This paper considers a stochastic chemostat model with degenerate diffusion. Firstly, the Markov semigroup theory is used to establish sufficient criteria for the existence of a unique stable stationary distribution. The authors show that the densities of the distributions of the solutions can converge in L¹ to an invariant density. Then, conditions are obtained to guarantee the washout of the microorganism. Furthermore, through solving the corresponding Fokker-Planck equation, the authors give the exact expression of density function around the positive equilibrium of deterministic system. Finally, numerical simulations are performed to illustrate the theoretical results.

本文考虑具有简并扩散的随机恒化器模型。首先，使用马尔可夫半群理论为唯一稳定平稳分布的存在建立充分的标准。作者表明，解分布的密度可以在L ^{1 中}收敛到一个不变的密度。然后，获得条件以保证微生物被冲洗掉。此外，通过求解相应的Fokker-Planck方程，作者给出了确定性系统正平衡附近密度函数的精确表达式。最后，进行数值模拟以说明理论结果。