Dynamics of a chemostat model with Ornstein–Uhlenbeck process and general response function

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.