The bifurcation conditions of Sokol-Howell prey-predator model involving fear and toxin

Abstract

In this paper, the conditions of the occurrence of local bifurcation (LB) have been founded for a food- chain eco-toxicant model involving three species: prey, first and second predators, incorporating factors such as fear, as well as linear and nonlinear harvesting strategies, including two various functional responses (Lotka-Volterra and Sokol-Howell). This model features six equilibrium points (EPS), all of which are stable under appropriate conditions, the saddle-node bifurcation (SNB) appears near the positive point E₅, transcritical bifurcation (TB) and pitchfork bifurcation (PB) occur near the points E₂ and E₃, while only a transcritical bifurcation takes place near the point E₀, E₁, and E₄. Additionally, the Hopf bifurcation (HB) conditions near the positive equilibrium point (PEP) have been discussed. Finally, the numerical simulation for the hypothetical parameter set confirm our analytical findings about (LB) of this model.