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Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both delay digestion and stochastic fluctuation. In the literature, due to the computational complexity and the lack of available methods, few papers have directly considered the predator–prey model for the combination of general infinite time delays with stochasticity. In this sense, this work is devoted to studying the long-term qualitative behavior of a stochastic predator–prey system with general infinite distributed delay. First, we establish sufficient and necessary conditions for exponential extinction and persistence of species. Notably, not only do our results cover the predator–prey model without stochastic noises and time delays, but also reveal the impact of environmental noises on population dynamics. Moreover, there are some challenges to analyze the stationary distribution of the numbers of prey and predator under species persistence. Based on the recent work (Zhou et al., 2020) where the density function of a three-dimensional avian influenza model with non-degenerate diffusion is concerned, we further generalize the relevant theories to a five-dimensional population system with degenerate diffusion. Via solving the corresponding Fokker–Planck equations, several approximate expressions of the density function of the stochastic predator–prey system with special distribution delay kernels are obtained. Finally, we provide some numerical simulations to supplement the analytical results and study the impact of stochastic noises. • A high-dimensional stochastic prey–predator system with general time delays is studied. • If ℛ 01 < 0 , both prey and predator will be eradicated exponentially in a long term. • If ℛ 01 > 0 and ℛ 02 < 0 , the prey will persist but the predator will be extinct in a long term. • If ℛ 02 > 0 , the prey and predator will be coexistent. • We obtain some approximate expressions of local density functions of the stochastic system. [ABSTRACT FROM AUTHOR]

In this paper, we present a model of infection in which a virus can simultaneously infect two types of target cells. The model has a general form of infection rates in deterministic and stochastic settings, incorporating bilinear infection rates, saturated incidences and half-saturated incidences. In the deterministic case, we investigate the existence and stability of infection-free steady state and infection steady state, respectively. Considering that the infection rate coefficient is affected by random noise, we built the corresponding stochastic model using the Ornstein–Uhlenbeck process. In the stochastic case, by constructing suitable Lyapunov functions, we established sufficient conditions for the stationary distribution and extinction of the model, respectively. In addition, the covariance matrix in the probability density function of the model near the infection steady state is determined. Finally, we consider the three most common forms of virus infection and perform comprehensive numerical simulations to support our theoretical results. • A viral co-infection model with general infection rate in deterministic and stochastic environments is developed. • We derive steady states and asymptotic stability for the deterministic model. • We obtain sufficient conditions for the stationary distribution and extinction of the stochastic model, respectively. • We conduct extensive numerical simulations to illustrate the effect of different infection functions. [ABSTRACT FROM AUTHOR]

• A chemostat model with single-species growth on two nutrients under telegraph noise is studied. • We obtain the threshold between extinction and persistence in the mean of the microorganism. • In the case of persistence, sufficient conditions for the existence of positive recurrence are investigated. • The results show the threshold is not only related to the value of dilution rate, but also depends on the probability distribution of the Markov chain. In this paper, we investigate the dynamics of a chemostat model with single-species growth on two nutrients, which is disturbed by the telegraph noise. Switching between different environmental states is achieved by Markov chain. Firstly, we prove the existence and uniqueness of the positive solution. Then the threshold between extinction and persistence in the mean of the microorganism is obtained. Moreover, in the case of persistence, we establish sufficient conditions for the existence of positive recurrence. Finally, some simulations are carried out to demonstrate our theoretical results. Our main effort is to build the suitable Lyapunov functions with regime switching. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic HIV infection model with the Ornstein–Uhlenbeck process is established and studied. It is assumed that the death rate of healthy CD4 + T cells satisfies the Ornstein–Uhlenbeck process. First, we verify that the stochastic model has an unique global solution for any initial value. Then, using Markov semigroup technique, we obtain the sufficient criteria for the existence of a stable stationary distribution to the stochastic system, which reflects the strong persistence of all CD4 + T cells and HIV virus particles. Furthermore, it is deduced that the above global solution around the endemic equilibrium follows an unique probability density function by solving the corresponding Fokker–Planck equation and having to apply our developed algebraic equation theory. The necessary conditions for the exponential extinction of HIV infection are established for completeness. Finally, several numerical simulations are provided to validate our theories. • A stochastic HIV infection model with the Ornstein–Uhlenbeck process is developed. • Sufficient conditions for the stationary distribution are obtained. • Exact expression of density function around a quasi-equilibrium is derived. • Sufficient criteria for extinction exponentially of the model are established. • Numerical results are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, two stochastic vegetation-water dynamic systems are proposed and studied. First, for the deterministic system, the possible equilibria and the related local stability are analyzed. Then for the stochastic system perturbed by Gaussian white noise, we establish sufficient conditions for the existence and uniqueness of ergodic stationary distribution, which reflects the long-term persistence of vegetation. By defining a stochastic quasi-positive equilibrium E ¯ ∗ and solving the Kolmogorov–Fokker–Planck equation, an approximate expression of probability density function of the stationary distribution around E ¯ ∗ is derived. Besides, we obtain some sufficient criteria for vegetation extinction. In terms of the stochastic system perturbed by both white and colored noises, the related extinction law and stationary distribution are investigated. Finally, several numerical examples are performed to substantiate our theoretical results and analyze the mean first passage time (MFPT). • We formulate and examine two stochastic vegetation-water systems for the ecological stability in semi-arid areas. • Two possible equilibria and the related local stability in the deterministic vegetation system are studied. • We establish sufficient conditions for the existence and uniqueness of ergodic stationary distribution of the stochastic systems. • We obtain the approximate expression of density function of the stochastic system around the quasi-positive equilibrium. • We give sufficient conditions for vegetation extinction in the long term. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic mutualism model with Lévy jumps. First of all, we show that the positive solution of the system is stochastically ultimate bounded. Then under a simple assumption, we establish sufficient and necessary conditions for the stochastic permanence and extinction of the system. The results show an important property of the Lévy jumps: they are unfavorable for the permanence of the species. Moreover, when there are no Lévy jumps, we show that there is a unique ergodic stationary distribution of the corresponding system under certain conditions. Some numerical simulations are introduced to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a stochastic delayed SIR epidemic model with nonlinear incidence. We firstly show that the system has a unique global positive solution with any positive initial value, then by constructing some suitable Lyapunov functionals, we investigate the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium, respectively. [ABSTRACT FROM AUTHOR]

Due to the continuous interference of environmental white noise, the dynamical behaviors of a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck process and standard incidence are considered. Several conclusions can be verified after dimensionality reduction. We study the existence and uniqueness of positive solution by construct a nonnegative Lyapunov function at first. Then, a sufficient condition for extinction of the diseases is derived by constructing a suitable Lyapunov function. In addition, we also obtain the stationary distribution of the model by constructing a complex Lyapunov function. Particularly, we propose the exact local expression of the density function of the random model near the unique endemic equilibrium. Finally, the numerical simulations illustrate our above theoretical results. • System of a SIQR model with Ornstein–Uhlenbeck process and standard incidence rate is studied. • In order to get more meaningful conclusions and reduce the dimension of system SIQR, it has also been shown. • Existence of stationary distribution is proved. • New method in constructing Lyapunov function on high-dimensional system is given. • Density function of the stochastic model around the unique endemic equilibrium is derived. [ABSTRACT FROM AUTHOR]

A stochastic chemostat model with periodic washout rate is proposed in this paper. First, sufficient conditions are established for the existence of stochastic nontrivial positive periodic solution for the system, on the basis of Khasminskii’s theory for periodic Markov processes. Furthermore, we observe that there exists a unique boundary periodic solution of the stochastic model which is globally attractive. Numerical simulations are carried out to illustrate our main conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of the stochastic autonomous and non-autonomous predator-prey systems with nonlinear predator harvesting respectively. For the autonomous system, we first give the existence of the global positive solution. Then, in the case of persistence, we prove that there exists a unique stationary distribution and it has ergodicity by constructing a suitable Lyapunov function. The result shows that, the relatively weaker white noise will strengthen the stability of the system, but the stronger white noise will result in the extinction of one or two species. Particularly, for the non-autonomous periodic system, we show that there exists at least one nontrivial positive periodic solution according to the theory of Khasminskii. Finally, numerical simulations illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

A stochastic Lotka–Volterra predator–prey model with regime switching is investigated. We examine when the system is persistence in mean and the extinction under some appropriate conditions, and we discuss the threshold between them. Sufficient conditions for the stationary distribution which is ergodic and positive recurrent of the solution is established using stochastic Lyapunov functions. Simulations are also carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

Considering the survival regulation mechanisms of many groups of animals and the complexity of random variations in ecosystem, in this paper, we mainly formulate and study a stochastic non-autonomous population model with Allee effects and two mean-reverting Ornstein–Uhlenbeck processes. First, the biological implication of introducing the Ornstein–Uhlenbeck process is illustrated. After that, we give the existence and moment estimate of a global solution of the stochastic model. Then the sufficient criteria for exponential extinction and the existence of a stationary distribution of the stochastic model are established. Moreover, there are some challenges to give the explicit expression of probability density function of the stationary distribution. By solving the relevant Fokker–Planck equation, we derive the approximate expression of the density function of the stochastic model. Finally, some numerical simulations are provided to verify our analytical results and study the impact of stochastic noises on population dynamics. • A stochastic population model with Ornstein-Uhlenbeck processes is established. • We give the existence and moment estimate of a global solution. • The exponential extinction and the existence of stationary distribution are studied. • We obtain the approximate expression of density function of the stochastic model. • Some actual computer simulations are provided for better population stability. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the multi-group SEIR and SIR model with random perturbation, allowing random fluctuation around the endemic equilibrium of deterministic SEIR and SIR models. We prove that the endemic equilibriums of the two models are both stochastic asymptotically stable. In addition, the stability condition is obtained by the construction of the Lyapunov function and graph theory. Finally, numerical simulations are presented to illustrate our mathematical findings. [Copyright &y& Elsevier]

This paper concerns with a stochastic system modeling the population dynamical behavior of one prey and two predators. In this paper, we adopt a special method to simulate the effect of the environmental interference to the system instead of using the linear functions of white noise, i.e., the growth rate of the prey and the death rates of the two predators are all governed by Ornstein–Uhlenbeck processes, which is more practical and interesting. The dynamical behavior among the three species in the model is discussed in detail. The existence and uniqueness, moment boundedness and long time asymptotic behavior of the unique global solution are investigated. Through rigorous proof and theoretical derivation, some sharp sufficient conditions for the persistence and extinction of the three species have been established. Furthermore, we obtain that when one or two predators die out, the remaining species can be stable in time average under same conditions. Moreover, by constructing some suitable Lyapunov functions, the sufficient conditions for the existence of the stationary distribution are explicitly determined. Under the same parametric restrictions, it is worth noting that we obtain the six-dimensional probability density function of the stochastic one-prey two-predator model around the quasi-equilibrium E ∗ , which can reflect most statistical properties. Finally, several numerical simulations are carried out to illustrate the theoretical results. • A stochastic system modeling the dynamical behavior of the species is considered. • The growth rate of the prey and the death rates of the two predators are all governed by OU processes. • The stability of the equilibria of deterministic model is discussed. • The persistence and extinction properties of each species of stochastic system are analyzed. • The expression of the density function for the stochastic system around the quasi-equilibrium point is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamics of stochastic predator–prey models with Holling II functional response. For the stochastic systems, we firstly establish sufficient conditions for the existence of the globally positive solutions. Then we investigate the asymptotic moment estimations of the positive solutions and study the ultimately bounded in the mean of them. Thirdly, by constructing some suitable Lyapunov functions, we verify that there are unique stationary distributions and they are ergodic. The obtained results show that the systems still retain some stability in the sense of weak stability provided that the intensity of the white noise is relatively small. Finally, some numerical simulations are introduced to illustrate our main results. [ABSTRACT FROM AUTHOR]

Considering the profound ecological implication of Allee effect, and the effects after incorporating it into models of population dynamics, a stochastic predator–prey model with Holling-(n+1) functional response and weak Allee effect is mainly investigated in this paper. Firstly, we verify the existence of unique global positive solution to the model, and then the boundedness of θ t h moment of the solution is studied. Secondly, we investigate the corresponding one-dimensional model, and obtain the explicit density function of the solution to the model. Then, based on the results of the one-dimensional system and a series of appropriate Lyapunov functions, we adopt a new technique to obtain a sufficient and almost necessary condition for the existence of the unique ergodic stationary distribution and extinction. Next, we further study the dynamical behavior of the model with Markovian switching and give some main conclusions. Finally, several numerical simulations are carried out to illustrate the theoretical results. • A stochastic predator–prey model with weak Allee effect is investigated. • A novel method is adopted to study the extinction and persistence of the model. • The dynamical behavior of the model with Markovian switching is studied. • Numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]