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As the prevalence of viral infection in body, human T‐cell leukemia virus type 1 (HTLV‐1) is receiving increasing attention. Research on the corresponding virus models is of great significance to tackle the challenges of understanding HTLV‐1 development and treatment. This paper focuses on the dynamic analysis for a stochastic model with nonlinear cytotoxic T lymphocyte (CTL) response, which is driven by Ornstein–Uhlenbeck (OU) process to model the progression of HTLV‐1 in vivo. Rich dynamic behaviors such as the extinction of infected CD4+ T cells (ITCs), stationary distribution (SD), probability density, and finite‐time stability (FTS) of the model are established to reveal the interaction of cell populations. The optimal therapeutic strategy based on the cost‐benefit viewpoint is further obtained. Finally, illustrative numerical simulations are represented to corroborate the effectiveness of treatment and the ambient perturbation's impact that strengthening the noise strength can lead to rapid virus clearance. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito’s formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIQS epidemic model perturbed by both white and telephone noises is investigated. By constructing several suitable Lyapunov functions, we obtain sufficient conditions for the existence of ergodic stationary distribution of the positive solution. Moreover, by solving the Fokker–Planck equation, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. In addition, sufficient conditions for the extinction are established. Finally, the results of this paper are further verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we study three stochastic two-species models. We construct the stochastic models corresponding to its deterministic model by introducing stochastic noise into the equations. For the first model, we show that the system has a unique global solution starting from the positive initial value. In addition, we discuss the extinction and the existence of stationary distribution under some conditions. For the second system, we explore the existence and uniqueness of the solution. Then we obtain sufficient conditions for global asymptotic stability of the equilibrium point and the positive recurrence of solution. For the last model, the existence and uniqueness of solution, the sufficient conditions for extinction and asymptotic stability and the positive recurrence of solution and weak persistence are derived. And numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

This paper delves into the analysis of a stochastic epidemic model known as the susceptible–vaccinated–exposed–infectious–recovered (SVEIR) model, where transmission dynamics are governed by a nonlinear function. In the theoretical analysis section, by suitable stochastic Lyapunov functions, we establish that when the threshold value, denoted as R 0 s , falls below 1, the epidemic is destined for extinction. Conversely, if the reproduction number R 0 of the deterministic model surpasses 1, the model manifests an ergodic endemic stationary distribution. In the numerical simulations and data interpretation section, leveraging a graphical analysis with COVID-19 data, we illustrate that random fluctuations possess the capacity to quell disease outbreaks, underscoring the role of vaccines in curtailing the spread of diseases. This study not only contributes to the understanding of epidemic dynamics but also highlights the pivotal role of stochasticity and vaccination strategies in epidemic control and management. The inherent balance and patterns observed in epidemic spread and control strategies, reflect a symmetrical interplay between stochasticity, vaccination, and disease dynamics. [ABSTRACT FROM AUTHOR]

This paper presents a stochastic vector-borne epidemic model with direct transmission and media coverage. It proves the existence and uniqueness of positive solutions through the construction of a suitable Lyapunov function. Immediately after that, we study the transmission mechanism of vector-borne diseases and give threshold conditions for disease extinction and persistence; in addition we show that the model has a stationary distribution that is determined by a threshold value, i.e., the existence of a stationary distribution is unique under specific conditions. Finally, a stochastic model that describes the dynamics of vector-borne diseases has been numerically simulated to illustrate our mathematical findings. [ABSTRACT FROM AUTHOR]

The stochastic fluctuation of information induces the dynamic behavior and disease control strategy to change in the HBV epidemic model, but how stochastic information dissemination affects them is vague. Here, we consider an HBV epidemic model with stochastic information intervention. Using the Lyapunov function, we demonstrate that the system has a unique global positive solution, in addition, we also derive the threshold dynamics to obtain sufficient conditions that can ensure the extinction and persistence, as well as stationarity of the system. Moreover, we qualify the impact of stochasticity of the information on the optimal control strategy of the HBV, indicating that while information intervention could effectively reduce the disease peak, the fluctuation will attenuate this effect, i.e., the stable information is better than the noisy one. Finally, various stochastic and optimal control simulations are performed to verify the theoretical results and verify that optimal control can accelerate the extinction of the disease.

The stochastic fluctuation of information induces the dynamic behavior and disease control strategy to change in the HBV epidemic model, but how stochastic information dissemination affects them is vague. Here, we consider an HBV epidemic model with stochastic information intervention. Using the Lyapunov function, we demonstrate that the system has a unique global positive solution, in addition, we also derive the threshold dynamics to obtain sufficient conditions that can ensure the extinction and persistence, as well as stationarity of the system. Moreover, we qualify the impact of stochasticity of the information on the optimal control strategy of the HBV, indicating that while information intervention could effectively reduce the disease peak, the fluctuation will attenuate this effect, i.e., the stable information is better than the noisy one. Finally, various stochastic and optimal control simulations are performed to verify the theoretical results and verify that optimal control can accelerate the extinction of the disease. [ABSTRACT FROM AUTHOR]

In this article, we mainly consider the dynamic analysis of a stochastic infectious disease model with negative feedback, a symmetric and compatible distribution family. Based on the sir epidemic model taking into account the isolation (y) and the death (v), we consider adding a new variable (w) to control the information of non-drug interventions, which measures transformations in isolation performance that determine the epidemic, and establish a new model. We have demonstrated various properties of the model solution using Lyapunov functions for this model. To begin with, we demonstrate the existence and uniqueness of the global positive solution. After that, we obtained the conditions that need to be met for the extinction of the disease and verified the correctness of the conclusion by simulating numerical values. Afterwards, we prove the stochastic boundedness and stationary distribution of the model solution. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic COVID-19 model with large-scale nucleic acid detection and isolation measures is proposed. Firstly, the existence and uniqueness of the global positive solution is obtained. Secondly, threshold criteria for the stochastic extinction and persistence in the mean with probability one are established. Moreover, a sufficient condition for the existence of unique ergodic stationary distribution for any positive solution is also established. Finally, numerical simulations are carried out in combination with real COVID-19 data from Urumqi, China and the theoretical results are verified. [ABSTRACT FROM AUTHOR]

The paper mainly investigates a stochastic SIRS epidemic model with Logistic birth and nonlinear incidence. We obtain a new threshold value ( R 0 m ) through the Stratonovich stochastic differential equation, different from the usual basic reproduction number. If R 0 m < 1 , the disease-free equilibrium of the illness is globally asymptotically stable in probability one. If R 0 m > 1 , the disease is permanent in the mean with probability one and has an endemic stationary distribution. Numerical simulations are given to illustrate the theoretical results. Interestingly, we discovered that random fluctuations can suppress outbreaks and control the disease. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic differential equation model which is used to explore how the environmental noise affects the spread of hepatitis B virus. Firstly, we show that there exists a unique global positive solution of the stochastic system with any positive initial value. Then, we adopt a stochastic Lyapunov function method to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model. Especially, under the same conditions as the existence of a stationary distribution, it is worth noting that we obtain the specific form of probability density around the quasi-infected steady state of the stochastic system. Thirdly, we obtain sufficient criteria for extinction of the infected hepatocytes. Finally, numerical simulations are introduced to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we mainly present the dynamic properties of a stochastic SICA model with general incidence rate. Through rigorous analysis and reasoning of the stochastic model, we obtain that the solution of the model is global, positive, and unique. By constructing the threshold value R0s$$ {R}_0^s $$, which includes the influence of white noise, we obtain a sufficient condition for the ergodicity of this model. Furthermore, we show that the model has a unique ergodic stationary distribution while R0s>1$$ {R}_0^s>1 $$ by adopting a novel method. The extinction of the system is also established. Besides, the expression of probability density function of the stochastic model with bilinear incidence rate around the unique stable positive equilibrium of the deterministic model is derived. Finally, numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper proposes a stochastic intestinal chemostat model considering microbial migration, intraspecific competition and stochastic perturbation. First, the extinction and persistence in mean of the intestinal microbe of the chemostat model are investigated by constructing the appropriate Lyapunov functions. Second, we explore and obtain suffcient conditions for the existence and uniqueness of an ergodic stationary distribution of the model by using ergodic theory. The results show stochastic interference has a critical impact on the extinction and sustainable survival of the intestinal microbe. Eventually, numerical simulations are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a detailed investigation of a stochastic model that rules the spreading behavior of the measles virus while accounting for the white noises and the influence of immunizations. It is hypothesized that the perturbations of the model are nonlinear, and that a person may lose the resistance after vaccination, implying that vaccination might create temporary protection against the disease. Initially, the deterministic model is formulated, and then it has been expanded to a stochastic system, and it is well-founded that the stochastic model is both theoretically and practically viable by demonstrating that the model has a global solution, which is positive and stochastically confined. Next, we infer adequate criteria for the disease's elimination and permanence. Furthermore, the presence of a stationary distribution is examined by developing an appropriate Lyapunov function, wherein we noticed that the disease will persist for R 0 s > 1 and that the illness will vanish from the community when R 0 s < 1 . We tested the model against the accessible data of measles in Pakistan during the first ten months of 2019, using the conventional curve fitting methods and the values of the parameters were calculated accordingly. The values obtained were employed in running the model, and the conceptual findings of the research were evaluated by simulations and conclusions were made. Simulations imply that, in order to fully understand the dynamic behavior of measles epidemic, time-delay must be included in such analyses, and that advancements in every vaccine campaign are inevitable for the control of the disease. [ABSTRACT FROM AUTHOR]

The present article formulates and dissects a single-species model with intraspecific cooperation and stochastic perturbations. Sufficient criteria for extinction and permanence are provided. In addition, the existence of a unique ergodic stationary distribution (UESD) is dissected, and the explicit form of the density function of the UESD is obtained under some conditions. Finally, the theoretical outcomes are used to study the growth of gypsy moth (Lymantria dispar) in North America, and some critical impacts of stochastic perturbations are uncovered and numerically illustrated. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior of a stochastic SQEIAR epidemic model is investigated. First of all, we establish sufficient conditions for extinction of the diseases. Then the sufficient conditions for the existence of an ergodic stationary distribution of the positive solutions to the model are obtained by constructing a suitable stochastic Lyapunov function. The existence of stationary distribution implies stochastic weak stability. Furthermore, the optimal control problem is considered to provide a theoretical basis for the prevention and control of the disease. Finally, the theoretical results are verified by numerical simulations. [ABSTRACT FROM AUTHOR]

A stochastic SIAM (Susceptible individual-Infected individual-Aware individual-Media coverage) epidemic model with nonlinear disturbances is constructed, where awareness dissemination rate satisfies the mean-reverting Ornstein–Uhlenbeck process. Hybrid dynamic effects of Lévy jump and Ornstein–Uhlenbeck process on infectious disease transmission are discussed. By constructing appropriate stochastic Lyapunov functionals, sufficient conditions for existence of stationary distribution of the proposed stochastic system are investigated. • A stochastic SIAM model with media coverage and nonlinear disturbances is proposed. • Hybrid dynamic effects of Ornstein–Uhlenbeck process and Lévy jumps are studied. • Sufficient conditions for existence of stationary distribution are investigated. [ABSTRACT FROM AUTHOR]

Birth vaccinations are becoming more common in society. In this paper, we describe the developed stochastic susceptible‐vaccinated‐infected‐recovered (SVIR) epidemic model with vaccination of newborns that enable us to concern the stationary distribution and further density function. By constructing a series suitable Lyapunov function, we derive the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution. More importantly, under the same conditions, we creatively find further the density function which is based on solving corresponding Fokker–Planck equation. The results of numerical simulation, which is supported by pertussis disease data, show that our conclusion accords with reality. The density function throws light on the property of an epidemic after being stationary and furnishes more information about the disease. [ABSTRACT FROM AUTHOR]

This work is concerned with a novel stochastic competitive model with saturation effect and distributed delay, in which two coupling noise sources are incorporated and the interspecific competition delayed terms show saturation effect. A good understanding of exponential extinction, extinction, persistence in the mean and permanence in time average of two species are gained. Also, with the help of Lyapunov function and the global attraction of positive solution, we derive the existence and uniqueness of stationary distribution. Our main results reveal that the coupling noise sources can significantly change the survival results of two species and affect the existence of a unique stationary distribution. [ABSTRACT FROM AUTHOR]

In this paper, we establish stochastic differential equations on the basis of a nonlinear deterministic model and study the global dynamics. For the deterministic model, we show that the basic reproduction number ℜ0 determines whether there is an endemic outbreak or not: if ℜ0 ℜ0 < 1, the disease dies out; while if ℜ0 > 1, the disease persists. For the stochastic model, we provide analytic results regarding the stochastic boundedness, perturbation, permanence and extinction. Finally, some numerical examples are carried out to confirm the analytical results. One of the most interesting findings is that stochastic fluctuations introduced in our stochastic model can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. [ABSTRACT FROM AUTHOR]

In the present work, we consider a stochastic predator-prey model with disease in prey and distributed delay. Firstly, we establish sufficient conditions for the extinction of the disease and also permanence of healthy prey and predator. Besides, we obtain the condition for the existence of an ergodic stationary distribution through the stochastic Lyapunov function. Finally, we provide some numerical simulations to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, stochastic dynamics with Lévy noise of two-consumers-one-resource competing systems with Beddington–DeAngelis functional response are considered. We first show the existence of the global positive solution, then discuss the effects of noises on the extinction of the species and the stochastic persistence of the species. In the meanwhile, numerical simulations are carried to support results. Finally, we show the existence of the stationary distribution for a special case. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate a stochastic Holling type-II predator–prey model with prey refuge and fear effect. We first prove the existence and uniqueness of the global positive solution. Then we perform the survival analysis of the model, including the existence of a unique ergodic stationary distribution and the extinction of the model. Numerical simulations are carried out to validate our analytical results. Our findings indicate that the white noise is adverse to the growth of predator and prey populations, and the increase of fear effect will lead to the decrease of predator density, but with no obvious effect on prey density. [ABSTRACT FROM AUTHOR]

• Classify the asymptotic behavior SIR models with noise perturbing to linear terms. • Construct a threshold value based on coefficients. • The negativity of threshold implies the extinction of the model at exponential rate. • If it is positive, the solution converges to unique stationary measure in total variation. • We can use this technique to classify models with noise perturbing to non linear terms. In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and to initiate a study of a predator–prey system with Holling type II functional response, which is disturbed by both Markov switching and white noise. Our main aim is to investigate the dynamical properties of this model. First, we establish sufficient conditions of the strong persistence in the mean and the extinction of the predator population and obtain the threshold between them. Then, we show the existence of a unique ergodic stationary distribution under some natural conditions. Finally, a numerical example and figures are presented to illustrate the results. [ABSTRACT FROM AUTHOR]

This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing, a real-valued threshold. We show that if < 0, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value > 0, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all t 2 [0;1), approximation in a long but finite interval [0; T] can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic epidemic system with both switching noise and white noise is proposed to research the dynamics of the diseases. Nonlinear incidence and vaccination strategies are also considered in the proposed model. By using the method of stochastic analysis, we point out the key parameters that determine the persistence and extinction of the diseases. Specifically, if Rs 0 is greater than 0, the stochastic system has a unique ergodic stationary distribution; while if R* 0 is less than 0, the diseases will be extinct at an exponential rate. [ABSTRACT FROM AUTHOR]

This paper focuses on a predator-prey system with foraging arena scheme incorporating stochastic noises. This SDE model is generated from a deterministic framework by the stochastic parameter perturbation. We then study how the correlations of the environmental noises affect the long-time behaviours of the SDE model. Later on the existence of a stationary distribution is pointed out under certain parametric restrictions. Numerical simulations are carried out to substantiate the analytical results. [ABSTRACT FROM AUTHOR]

In this article, we consider a stochastic plant disease model with logistic growth and saturated incidence rate. We analyze long-term behaviors of densities of the distributions of the solution. On the basis of the theory of Markov semigroup, we obtain the existence of asymptotically stable stationary distribution density of the stochastic system. We demonstrate that the densities can converge in L¹ to an invariant density under appropriate conditions. Moreover, we obtain the sufficient conditions for extinction of the disease. Also, we present a series of numerical simulations to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic model is formulated to describe the transmission dynamics of tuberculosis. The model incorporates vaccination and treatment in the intervention strategies. Firstly, sufficient conditions for persistence in mean and extinction of tuberculosis are provided. In addition, sufficient conditions are obtained for the existence of stationary distribution and ergodicity. Moreover, numerical simulations are given to illustrate these analytical results. The theoretical and numerical results show that large environmental disturbances can suppress the spread of tuberculosis. [ABSTRACT FROM AUTHOR]

Abstract A stochastic generalized logistic model is considered in this paper. The condition of the existence of its stationary distribution is generalized. Recurrence and strong stochastic persistence are obtained. Finally some numerical simulations are carried out to support our results. [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]

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. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the asymptotic behaviors of one classical chemostat model in a stochastic environment. Based on the Feller property, sharp conditions are derived for the existence of a stationary distribution by using the mutually exclusive possibilities known in [11, 12] (See Lemma 2.4 for details), which closes the gap left by the Lyapunov function. Further, we obtain a sufficient condition for the extinction of the organism based on two noise-induced parameters: an analogue of the feed concentration S * and the break-even concentration λ. Results indicate that both noises have negative effects on persistence of the microorganism. [ABSTRACT FROM AUTHOR]

Abstract In this work we investigate the existence of a unique global positive solution for a stochastic SIRS epidemic model with a general awareness-induced incidence. We also study the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Numerical simulations are presented to confirm the theoretical results. Highlights • We formulate a stochastic epidemic model with a general awareness-induced incidence. • This model admits a unique global positive solution. • Sufficient conditions for fluctuations around equilibriums are obtained. • sufficient conditions for the existence of an ergodic stationary distribution are obtained. • Numerical simulations are introduced to illustrate the theoretical results. [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]

This paper is devoted to studying the dynamics of a susceptible-infective-latent-infective (SILI) epidemic model that is subject to the combined effects of environmental noise and intervention strategy. We extend the classical SILI epidemic model from a deterministic framework to a stochastic one. For the deterministic case, the global stability analysis of the solution is carried out in terms of the basic reproduction number. For the stochastic case, sufficient conditions for the extinction of diseases are obtained. Then, the existence of stationary distribution and asymptotic behavior of the solution are further studied to illustrate the cycling phenomena of recurrent diseases. Numerical simulations are conducted to verify these analytical results. It is shown that both stochastic noise and intervention strategy contribute to the control of diseases. [ABSTRACT FROM AUTHOR]

This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we develop and study a stochastic predator-prey model with stage structure for predator and Holling type II functional response. First of all, by constructing a suitable stochastic Lyapunov function, 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 predator populations in two cases, that is, the first case is that the prey population survival and the predator populations extinction; the second case is that all the prey and predator populations extinction. The existence of a stationary distribution implies stochastic weak stability. Numerical simulations are carried out to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

In this paper, by linking Fokker-Planck equations with stochastic coupled systems, a new method is provided to investigate the existence of a stationary distribution of stochastic coupled systems. Based on the graph theory and the Lyapunov method, an appropriate Lyapunov function associated with stationary Fokker-Planck equations is constructed. Moreover, a Lyapunov-type theorem and a coefficients-type criterion are obtained to guarantee the existence of a stationary distribution. Furthermore, theoretical results are applied to explore the existence of a stationary distribution of stochastic predator-prey models with dispersal and a sufficient criterion is presented correspondingly. Finally, a numerical example is given to illustrate the effectiveness of our results. [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]