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The density of a population in a natural state often fluctuates greatly, but it is not an unlimited change. Throughout the full text, we consider a stochastic predator–prey system with fear, Holling-II response function, distributed delay and mean-reverting Ornstein–Uhlenbeck process, which can better reflect the actual situation and provide theoretical basis for species research in the actual environment. And we drew some significant conclusions. The first step is to obtain the stability of equilibrium point. Then, the existence and uniqueness of positive solution for stochastic system is got and we also reach the stationary distribution of the stochastic system. In addition, we present an exact local expression for the density function of the random system near the unique positive equilibrium. Then we obtain the pth moment boundedness and the asymptotic behavior of the solution. At last, we give the extinction condition of the population system by the asymptotic behavior of the solution. In particular, we confirm the theoretical results by numerical simulation in the corresponding section. [ABSTRACT FROM AUTHOR]

In this paper, we study an SICA model with a standard incidence rate, where the contact rate β is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when R 0 s > 1 , the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value R 0 e for disease extinction, and when R 0 e < 1 , the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate a novel stochastic SIHR epidemic model, which includes a general incidence rate and mean-reversion Ornstein–Uhlenbeck process. Firstly, the existence of global positivity of the solution is testified by Lyapunov function. Secondly, this disease will be eradicated if the reproduction number R 0 s < 1 . Otherwise, if the reproduction number R 0 ∗ > 1 , then the system has a stationary distribution, which means that the pandemic will persist. In addition, an explicit expression of the probability density function for a linear system near quasi-endemic equilibrium is obtained under certain conditions. Finally, a series of numerical simulations is carried out to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

Considering the transmission characteristics of COVID-19, we formulate a Susceptible-Exposed-Quarantine-Infected-Recovered epidemic model by five first-order differential equations to study the dynamical behaviors of diseases that have a latent period, quarantine strategy, governmental intervention and general incidence rate. After giving the basic reproduction number R 0 , conditions for the existence of equilibria and their local asymptotic stability are both investigated. However, environmental perturbations always have influence on the epidemic in the natural world. With the assumption that the transmission rate is driven by the log-normal Ornstein–Uhlenbeck process, we construct a corresponding stochastic epidemic model that incorporates environmental impacts. Based on the proof of existence and uniqueness of the global positive solution, two critical values R 0 e and R 0 s are established that can determine the extinction and persistence of disease, which are completely constituted by the basic reproduction number and random factors. By solving a changing four-dimensional Fokker–Planck equation, we calculate the exact analytical expression of the probability density function of stationary distribution near the quasi-endemic equilibrium. Finally, some numerical simulations are performed to support obtained theoretical results, and we show the sensitivity index to study the impact of each parameter on disease transmission. [ABSTRACT FROM AUTHOR]

In this paper, assuming the certain variable satisfies the Ornstein–Uhlenbeck process, we formulate a stochastic within-host dengue model with immune response to obtain further understanding of the transmission dynamics of dengue fever. Then we analyze the dynamical properties of the stochastic system in detail, including the existence and uniqueness of the global solution, the existence of a stationary distribution, and the extinction of infected monocytes and free viruses. In particular, it is worth revealing that we get the specific form of covariance matrix in its probability density around the quasi-endemic equilibrium of the stochastic system. Finally, numerical illustrative examples are depicted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Two different approaches to incorporate environmental perturbations in stochastic systems are compared analytically and computationally. Then we present a stochastic model for COVID-19 that considers susceptible, exposed, infected, and recovered individuals, in which the contact rate between susceptible and infected individuals is governed by the Ornstein–Uhlenbeck process. We establish criteria for the existence of a stationary distribution of the system by constructing a suitable Lyapunov function. Next, we derive the analytical expression of the probability density function of the model near the quasi-equilibrium. Additionally, we establish sufficient conditions for the extinction of disease. Finally, we analyze the effect of the Ornstein–Uhlenbeck process on the dynamic behavior of the stochastic model in the numerical simulation section. Overall, our findings shed light on the underlying mechanisms of COVID-19 dynamics and the influence of environmental factors on the spread of the disease, which can inform policy decisions and public health interventions. [ABSTRACT FROM AUTHOR]

In this paper, we construct a general four-dimensional delayed HIV infection model with virus-to-cell infection, cell-to-cell transmission, CTL immune response and parameter perturbations. By substitution, the four-dimensional delayed stochastic differential equations can be transformed into a degenerate eight-dimensional stochastic differential equations. The existence of the global positive solution of the system is obtained rigorously. By constructing appropriate Lyapunov functions, the existence of a stationary Markov process is derived when the stochastic reproduction number R 0 s is greater than one. Finally, we investigate the effects of noise level and the cell-to-cell transmission on the dynamic behavior of the model. Our model is a general model of the existing stochastic virus infection model, and the theoretical results improve and generalize the existing conclusion. [ABSTRACT FROM AUTHOR]

In this paper, considering the inevitable effects of environmental perturbations on disease transmission, we mainly study a stochastic SVEIR epidemic model in which the transmission rate satisfies the log-normal Ornstein–Uhlenbeck process and the incidence rate is general. To analyze the dynamic properties of the stochastic model, we firstly verify that there is a unique positive global solution. By constructing several suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. The sufficient condition for disease extinction is also given. Next, as a special case, we investigate the asymptotic stability of equilibria for the deterministic model and establish the exact expression of the probability density function of stationary distribution for the stochastic model. Finally, we calculate the mean first passage time from the initial value to the stationary state or extinction state to study the influence of environmental perturbations; meanwhile, some numerical simulations are carried out to demonstrate theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a stochastic SIRC model with Ornstein–Uhlenbeck process. Firstly, we give the existence and uniqueness of global solution of stochastic SIRC model and prove it. In addition, the existence of ergodic stationary distributions for stochastic SIRC system is proved by constructing a suitable series of Lyapunov functions. A quasi-endemic equilibrium related to endemic equilibrium of deterministic systems is defined by considering randomness. And we obtain the probability density function of the linearized system near the equilibrium point. After the proof of probability density function, the sufficient condition of disease extinction is given and proved. We prove the theoretical results in the paper by numerical simulation at the end of the paper. [ABSTRACT FROM AUTHOR]

Due to many uncertain factors, parameter values in many microorganism cultivation systems are affected to a greater or lesser extent by environmental fluctuation. In this paper, the authors develop a stochastic turbidostat model that considers white noise, formulate and analyze dynamical behavior for the stochastic model. The authors obtain the existence and uniqueness of globally positive equilibrium. Sufficient conditions of the extinction for the system are established. Since the uniformly elliptic condition can fail to satisfy, the existence of the stationary distribution of the system is proved by using Markov semigroups theory. Biologically speaking, the existence of stationary distribution suggests that microorganism will exist for a long time. The analytical results are tested and verified by numerical simulation, which evaluate the influence of white noise on the dynamics of microorganism. [ABSTRACT FROM AUTHOR]

In this paper, a generalized n-species Gilpin–Ayala competition system with saturation effect and nonlinear perturbations is proposed and examined. We first develop a new mathematical technique called “stochastic ϵ -threshold method”, to tackle the nonlinear perturbations and study the competitive coexistence, which includes the existence of ergodic stationary distribution as well as stochastic positive periodic solution. It should be mentioned that the method can be successfully applied to the species coexistence of other biological models. Then, we establish sufficient conditions for exponential extinction of competing species. Finally, several numerical examples are provided to support our theoretical results and analyze the impact of key parameters. [ABSTRACT FROM AUTHOR]

In this paper, we examine a stochastic avian influenza model with a nonlinear incidence rate within avian populations and the psychological effect within the human population, where susceptible humans reduce their contact with infected avians as the number of infected humans increases. For the deterministic model, the basic reproduction number R 0 , possible equilibria, and related asymptotic stability are first studied. Then, for the stochastic model, we obtain a critical value R 0 S , which can determine the persistence and extinction of avian influenza. It is theoretically proved that the stochastic model has a unique stationary distribution ϖ (·) if R 0 S > 1 , but the disease will go to extinction when R 0 S < 1 . Taking stochasticity into account, a quasi-endemic equilibrium T ¯ ∗ related to the endemic equilibrium of the deterministic model is defined. We develop an important lemma for solving the special Fokker–Planck equation and derive the explicit expression of the density function of the distribution ϖ (·) around the equilibrium T ¯ ∗ . Numerical simulations verify our theoretical results, and we study the impact of noise and the psychological effect on the transmission dynamics of avian influenza. [ABSTRACT FROM AUTHOR]

In this paper, we consider a high-dimensional stochastic HIV/AIDS model that incorporates both multiple stages treatment and higher order perturbation. Firstly, we establish sufficient criteria for the existence of a unique ergodic stationary distribution by making use of stochastic Lyapunov analysis method. Stationary distribution shows that the disease will be persistent in the long term. Then, conditions for extinction of the disease are obtained. Theoretical analysis indicates that large noise intensity can suppress the prevalence of HIV/AIDS epidemic. Finally, we provide some numerical simulations to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

Time delay, where it depends on the current state and on the past situation, is often occurred in biological activities, for example, the process by which microorganism consume nutrients into their available biomass is not instantaneous. This investigation inspects the dynamic behavior of stochastic turbidostat model coupled with distributed delay and degenerate diffusion, including sufficient conditions of the extinction and the existence of a unique stationary distribution. What's more, the existence and uniqueness of globally positive equilibrium of the exploited model are studied. The findings manifest that the turbidostat system is ergodic only when the intensity of white noise is very small. Finally, some numerical examples are proposed to indicate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

Phytoplankton is an important indicator organism to evaluate the quality of water environment, which may reflect the nutritional level of the sea area. Conversely, environmental conditions can directly affect the community structure of phytoplankton. A stochastic phytoplankton–zooplankton model considering non-degenerate and degenerate diffusions is formulated in this paper. What’s more, in both systems, we obtain sufficient conditions of extinction and ergodicity. Our results demonstrate that the weaker white noise can ensure the permanence of zooplankton and the stronger white noise can lead to the disappearance of zooplankton. Moreover, the threshold value of extinction and persistence can serve as a theoretical basis for controlling phytoplankton and zooplankton. Numerical examples are performed on the analysis results of the two cases to confirm our theoretical results. In addition, we also provide a real-life case study, the validity of the model is verified based on experimental data, and it is shown that fluctuation of external environment and the consumption of phytoplankton by zooplankton will affect the growth and number of phytoplankton. Effectively controlling the quantity of phytoplankton can delay the occurrence of water bloom or red tide. [ABSTRACT FROM AUTHOR]

A stochastic waterborne disease model is analyzed to reveal the effects of the white noise and telegraph noise. For this stochastic model, a critical quantity R 0 s is derived, which depends on not only model parameters but also environmental noises. If R 0 s > 1 , then this model admits an ergodic stationary distribution. One of the most important contributions is the suitable Lyapunov function depending on the state of Markov chains is constructed. Theoretical results and numerical simulations provide a clear insight of the impact of environmental noises on the dynamical behavior of the model. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic chemostat model with degenerate diffusion. Firstly, the Markov semigroup theory is used to establish sufficient criteria for the existence of a unique stable stationary distribution. The authors show that the densities of the distributions of the solutions can converge in L1 to an invariant density. Then, conditions are obtained to guarantee the washout of the microorganism. Furthermore, through solving the corresponding Fokker-Planck equation, the authors give the exact expression of density function around the positive equilibrium of deterministic system. Finally, numerical simulations are performed to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Considering the important role that food chains play in ecosystems, in this paper, we study a three-species stochastic food chain model in which the growth rates and death rates are governed by Ornstein–Uhlenbeck process. The main purpose of this paper is to study the dynamic properties of the model. We first prove the existence and uniqueness of the global solution to the model. The moment boundedness and asymptotic behavior of the solution are also verified. Secondly, we get the result of extinction of the predators. Then, a sufficient criteria for the existence of the stationary distribution to the system are established by constructing a suitable Lyapunov function. Under the same conditions, it is worth noting that we further obtain the explicit formulas for the mean and the covariance of the probability density function for a linearized system around a equilibrium point. Finally, several numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a three-dimensional stochastic vegetation–water model in arid ecosystems, where the soil water and the surface water are considered. First, for the deterministic model, the possible equilibria and the related local asymptotic stability are studied. Then, for the stochastic model, by constructing some suitable stochastic Lyapunov functions, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution ϖ (·) . In a biological interpretation, the existence of the distribution ϖ (·) implies the long-term persistence of vegetation under certain conditions. Taking the stochasticity into account, a quasi-positive equilibrium D ¯ ∗ related to the vegetation-positive equilibrium of the deterministic model is defined. By solving the relevant Fokker–Planck equation, we obtain the approximate expression of the distribution ϖ (·) around the equilibrium D ¯ ∗ . In addition, we obtain sufficient condition R 0 E < 1 for vegetation extinction. For practical application, we further estimate the probability of vegetation extinction at a given time. Finally, based on some actual vegetation data from Wuwei in China and Sahel, some numerical simulations are provided to verify our theoretical results and study the impact of stochastic noise on vegetation dynamics. [ABSTRACT FROM AUTHOR]

Due to many uncertain factors, the microorganism flocculation models could be affected by environmental noise. The paper aims to discuss the dynamical behavior of a stochastic microorganism flocculation model, including the extinction and the persistence. Moreover, the expression of density function near the positive equilibrium point is explicitly obtained. Our results indicate that a larger white noise can accelerate the extinction of microorganism, while a weaker white noise can guarantee the existence of stationary distribution. In addition, our theory is confirmed by some numerical examples. [ABSTRACT FROM AUTHOR]

This paper develops the spread dynamics of a 11-dimensional stochastic multi-host zoonotic model for the dog-CFB-human transmission of rabies, which is formulated as a piecewise deterministic Markov process. We firstly prove the existence of the global unique positive solution. Then we obtain sufficient conditions for the extinction and persistence of disease. One of the distinct features of this paper is that we prove the positive recurrence of the solution to the model by constructing a series of appropriate Lyapunov functions. Finally, numerical simulations are carried out to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Recently, the stochastic θ -threshold method has been proposed to analyze the impact of second-order perturbations on disease persistence. In this paper, considering the complexity of the environmental variations in the actual situation, we construct and study a stochastic staged progression HIV/AIDS infection model with third-order perturbations. First, we generalize the stochastic θ -threshold method to eliminate the effect of third-order perturbation of the stochastic model and obtain a critical value R 0 S related to the basic reproduction number R 0 of its deterministic model. By constructing several suitable θ -Lyapunov functions, it is theoretically proved that the stochastic model has a unique ergodic stationary distribution if R 0 S > 1 . In a biological interpretation, the existence of a stationary distribution implies the long-term persistence of HIV/AIDS. Moreover, for some small perturbations, by means of the exponential martingale inequality, we establish sufficient condition ϱ 0 < 0 for disease extinction. Then, for some large perturbations, another sufficient condition R 0 C < 1 for disease extinction of the stochastic model is also derived. For completeness, we provide some examples and numerical simulations to verify our analytical results. What is more, the generalized stochastic θ -threshold method can be successfully applied to the dynamical analysis of other complex high-dimensional epidemic models with nonlinear perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIR epidemic model with general disease incidence rate and perturbation caused by nonlinear white noise and L e ´ vy jumps. First of all, we study the existence and uniqueness of the global positive solution of the model. Then, we establish a threshold λ by investigating the one-dimensional model to determine the extinction and persistence of the disease. To verify the model has an ergodic stationary distribution, we adopt a new method which can obtain the sufficient and almost necessary conditions for the extinction and persistence of the disease. Finally, some numerical simulations are carried out to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

Many turbidostat models are affected by environmental noise due to various complicated and uncertain factors, and Ornstein-Uhlenbeck process is a more effective and precise way. We formulate a stochastic turbidostat system incorporating Ornstein-Uhlenbeck process in this paper and develop dynamical behavior for the stochastic model, which includes the existence and uniqueness of globally positive equilibrium, sufficient conditions of the extinction, the existence of a unique stationary distribution and an expression of density function of quasi-stationary distribution around the positive solution of the deterministic model. The results indicate that the weaker volatility intensity can ensure the existence and uniqueness of the stationary distribution, and the stronger reversion speed can lead to the extinction of microorganisms. Numerical simulations verify the validity of the analysis results, which assess the influence of the speed of reversion and the intensity of volatility on the long-term behavior of microorganisms. [ABSTRACT FROM AUTHOR]

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas'minskii theory and Lyapunov function approach, we construct a critical value R 0 S corresponding to the basic reproduction number R 0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if R 0 S > 1 . Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under R 0 S > 1 . It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dynamical behaviors of a model of syphilis transmission disturbed by both white noises and telegraph noises. Multiple infections and treatment stages are considered, which include and extend the existing ones. The existence and ergodicity of the stationary distribution are obtained by constructing a suitable Lyapunov function, which determines a critical value R 0 ∗ corresponding to the control reproduction number R c of the corresponding determined system. In addition, a sufficient criteria for extinction of the diseases is derived. Finally, the numerical simulations illustrate our theoretical results, which show that, the stronger white noises can result in the extinction of the diseases and telegraph noises can strength the stability of the system. [ABSTRACT FROM AUTHOR]

The mathematical model with time delay is often more practical because it is subject to current and past state. What remains unclear are the details, such as how time delay and sudden environmental changes influence the dynamic behavior of systems. The purpose of this paper is to analyze the long-time behavior of a stochastic Nicholson's blowflies model, which includes distributed delay and degenerate diffusion. The application of the Markov semigroups theory is to prove that there exists a unique stationary distribution. What's more, the expression of probability density function around the unique positive equilibrium of the deterministic model is briefly described under a certain condition. The results of this paper can be used to find that the weaker white noise can guarantee the existence of a unique stationary distribution and the stronger mortality rate can cause the extinction of Nicholson's blowflies. Some numerical examples are also given to explain the effect of the white noise. [ABSTRACT FROM AUTHOR]

We consider a model of multi-species competition in the chemostat that includes demographic stochasticity and discrete delays. We prove that for any given initial data, there exists a unique global positive solution for the stochastic delayed system. By employing the method of stochastic Lyapunov functionals, we determine the asymptotic behaviors of the stochastic solution and show that although the sample path fluctuate, it remains positive and the expected time average of the distance between the stochastic solution and the equilibrium of the associated deterministic delayed chemostat model is eventually small, i.e. we obtain an analogue of the competition exclusion principle when the noise intensities are relatively small. Numerical simulations are carried out to illustrate our theoretical results. [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]

In this paper, we consider a stochastic food-limited mutualism model with periodic coefficients. Using Has'minskii theory of periodic Markov processes, sufficient conditions for the existence of the stochastic nontrivial periodic solutions to the model are established. Then we obtain conditions for the existence and global attractivity of the boundary periodic solutions. Finally, examples are introduced to illustrate the results developed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic HIV-1 infection model with distributed delay and logistic growth. Firstly, we transfer the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Then, we establish sufficient conditions for the existence of a stationary distribution of the model by constructing a suitable stochastic Lyapunov function. Moreover, we obtain sufficient criteria for extinction of the infected cells; that is, the uninfected cells are survival and the infected cells are extinct. Our results show that the smaller white noise can ensure the existence of a stationary distribution when the basic reproduction number R 0 S of the stochastic system is bigger than one, while the larger white noise can lead to the extinction of the infected cells when the basic reproduction number R 0 of the deterministic system is smaller than one. [ABSTRACT FROM AUTHOR]

A biological population may be subjected to stochastic disturbance and exhibit periodicity. In this paper, a stochastic non-autonomous predator-prey system with Holling II functional response is proposed, and the existence of a unique positive solution is derived. We give sufficient conditions for extinction and strong persistence in the mean by analyzing a corresponding one-dimensional stochastic system. Also we establish the existence of positive periodic solutions for this stochastic non-autonomous predator-prey system. Finally, we use numerical simulations to illustrate our results and we present some conclusions and future directions. The results of this paper provide methods for other stochastic population models, which we hope to analyze in the future. [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, two stochastic predator-prey models with general functional response and higher-order perturbation are proposed and investigated. For the nonautonomous periodic case of the system, by using Khasminskii’s theory of periodic solution, we show that the system admits a nontrivial positive T-periodic solution. For the system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent. [ABSTRACT FROM AUTHOR]

We consider a single-species stochastic logistic model with the population's nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic predator-prey model with hyperbolic mortality and Holling type II response. Firstly, we show that there is a critical value which can easily determine the extinction and persistence in the mean of the predator population. Then by constructing appropriate Lyapunov functions, we prove that there is a stationary distribution to this model and it has the ergodic property. Finally, a numerical example is introduced to illustrate the results developed. [ABSTRACT FROM AUTHOR]

The objective of this paper is to explore the long time behavior of a stochastic SIR model. We establish a threshold condition called the basic reproduction number under stochastic perturbation for persistence or extinction of the disease. Especially, some numerical simulations are applied to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss a stochastic enzymatic reaction model under negative feedback process. We will prove that the solution of the system is positive and global. Furthermore, we draw a conclusion that there is a stationary distribution for the stochastic system and it has ergodicity without any additional conditions. Finally, we test our theory conclusion by simulations. It is interesting that our stochastic model is always ergodic. [ABSTRACT FROM AUTHOR]

The present paper deals with the problem of a chemostat model with Holling type II functional response by stochastic perturbation. The main objective of the work is to find out sufficient conditions which guarantee that the principle of competitive exclusion holds for this perturbed model. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

The present paper deals with the problem of an ecoepidemiological model with linear mass-action functional response perturbed by white noise. The essential mathematical features are analyzed with the help of the stochastic stability, its long time behavior around the equilibrium of deterministic ecoepidemiological model, and the stochastic asymptotic stability by Lyapunov analysis methods. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a stochastic SIS epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number $R_{0}$. When the perturbation is large, the number of infected decays exponentially to zero and the solution converges to the disease-free equilibrium regardless of the magnitude of $R_{0}$. Moreover, we get the same exponential stability and the convergence if $R_{0}<1$. When the perturbation and the disease-related death rate are small, we derive that the disease will persist, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model on average in time if $R_{0}>1$. Furthermore, we prove that the system is persistent in the mean. Finally, the results are illustrated by computer simulations. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic SIS model with saturated incidence rate. We investigate the existence and uniqueness of the positive solution to the system, and we show the condition for the infectious individuals to be extinct. Moreover, we prove that the system has the ergodic property and derive the expression for its invariant density. The simulation results are illustrated finally. [ABSTRACT FROM AUTHOR]

In this paper, we introduce stochasticity into a multigroup SIS model. We present the sufficient condition for the exponential extinction of the disease and prove that the noises significantly raise the threshold of a deterministic system. In the case of persistence, we prove that there exists an invariant distribution which is ergodic. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic SIS epidemic model with perturbed disease transmission coefficient. We discuss the threshold behavior in the sense that if $$\mathcal {R}_0^s<1$$ , the disease dies out in probability; whereas if $$\mathcal {R}_0^s>1$$ , the densities of the distributions of the solution can converge in $$L^1$$ to an invariant density. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce a low concentration trimolecular stochastic chemical reaction model. We will prove that the solution of the system is positive and global. And then we draw a conclusion that there is a stationary distribution for the stochastic system and it has ergodicity under appropriate conditions. Finally, we test our theory conclusion by simulations. It is interesting that no matter what states the unique equilibrium of ordinary differential equation model appears, and regardless of whether the limit cycle exists, our stochastic model is always ergodic. [ABSTRACT FROM AUTHOR]

We establish the existence of positive solutions for the second order singular semipositone coupled Dirichlet systems The proof relies on Schauder's fixed point theorem. [ABSTRACT FROM AUTHOR]

The paper introduces the dynamics of a stochastic multi-molecule biochemical reaction model.First, we show that there is a unique positive solution of the stochastic model. Furthermore, we deduce the conditions when the reaction will end and when the reaction being proceed. At last, we derive that the solution of (1.5) oscillates around the endemic proportion equilibrium $$P^*(x^*,y^*)$$ , and the intensity of fluctuation is proportional to white noise. The key to the analysis in this paper is choosing appropriate Lyapunov function. The outcomes are illustrated by computer simulations throughout this paper. [ABSTRACT FROM AUTHOR]

Taking both white noise and colored environmental noise into account, a single-species logistic model with population’s nonlinear diffusion among two patches is proposed and investigated. The sufficient conditions of the existence of positive solutions, stochastic permanence, persistence in mean and extinction are established. Moreover, we use an example and simulation figures to illustrate our main results. [ABSTRACT FROM AUTHOR]

The singular boundary-value problemis studied. The singularity may appear at u = 0, and the function g may change sign. An existence theorem for solutions to the above boundary-value problem is proposed, and it is proved via the method of upper and lower solutions. [ABSTRACT FROM AUTHOR]