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This paper focuses on the dynamics of a chemostat model with general response function, in which the maximum growth rate of microorganisms is assumed to satisfy the Ornstein–Uhlenbeck process. Under the weak assumption of response function, we first show the existence and uniqueness of the global solution. Then, using the Markov semigroup theory, we establish sufficient condition for the existence of a unique stable stationary distribution. Biologically, the existence of stationary distribution implies the microorganism can survive for a long time. It should be emphasized that we further prove the positive definiteness of the covariance matrix and give the exact expression of probability density function for the distribution. Moreover, sufficient condition for extinction of the microorganism is derived. Finally, some numerical examples are carried out to support the theoretical analysis results. • A chemostat model with Ornstein–Uhlenbeck process is studied. • We obtain conditions for persistence and extinction of microorganisms. • The exact expression of density function for the distribution is given. • We analyze the impact of volatility intensity and reversion speed on the dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a nutrient–phytoplankton–zooplankton model with nutrient recycling, in which the maximal nutrient uptake rate and maximal zooplankton ingestion rate are given by a continuous, mean-reverting, stochastic process. We first prove the existence and uniqueness of the global solution. Then conditions for the extinction of plankton are derived in two cases. Moreover, we establish sufficient condition for the existence of stationary distribution by constructing appropriate Lyapunov functions. It is worth noting that we further give the exact expression of density function around the positive equilibrium of deterministic system. Finally, some simulations are carried out to demonstrate our theoretical results. • A nutrient–plankton model with Ornstein–Uhlenbeck process is studied. • We obtain conditions for extinction and persistence of plankton. • The exact expression of density function for the distribution is investigated. • We analyze the impact of volatility intensity and reversion speed on the dynamics. [ABSTRACT FROM AUTHOR]