Resumen de Dynamical Properties of a Chemostat Model with Log-Normal Ornstein–Uhlenbeck Process and Distributed Delay
Miaomiao Gao, Daqing Jiang
This paper centers on a chemostat model with distributed delay, where the maximal growth rate of microorganisms adheres to the log-normal Ornstein–Uhlenbeck process. Our analysis relies on the linear chain technique. We first show the existence and uniqueness of the global positive solution. Subsequently, via constructing appropriate Lyapunov functions, we determine sufficient condition for the existence of stationary distribution. From a biological point of view, the existence of stationary distribution suggests that the microorganisms have the ability to survive over an extended period.
Furthermore, for the case of weak kernel and strong kernel, we derive the exact expression of local probability density function through matrix transformation respectively.
In addition, the condition for extinction of the microorganisms is given. Theoretical analysis underscores that the dynamics of the model under consideration are governed by two pivotal parameters Rs 0 and Re 0, which are interconnected with R0 of corresponding deterministic system. Finally, several numerical examples are provided to corroborate the theoretical analysis results and illustrate the impact of main parameters on the dynamics.
