Rich dynamics and density function analysis of a hybrid stochastic predator-prey model with Logistic growth and SIS parasite infection

Based on the uncertainty of changes in population survival environment and the complexity of epidemic transmission dynamics in ecosystems, we formulate a hybrid stochastic predator-prey model with Logistic growth and SIS parasite infection in the prey. Firstly, the local stability of the endemic equilibria is discussed with the Routh-Hurwitz criterion. Then we illustrate the coexistence of disease and populations from the perspective of stationary distribution. Next, we prove the persistence in the mean and the extinction of the predator and obtain the threshold value $$ \mathscr {R}_1$$ between them. $$ \mathscr {R}_1$$ is a very important index for the population dynamics. Moreover, sufficient conditions of the weak persistence of the predator and the extinction of the infected prey are derived. The extinction of the infected prey also reflects the extinction of parasitic disease. Further, it is worth noting that the accurate expression of the density function $$ \Psi $$ of the stochastic system is given, which reveals the nature of the epidemic after it has stabilized. Finally, the results of numerical simulations verify the correctness of our conclusion.