Ergodic stationary distribution and extinction of stochastic delay chemostat system with Monod–Haldane functional response and higher-order Lévy jumps

Abstract

In this work, the case of the stochastic delayed chemostat model with Monod–Haldane functional response and nonlinear stochastic perturbation is investigated, in which both higher-order Gaussian white noise and Lévy jumps are introduced. Firstly, the condition for the occurrence of Hopf bifurcation in the deterministic system is derived. Secondly, the existence and uniqueness of the positive global solution of the stochastic system are established, and the stochastically ultimate boundedness of the solution is studied. To understand the statistical characteristics of the stochastic system, the key threshold is established by investigating the auxiliary equation of the corresponding stochastic system. Then, the existence of an ergodic stationary distribution of the stochastic system, which is indicative of the long-term persistence of the microbial population from a biological perspective, is proved. Furthermore, the extinction of the microorganism x for the stochastic system is studied. Finally, some important numerical experiments are provided to further support the theoretical results. The impact of various noises and time delays on the stochastic system is also analyzed using 2D and 3D graphs of the joint two-dimensional densities.