Dynamical Analysis of a Stochastic Delayed Two-Species Competition Chemostat Model
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Dynamical Analysis of a Stochastic Delayed Two-Species Competition Chemostat Model Xiaofeng Zhang1 · Shulin Sun2 Received: 28 June 2019 / Revised: 24 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we consider a stochastic delayed two-species competition chemostat model with the Monod growth response function. First, we verify that there is a unique global positive solution for any given initial conditions for this stochastic delayed system. Second, we give the dynamical behavior of the solution of stochastic delay system around the washout equilibrium of deterministic system; moreover, we discuss the competition exclusion and coexistence of microorganisms x1 and x2 . Finally, computer simulations are carried out to illustrate the obtained results; in addition, results show that time delay has critical effects on the survival of the microorganisms. Keywords Stochastic delayed chemostat model · Competition exclusion · Coexistence · Itô formula Mathematics Subject Classification 39A50 · 60H10 · 60H35
1 Introduction The chemostat is an instrument for continuous cultivation of microorganisms. It is a dynamic system with continuous material inputs and outputs, and it plays an important role in biological mathematics. Smith and Waltman [1] introduced many chemostat
Communicated by See Keong Lee. This work is supported by the Natural Science Foundation of Shanxi Province (201801D121011).
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Xiaofeng Zhang [email protected] Shulin Sun [email protected]
1
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
2
School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004, China
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X. Zhang, S. Sun
models and many mathematical methods for analyzing and studying various chemostat models. The theoretical research of the chemostat models was initiated by the works [2–6], etc. Smith and Waltman discussed the deterministic chemostat competitive model with a single nutrient and two different microorganisms in [1]. The model is as follows: ⎧ 1 (t) S (t) = (S 0 − S(t))D − m 1a1S(t)x ⎪ ⎪ +S(t) − ⎪ ⎨ 1 (t) x1 (t) = − Dx1 (t) + m 1a1S(t)x +S(t) , ⎪ ⎪ ⎪ ⎩ 2 (t) x2 (t) = − Dx2 (t) + m 2a2S(t)x +S(t) ,
m 2 S(t)x2 (t) a2 +S(t) ,
(1.1)
where S(t), x1 (t) and x2 (t) stand for the concentrations of the nutrient and two microorganisms at time t in chemostat, respectively. S 0 and D are positive constants, which, respectively, represent the input concentration of the nutrient and the washout i S(t) rate. ami +S(t) (i = 1, 2) denote the Monod growth functional response, where m i > 0 (i = 1, 2) are called the maximal growth rates and ai > 0(i = 1, 2) are the Michaelis– Menten ( or half-saturation ) constants [2]. On the other hand, in the process of continuous microbial culture, the chemostat model will inevitably be disturbed by environmental factors. Therefore, it is obvious that there are some limitations in the study of determining the chemostat model. Therefore, it is necessary to study how
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