In this paper, we propose and analyze a higher-order stochastically perturbed two-species Lotka-Volterra competition system in a patchy environment. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we also obtain the asymptotic behavior of the solutions around the boundary equilibrium of the deterministic system.

In this paper, we study a stochastic cholera model with differential infectivity which is disturbed by both white noise and telegraph noise. Firstly, we prove that there exists a unique global positive solution to the system with any positive initial value. Then we obtain sufficient criteria for extinction of the diseases. Finally, we establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The stationary distribution implies that all the individuals can coexist and persist in the long term.