Dynamical behavior of a stochastic two-species Monod competition chemostat model

This paper studies a stochastic two-species Monod competition chemostat model which is subject to environment noises. Such noises are described by independent standard Brownian motions. It proves that the initial value problem of the model has a unique positive global solution. However, unlike the corresponding deterministic model, the stochastic model no longer has positive equilibrium points. The asymptotic behaviors and the steady state distributions are established by using Itô's formula, Lyaponov method and Gronwall inequality. In addition, numerical simulations are given to illustrate the theoretical results.