Assessment of rabbit hemorrhagic disease in controlling the population of red fox: A measure to preserve endangered species in Australia

•Dissemination of RHD in the rabbit population and its consequences on red foxes has been studied.•Analyses of an eco-epidemiological model with sigmoid functional response are carried out.•Global stability of endemic equilibrium is established by geometric approach.•Turing patterns formation selected by RD system under zero flux boundary conditions are presented.•Red foxes might be controlled if rabbit hemorrhagic disease is introduced into rabbit population. Predator's management requires a detailed understanding of the ecological circumstances associated with predation. Predation by foxes has been a significant contributor to the Australian native animal reduction. This paper mainly focuses on the dissemination of rabbit hemorrhagic disease in the rabbit population and its subsequences on red fox (Vulpes vulpes) population, by qualitative and quantitative analyses of a designed eco-epidemiological model with simple law of mass action and sigmoid functional response. Existence of solution has been analyzed and shown to be uniformly bounded. The basic reproduction number (R0) is obtained and the occurrence of a backward bifurcation at R0=1 is shown to be possible using central manifold theory. Global stability of endemic equilibrium is established by geometric approach. Criteria for diffusion-driven ecological instability caused by local random movements of European rabbits and red fox are obtained. Detailed analyses of Turing patterns formation selected by reaction-diffusion system under zero flux boundary conditions are presented. We found that transmission rate, self and cross-diffusion coefficients have appreciable influence on spatial spread of epidemics. Numerical simulation results confirm the analytical finding and generate patterns which indicate that population of red foxes might be controlled if rabbit hemorrhagic disease (RHD) is introduced into the rabbit population and thus ecological balance can be maintained.