Unique Ergodic Stationary Distribution Research Articles

Stationary distribution of a stochastic food chain chemostat model with general response functions

In this paper, we focus on a food chain chemostat model with general response functions, perturbed by white noise. Under appropriate assumptions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution by using stochastic Lyapunov analysis method. Our main effort is to construct the suitable Lyapunov function.

Analysis of a stochastic HIV-1 infection model with degenerate diffusion

This paper studies a stochastic HIV-1 infection model with degenerate diffusion. The asymptotic dynamics of the stochastic model are shown to be governed by a threshold parameter. When the parameter is negative, the infection is predicted to go extinct exponentially while the level of healthy cells converges weakly to a unique invariant measure. When the threshold parameter is positive, the solution of the stochastic model converges polynomially to a unique invariant probability measure, indicating that the system admits a unique ergodic stationary distribution. Numerical simulations are conducted to show the analytical results. These results highlight the role of environmental noise in the spread of HIV-1. The method can also be applied to the non-degenerate systems.

Stationary distribution and extinction of a stochastic HIV-1 model with Beddington–DeAngelis infection rate

In this paper, we consider a stochastic HIV-1 model with Beddington–DeAngelis infection rate. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to the model. Then we obtain sufficient conditions for extinction of the disease. The existence of a stationary distribution implies stochastic weak stability.

Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV

In this paper, we study two stochastic multigroup S-DI-A epidemic models for the transmission of HIV. For the stochastic S-DI-A epidemic model with periodic coefficients, we first obtain sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we show that the model admits a positive T-periodic solution by using Khasminskii theory of periodic solution. Moreover, we establish sufficient conditions for exponential extinction of the infectious disease. For the stochastic S-DI-A epidemic model disturbed by both white and telegraph noises, we first establish sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we obtain sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions by constructing a suitable stochastic Lyapunov function with regime switching and we also obtain sufficient conditions for exponential extinction of the system with regime switching.

Dynamics of a stochastic delay competitive model with harvesting and Markovian switching

This work is concerned with a two-species stochastic state-switching competitive population model with distributed delays and harvesting. First, necessary and sufficient criteria for the existence of a unique ergodic stationary distribution of the system are established. Then necessary and sufficient criteria for the existence of the optimal harvesting policy are given, and the explicit expression of the optimal harvesting policy is obtained. Finally, some effects of the state-switching on the persistence, extinction and optimal harvesting strategy of the system are discussed with the help of several simulations.

Dynamics Analysis of a Nonlinear Stochastic SEIR Epidemic System with Varying Population Size.

This paper considers a stochastic susceptible exposed infectious recovered (SEIR) epidemic model with varying population size and vaccination. We aim to study the global dynamics of the reduced nonlinear stochastic proportional differential system. We first investigate the existence and uniqueness of global positive solution of the stochastic system. Then the sufficient conditions for the extinction and permanence in mean of the infectious disease are obtained. Furthermore, we prove that the solution of the stochastic system has a unique ergodic stationary distribution under appropriate conditions. Finally, the discussion and numerical simulation are given to demonstrate the obtained results.

Analysis of a Three-Species Stochastic Delay Predator-Prey System with Imprecise Parameters

In this paper, a stochastic delay three-species food chain model with imprecise biological parameters has been developed. For this model, the sharp sufficient conditions for the existence of a unique ergodic stationary distribution and the extinction are established. We also discuss the effects of imprecise parameters on the persistence, extinction and existence of the stationary distribution.

Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation

This paper investigates a new impulsive stochastic chemostat model with nonlinear perturbation in a polluted environment. We present the analysis and the criteria of the extinction of the microorganisms, and establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model via Lyapunov functions method. The results show that both stochastic noise and impulsive toxicant input have great effects on the survival and extinction of the microorganisms. Moreover, we provide a series of numerical simulations to illustrate the analytical results.

Stability of a budworm growth model with random perturbations

A budworm growth model perturbed by both white noises and regime switchings is proposed and analyzed. It is proven that there is a threshold. If this threshold is positive, then the model has a unique ergodic stationary distribution; if this threshold is negative, then the zero solution of the model is stable. The results show that both white noises and regime switchings can change the stability of the model greatly. Several numerical simulations based on realistic data are also introduced to illustrate the main results.

Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation

In this paper, we analyze a stochastic SIR model with nonlinear perturbation. By the Lyapunov function method, we establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model. Moreover, sufficient conditions for extinction of the disease are also obtained.

Stochastic mutualism model with Lévy jumps

In this paper, we consider a stochastic mutualism model with Lévy jumps. First of all, we show that the positive solution of the system is stochastically ultimate bounded. Then under a simple assumption, we establish sufficient and necessary conditions for the stochastic permanence and extinction of the system. The results show an important property of the Lévy jumps: they are unfavorable for the permanence of the species. Moreover, when there are no Lévy jumps, we show that there is a unique ergodic stationary distribution of the corresponding system under certain conditions. Some numerical simulations are introduced to validate the theoretical results.

Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching

This paper addresses a stochastic SIS epidemic model with vaccination under regime switching. The stochastic model in this paper includes white and color noises. By constructing stochastic Lyapunov functions with regime switching, we establish sufficient conditions for the existence of a unique ergodic stationary distribution.

Stability of a stochastic logistic model under regime switching

In this letter, we consider a stochastic generalized logistic equation with Markovian switching. We obtain a critical value which has the property that if the critical value is negative, then the trivial solution of the model is stochastically globally asymptotically stable; if the critical value is positive, then the solution of the model is positive recurrent and has a unique ergodic stationary distribution. We find out that the critical value has a close relationship with the stationary probability distribution of the Markov chain.

Stationary distribution of stochastic population systems under regime switching

This paper is concerned with n-species model of facultative mutualism in random environments. The environment variability in this study is characterized with both white noise and color noise modeled by Markovian switching. We established new sufficient conditions that ensuring that the system model is positive recurrent. We also showed the existence of a unique ergodic stationary distribution. The presented results are demonstrated by numerical simulations.