Dynamics of a Food Chain Model with Two Infected Predators

In this paper, the dynamics of a three-species food chain model with two predators infected by an infectious disease is investigated. The positivity and boundedness of the system, the existence of the equilibria and the basic reproductive number are given. Sufficient conditions for the local stability of all equilibria are obtained by analyzing the corresponding characteristic equations. By constructing suitable Lyapunov functions and taking the geometric approach, the global stability of all equilibria is proved. According to the center manifold theory, this model undergoes the phenomenon of backward and forward bifurcations in a certain range of the basic reproductive number R0. By taking the disease transmission coefficient of predator as bifurcation parameter, Hopf bifurcation emerges in the neighborhood of the endemic equilibrium. Furthermore, the optimal control of the disease is discussed by the Pontryagin's maximum principle. Various simulations are given to support the analytical results.

Automated summary

This paper investigates a three-species food chain model with two predators infected by an infectious disease, analyzing the positivity, boundedness, existence of equilibria, and basic reproductive number of the system. The local stability of all equilibria is obtained, and the global stability is proved by constructing suitable Lyapunov functions and using the geometric approach. The model undergoes backward and forward bifurcations in a certain range of the basic reproductive number R0, with Hopf bifurcation arising near the endemic equilibrium when the disease transmission coefficient of predators is taken as a bifurcation parameter. Furthermore, the optimal control of the disease is discussed using Pontryagin's maximum principle, and various simulations are provided to support the analytical results.