Dynamics analysis of a discrete eco-epidemiological model with diseases in prey

This paper investigates a discrete-time eco-epidemiological model that describes the spread of disease within the prey population in a predator-prey system. We demonstrate the existence of multiple equilibrium points in the system, including a disease-free equilibrium, an equilibrium with predators but without disease transmission, and an equilibrium where the disease coexists. Through linear stability analysis, we determine the local stability conditions for each equilibrium point and prove the global stability of the positive equilibrium. Moreover, we have also demonstrated the consistent persistence of the system. Through numerical analysis, we identify complex dynamic behaviors, encompassing periodic oscillations and chaotic phenomena. Our findings reveal that when the disease transmission rate exceeds a critical threshold, the system may undergo a Neimark-Sacker bifurcation, transitioning from a stable equilibrium to periodic or chaotic dynamics. Furthermore, we identify that the predator's birth rate can induce a flip bifurcation within the system. The research findings underscore that disease dissemination in the predator-prey system not only influences the stable state of populations but also may precipitate complex dynamic behaviors.