Dynamics of microorganism cultivation with delay and stochastic perturbation

In the microorganism cultivation process, delay and stochastic perturbations are inevitably accompanied, which results in complicated dynamical behaviors for microorganisms. In this paper, a mathematical model with discrete delay and random perturbation is constructed to understand how the dynamics of microorganisms in the turbidostat can be characterized. The existence, uniqueness and boundedness of the positive solution are first determined for the mathematical model. Furthermore, sufficient conditions for microorganism extinction and permanence in the turbidostat are obtained with the theory of stochastic differential equations. The system has the stationary distribution under a low-level intensity of stochastic perturbation from the environment; that is, microorganism in the turbidostat is persistent fluctuating around a positive value. On the contrary, microorganisms will be extinct with a strong enough intensity of noise. Several numerical simulations are applied to validate the theoretical results for the dynamics of the system.