Local Stability Dynamics of Equilibrium Points in Predator-Prey Models with Anti-Predator Behavior

Abstract

This article describes the dynamics of local stability equilibrium point models of interaction between prey populations and their predators. The model involves response functions in the form of Holling type III and anti-predator behavior. The existence and stability of the equilibrium point of the model can be obtained by reviewing several cases. One of the factors that affect the existence and local stability of the model equilibrium point is the carrying capacity (k) parameter. If x₃^∗, y₃^∗  > 0 is a constant solution of the model and ∈ (0,x₃^∗), then there is a unique boundary equilibrium point E_k (k , 0). Whereas, if k ∈ (x₄^∗, y₄^∗], then E_k (k, 0) is unstable and E₃ (x₃^∗, y₃^∗) is stable. Furthermore, if k ∈ ( x₄^∗, ∞), then E_k ( k, 0) remains stable and E₄ (x₄^∗, y₄^∗) is unstable, but the stability of the equilibrium point E₃ (x₃^∗, y₃^∗) is branching. The equilibrium point E₃ (x₃^∗, y₃^∗) can be stable or unstable depending on all parameters involved in the model. Variations of k parameter values are given in numerical simulation to verify the results of the analysis. Numerical simulation indicates that if k = 0,92 then nontrivial equilibrium point E_k (0,92 ; 0) stable. If k = 0,93 then E_k (0,93 ; 0) unstable and E₃^∗(0,929; 0,00003) stable. If k = 23,94, then E_k (23,94 ; 0) and E₃^∗(0,929; 0,143) stable, but E₄^∗(23,93 ; 0,0005) unstable. If k = 38 then E_k(38,0) stable, but E₃^∗(0,929; 0,145) and E₄^∗(23,93 ; 0,739) unstable.
Keywords: anti-predator behavior, carrying capacity, and holling type III.