In this paper, a stochastic chemostat model with two distributed delays and nonlinear perturbation is proposed. We first transform the stochastic model into an equivalent high-dimensional system. Then we prove the existence and uniqueness of global positive solution of the model. Based on Khasminskii's theory, we study the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Then we also establish sufficient conditions for the extinction of the plankton. Finally, numerical simulations are carried out to illustrate the theoretical results and to conclude our study, which shows that environmental noise experienced by limiting nutrient completely determines the persistence and extinction of the plankton.

To understand the transmission dynamics of diarrhea in random environment, in this paper we propose a stochastically perturbed dysentery diarrhoea epidemic model with treatment and sanitation. Using the theory of stopping time, we first show the existence of global positive solution of the model. Then, we study the stochastic dynamics of the model and present a stochastic threshold $\mathscr {R}_{0}^{S}$ which determines the extinction and persistence of the disease. Based on Khasminskii’s theory, we further prove that the model has a unique ergodic stationary distribution under the condition of $\mathscr {R}_{0}^{S}>1$ . Numerical simulations are carried out to verify the analytical results, showing that the white noise, and the constant treatment and sanitation may have certain inhibitory effects on disease transmission. Lastly, the model is further extended to include colored noise and seasonal fluctuation to study the long-term transmission dynamics of disease. It is found that the method proposed in this paper is universal.