Equilibrium points and their stability of food - chain prey - predator model involving fear and toxin

Abstract

This paper presents and analyses a food chain eco-toxicant model involving one prey and two predators, incorporating a fear with linear and nonlinear harvesting, utilizing two distinct functional responses (Lotka-Volterra and Sokol-Howell). Prey species are directly exposed to environmental toxins, while first and second predators encounter these toxins through consumption. The model's boundedness and uniqueness have been established. All the equilibrium points of this model have been found. All requisite conditions for achieving the local stability of equilibrium points have been identified, leading to in the conclusion that all points where are locally stable. Additionally, a global stability study has been conducted using appropriate Lyapunov functions. Ultimately, numerical simulations for a one set of parameters and varying initial conditions have been conducted to validate the analytical results and to assess the effect of fear, toxins, linear and nonlinear harvesting on the dynamics of the proposed model.