Dynamic analysis of a stochastic eco-epidemiological model with disease in predators

It is widely accepted that epidemics in biological populations and stochasticity in the environments are two important components significantly influencing the real ecosystems. To more accurately reveal developmental changes of the population dynamics and explore effective control methods for infectious disease, in this paper, we first construct a stochastic eco-epidemiological model with disease in predators, and mathematically analyze the fundamental properties of the model. Then, we establish sufficient criterion for the existence of a unique ergodic stationary distribution of the positive solution. Also, we derive sufficient conditions, respectively, for three different extinction scenarios of populations: (i) The predators go extinct, only the prey persists; (ii) both the predators and prey go extinct; and (iii) the disease is eradicated from the system, both the predators and prey persist. Finally, we perform some numerical simulations under different interferences of environment to illustrate the main theoretical results. Especially, we observe that once the disease is eradicated from the system, then the predators and prey will coexist in an oscillatory manner. Our research indicates that the environmental noises have significant impacts on the determination of the persistence and extinction of the disease and populations.