Dynamic behavior on a multi-time scale eco-epidemic model with stochastic disturbances

In this paper, a multi-time scale stochastic eco-epidemic model where the prey population was infected with disease was proposed. The stochastic factors in the ecological environment and the fact that the growth and loss rates of predators were much smaller than those of prey were considered. First, the dynamical behavior of the deterministic model was analyzed, including the existence and the stability of the equilibrium points and the bifurcation phenomena. Second, the existence and uniqueness of global positive solutions and the ergodic property of stochastic model were discussed. Meanwhile, the solution trajectory which was perturbed was also analyzed by using random center-manifold and random averaging method. Finally, the stochastic P-bifurcation is shown by applying singular boundary theory and invariant measure theory. Numerical simulation also verified the correctness of the theoretical analysis.