Critical result on the break-even concentration in a single-species stochastic chemostat model

Abstract

In this paper, the strictly critical parameter λ˜ is identified as the break-even concentration for the stochastic chemostat model. λ˜>S0 suffices for extinction of the microorganism for any noise intensity σ2>0 is proved firstly, which completely solves the open problem proposed in [10]. Then, by using the Feller test, we show that the microorganism will go extinct in probability if λ˜=S0, which has not been studied in the known literature. When λ˜<S0, in addition to the previously discussed persistence in mean, the stochastic persistence of the microorganism is studied by use of Chebyshev's inequality. Besides, the existence of the stationary distribution is proved for the considered model. Numerical simulations are introduced finally to support the obtained results.