Long-time behaviour of a stochastic chemostat model with distributed delay

In this paper, we analyse a stochastic chemostat model with distributed delay and degenerate diffusion. We transform the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the positive solutions can converge in to an invariant density. The existence of a stable stationary distribution implies stochastic persistence of the microorganism.