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Long-Time Behavior and Density Function of a Stochastic Chemostat Model with Degenerate Diffusion

GAO Miaomiao1, JIANG Daqing1,2,3, WEN Xiangdan4   

  1. 1. College of Science, China University of Petroleum (East China), Qingdao 266580, China; 2. Key Laboratory of Unconventional Oil and Gas Development, China University of Petroleum (East China), Ministry of Education, Qingdao 266580, China; 3. Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia; 4. Department of Mathematics, Yanbian University, Yanji 133002, China
  • Received:2020-07-28 Revised:2020-12-09 Published:2022-06-20
  • Supported by:
    This work was supported by the National Natural Science Foundation of China under Grant Nos. U1808205 and 62173079, the Natural Science Foundation of Hebei Province under Grant No. F2020501018, and the Youth Foundation of Hebei Educational Committee under Grant No. QN2020522.

GAO Miaomiao, JIANG Daqing, WEN Xiangdan. Long-Time Behavior and Density Function of a Stochastic Chemostat Model with Degenerate Diffusion[J]. Journal of Systems Science and Complexity, 2022, 35(3): 931-952.

This paper considers a stochastic chemostat model with degenerate diffusion. Firstly, the Markov semigroup theory is used to establish sufficient criteria for the existence of a unique stable stationary distribution. The authors show that the densities of the distributions of the solutions can converge in L1 to an invariant density. Then, conditions are obtained to guarantee the washout of the microorganism. Furthermore, through solving the corresponding Fokker-Planck equation, the authors give the exact expression of density function around the positive equilibrium of deterministic system. Finally, numerical simulations are performed to illustrate the theoretical results.
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