Bifurcation and persistence conditions of Sokol-Howell prey-predator model with toxins and additional food for predators

Abstract

This work investigates the conditions for local bifurcation in a three-species food chain model, comprising prey, predator, and top predator, incorporating factors such as fear, a weak Allee effect, toxins, and supplementary food in the predator population, utilizing the Sokol-Howell functional response. This system possesses thirteen equilibrium points. Transcritical and pitchfork bifurcations occur near the extinction equilibrium point (Q₀), the predator-top predator-free equilibrium point (Q₁), the prey-top predator-free equilibrium point (Q₃), while near and the top predator-free equilibrium point (Q₄), while saddle node bifurcations are found near the prey-free equilibrium point (Q₈), and the interior equilibrium point (Q₁₀). Moreover, the persistence conditions have been derived to ensure the coexistence and stability of all species. Numerical simulation performed using matlab support the analytic finding and illustrate the complex dynamic behaviors near the equilibrium points. This study provides deeper insight into how Allee effect, toxin effects and additional food sources influence the bifurcation structure and persistence of the three-species ecological system.