Asymptotic behavior of a chemostat model with stochastic perturbation on the dilution rate

We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise.The long time behavior of the system is studied.Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in themean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results