Asymptotic properties of a stochastic chemostat including species death rate

Abstract

This paper deals with a stochastic system which models the population dynamics of a chemostat including species death rate. On the basis of the theory on Markov semigroup, we demonstrate that the probability densities of the distributions for the solutions are absolutely continuous. The densities will convergence in L 1 to an invariant density or weakly convergence to a singular measure under appropriate conditions. We also give the sufficient criteria for extinction exponentially of the species. To be specific, when D 1 > D and the strength of perturbation is relatively small, we derive a precise threshold for the species survival.

Stationary probability density of a stochastic chemostat model with Monod growth response function

“…They employed the stochastic Lyapunov analysis method to derive the existence condition of a stationary distribution. With the help of Markov semigroup theory, Wang and Jiang 10 proved the existence of stationary distribution for a single‐species chemostat model in which the maximal growth rate of the microorganism is subject to the environmental white noise. Besides, the existence of stationary distribution was studied for some other stochastic chemostat models 11–13 .…”

Ergodicity and threshold behaviors of a predator‐prey model in stochastic chemostat driven by regime switching

“…In view of the environmental random effect on the natural growth for the species in chemostat model, extensive results are concerned with the stochastic chemostat systems. Here, we only mention Imhof and Walcher, 8 Campillo et al, 9 Xu and Yuan, 10 Zhang and Jiang, 11 Wang and Jiang, 12,13 Sun et al, 14 and Meng et al 15 Especially, Imhof and Walcher 8 initially formulated a competition chemostat model under linear random perturbation and obtained the persistence for both the deterministic system and the stochastic version of the chemostat model. Wang and Jiang 12 investigated a classic chemostat model involving random fluctuation with general functional response and analyzed the stationary distribution.…”

Global Asymptotic Behavior of a Multi-species Stochastic Chemostat Model with Discrete Delays

“…Wang and Jiang [46] introduced demographic stochasticity into chemostat model (1.1) in the case of single-species, and investigated the ergodic property of the stochastic chemostat model under regime switching. The authors in [47] further established a stochastic chemostat model with growth response of the Monod form and suppose the maximal growth rate is influenced by Brownian motion, and studied the asymptotic properties of the stochastic system including species death rate.…”