Stochastic Chemostat Model Research Articles

Ultimate Boundedness of a Stochastic Chemostat Model with Periodic Nutrient Input and Random Disturbance

Ultimate Boundedness of a Stochastic Chemostat Model with Periodic Nutrient Input and Random Disturbance

Competitive exclusion and coexistence in a general two-species stochastic chemostat model with Markov switching

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

Geometric ergodicity and ultimate boundedness of a stochastic chemostat model with general nutrient uptake function

The ergodicity of solutions for stochastic systems has been widely concerned by many scholars in recent years. Thus, in this paper, we further analyze the geometric ergodicity of a stochastic chemostat model with general nutrient uptake function. By verifying minorization condition and Lyapunov condition, geometric ergodicity of stochastic chemostat system is presented, which means the transfer probability of Markov process will converge to its limiting distribution at an exponentially rate. Meanwhile, we prove that the solution of stochastic system is stochastically ultimately bounded.

Ultimate boundedness of a stochastic chemostat model with periodic nutrient input and discrete delay

Stochastically ultimate boundedness is a very important property, which plays an important role in the study of stochastic models. Especially, for the stochastic biological mathematical model with discrete delay, we urgently need to solve this problem through some new mathematical methods. Thus, in this paper, we will study a stochastic periodic chemostat system with discrete delay, and we assume that the nutrient input concentration and noise intensities are periodic. In order to make the stochastic periodic model with discrete delay have mathematical and biological significance, we will study a very important issue: the existence, uniqueness and ultimate boundedness of a global positive solution.

A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay

A Note on the Stationary Probability Density Function and Covariance Matrix of a Stochastic Chemostat Model with Distributed Delay

Long-time analysis of a stochastic chemostat model with instantaneous nutrient recycling

<abstract><p>This paper presents long-time analysis of a stochastic chemostat model with instantaneous nutrient recycling. We focus on the investigation of the sufficient and almost necessary conditions of the exponential extinction and persistence for the model. The convergence to the invariant measure is also established under total variation norm. Our work generalizes and improves many existing results. One of the interesting findings is that random disturbance can suppress microorganism growth, which can provide us some useful control strategies to microbiological cultivation. Finally, some numerical simulations partly based on the stochastic sensitive function technique are given to illustrate theoretical results.</p></abstract>

Statistical property analysis for a stochastic chemostat model with degenerate diffusion

<abstract><p>By considering the fact that the growth of microorganisms in a chemostat is subject to white noise, we construct a stochastic chemostat model with degenerate diffusion by using a discrete Markov chain. By solving the corresponding Fokker-Planck equation, we derive the explicit expression of the stationary joint probability density, which peaks near the deterministic equilibrium. Next, we simulate the the marginal probability density functions for different noise intensities and further discuss the relationship of the marginal probability density function and noise intensities. For the statistical properties of the stochastic model, we mainly investigate the effect of white noise on the variance and skewness of the concentration of microorganisms.</p></abstract>

Threshold dynamics in a stochastic chemostat model under regime switching

A stochastic chemostat model in random environments that is driven by Brownian motions and subjected to Markov regime switching is considered. The new break-even concentration, i.e., critical value between persistence in mean and extinction is explored for the microorganism species. Moreover, sufficient conditions for ergodicity and positive recurrence is established by using stochastic Lyapunov analysis. Numerical simulations are accomplished to verify the analytical results.

Dynamics of stochastic chemostat models with mixed nonlinear incidence

In this paper, we proposes a stochastic chemostat model with mixed nonlinear incidence. Firstly, the existence and uniqueness of the global positive solutions are proved. Secondly, we demonstrate that the chemostat model is persistence in mean and the solution of this stochastic chemostat model is bounded for any initial condition by constructing the Lyapunov function. Then we obtain the sufficient condition for the existence of an ergodic stationary distribution in this system. Finally, the numerical simulation results of the model are given. The simulation results show that a particular random perturbation can change the fate of microorganisms.

REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS

In this paper, we mechanistically formulate a type of stochastic chemostat model with two complementary nutrients, which is affected by seasonal variations and flocculation effect. The phase transition properties of the model are investigated by theoretical analysis and numerical simulation. The well-posedness of the model is considered. Further, by utilizing Khasminskii’s theory, sufficient conditions for the existence of the stochastic nontrivial positive periodic solution are obtained. The existence of the stochastic nontrivial positive periodic solution implies periodic change of microorganism’s density. Some sufficient conditions for the global attractivity of the boundary periodic solution of the model are also derived. At last, numerical simulations are performed to illustrate our theoretical results. It is found numerically that the stable positive periodic solution and a stable boundary periodic solution of the model may coexist. Especially, for appropriate random perturbations, the population of the microorganisms changes from an endangered state to an oscillatory persistence state in some regions.

Forward attractor for stochastic chemostat model with multiplicative noise

In this paper, we consider a stochastic chemostat model with multiplicative noise. By appropriate variable substitution, we get a random chemostat system driven by Ornstein-Uhlenbeck process, which will no longer contain white noise. Firstly, we prove the existence and uniqueness of the global positive solution for any positive initial value for random chemostat system. Secondly, we state some results regarding the existence of a compact forward absorbing set as well as a forward attracting one, its internal structure will provide us some useful information about the long-time behavior of microorganism in random chemostat model. Finally, we make some comparison and analysis between both ways of modeling randomness and stochasticity in the chemostat model and show some numerical simulations to support our theoretical results.

PERIODIC SOLUTIONS FOR A STOCHASTIC CHEMOSTAT MODEL WITH IMPULSIVE PERTURBATION ON THE NUTRIENT

In this paper, we investigate a stochastic chemostat model with impulsive perturbation on the nutrient. First, the existence and uniqueness of solutions are proved by constructing a pulseless equivalent system. Second, based on Khasminskii’s Markov periodic process theory, we give sufficient conditions for the existence of positive [Formula: see text]-periodic solutions. Third, under certain conditions, the existence and global attractivity of boundary periodic solutions are established by using comparison theorem. Finally, numerical simulations are provided to verify our main results.

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

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

Stationary probability density of a stochastic chemostat model with Monod growth response function

The stationary probability density (SPD) is derived and studied for a stochastic chemostat model with Monod growth response function. First, with the help of polar coordinate transformation and stochastic averaging method, we derive a two‐dimensional diffusion process of averaged amplitude and phase angle. Furthermore, the SPD of the diffusion process is obtained by the corresponding Fokker Planck‐Kolmogorov equation. We also analyze the effects of noise intensities on the geometric property of the SPD.

Noise-induced stochastic transition: A stochastic chemostat model with two complementary nutrients and flocculation effect

A novel stochastic chemostat model with two complementary nutrients and flocculation effect is considered in this paper. Firstly, the well-posedness of the stochastic chemostat model is considered. Then, by constructing appropriate stochastic Lyapunov functions, some sufficient conditions for the existence of an ergodic stationary distribution and persistence of the stochastic model are given. The results show that the microorganisms in chemostat can be collected continuously. Furthermore, based on sensitivity analysis techniques, some control strategies are discussed. Finally, we carry out some numerical simulations to illustrate the applications of theoretical results and give the empirical probability densities in numerical forms. In particular, the numerical simulations show that, when the random fluctuation of the environment is large, the growth of the microorganisms in chemostat can be transformed from the state of tending to extinction to the state of persistence. The interesting observation reveals that the random fluctuation may have positive biological effects and we call it the noise-induced stochastic transition phenomenon.

DYNAMICAL BEHAVIOR OF STOCHASTIC COMPETITION BETWEEN PLASMID-BEARING AND PLASMID-FREE ORGANISMS IN A CHEMOSTAT MODEL

In this paper, a model of stochastic competition between plasmid-bearing and plasmid-free organisms in a chemostat is investigated. First, we show that there is a unique global positive solution for the stochastic system. Second, by employing stochastic Lyapunov functions, Itô formula, strong law of large number and some other important inequalities, stochastic characteristics of the stochastic competition chemostat model are studied such as the stochastic asymptotic behaviors of the system. Finally, some numerical simulations are given.

A stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function

In this paper, we mainly construct and analyze a stochastic chemostat model with mean-reverting Ornstein-Uhlenbeck process and Monod-Haldane response function, which is a stochastic non-autonomous system. We first study the existence of global unique positive solution with any initial value for stochastic chemostat system. After that, the sufficient conditions are established for extinction exponentially and persistence in the mean of microorganism. Finally, we also give numerical simulations to illustrate our main conclusions. Our results show that the mean-reverting process is an effective and reasonable method to introduce environmental noise into the continuous culture model of microorganism, and we also find that the reversion speed and volatility intensity have an important influence on the extinction and persistence of microorganism.

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.

Stationary distribution of a stochastic chemostat model with Beddington–DeAngelis functional response

In this paper, we study a stochastic chemostat model with Beddington–DeAngelis functional response. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to the model. The existence of a stationary distribution implies stochastic weak stability.

Stochastic Characteristics and Optimal Control for a Stochastic Chemostat Model with Variable Yield

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.