The bifurcation analysis of food chain prey - predator model involving hunting cooperation with toxin and disease in top predator

Abstract

This paper establishes the local bifurcation (LB) conditions for an eco-epidemiological food chain model involving prey, predator, and a top predator (susceptible and infected), incorporating fear, hunting cooperation, toxicity and anti-predator via Holling type IV and Sokol-Howell functional responses. The analysis identifies twelve equilibrium points (EPs), characterizing their asymptotic stability and identifying transcritical bifurcation (TB) occurs at \(M_0\), \(M_6\) and \(M_7\), whereas both (TB) and pitchfork bifurcation (PB) are observed near \(M_1\), \(M_2\), \(M_3\), \(M_4\), \(M_5\) and \(M_8\). Furthermore saddle-node bifurcation (SNB) manifests at \(M_9\), \(M_{10}\) and \(M_{11}\), In addition to the analytical determination of the thresholds that lead to the occurrence of Hopf branching (HB) near the positive steady state. Finally, a thorough numerical simulation verifies the analytical results. These findings shed light on how biological variables and disease affect biological community survival and pattern alterations.