Dynamics of a stochastic delayed chemostat model with nutrient storage and Lévy jumps

In this paper, a stochastic delayed chemostat model with nutrient storage and Levy jumps is proposed. Firstly, the existence and uniqueness of the positive global solution of the model are discussed. Then, the threshold (lambda) over bar. and optimal control conditions for the persistence in the mean and extinction of the microorganism x are obtained. Besides, the ergodic stationary distribution of the SDDE model under a low-level intensity of stochastic noise is deduced. Finally, some numerical examples are given to support the theoretical analysis results. The simulation results show that stochastic noise and time delay play a vital role in controlling the persistence and extinction of microorganisms, respectively. On the one hand, high-intensity noise can inhibit the growth of microorganisms. On the other hand, if tau > iota*, the corresponding deterministic model will become unstable and produce a Hopf bifurcation. Moreover, the solutions of the SDDE model will oscillate around the non-constant. F - periodic solution of the corresponding deterministic model.