Existence Of Ergodic Stationary Distribution Research Articles

Dynamic for a Stochastic Multi-Group AIDS Model with Saturated Incidence Rate

In this paper, a stochastic multi-group AIDS model with saturated incidence rate is studied. We prove that the system is persistent in the mean under some parametric restrictions. We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function. Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system, which greatly improves upon previous results.

Dynamical analysis of a stochastic toxin-producing phytoplankton\u2013fish system with harvesting

In this paper, we analytically and numerically study the dynamics of a stochastic toxin-producing phytoplankton–fish system with harvesting. Mathematically, we give the existence and stability of the positive equilibrium in the deterministic system (i.e., the system without environmental noise fluctuations). In the case of the stochastic system (i.e., the system with environmental noise fluctuations), in addition to the existence and uniqueness of the positive solution, we provide the properties of the stochastic dynamics including the stochastic extinction and persistence in the mean, almost sure permanence and uniform boundedness, and the existence of ergodic stationary distribution for the phytoplankton and fish. Ecologically, via numerical analysis, we find that (1) the small random environmental fluctuations can ensure the persistence of phytoplankton and fish, but the larger one can result in the extinction of these populations; (2) an appropriate increase in harvest rate can reduce the irregular random variation of phytoplankton and fish; (3) the increase of toxin liberate rate is capable to decrease the height of probability density function of phytoplankton. These results may help us to better understand the phytoplankton–fish dynamics.

Ergodic stationary distribution and extinction of a stochastic SIRS epidemic model with logistic growth and nonlinear incidence

A stochastic SIRS model with logistic growth and nonlinear incidence rate is probed in this paper. We exemplify that the proposed stochastic SIRS model reveals a global and positive solution. By applying suitable Lyapunov functions, the sufficient conditions for the existence of ergodic stationary distribution of the solution to the stochastic SIRS model are derived. Furthermore, we acquire the sufficient conditions for extinction of the infectious disease.

The effect of environmental noise on threshold dynamics for a stochastic viral infection model with two modes of transmission and immune impairment

Abstract A stochastic viral infection model with two modes of transmission and immune impairment is proposed in this paper, which contains four variables; uninfected cells, infected cells, virus particles and cytotoxic T lymphocytes. We first investigate the exponential stability of the model, and further give sufficient conditions for the extinction and persistence of the disease. By constructing a suitable stochastic Lyapunov function, we then prove the existence of a unique ergodic stationary distribution of the model. More importantly, a stochastic threshold R 0 W is presented, which plays the similar role as R 0 in determining the persistence and extinction of the infected cells. As an application of the method proposed in this paper, the existence of an ergodic stationary distribution and a stochastic positive periodic solution of the model under the influence of colored noise and seasonal fluctuations are studied respectively. Finally, we carry out some numerical simulations, showing that environmental noise may suppress the spread of disease.

Stationary distribution of stochastic NP ecological model under regime switching

Marine ecosystem is inevitably disturbed by random factors. This article discusses a nutrient–phytoplankton (NP) model with TPP which is disturbed by both white and telegraph noises. Firstly, we get sufficient conditions for the existence of ergodic stationary distribution by constructing a suitable stochastic Lyapunov function with regime switching. And then we establish sufficient conditions for extinction of the phytoplankton. Finally, the results of this paper are verified by numerical simulation.

The stationary distribution and extinction of a double thresholds HTLV-I infection model with nonlinear CTL immune response disturbed by white noise

This paper investigates the stochastic HTLV-I infection model with CTL immune response, and the corresponding deterministic model has two basic reproduction numbers. We consider the nonlinear CTL immune response for the interaction between the virus and the CTL immune cells. Firstly, for the theoretical needs of system dynamical behavior, we prove that the stochastic model solution is positive and global. In addition, we obtain the existence of ergodic stationary distribution by stochastic Lyapunov functions. Meanwhile, sufficient condition for the extinction of the stochastic system is acquired. Reasonably, the dynamical behavior of deterministic model is included in our result of stochastic model when the white noise disappears.

Stationary Distribution and Extinction of a Stochastic SIQR Model with Saturated Incidence Rate

In this paper, we consider a stochastic SIQR epidemic model with saturated incidence rate. By constructing a proper Lyapunov function, we obtain the existence and uniqueness of positive solution for this SIQR model. Furthermore, we study the dynamical properties of this stochastic SIQR model; that is, (i) we establish the sufficient condition for the existence of ergodic stationary distribution of the model; (ii) we obtain the extinction of the disease under some conditions. At last, numerical simulations are introduced to illustrate our theoretical results.

Ergodic stationary distribution of a stochastic chemostat model with regime switching

In this paper, a stochastic chemostat model with single-species growth on two perfectly substitutable nutrients under telegraph noise is investigated. Firstly, we prove the solution of the system is positive and global. Then we establish sufficient conditions for the existence of ergodic stationary distribution of the model by constructing suitable Lyapunov functions with regime switching. Finally, numerical simulations are carried out to demonstrate our theoretical results.

Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size

We investigate a stochastic heroin epidemic model with bilinear incidence and varying population size. Sufficient criteria for the extinction of the drug abusers and the existence of ergodic stationary distribution for the model are established by constructing suitable stochastic Lyapunov functions. By analyzing the sensitivity of the threshold of spread, we obtain that prevention is better than cure. Numerical simulations are carried out to confirm the analytical results.

Impact of Cell Size Effect on Nutrient‐Phytoplankton Dynamics

In this paper, a nutrient‐phytoplankton model, which is described by a system of ordinary differential equations incorporating the effect of cell size, and its corresponding stochastic differential equation version are studied analytically and numerically. A key advantage of considering cell size effect is that it can more accurately reveal the intrinsic law of interaction between nutrient and phytoplankton. The main purpose of this paper is to research how cell size affects the nutrient‐phytoplankton dynamics within the deterministic and stochastic environments. Mathematically, we show that the existence and stability of the equilibria in the deterministic model can be determined by cell size: the smaller or larger cell size can lead to the disappearance of the positive equilibrium, but the boundary equilibrium always exists and is globally asymptotically stable; the intermediate cell size is capable to drive the positive equilibrium to appear and be globally asymptotically stable, whereas the boundary equilibrium becomes unstable. In the case of the stochastic model, the stochastic dynamics including the stochastic extinction, persistence in the mean, and the existence of ergodic stationary distribution is found to be largely dependent on cell size and noise intensity. Ecologically, via numerical simulations, it is found that the smaller cell size or larger cell size can result in the extinction of phytoplankton, which is similar to the effect of larger random environmental fluctuations on the phytoplankton. More interestingly, it is discovered that the intermediate cell size is the optimal size for promoting the growth of phytoplankton, but increasing appropriately the cell size can rapidly reduce phytoplankton density and nutrient concentrations at the same time, which provides a possible strategy for biological control of algal blooms.

Dynamics of a stochastic heroin epidemic model

In this paper, we investigate a stochastic heroin epidemic model. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the extinction of the drug users. Moreover, we also establish sufficient conditions for the existence of ergodic stationary distribution to the model, which reveals that the drug users will persist. The sensitivity analysis indicates that prevention would be more important than cure.

Dynamics of a hepatitis B model with saturated incidence

In this article, we present a hepatitis B epidemic model with saturated incidence. The dynamic behaviors of the deterministic and stochastic system are studied. To this end, we first establish the local and global stability conditions of the equilibrium of the deterministic model. Second, by constructing suitable stochastic Lyapunov functions, the sufficient conditions for the existence of ergodic stationary distribution as well as extinction of hepatitis B are obtained.

Periodic Solution and Ergodic Stationary Distribution of SEIS Dynamical Systems with Active and Latent Patients

This paper proposes two new stochastic non-autonomous SEIS epidemic dynamical models with latent and active patients. For the non-autonomous periodic system, we first obtain the sufficient conditions for the existence of nontrivial positive periodic solution by constructing some suitable stochastic Lyapunov functions with regime switching. Then we prove the existence of ergodic stationary distribution of the stochastic SEIS epidemic model with Markov conversion by using the stochastic qualitative theory. At last, some rigorous computer numerical simulations illustrate our theoretical results. The results show that the large random disturbance can destroy the periodic solution and the ergodic stationary distribution.

Threshold behavior in a stochastic HTLV‐I infection model with CTL immune response and regime switching

We investigate the stochastic HTLV‐I infection model with CTL immune response, which is disturbed by both white and telegraph noises. First, we obtain that the model solution is positive and global. Furthermore, we obtain the existence of ergodic stationary distribution of the solution by stochastic Lyapunov functions with regime switching, and we also establish sufficient conditions for the extinction of system with regime switching. Meanwhile, the threshold between ergodic property and extinction of the stochastic system is acquired. Finally, our theoretical results are verified by numerical simulation.

Periodic solution and ergodic stationary distribution of two stochastic SIQS epidemic systems

This paper proposes two stochastic SIQS epidemic models with periodic parameters and Markov switching. We first prove that the stochastic non-autonomous periodic system has a nontrivial positive periodic solution by using the Khasminskii’s theory. Then the sufficient conditions for extinction of the disease are obtained. Furthermore, we construct suitable stochastic Lyapunov functions with regime switching to prove the existence of ergodic stationary distribution of the stochastic SIQS epidemic model. At last, some rigorous numerical simulations are presented to illustrate our theoretical results.

The effects of toxin-producing phytoplankton and environmental fluctuations on the planktonic blooms

Toxin-producing phytoplankton (TPP) and environmental fluctuations are important factors influencing the real aquatic ecosystems. Whether planktonic blooms are related to these factors is an interesting problem. In this paper, based on bottom-up mechanism, we first propose a nutrient–phytoplankton (NP) model with TPP under environmental fluctuations and study the extinction and persistence in mean. We also investigate the existence of ergodic stationary distribution. Taking into account seasonal variation, then we extend the NP model into a non-autonomous stochastic model and investigate the existence of periodic solution. Combined with the published data, we numerically illustrate the obtained results. From theoretical analysis and numerical simulations, periodic oscillatory succession of planktonic blooms can be seen clearly, which suggests that toxin-producing phytoplankton and environmental fluctuations play a key role in the termination of algal blooms.

A stochastic HIV infection model with T-cell proliferation and CTL immune response

A stochastic HIV infection model with T-cell proliferation and CTL immune response is formulated to investigate the effect of environmental fluctuations on the HIV viral dynamics. We obtain that the model solution is positive and global, and analyze the extinction of the model. We also derive a critical condition R0s, when R0s is greater than one, the existence of ergodic stationary distribution of the model solution is established by constructing suitable Lyapunov functions. Numerical simulations are performed to investigate the effect of white noises on model behavior, we investigate that the small intensities of white noise can maintain the irregular recurrence of HIV virus and CTL immune response, while the larger ones may be help to the elimination of the virus and CTL immune response, and the medium intensities of white noises may cause both the persistence and extinction on model dynamics behavior.

Dynamical behavior of a stochastic HBV infection model with logistic hepatocyte growth

This paper is concerned with a stochastic HBV infection model with logistic growth. First, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HBV infection model. Then we obtain sufficient conditions for extinction of the disease. The stationary distribution shows that the disease can become persistent in vivo.

Stationary distribution and extinction of the DS-I-A model disease with periodic parameter function and Markovian switching

This paper introduces the DS-I-A model with periodic parameter function and Markovian switching. First, we will prove that the solution of the system is positive and global. Furthermore, we draw a conclusion that there exists nontrivial positive periodic solution for the stochastic system and we establish sufficient conditions for extinction of system. Moreover, we construct stochastic Lyapunov functions with regime switching to obtain the existence of ergodic stationary distribution of the solution to DS-I-A model perturbed by white and telephone noises and we also establish sufficient conditions for extinction of system with regime switching. Finally, we test our theory conclusion by simulations.