Stochastic Characteristics and Optimal Control for a Stochastic Chemostat Model with Variable Yield

Rong Yan,Shulin Sun

doi:10.1155/2020/5065172

Abstract

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

Highlights

Where all the parameters are positive constants
Experimental data indicate that a constant yield may fail to explain the observed oscillatory behavior in the vessel. is leads to the formulation of the variable yield model, for example [7,8,9]. e chemostat model with variable yield takes the following form: Complexity
The above models are all deterministic models, but almost all ecosystems are inevitably perturbed by various types of environmental noises

Summary

Preliminaries

The notations are described for the whole paper as follows:. (i) Ω: a set of the elementary events (ii) F: a family of the subsets of Ω (iii) 􏼈Ft􏼉t≥0: a family of increasing sub− σ− algebras of F (iv) P(ω): the probability of events ω (v) EX: the expectation of X (vi) ∅: the empty set (vii) IA: the indicator function of a set A, i.e., IA(x) 1 if x ∈ A or otherwise 0. To further study the stochastic chemostat model with variable yield and Contois growth function (4), the first problem to be solved is the existence of the unique global positive solution, namely, there is no explosion in a finite time under the initial value (5). If the coefficients of the equations are generally required to satisfy the linear growth condition and the local Lipschitz condition (see [28, 30]), the stochastic differential equations for any given initial value have a unique global solution. E following theorem assures that the solution of the model (4) with the initial value (5) is unique, global, and positive. For any initial value (S0, x0) ∈ R2+, there exists a unique positive solution (S(t), x(t)) to the model (4) for t ≥ 0, and the solution will remain in R2+ with probability one (i.e., (S(t), x(t)) ∈ R2+ for all t ≥ 0 a.s.). At is, the microorganism will be extinct at an exponential rate with probability one

Stochastic Characteristics

Stochastic Maximum Principle

Numerical Simulations