Stabilization in a 3D eco-epidemiological model: From the complete extinction of a predator population to their self-healing

In this paper, using the localization method of compact invariant sets, we examine the ultimate dynamics of the 3D prey–predator model containing two subpopulations of susceptible and infected predators. Our attention is focused to finding ultimate sizes of interacting populations, and, in addition, we show the existence of a global attracting set. Then, we derive various global conditions of ultimate extinction of at least one of the predators subpopulations and describe conditions under which all types of internal bounded dynamics are ruled out. In particular, we describe convergence conditions to omega-limit sets located (1) in the intersection of the prey-free plane with the infected predators-free plane and (2) in the infected predators-free plane. Based on the dynamical analysis of the 2D infection-free subsystem, we obtain conditions of global attraction to (i) the prey-only disease-free equilibrium point, (ii) the disease-free prey-predator equilibrium point (self-healing of the predator population), and (iii) the omega-limit set containing an equilibrium point or a periodic orbit. Main theoretical results are illustrated by numerical simulation. Tools and techniques developed in this work can be appropriated in the studies within predictive population ecology of more complex eco-epidemiological models.