ) ) PUNCT iajs-3505 251 34 , , PUNCT iajs-3505 251 35 629–652 629–652 NUM iajs-3505 251 36 . . PUNCT iajs-3505 252 1 https://doi.org/10.1111/j.1939-7445.2011.00116.x https://doi.org/10.1111/j.1939-7445.2011.00116.x ADJ iajs-3505 252 2 8 8 NUM iajs-3505 252 3 . . PUNCT iajs-3505 253 1 mondal mondal PROPN iajs-3505 253 2 , , PUNCT iajs-3505 253 3 s. s. PROPN iajs-3505 253 4 ; ; PUNCT iajs-3505 253 5 samanta samanta PROPN iajs-3505 253 6 , , PUNCT iajs-3505 253 7 g. g. PROPN iajs-3505 253 8 ; ; PUNCT iajs-3505 253 9 de de PROPN iajs-3505 253 10 la la X iajs-3505 253 11 sen sen PROPN iajs-3505 253 12 , , PUNCT iajs-3505 253 13 m. m. NOUN iajs-3505 253 14 dynamics dynamic NOUN iajs-3505 253 15 of of ADP iajs-3505 253 16 oxygen oxygen NOUN iajs-3505 253 17 - - PUNCT iajs-3505 253 18 plankton plankton NOUN iajs-3505 253 19 model model NOUN iajs-3505 253 20 with with ADP iajs-3505 253 21 variable variable ADJ iajs-3505 253 22 zooplankton zooplankton NOUN iajs-3505 253 23 search search NOUN iajs-3505 253 24 rate rate NOUN iajs-3505 253 25 in in ADP iajs-3505 253 26 deterministic deterministic ADJ iajs-3505 253 27 and and CCONJ iajs-3505 253 28 fluctuating fluctuate VERB iajs-3505 253 29 environments environment NOUN iajs-3505 253 30 . . PUNCT iajs-3505 254 1 mathematics mathematic NOUN iajs-3505 254 2 , , PUNCT iajs-3505 254 3 2022 2022 NUM iajs-3505 254 4 , , PUNCT iajs-3505 254 5 10(10 10(10 NUM iajs-3505 254 6 ) ) PUNCT iajs-3505 254 7 , , PUNCT iajs-3505 254 8 1641 1641 NUM iajs-3505 254 9 . . PUNCT iajs-3505 255 1 https://doi.org/10.3390/math10101641 https://doi.org/10.3390/math10101641 PROPN iajs-3505 255 2 9 9 NUM iajs-3505 255 3 . . X iajs-3505 255 4 turner turner PROPN iajs-3505 255 5 , , PUNCT iajs-3505 255 6 j. j. PROPN iajs-3505 255 7 ; ; PUNCT iajs-3505 255 8 vollrath vollrath PROPN iajs-3505 255 9 , , PUNCT iajs-3505 255 10 f. f. PROPN iajs-3505 255 11 ; ; PUNCT iajs-3505 255 12 hesselberg hesselberg PROPN iajs-3505 255 13 , , PUNCT iajs-3505 255 14 t. t. NOUN iajs-3505 255 15 wind wind NOUN iajs-3505 255 16 speed speed NOUN iajs-3505 255 17 affects affect VERB iajs-3505 255 18 prey prey NOUN iajs-3505 255 19 - - PUNCT iajs-3505 255 20 catching catch VERB iajs-3505 255 21 behaviour behaviour NOUN iajs-3505 255 22 in in ADP iajs-3505 255 23 an an DET iajs-3505 255 24 orb orb NOUN iajs-3505 255 25 web web NOUN iajs-3505 255 26 spider spider NOUN iajs-3505 255 27 . . PUNCT iajs-3505 256 1 naturwissenschaften naturwissenschaften PROPN iajs-3505 256 2 , , PUNCT iajs-3505 256 3 2011 2011 NUM iajs-3505 256 4 , , PUNCT iajs-3505 256 5 98 98 NUM iajs-3505 256 6 , , PUNCT iajs-3505 256 7 1063–1067 1063–1067 NUM iajs-3505 256 8 . . PUNCT iajs-3505 257 1 https://doi.org/10.1007/s00114-011-0854-4 https://doi.org/10.1007/s00114-011-0854-4 NUM iajs-3505 257 2 10 10 NUM iajs-3505 257 3 . . PUNCT iajs-3505 258 1 das das PROPN iajs-3505 258 2 , , PUNCT iajs-3505 258 3 a. a. NOUN iajs-3505 258 4 ; ; PUNCT iajs-3505 258 5 samanta samanta PROPN iajs-3505 258 6 , , PUNCT iajs-3505 258 7 g. g. PROPN iajs-3505 258 8 p. p. NOUN iajs-3505 258 9 modeling model VERB iajs-3505 258 10 the the DET iajs-3505 258 11 fear fear ADJ iajs-3505 258 12 effect effect NOUN iajs-3505 258 13 on on ADP iajs-3505 258 14 a a DET iajs-3505 258 15 stochastic stochastic ADJ iajs-3505 258 16 prey prey NOUN iajs-3505 258 17 – – PUNCT iajs-3505 258 18 predator predator NOUN iajs-3505 258 19 system system NOUN iajs-3505 258 20 with with ADP iajs-3505 258 21 additional additional ADJ iajs-3505 258 22 food food NOUN iajs-3505 258 23 for for ADP iajs-3505 258 24 the the DET iajs-3505 258 25 predator predator NOUN iajs-3505 258 26 . . PUNCT iajs-3505 259 1 journal journal PROPN iajs-3505 259 2 of of ADP iajs-3505 259 3 physics physics PROPN iajs-3505 259 4 a a PRON iajs-3505 259 5 : : PUNCT iajs-3505 259 6 mathematical mathematical ADJ iajs-3505 259 7 and and CCONJ iajs-3505 259 8 theoretical theoretical ADJ iajs-3505 259 9 , , PUNCT iajs-3505 259 10 2018 2018 NUM iajs-3505 259 11 , , PUNCT iajs-3505 259 12 51(46 51(46 NUM iajs-3505 259 13 ) ) PUNCT iajs-3505 259 14 , , PUNCT iajs-3505 259 15 465601 465601 NUM iajs-3505 259 16 . . PUNCT iajs-3505 260 1 https://doi.org/10.1088/1751-8121/aae4c6 https://doi.org/10.1088/1751-8121/aae4c6 PROPN iajs-3505 260 2 11 11 NUM iajs-3505 260 3 . . PUNCT iajs-3505 261 1 arditi arditi PROPN iajs-3505 261 2 , , PUNCT iajs-3505 261 3 r. r. PROPN iajs-3505 261 4 ; ; PUNCT iajs-3505 261 5 ginzburg ginzburg NOUN iajs-3505 261 6 , , PUNCT iajs-3505 261 7 l. l. PROPN iajs-3505 261 8 r. r. 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PUNCT iajs-3505 263 1 2024 2024 NUM iajs-3505 263 2 , , PUNCT iajs-3505 263 3 37(4 37(4 PROPN iajs-3505 263 4 ) ) PUNCT iajs-3505 263 5 380 380 NUM iajs-3505 263 6 theoretical theoretical ADJ iajs-3505 263 7 biology biology NOUN iajs-3505 263 8 , , PUNCT iajs-3505 263 9 1989 1989 NUM iajs-3505 263 10 , , PUNCT iajs-3505 263 11 139(3 139(3 NUM iajs-3505 263 12 ) ) PUNCT iajs-3505 263 13 , , PUNCT iajs-3505 263 14 311–326 311–326 NUM iajs-3505 263 15 . . PUNCT iajs-3505 264 1 https://doi.org/10.1016/s0022-5193(89)80211-5 https://doi.org/10.1016/s0022-5193(89)80211-5 VERB iajs-3505 264 2 12 12 NUM iajs-3505 264 3 . . PUNCT iajs-3505 264 4 sajan sajan PROPN iajs-3505 264 5 ; ; PUNCT iajs-3505 264 6 sasmal sasmal PROPN iajs-3505 264 7 , , PUNCT iajs-3505 264 8 s. s. PROPN iajs-3505 264 9 k. k. PROPN iajs-3505 264 10 ; ; PUNCT iajs-3505 264 11 dubey dubey PROPN iajs-3505 264 12 , , PUNCT iajs-3505 264 13 b. b. PROPN iajs-3505 264 14 a a DET iajs-3505 264 15 phytoplankton phytoplankton NOUN iajs-3505 264 16 – – PUNCT iajs-3505 264 17 zooplankton zooplankton NOUN iajs-3505 264 18 – – PUNCT iajs-3505 264 19 fish fish NOUN iajs-3505 264 20 model model NOUN iajs-3505 264 21 with with ADP iajs-3505 264 22 chaos chaos NOUN iajs-3505 264 23 control control NOUN iajs-3505 264 24 : : PUNCT iajs-3505 264 25 in in ADP iajs-3505 264 26 the the DET iajs-3505 264 27 presence presence NOUN iajs-3505 264 28 of of ADP iajs-3505 264 29 fear fear NOUN iajs-3505 264 30 effect effect NOUN iajs-3505 264 31 and and CCONJ iajs-3505 264 32 an an DET iajs-3505 264 33 additional additional ADJ iajs-3505 264 34 food food NOUN iajs-3505 264 35 . . PUNCT iajs-3505 265 1 chaos chaos NOUN iajs-3505 265 2 : : PUNCT iajs-3505 265 3 an an DET iajs-3505 265 4 interdisciplinary interdisciplinary ADJ iajs-3505 265 5 journal journal NOUN iajs-3505 265 6 of of ADP iajs-3505 265 7 nonlinear nonlinear ADJ iajs-3505 265 8 science science NOUN iajs-3505 265 9 , , PUNCT iajs-3505 265 10 2022 2022 NUM iajs-3505 265 11 , , PUNCT iajs-3505 265 12 32(1 32(1 NUM iajs-3505 265 13 ) ) PUNCT iajs-3505 265 14 , , PUNCT iajs-3505 265 15 13114 13114 NUM iajs-3505 265 16 . . PUNCT iajs-3505 266 1 https://doi.org/10.1063/5.0069474 https://doi.org/10.1063/5.0069474 PROPN iajs-3505 266 2 13 13 NUM iajs-3505 266 3 . . PUNCT iajs-3505 266 4 gard gard NOUN iajs-3505 266 5 , , PUNCT iajs-3505 266 6 t. t. PROPN iajs-3505 266 7 c. c. PROPN iajs-3505 266 8 ; ; PUNCT iajs-3505 266 9 hallam hallam PROPN iajs-3505 266 10 , , PUNCT iajs-3505 266 11 t. t. PROPN iajs-3505 266 12 g. g. PROPN iajs-3505 266 13 persistence persistence NOUN iajs-3505 266 14 in in ADP iajs-3505 266 15 food food NOUN iajs-3505 266 16 webs webs NOUN iajs-3505 266 17 β€” β€” PUNCT iajs-3505 266 18 i i PROPN iajs-3505 266 19 lotka lotka PROPN iajs-3505 266 20 - - PUNCT iajs-3505 266 21 volterra volterra PROPN iajs-3505 266 22 food food NOUN iajs-3505 266 23 chains chain NOUN iajs-3505 266 24 . . PUNCT iajs-3505 267 1 bulletin bulletin NOUN iajs-3505 267 2 of of ADP iajs-3505 267 3 mathematical mathematical ADJ iajs-3505 267 4 biology biology NOUN iajs-3505 267 5 , , PUNCT iajs-3505 267 6 1979 1979 NUM iajs-3505 267 7 , , PUNCT iajs-3505 267 8 41(6 41(6 NOUN iajs-3505 267 9 ) ) PUNCT iajs-3505 267 10 , , PUNCT iajs-3505 267 11 877–891 877–891 NUM iajs-3505 267 12 . . PUNCT iajs-3505 268 1 https://doi.org/10.1016/s0092-8240(79)80024-5 https://doi.org/10.1016/s0092-8240(79)80024-5 NOUN iajs-3505 268 2 14 14 NUM iajs-3505 268 3 . . PUNCT iajs-3505 269 1 paine paine PROPN iajs-3505 269 2 , , PUNCT iajs-3505 269 3 r. r. PROPN iajs-3505 269 4 t. t. PROPN iajs-3505 269 5 food food PROPN iajs-3505 269 6 web web NOUN iajs-3505 269 7 complexity complexity NOUN iajs-3505 269 8 and and CCONJ iajs-3505 269 9 species species NOUN iajs-3505 269 10 diversity diversity NOUN iajs-3505 269 11 . . PUNCT iajs-3505 270 1 the the DET iajs-3505 270 2 american american PROPN iajs-3505 270 3 naturalist naturalist PROPN iajs-3505 270 4 , , PUNCT iajs-3505 270 5 1966 1966 NUM iajs-3505 270 6 , , PUNCT iajs-3505 270 7 100(910 100(910 NUM iajs-3505 270 8 ) ) PUNCT iajs-3505 270 9 , , PUNCT iajs-3505 270 10 65 65 NUM iajs-3505 270 11 – – PUNCT iajs-3505 270 12 75 75 NUM iajs-3505 270 13 . . PUNCT iajs-3505 270 14 15 15 NUM iajs-3505 270 15 . . PUNCT iajs-3505 271 1 meng meng PROPN iajs-3505 271 2 , , PUNCT iajs-3505 271 3 x.-y x.-y PROPN iajs-3505 271 4 . . 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PUNCT iajs-3505 275 1 modeling model VERB iajs-3505 275 2 earth earth NOUN iajs-3505 275 3 systems system NOUN iajs-3505 275 4 and and CCONJ iajs-3505 275 5 environment environment NOUN iajs-3505 275 6 , , PUNCT iajs-3505 275 7 2016 2016 NUM iajs-3505 275 8 , , PUNCT iajs-3505 275 9 2(1 2(1 NUM iajs-3505 275 10 ) ) PUNCT iajs-3505 275 11 , , PUNCT iajs-3505 275 12 1–15 1–15 NUM iajs-3505 275 13 . . PUNCT iajs-3505 276 1 https://doi.org/10.1007/s40808-015-0061y https://doi.org/10.1007/s40808-015-0061y PROPN iajs-3505 276 2 17 17 NUM iajs-3505 276 3 . . PUNCT iajs-3505 277 1 hirsch hirsch PROPN iajs-3505 277 2 , , PUNCT iajs-3505 277 3 m. m. PROPN iajs-3505 277 4 w. w. PROPN iajs-3505 277 5 ; ; PUNCT iajs-3505 277 6 smale smale PROPN iajs-3505 277 7 , , PUNCT iajs-3505 277 8 s. s. PROPN iajs-3505 277 9 ; ; PUNCT iajs-3505 277 10 devaney devaney PROPN iajs-3505 277 11 , , PUNCT iajs-3505 277 12 r. r. PROPN iajs-3505 277 13 l. l. PROPN iajs-3505 277 14 differential differential PROPN iajs-3505 277 15 equations equation NOUN iajs-3505 277 16 , , PUNCT iajs-3505 277 17 dynamical dynamical ADJ iajs-3505 277 18 systems system NOUN iajs-3505 277 19 , , PUNCT iajs-3505 277 20 and and CCONJ iajs-3505 277 21 an an DET iajs-3505 277 22 introduction introduction NOUN iajs-3505 277 23 to to PART iajs-3505 277 24 chaos chaos VERB iajs-3505 277 25 . . PUNCT iajs-3505 278 1 2012 2012 NUM iajs-3505 278 2 , , PUNCT iajs-3505 278 3 academic academic ADJ iajs-3505 278 4 press press NOUN iajs-3505 278 5 . . PUNCT iajs-3505 279 1 18 18 NUM iajs-3505 279 2 . . X iajs-3505 279 3 perko perko PROPN iajs-3505 279 4 , , PUNCT iajs-3505 279 5 l. l. PROPN iajs-3505 279 6 differential differential PROPN iajs-3505 279 7 equations equation NOUN iajs-3505 279 8 and and CCONJ iajs-3505 279 9 dynamical dynamical ADJ iajs-3505 279 10 systems system NOUN iajs-3505 279 11 ( ( PUNCT iajs-3505 279 12 vol vol NOUN iajs-3505 279 13 . . PROPN iajs-3505 279 14 7 7 NUM iajs-3505 279 15 ) ) PUNCT iajs-3505 279 16 . . PUNCT iajs-3505 280 1 2013 2013 NUM iajs-3505 280 2 , , PUNCT iajs-3505 280 3 springer springer NOUN iajs-3505 280 4 science science PROPN iajs-3505 280 5 & & CCONJ iajs-3505 280 6 business business NOUN iajs-3505 280 7 media medium NOUN iajs-3505 280 8 . . PUNCT iajs-3505 281 1 19 19 NUM iajs-3505 281 2 . . X iajs-3505 281 3 lasalle lasalle PROPN iajs-3505 281 4 , , PUNCT iajs-3505 281 5 j. j. PROPN iajs-3505 281 6 p. p. PROPN iajs-3505 281 7 stability stability PROPN iajs-3505 281 8 theory theory NOUN iajs-3505 281 9 and and CCONJ iajs-3505 281 10 invariance invariance NOUN iajs-3505 281 11 principles principle NOUN iajs-3505 281 12 . . PUNCT iajs-3505 282 1 in in ADP iajs-3505 282 2 dynamical dynamical ADJ iajs-3505 282 3 systems system NOUN iajs-3505 282 4 , , PUNCT iajs-3505 282 5 1976 1976 NUM iajs-3505 282 6 , , PUNCT iajs-3505 282 7 211–222 211–222 NUM iajs-3505 282 8 . . PUNCT iajs-3505 282 9 elsevier elsevier NOUN iajs-3505 282 10 . . PUNCT iajs-3505 283 1 https://doi.org/10.1016/b978-0-12-164901-2.50021-0 https://doi.org/10.1016/b978-0-12-164901-2.50021-0 PRON iajs-3505 283 2 20 20 NUM iajs-3505 283 3 . . PUNCT iajs-3505 284 1 liu liu PROPN iajs-3505 284 2 , , PUNCT iajs-3505 284 3 y. y. PROPN iajs-3505 284 4 ; ; PUNCT iajs-3505 284 5 zhao zhao PROPN iajs-3505 284 6 , , PUNCT iajs-3505 284 7 l. l. PROPN iajs-3505 284 8 ; ; PUNCT iajs-3505 284 9 huang huang PROPN iajs-3505 284 10 , , PUNCT iajs-3505 284 11 x. x. PROPN iajs-3505 284 12 ; ; PUNCT iajs-3505 284 13 deng deng PROPN iajs-3505 284 14 , , PUNCT iajs-3505 284 15 h. h. PROPN iajs-3505 284 16 stability stability PROPN iajs-3505 284 17 and and CCONJ iajs-3505 284 18 bifurcation bifurcation NOUN iajs-3505 284 19 analysis analysis NOUN iajs-3505 284 20 of of ADP iajs-3505 284 21 two two NUM iajs-3505 284 22 species specie NOUN iajs-3505 284 23 amensalism amensalism NOUN iajs-3505 284 24 model model NOUN iajs-3505 284 25 with with ADP iajs-3505 284 26 michaelis michaeli NOUN iajs-3505 284 27 – – PUNCT iajs-3505 284 28 menten menten NOUN iajs-3505 284 29 type type NOUN iajs-3505 284 30 harvesting harvesting NOUN iajs-3505 284 31 and and CCONJ iajs-3505 284 32 a a DET iajs-3505 284 33 cover cover NOUN iajs-3505 284 34 for for ADP iajs-3505 284 35 the the DET iajs-3505 284 36 first first ADJ iajs-3505 284 37 species specie NOUN iajs-3505 284 38 . . PUNCT iajs-3505 285 1 advances advance NOUN iajs-3505 285 2 in in ADP iajs-3505 285 3 difference difference NOUN iajs-3505 285 4 equations equation NOUN iajs-3505 285 5 , , PUNCT iajs-3505 285 6 2018 2018 NUM iajs-3505 285 7 , , PUNCT iajs-3505 285 8 2018(1 2018(1 NUM iajs-3505 285 9 ) ) PUNCT iajs-3505 285 10 , , PUNCT iajs-3505 285 11 1–19 1–19 PROPN iajs-3505 285 12 . . PUNCT iajs-3505 285 13 https://doi.org/10.1186/s13662-018-1752-2 https://doi.org/10.1186/s13662-018-1752-2 NUM iajs-3505 285 14 21 21 NUM iajs-3505 286 1 . . PUNCT iajs-3505 287 1 kuznetsov kuznetsov PROPN iajs-3505 287 2 , , PUNCT iajs-3505 287 3 v. v. PROPN iajs-3505 287 4 a. a. NOUN iajs-3505 287 5 ; ; PUNCT iajs-3505 287 6 makalkin makalkin X iajs-3505 287 7 , , PUNCT iajs-3505 287 8 i. i. PROPN iajs-3505 287 9 a. a. PROPN iajs-3505 287 10 ; ; PUNCT iajs-3505 287 11 taylor taylor PROPN iajs-3505 287 12 , , PUNCT iajs-3505 287 13 m. m. NOUN iajs-3505 287 14 a. a. 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