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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]

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 analyze a stochastic multigroup DS-DI-A model for the transmission of HIV. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by constructing a series of suitable Lyapunov functions. Moreover, we make up adequate conditions for complete eradication and wiping out of the diseases. In a biological viewpoint, the existence of a stationary distribution shows that the diseases will be prevalent and persistent in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the salient features of a stochastic multigroup SEI epidemic model. 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 indicates that the diseases will be prevalent and persistent in the long term. In addition, we make up adequate conditions for complete eradication and wiping out of the diseases. Some numerical simulations are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions 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 point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate conditions for complete eradication and wiping out of the infectious disease. [ABSTRACT FROM AUTHOR]

This paper is concerned with a stochastic multigroup S-DI-A model for the transmission of HIV with general incidence rate. We obtain sufficient criteria for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model by constructing a proper stochastic Lyapunov function. The stationary distribution implies that the diseases will be persistent in the long term. Some numerical simulations are introduced to illustrate our theoretical result. [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 paper, we investigate a stochastic susceptible-infective-removed-infective (SIRI) epidemic model with relapse. We show that the densities of the distributions of the solutions can converge in L 1 to an invariant density or can converge weakly to a singular measure under certain condition. We also find the support of the invariant density. Moreover, we establish sharp sufficient criteria for the extinction of the disease in two cases. The results show that the smaller white noise can assure the existence of a stationary distribution which implies the persistence of the disease while the larger white noise can lead to the extinction of the disease. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a multigroup SIRS epidemic model with random perturbations and varying total population size. By utilizing the stochastic Lyapunov function method, we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. Since our model is multidimensional, it is extremely difficult to construct an appropriate stochastic Lyapunov function to prove the existence of the stationary distribution, which implies stochastic weak stability. Then we establish sufficient conditions for extinction of the diseases. These conditions are related to the basic reproduction number in its corresponding deterministic system. [ABSTRACT FROM AUTHOR]

In this article, we investigate a stochastic one-prey two-predator model with Holling type II functional response. We first establish sufficient conditions for persistence and extinction of prey and predator populations, then by constructing a suitable stochastic Lyapunov function, we establish sharp sufficient criteria for the existence of a unique ergodic stationary distribution of the positive solutions to the model. The results show that the smaller white noise can ensure the persistence of prey and predator populations while the larger white noise can lead to the extinction of prey and predator populations. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a regime-switching predator–prey model with anti-predator behaviour and higher-order perturbations. We obtain the ergodic property by constructing a suitable stochastic Lyapunov function with regime-switching, which provides us a biological perspective of cycling phenomena of a population system, and can better describe the stochastic persistence of a population system in practice. We find that these restrictive assumptions on the functional response are relative weak and valid for many types of response functions. Highlights • A regime-switching predator–prey model with anti-predator behaviour is studied. • We obtain the ergodic property by constructing a suitable stochastic Lyapunov function. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study a stochastic predator–prey model with distributed delay and general functional response. We transform the model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition does not hold. The Markov semigroup theory is used to derive the existence of a unique stable stationary distribution. We verify the densities of the distributions of the positive solutions can converge in L 1 to an invariant density. Highlights • Stochastic predator–prey model with distributed delay and general functional response is studied. • We transform the model with weak kernel case into an equivalent system. • The densities of the distributions of the solutions can be proved converge to an invariant density. [ABSTRACT FROM AUTHOR]

Abstract 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. Highlights • A stochastic HIV-1 model with Beddington–DeAngelis infection rate is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • We obtain sufficient conditions for extinction of the disease. • The existence of a stationary distribution implies stochastic weak stability. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic logistic model with distributed delay and nonlinear perturbation. We first use the linear chain technique to transfer the one-dimensional stochastic model with strong kernel or weak kernel into an equivalent system with degenerate diffusion. Then we establish sufficient and necessary criteria for extinction of the population with probability one. Moreover, 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 show that the densities of the distributions of the solutions can converge in L 1 to an invariant density. The results show that the larger white noise can lead to the extinction of the population while the smaller white noise can guarantee the existence of a stable stationary distribution which leads to the stochastic persistence of the population. Numerical simulations are provided to demonstrate the theoretical results. [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 focus on a stochastic predator–prey model with distributed delay. We first obtain the existence of a stationary distribution to the positive solutions by stochastic Lyapunov function method. Then we establish sufficient conditions for extinction of the predator population, that is, the prey population is survival and the predator population is extinct. [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]

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. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SEIR epidemic model with standard incidence. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of ergodic stationary distribution to the model. Our result does not depend on the positive endemic equilibrium E ∗ of the deterministic system, which improves the previous result to a greater extent. Moreover, we also establish sufficient conditions for extinction of the disease. [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 develop a mathematical model for a tuberculosis model with constant recruitment and varying total population size by incorporating stochastic perturbations. By constructing suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence of an ergodic stationary distribution as well as extinction of the disease to the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a mathematical model of cell-to-cell HIV-1 infection to C D 4 + T cells perturbed by stochastic perturbations. First of all, we investigate that there exists a unique global positive solution of the system for any positive initial value. Then by using Lyapunov analysis methods, we study the asymptotic property of this solution. Moreover, we discuss whether there is a stationary distribution for this system and if it owns the ergodic property. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic Susceptible–Infective (SI) epidemic model under regime switching. Firstly, by constructing suitable Lyapunov functions, we establish sufficient criteria for the existence and uniqueness of an ergodic stationary distribution. Then we obtain the threshold which guarantees the extinction and the existence of the stationary distribution of the epidemic. Finally, some numerical simulations are introduced to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a stochastic SIR epidemic model with saturated incidence is investigated. Firstly, we prove that the system has a unique global positive solution with any positive initial value. Then we verify that random effect may lead the disease to extinction under a simple condition. Thirdly, we establish a sufficient condition for persistence in the mean of the disease. Moreover, we show that there is a stationary distribution to the stochastic system under certain parametric restrictions. Finally, some numerical simulations are carried out to confirm the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, the deterministic and stochastic SIRS epidemic models with nonlinear incidence are introduced and investigated. For deterministic system, the basic reproductive number R 0 is obtained. Furthermore, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable and if R 0 > 1 , then there is a unique endemic equilibrium which is globally asymptotically stable. For stochastic system, to begin with, we verify that there is a unique global positive solution starting from the positive initial value. Then when R 0 > 1 , we prove that stochastic perturbations may lead the disease to extinction in scenarios where the deterministic system is persistent. When R 0 ≤ 1 , a result on fluctuation of the solution around the disease-free equilibrium of deterministic model is obtained under appropriate conditions. At last, if the intensity of the white noise is sufficiently small and R 0 > 1 , then there is a unique stationary distribution to stochastic system. [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]

• 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 consider a stochastic SIR epidemic model with Logistic birth. By using the stochastic Lyapunov function method, we show that the stochastic basic reproduction number R 0 S can be used to determine the threshold dynamics of the stochastic system. If R 0 S > 1 , we establish sufficient conditions for the existence of a stationary distribution of the positive solutions to the model. While if R 0 S < 1 , under some extra conditions, we obtain sufficient conditions for extinction of the disease. Finally, some examples and numerical simulations are provided to illustrate the theoretical results. • A stochastic SIR epidemic model with logistic birth is studied. • We establish sufficient conditions for the existence of a stationary distribution. • We obtain sufficient conditions for extinction of the disease. • Some examples and numerical simulations are provided to illustrate the theoretical results. [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 develop a mathematical model of viral infection of monocytes population by dengue virus with immune response which is perturbed by both white and telegraph noises. By constructing a suitable stochastic Lyapunov function with regime switching, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. The existence of a stationary distribution implies stochastic weak stability. • A stochastic dengue infection model with immune response and regime switching is studied. • We establish sufficient conditions for the existence of a unique ergodic stationary distribution. • The existence of a stationary distribution implies 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]

In this paper, we study a stochastic SIS epidemic model with nonlinear incidence rate. By employing the Markov semigroups theory, we verify that the reproduction number R 0 − σ 2 Λ 2 f 2 (Λ μ , 0) 2 μ 2 (μ + γ + α) can be used to govern the threshold dynamics of the studied system. If R 0 − σ 2 Λ 2 f 2 (Λ μ , 0) 2 μ 2 (μ + γ + α) > 1 , we show that there is a unique stable stationary distribution and the densities of the distributions of the solutions can converge in L 1 to an invariant density. If R 0 − σ 2 Λ 2 f 2 (Λ μ , 0) 2 μ 2 (μ + γ + α) < 1 , under mild extra conditions, we establish sufficient conditions for extinction of the epidemic. Our results show that larger white noise can lead to the extinction of the epidemic while smaller white noise can ensure the existence of a stable stationary distribution which leads to the stochastic persistence of the epidemic. • A stochastic SIS epidemic model with nonlinear incidence rate is studied. • There is a unique stable stationary distribution and the densities of the distributions of solutions can converge in L 1. • We establish sufficient conditions for extinction of the epidemic. [ABSTRACT FROM AUTHOR]