A stochastic turbidostat model with Ornstein-Uhlenbeck process: dynamics analysis and numerical simulations

Many turbidostat models are affected by environmental noise due to various complicated and uncertain factors, and Ornstein-Uhlenbeck process is a more effective and precise way. We formulate a stochastic turbidostat system incorporating Ornstein-Uhlenbeck process in this paper and develop dynamical behavior for the stochastic model, which includes the existence and uniqueness of globally positive equilibrium, sufficient conditions of the extinction, the existence of a unique stationary distribution and an expression of density function of quasi-stationary distribution around the positive solution of the deterministic model. The results indicate that the weaker volatility intensity can ensure the existence and uniqueness of the stationary distribution, and the stronger reversion speed can lead to the extinction of microorganisms. Numerical simulations verify the validity of the analysis results, which assess the influence of the speed of reversion and the intensity of volatility on the long-term behavior of microorganisms.

Automated summary

This paper investigates the Ornstein-Uhlenbeck process in turbidostat systems and studies the dynamic behavior of the stochastic model, including the existence and uniqueness of globally positive equilibrium, conditions for extinction, the existence of a unique stationary distribution, and the expression of density function for quasi-stationary distribution. The results indicate that weaker volatility intensity ensures the existence and uniqueness of the stationary distribution, while stronger reversion speed leads to the extinction of microorganisms.