Abstract
A new stochastic chemostat model with two substitutable nutrients and one microorganism is proposed and investigated. Firstly, for the corresponding deterministic model, the threshold for extinction and permanence of the microorganism is obtained by analyzing the stability of the equilibria. Then, for the stochastic model, the threshold of the stochastic chemostat for extinction and permanence of the microorganism is explored. Difference of the threshold of the deterministic model and the stochastic model shows that a large stochastic disturbance can affect the persistence of the microorganism and is harmful to the cultivation of the microorganism. To illustrate this phenomenon, we give some computer simulations with different intensity of stochastic noise disturbance.
Highlights
Chemostat is commonly used to describe the dynamics of a microbial population in a continuous bioreactor in which microorganisms grow on a substrate and has attracted great interest of many scholars [1–8], since it was first introduced by Monod [9]
Our main objective in the rest of this paper is to investigate the threshold dynamics of stochastic chemostat model (5) and explore the conditions under which microorganisms will die out or exist
This paper proposes and investigates a new stochastic chemostat model with two substitutable nutrients and one microorganism
Summary
Chemostat is commonly used to describe the dynamics of a microbial population in a continuous bioreactor in which microorganisms grow on a substrate and has attracted great interest of many scholars [1–8], since it was first introduced by Monod [9]. A single simple species chemostat model with Michaelis-Menten-Monod functional response was proposed by [9] as follows: dS (t) dt. A model of single-species growth in the chemostat on two substitutable resources with Michaelis-Menten-Monod functional response was proposed by [14] as follows: dS1 (t) dt. Mathematical Problems in Engineering proposed a stochastic chemostat model for a single microorganism species consuming a single nutrient They found that random effects may lead to extinction in scenarios where the deterministic model predicts persistence. Xu and Yuan [35] established a stochastic chemostat model in which the maximal growth rate is influenced by the white noise in environment as follows: dS (t). They got an analogue break-even concentration involving the white noise which can determine the exclusion and persistence of the microorganism. Our main objective in the rest of this paper is to investigate the threshold dynamics of stochastic chemostat model (5) and explore the conditions under which microorganisms will die out or exist
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