id author title date pages extension mime words sentence flesch summary cache txt iajs-3461 Matlob, Intsar; Naji, Saad The local Bifurcation of Dynamic Behavior of Predator-Prey System with Refuge for both SpeciesΒ  2025 17 .pdf application/pdf 7733 461 78 Where: �̃�𝑖𝑗 = πœ‚π‘–π‘— for all i , j = 1,2,3 except �̃�𝑖𝑗 = π‘Ÿ3β„ŽΜΏ βˆ’ �̈�4𝑗̿ βˆ’ π‘Ÿ4 = 0. 4.3 Local4 bifurcation4 analysis4 near π‘¨πŸ‘ = (οΏ½ΜΏοΏ½, 𝟎, 𝒋 ΜΏ): Theorem (4): Assume that the following criteria are fulfilled: π‘Ÿ4 > π‘Ÿ3 β„Ž ΜΏ βˆ’ π‘Ÿ5 𝑗 ΜΏ , (27) νœ€2Μ€ β‰  0, (28) ℝ β‰  0 (29) where: νœ€1Μ€ = π‘Ÿ1π‘Ÿ6β„ŽΜΏ + π‘Ÿ7𝑗(ΜΏ1 βˆ’ 2β„ŽΜΏ βˆ’ π‘Ÿ2) β„ŽΜΏ[π‘Ÿ2 + π‘Ÿ1(π‘Ÿ6β„ŽΜΏ βˆ’ π‘Ÿ8)] , νœ€2Μ€ = 1 βˆ’ π‘Ÿ2 𝑗 ΜΏ βˆ’ (2 βˆ’ π‘Ÿ2νœ€1Μ€)β„ŽΜΏ π‘Ÿ1 β„ŽΜΏ , νœ€3Μ€ = π‘Ÿ7𝑗̿ π‘Ÿ1β„ŽΜΏ , ℝ = Γ‡3 cache/iajs-3461.pdf txt/iajs-3461.txt