## 具有一般反应函数与贴壁生长现象的随机恒化器模型的全局动力学行为

1 中国石油大学(华东)石油工程学院非常规油气开发教育部重点实验室 山东青岛 266580

2 中国石油大学(华东)理学院 山东青岛 266580

## Global Dynamics of a Stochastic Chemostat Model with General Response Function and Wall Growth

Liu Liya,1, Jiang Daqing,1,2

1 School of Petroleum Engineering, Key Laboratory of Unconventional Oil & Gas Development, China University of Petroleum (East China), Ministry of Education, Shandong Qingdao 266580

2 College of Science, China University of Petroleum, Shandong Qingdao 266580

 基金资助: 国家自然科学基金.  11871473中央高校基本科研业务费专项资金.  15CX08011A

 Fund supported: the NSFC.  11871473the Fundamental Research Funds for the Central Universities.  15CX08011A

Abstract

This paper deals with problems of a stochastic chemostat model with general response function and wall growth. We show the conditions for the microorganism to be extinct. On the other hand, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the model which means the microorganism can become persistent. Finally, example and numerical simulations are introduced to illustrate the analytical results.

Keywords： Chemostat ; Wall growth ; Extinction ; Stationary distribution ; General response functions

Liu Liya, Jiang Daqing. Global Dynamics of a Stochastic Chemostat Model with General Response Function and Wall Growth. Acta Mathematica Scientia[J], 2021, 41(6): 1912-1924 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} { } \frac{{\rm d}S(t)}{{\rm d}t} = D\left ( S^{0}-S(t) \right )-\frac{1}{\delta }P(S(t))x(t), \\ { } \frac{{\rm d}x(t)}{{\rm d}t} = -Dx(t)+P(S(t))x(t), \end{array}\right. \end{eqnarray}$

## 4 举例讨论与数值模拟

$\begin{eqnarray} \left\{\begin{array}{ll} { } {\rm d}S(t) = \left [D\left ( S^{0}-S(t) \right )-\frac{1}{\delta }\frac{mS}{a+S}x_1(t)-\frac{1}{\delta }\frac{mS}{a+S}x_2(t)+bvx_1(t) \right ]{\rm d}t+\alpha S(t){\rm d}B_1(t), \\ { } {\rm d}x_1(t) = \left [-(v+D)x_1(t)+\frac{mS}{a+S}x_1(t)-r_1x_1(t)+r_2x_2(t) \right ]+\beta x_1(t){\rm d}B_2(t), \\ { } {\rm d}x_2(t) = \left [-vx_2(t)+\frac{mS}{a+S}x_2(t)+r_1x_1(t)-r_2x_2(t) \right ]+\beta x_2(t){\rm d}B_2(t). \end{array}\right. \end{eqnarray}$

## 附录A 定理2.1的证明

$\begin{eqnarray} P\left \{ \tau _{n}\leq T \right \}\geq\epsilon . \end{eqnarray}$

## 附录B 定理3.1的证明

$M>0$满足

$\begin{eqnarray} -M\lambda +H\leq -2, \end{eqnarray}$

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