Abstract
The stochastic chemostat model with Monod-Haldane response function is perturbed by environmental white noise. This model has a global positive solution. We demonstrate that there is a stationary distribution of the stochastic model and the system is ergodic under appropriate conditions, on the basis of Khasminskii’s theory on ergodicity. Sufficient criteria for extinction of the microbial population in the stochastic system are established. These conditions depend strongly on the Brownian motion. We find that even small scale white noise can promote the survival of microorganism populations, while large scale noise can lead to extinction. Numerical simulations are carried out to illustrate our theoretical results.
Highlights
The chemostat, known as a continuous stir tank reactor (CSTR) in the engineering literature, is a basic piece of laboratory apparatus used for the continuous culture of microorganisms
Where S(t), x(t) stand for the concentrations of nutrient and microbial population at time t respectively; S0 denotes the imput concentration of nutrient and Q represents the volumetric flow rate of the mixture of nutrient and microorganism; the coefficient δ is the ratio of the biomass of the microbial population produced by the nutrient consumed
Remark 4.1 We refer to the condition (ii) in Theorem 4.1, which tells us that the microorganism species may die out when dilution rate Q and white noise are not large
Summary
Before investigating the dynamical behavior of system (1.5), the existence of a global positive solution is proved. Throughout this paper, unless otherwise specified, let (Ω, , { t}t≥0, P) be a complete probability space with filtration { t}t≥0 satisfying the usual conditions (i.e. it is right continuous and 0 contains all P-null sets). Using the Lyapunov analysis method[26], we prove that the solution of the system (1.5) is positive and global. Theorem 2.1 For given initial value (S(0), x(0)) ∈ +2 , there is a unique solution (S(t), x(t)) of system (1.5) defined for all t ≥ 0, and the solution remains in +2 with probability one, i.e. Let S = eμ, x = eμ, Itô’s formula (given in Section 3) implies that system (1.5) has a unique local positive solution. It suffices to prove that this unique local positive solution of system (1.5) is global. Using Itô’s formula (see Theorem 3.1), we m dV(S, x)
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