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

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 L^(1)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.

Confidence domain in the stochastic competition chemostat model with feedback control

This paper studies a stochastically forced chemostat model with feedback control in which two organisms compete for a single growth-limiting substrate. In the deterministic counterpart, previous researches show that the coexistence of two competing organisms may be achieved as a stable positive equilibrium or a stable positive periodic solution by different feedback schedules. In the stochastic case, based on the stochastic sensitivity function technique,we construct the confidence domains for different feedback schedules which allow us to find the configurational arrangements of the stochastic attractors and analyze the dispersion of the random states of the stochastic model.

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper,stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied,and we use distribution delay to simulate the delay in nutrient conversion.By the linear chain technique,we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations.First,we state that this model has a unique global positive solution for any initial value,which is helpful to explore its stochastic properties.Furthermore,we prove the stochastic ultimate boundness of the solution of system.Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established,which means the microorganism will be extinct.Moreover,we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions.Finally,we provide some numerical examples to illustrate theoretical results,and some conclusions and analysis are given.

Dynamical Analysis of Nonautonomous Trophic Cascade Chemostat Model with Regime Switching and Nonlinear Perturbations in a Polluted Environment

This paper investigates the stochastic dynamics of trophic cascade chemostat model perturbed by regime switching, Gaussian white noise and impulsive toxicant input. For the system with only white noise interference, sufficient conditions for stochastically ultimate boundedness and stochastically permanence are obtained, and we demonstrate that the stochastic system has at least one nontrivial positive periodic solution. For the system with Markov regime switching, sufficient conditions for extinction of the microorganisms are established. Then we prove the system is ergodic and has a stationary distribution. The results show that both impulsive toxins input and stochastic noise have great effects on the survival and extinction of the microorganisms. Finally, a series of numerical simulations are presented to illustrate the theoretical analysis.