Hunting cooperation among predators effects on the persistence and bifurcation of food-web eco-epidemiological model with additional food to predators

Abstract

In this paper, the local bifurcation's occurrence conditions have been established for a food web eco-epidemiological model of four species, including first prey, second prey, susceptible predator, and infected predator with fear effect, and internal competition among prey populations, incorporating additional food resources and hunting cooperation among predators, and disease dynamics with treatment in the predator population with Holling type-II and Lotka-Volterra functional response. This model has fifteen equilibria, the saddle-node bifurcation have been shown close to the interior equilibrium points 𝑃₁₃ and 𝑃₁₄, also at the first-second prey-free equilibrium point 𝑃₇, and the second prey-free equilibrium points 𝑃₉ and 𝑃₁₀ a transcritical bifurcation occurred, while at the preys-infected predator-free equilibrium point 𝑃₃, the predators-free equilibrium point 𝑃₄, the second prey-infected predator-free equilibrium point 𝑃₅, the first prey-infected predator-free equilibrium point 𝑃₆, the infected predatorfree equilibrium point 𝑃₈, first prey-free equilibrium points 𝑃₁₁ and 𝑃₁₂ have a transcritical and pitchfork bifurcations. Furthermore, conditions for Hopf bifurcation close to positive points 𝑃₁₃ and 𝑃₁₄ have also been examined. Numerical results for the set of hypothetical parameters support our analytical results regarding the persistence of this model and the occurrence of bifurcation using Mathematica.