The local Bifurcation of Dynamic Behavior of Predator-Prey System with Refuge for both Species 

Abstract

The main purpose of this paper is to study a predator – prey dynamical system consisting of three species prey, specialized predator and generalist predator namely H (t), I (t) and J (t) respectively, w food web and refuge for the prey and specialized predator population. The consider system has five equilibrium points A0=(0, 0, 0), A1=(1, 0, 0), A2=(h, i, 0), A3=(h, 0, j), and the positive equilibrium point A4=(h, i, j) .The stability and bifurcation of the equilibrium points was studied and the main influence was the qualitative behavior of the solution. It was found that A0 was unstable while the other equilibrium points are stable under condition so we study their bifurcation and we show that A1,A2,and A3 are transcritical while A, is saddle node bifurcation. Numerical simulations were used to illustrate the occurrence of local bifurcation of this model.

Keywords

• Local bifurcation
• predator-pery
• stability analysis
• Lyapunov's function
• ecological
• Refuge