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In this paper, a stochastic predator–prey model with two competitive preys and Ornstein–Uhlenbeck process is formulated and analysed, which is used to obtain a better understanding of the population dynamics. At first, we validate that the stochastic system has a unique global solution with any initial value. Then we analyse the stochastic dynamics of the model in detail, including pth moment boundedness, asymptotic pathwise estimation in turn. After that, we obtain sufficient conditions for the existence of a stationary distribution of the system by adopting stochastic Lyapunov function methods. In addition, under some mild conditions, we derive the specific form of covariance matrix in the probability density near the quasi-positive equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We develop and study a stochastic dengue model with Ornstein–Uhlenbeck process, in which we assume that the transmission coefficients between vector and human satisfy the Ornstein–Uhlenbeck process. We first show that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient criteria for the existence of a stationary distribution of the system, which indicates the persistence of the disease. In particular, under some mild conditions which are applied to ensure the local asymptotic stability of the endemic equilibrium of the deterministic system, we obtain the specific form of covariance matrix in the probability density around the quasi-positive equilibrium of the stochastic system. In addition, we also establish sufficient criteria for wiping out of the disease. Finally, several numerical simulations are performed to illustrate our theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a stochastic human immunodeficiency virus (HIV) model with logistic growth and Ornstein-Uhlenbeck process, which is used to describe the pathogenesis and transmission dynamics of HIV in the population. We first validate that the stochastic system has a unique global solution with any initial value. Then we use a novel Lyapunov function method to establish sufficient conditions for the existence of a stationary distribution of the system, which shows the coexistence of all CD4+ T cells and free viruses. Especially, under some mild conditions which are used to ensure the local asymptotic stability of the quasi-chronic infection equilibrium of the stochastic system, we obtain the specific expression of covariance matrix in the probability density around the quasi-chronic infection equilibrium of the stochastic system. In addition, for completeness, we also obtain sufficient criteria for elimination of all infected CD4+ T cells and free virus particles. Finally, several examples together with comprehensive numerical simulations are conducted to support our analytic results. [ABSTRACT FROM AUTHOR]

In this paper, we establish and analyze a stochastic human immunodeficiency virus model with both virus-to-cell and cell-to-cell transmissions and Ornstein–Uhlenbeck process, in which we suppose that the virus-to-cell infection rate and the cell-to-cell infection rate satisfy the Ornstein–Uhlenbeck process. First, we validate that there exists a unique global solution to the stochastic model with any initial value. Then, we adopt a stochastic Lyapunov function technique to develop sufficient criteria for the existence of a stationary distribution of positive solutions to the stochastic system, which reflects the strong persistence of all CD4+ T cells and free viruses. In particular, under the same conditions as the existence of a stationary distribution, we obtain the specific form of the probability density around the quasi-chronic infection equilibrium of the stochastic system. Finally, numerical simulations are conducted to validate these analytical results. Our results suggest that the methods used in this paper can be applied to study other viral infection models in which the infected CD4+ T cells are divided into latently infected and actively infected subgroups. [ABSTRACT FROM AUTHOR]

In this article, a stochastic differential equation model is proposed to investigate how the environmental noise affects the transmission dynamics of the interaction between crime, criminality and victimization with the influence of case reporting through the short message service (SMS). We prove that there is a unique global positive solution of the stochastic system with any positive initial value which is fundamental in the dynamics of population modeling. Moreover, we get sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution indicates that both criminal and victim will be persistent and coexistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic logistic model by incorporating diffusion and two Ornstein–Uhlenbeck processes, which is a stochastic non-autonomous system. We first show the existence and uniqueness of the global solution of the system with any initial value. After that, we study the pth moment boundedness, asymptotic pathwise estimation, asymptotic behavior, and global attractivity of the solutions of the stochastic system in turn. Moreover, we establish sufficient criteria for the existence and uniqueness of a stationary distribution of positive solutions of the stochastic system with the help of Lyapunov function methods. It is worth mentioning that we derive the exact expression of the local probability density for the stochastic system by solving the relevant four-dimensional Fokker–Planck equation. We find that the smaller intensity of volatility or the bigger speed of reversion is helpful for preserving the biodiversity of the species. Finally, numerical simulations are performed to support our analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we analyze two stochastic predator–prey models with distributed delay and stage structure for prey. For the nonautonomous periodic case of the model, by using Khasminskii's theory of periodic solution, we show that the system has at least one positive T -periodic solution. For the model which is disturbed by both white and telegraph noises, we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The positive recurrence implies that both prey and predator populations will be persistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic tuberculosis model with two kinds of treatments, that is, treatment at home and treatment in hospital. Firstly, we obtain sufficient conditions for extinction and persistence in the mean of the diseases, then in the case of persistence, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. The theoretical results show that large white noise is beneficial for eliminating TB endemic while small white noise is not. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical behavior of a multigroup SIQS epidemic model, which is formulated as a piecewise deterministic Markov process. Sufficient conditions for extinction and persistence in the mean of the diseases are obtained. In addition, in the case of diseases persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by using the stochastic Lyapunov function with regime switching. [ABSTRACT FROM AUTHOR]

In the present paper, we focus on a stochastic predator-prey model with stage structure for prey. Firstly, by using the stochastic Lyapunov function method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the predator population in two cases. Some examples and numerical simulations are carried out to validate our analytical findings. [ABSTRACT FROM AUTHOR]

In this article, we study two stochastic predator-prey models with distributed delay and general functional response. For the nonautonomous periodic case of the system, by utilizing Khasminskii's theory of periodic solution, we prove that the system admits a positive T-periodic solution. For the system which is perturbed by both white and colored noises, we establish sufficient conditions for the existence of positive recurrence of the solutions to the system. The existence of positive recurrence implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

In this paper, we analyse a stochastic chemostat model with distributed delay and degenerate diffusion. We transform the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the positive solutions can converge in L 1 to an invariant density. The existence of a stable stationary distribution implies stochastic persistence of the microorganism. [ABSTRACT FROM AUTHOR]

This article is concerned with a stochastic model of gene expression with distributed delay and degenerate diffusion. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is of degenerate type, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the solutions can converge in L1 to an invariant density. The existence of the stationary distribution implies stochastic weak stability. Numerical simulation is introduced to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic cell-to-cell HIV-1 model with distributed delay. Firstly, we show that there is a global positive solution of this model before exploring its long-time behavior. Then sufficient conditions for extinction of the disease are established. Moreover, we obtain sufficient conditions for the existence of an ergodic stationary distribution of the model by constructing a suitable stochastic Lyapunov function. The stationary distribution implies that the disease is persistent in the mean. Finally, we provide some numerical examples to illustrate theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical behavior of a stochastic SIQR epidemic model with standard incidence which is disturbed by both white and telegraph noises. Firstly, we obtain sufficient conditions for persistence in the mean of the disease. Then we establish sufficient conditions for extinction of the disease. In addition, in the case of persistence, we get sufficient conditions for the existence of positive recurrence of the solutions by constructing a suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, some numerical simulations are introduced to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two stochastic SIR epidemic models with higher order perturbation. For the nonautonomous periodic case of the model, by using Has’minskii’s theory of periodic solution, we show that the system has at least one nontrivial positive T -periodic solution. For the system disturbed by both the white noise and telephone noise, we establish sufficient conditions for positive recurrence and the existence of ergodic stationary distribution of the positive solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic HIV-1 infection model with logistic growth. Firstly, by constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution of the solution to the HIV-1 infection model. Then we obtain sufficient conditions for extinction of the infection. The stationary distribution shows that the infection can become persistent in vivo. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIRS epidemic model with standard incidence. By constructing suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of ergodic stationary distribution of the model. Moreover, we also establish sufficient conditions for extinction of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic non-autonomous SISV epidemic model. For the non-autonomous periodic system, firstly, we get the threshold of the system which determines whether the epidemic occurs or not. Then in the case of persistence, we show that there exists at least one nontrivial positive periodic solution of the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the dynamics of a stochastic nonautonomous SIR epidemic model, in which population growth is subject to logistic growth in absence of disease. For the periodic system, we present sufficient conditions for persistence of the epidemic and in the case of persistence, by constructing some suitable Lyapunov functions, we show that there is at least one nontrivial positive periodic solution. One of the most important findings is that random perturbations may be beneficial to formate the periodic solution to the stochastic nonautonomous SIR epidemic model. [ABSTRACT FROM AUTHOR]

In this paper, we study the threshold dynamics of a stochastic delayed SIR epidemic model with vaccination. We obtain sufficient conditions for extinction and persistence in the mean of the epidemic. The threshold between persistence in the mean and extinction of the stochastic system is also obtained. Compared with the corresponding deterministic model, the threshold affected by the white noise is smaller than the basic reproduction number R ¯ 0 of the deterministic system. Results show that time delay has important effects on the persistence and extinction of the epidemic. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic properties of a stochastic delayed SIR epidemic model with temporary immunity. Sufficient conditions for extinction and persistence in the mean of the epidemic are established. The threshold between persistence in the mean and extinction of the epidemic is obtained. Compared with the corresponding deterministic model, the threshold affected by the white noise is smaller than the basic reproduction number R 0 of the deterministic system. [ABSTRACT FROM AUTHOR]

An n-species stochastic Gilpin-Ayala cooperative model was investigated in this study. The Lyapunov function and the M-matrix method were applied to study the stability of the solutions. Sufficient conditions for the existence of a global positive solution of the Gilpin-Ayala cooperative model were established. Certain asymptotically stable results of a global positive solution of the cooperative model and its domain of attraction were estimated. That main objective of this study is to provide corrections for errors in some theorems given in the work of . The errors of Theorems 2, 3, 5, and 6 from the published work appeared in the parameters θ and p. [ABSTRACT FROM AUTHOR]

In this paper, a general stochastic non-autonomous logistic model with delays and Lévy jumps is proposed and studied. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the model are established. The threshold between weak persistence and extinction is also obtained. Finally, numerical simulations are introduced to illustrate the theoretical analysis results. Our results illustrate that both persistence and extinction have close relationships with time-dependent delay and Lévy noise. [ABSTRACT FROM AUTHOR]

In this paper, a three-species food web stochastic model with generalist predator is proposed and studied. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological viewpoint, the existence of a stationary distribution implies that all species can coexist in a community. • A three-species food web stochastic model with generalist predator is studied. • We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. • The existence of a stationary distribution implies that all species can coexist in a community. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyze a stochastic predator–prey system where apart from direct predation the prey population is affected by the fear induced from predators. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological interpretation, the existence of a stationary distribution indicates that both the prey and predator species will be persistent and coexistent in the long term. • A stochastic predator–prey system with fear effect is proposed and studied. • Criteria for the existence of a unique ergodic stationary distribution are obtained. • A stationary distribution shows that both the prey and predator species will persist. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. We find that the hypothetical conditions on the nonlinear function are relative weak and valid for many forms of incidence rate. • A stochastic SIRS model with logistic growth and general nonlinear incidence rate is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • The conditions on the nonlinear function are relative weak and valid for many forms of incidence rate. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic cholera model with imperfect vaccination is proposed and investigated. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a suitable stochastic Lyapunov function. Our result may provide some new insights for elimination of cholera. Some numerical simulation is provided to demonstrate our main result. • A stochastic cholera model with imperfect vaccination is proposed and investigated. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • Our result may provide some new insights for elimination of cholera. • Some numerical simulation is provided to demonstrate our main result. [ABSTRACT FROM AUTHOR]

In this paper, we study two types of stochastic models consisting of three strains of influenza. For each type of the stochastic model, we establish sufficient conditions for extinction and persistence in the mean of these diseases by using the stochastic Lyapunov analysis method. Meanwhile, we obtain the threshold between persistence in the mean and extinction of the diseases. • Two types of stochastic models consisting of three strains of influenza are studied. • We obtain sufficient conditions for extinction and persistence in the mean. • We obtain the threshold between persistence in the mean and extinction. [ABSTRACT FROM AUTHOR]

• A stochastic cholera model between communities linked by migration is studied. • We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution. • Our result shows that the migration and the small white noise are not beneficial for eliminating cholera. • Some examples together with numerical simulations are introduced to support the theoretical result. In this paper, we develop and study a stochastic cholera model between communities linked by migration. Firstly, we show that there exists a unique global positive solution of the considered system with any positive initial value. Then by using the stochastic Lyapunov analysis method, we obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Our result shows that the migration and the small white noise are not beneficial for eliminating cholera. Finally, some examples together with numerical simulations are introduced to support the theoretical result. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic predator–prey model with distributed delay and Markovian switching. Firstly, we obtain sufficient criteria for weak persistence of the predator population. Then in the case of weak persistence, we establish sufficient conditions for the existence of positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. Finally, we establish sufficient conditions for extinction of the predator population. • A stochastic predator–prey model with distributed delay and Markovian switching is studied. • We obtain sufficient criteria for weak persistence of the predator population. • We establish sufficient conditions for the existence of positive recurrence of the solutions. • We establish sufficient conditions for extinction of the predator population. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic model with vaccination and treatment to describe the transmission dynamics of tuberculosis. Firstly, by using the stochastic Lyapunov analysis method, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model, then we obtain sufficient conditions for extinction of the diseases. Our work greatly improves the results of Feng and Qiu (2018). Finally, some examples and numerical simulations are introduced to illustrate our theoretical results. • A stochastic vaccinated tuberculosis model with treatment is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • We obtain sufficient conditions for extinction of the diseases. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic staged progression HIV model with imperfect vaccination. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by using the stochastic Lyapunov analysis method. • A stochastic staged progression HIV model with imperfect vaccination is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • The existence of a stationary distribution implies the stochastic weak stability. [ABSTRACT FROM AUTHOR]

In this paper, we study a multigroup SIR epidemic model in a stochastic environment. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for extinction of the diseases. The results reveal that the smaller white noise can ensure the persistence of the diseases while the larger white noise can lead to the extinction of the diseases. Finally, some examples and numerical simulations are provided to illustrate the theoretical results. • A multigroup SIR epidemic model in a stochastic environment is studied. • We establish sufficient conditions for the existence of a stationary distribution to the model. • We establish sufficient conditions for extinction of the diseases. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]