Stochastic Mutualism Research Articles

Stationary distribution and ergodicity of a stochastic multi-species model with Holling type II response function

In this paper, we consider a stochastic multi-species model with the Holling type II functional response, which includes the stochastic competition model, the stochastic mutualism model, and the stochastic hybrid model of competition and mutualism. Sufficient conditions for the stationary distribution and ergodicity of the model are established. We support our analytical findings through extensive simulation studies. We show that under relatively small noise, the model has a stationary distribution and ergodicity, while the relatively large noise leads to the non-persistent and non-stationary distribution of the model through the numerical simulation studies of three species.

Modeling and dynamic analysis of a stochastic mutualism model with distributed delays

A stochastic two-species mutualism model that incorporates nonlinear perturbations and distributed delays is proposed and analyzed in this paper, in where saturation effects and distributed delays with weak kernel are contained in the interspecies mutualism term. We first investigate that a nonlinear stochastic mutualism model possesses a unique global positive solution regarding arbitrarily given initial value. Therewith the sufficient conditions for two-species extinction and the existence of a unique stationary distribution (USD) are obtained. Whereafter, it is derived that the stationary solution near the quasi-positive equilibrium adheres a unique probability density function with the aid of tackling Fokker–Planck (FP) equation corresponding to the linearized system and applying the theory of algebraic equations we have developed. At last, several numerical examples are presented to validate the validity of our theoretical findings.

Survival analysis of an impulsive stochastic facultative mutualism system with saturation effect

A system of impulsive stochastic differential equations is proposed as a two-species facultative mutualism model subject to impulsive and two coupling noise source perturbations, in which the saturation effect is taken into account. A set of sufficient criteria for extinction (exponential extinction and extinction) and permanence (permanence in time average and stochastic permanence) of the system are established. Extensive simulation figures are demonstrated to support the theoretical findings. Meanwhile, we look at the effects of coupling white noises, impulses, intrinsic growth rates, intra-specific competition rates and inter-specific mutualism rates on the survival of populations.

A stochastic mutualism model with saturation effect and impulsive toxicant input in a polluted environment

In this paper, we use a mean-reverting Ornstein–Uhlenbeck process to model the stochastic perturbations in the environment, and then a stochastic mutualism model with saturation effect and pulse toxicant input in a polluted environment is proposed. A set of sufficient conditions including exponential extinction, persistence in the mean, permanence in time average and stochastic permanence are derived. Numerical simulations are worked out to support the analysis results. Also, we look at the effects of the volatile intensity, reversion rate, impulsive period and impulsive toxicant input amount on the survival of two species.

Permanence and extinction of stochastic regime-switching mutualism model

This paper is concerned with the permanence and extinction of a stochastic regime-switching mutualism model. We aim to find the difference between the stochastic mutualism model with regime-switching and without regime-switching. By studying ergodicity of regime-switching diffusion processes, we establish the sufficient conditions to estimate the permanence and extinction of a species in a random switching environment. Moreover, compared with the system without switching, the advantages of the stochastic regime-switching mutualism model are given.

Optimal harvesting of a stochastic mutualism model with regime-switching

This article researches optimal harvesting strategy (OHS) of a two-species stochastic Lotka–Volterra mutualism model with regime-switching. Traditional approach needs an additional hypothesis. In this article we drop this hypothesis. We obtain thresholds between stochastic persistence in probability and collapse of the model, and establish sharp sufficient conditions for the existence of an OHS of the model. We also present several critical impacts of environmental perturbations on stochastic persistence in probability, collapse and the existence of OHS of the model with the help of some numerical simulations.

一类带Lévy跳的随机混杂互惠系统的渐近性态

一类带Lévy跳的随机混杂互惠系统的渐近性态

Optimal harvesting strategy for a stochastic mutualism system in a polluted environment with regime switching

In this paper, we stochastically perturb the harvesting mutualism system into stochastic system in a polluted environment by white noise, telegraph noise and Lévy jump. We first prove the existence and uniqueness of the global positive solution of the stochastic system, and establish the sufficient condition for persistence in mean of the two cooperative species. Then using aggregation and ergodic methods, the explicit forms of optimal harvesting strategy and the maximum of sustainable yield are obtained. We also give some examples and numerical simulations to illustrate the theoretical results. The impacts of toxicant in organism, telegraph noise and Lévy jump on the persistence in mean and optimal harvesting strategy are revealed.

Stability of a stochastic discrete mutualism system

Under the assumption that stochastic white noise perturbations are directly proportional to the deviation of the state from the equilibrium states of a continuous mutualistic model, we use the Euler–Maruyama discretization method to obtain a two-species stochastic discrete mutualism model. For this stochastic model, we establish conditions on the asymptotic mean square stability of the positive equilibrium state and the almost sure asymptotic stability of the three boundary equilibrium states. The theoretical results are supported with numerical simulations.

一类非自治随机互惠系统的渐近性态

一类非自治随机互惠系统的渐近性态

Globally asymptotic stability of a stochastic mutualism model with saturated response

A two-species stochastic mutualism model with saturated response is proposed and investigated in this paper. We demonstrate that there exists a unique positive solution to the model for any positive initial value. Under some conditions, we show that the stochastic model is globally asymptotically stable. Finally, we work out some figures to illustrate our results.

Stochastic mutualism model under regime switching with Lévy jumps

This paper is concerned with a stochastic mutualism model under regime switching with Lévy jumps. To begin with, the existence and uniqueness of the global positive solution is proved with any given positive initial value. Then, the sufficient conditions for stochastic permanence are established. The critical value between extinction and persistence in mean is also obtained. In addition, under some suitable conditions, we proved that there is a unique stationary distribution for the system without Lévy jumps. Our method relies on the Lyapunov function analysis and the Fredholm alternative. The results demonstrate that regime switching may contribute to the permanence but jump noise may suppress the permanence.

The stationary distribution and ergodicity of a stochastic mutualism model

Abstract This paper is concerned with a stochastic mutualism system with toxicant substances and saturation terms. We obtain the sufficient conditions for the existence of a unique stationary distribution to the equation and it has an ergodic property. It is interesting and surprising that toxicant substances have no effect on the stationary distribution of the stochastic model. Simulations are also carried out to confirm our analytical results.

Asymptotic behavior and numerical simulations of a Lotka-Volterra mutualism system with white noises

In the present paper, a stochastic mutualism model subject to white noises is established. We first investigate the existence and uniqueness of globally positive solution of the stochastic model. Then we study its asymptotic behavior, such as stochastic permanence and extinction, and estimate the limit of the average in time of the sample paths of every component. We also show that the stochastic system is globally attractive under some appropriate conditions. Finally, numerical simulations are presented to justify the analytical results.

Stochastic mutualism model with Lévy jumps

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.

Optimal harvesting of a stochastic mutualism model with Lévy jumps

In this letter, a stochastic mutualism model with Lévy jumps and harvesting is considered. Under some simple assumptions, sufficient and necessary criteria for the existence of optimal harvesting policy are established. The optimal harvesting effort and the maximum of sustainable yield are also obtained. The effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results.

Analysis of a stochastic mutualism model

In this paper, we study a stochastic autonomous mutualism model. The local and global existence and uniqueness of the positive solution are derived with any positive initial value. Then we obtain that the positive solution of the system is stochastically ultimate bounded. Furthermore, we establish sufficient conditions for stochastic permanence and extinction of the system. The sufficient criterion for the system to be persistent in the mean are established as well.

Dynamical Behavior of the Stochastic Delay Mutualism System

We discuss the dynamical behavior of the stochastic delay three-specie mutualism system. We develop the technique for stochastic differential equations to deal with the asymptotic property. Using it we obtain the existence of the unique positive solution, the asymptotic properties, and the nonpersistence. Finally, we give the numerical examinations to illustrate our results.

Two types of permanence of a stochastic mutualism model

Abstract A stochastic mutualism model is proposed and investigated in this paper. We show that there is a unique solution to the model for any positive initial value. Moreover, we show that the solution is stochastically bounded, uniformly continuous and globally attractive. Under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean. Finally, we introduce some figures to illustrate our main results.

Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at timetare estimated byrit+σitB˙i(t), i=1,2,respectively. Viewing the different intensities of the noisesσi(t),i=1,2as two parameters at timet, we conclude that there exists a global positive solution and thepth moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.