Persistence and extinction of a single-species population system in a polluted environment with random perturbations and impulsive toxicant input

Taking both white noises and colored noises into account, a stochastic single-species model with Markov switching and impulsive toxicant input in a polluted environment is proposed and investigated. Sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The threshold between weak persistence and extinction is obtained. Some simulation figures are introduced to illustrate the main results.     

Long term behaviors of stochastic single-species growth models in a polluted environment II

This is a continuation of our paper [M. Liu, K. Wang, X. Liu. Long term behaviors of stochastic single-species growth models in a polluted environment. Appl Math Model 2011;35:752–62]. This work still devotes to studying three stochastic single-species models in a polluted environment. For the first system, sufficient criteria for extinction, stochastic non-persistence in the mean, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence of the population are established. The threshold between stochastic weak persistence in the mean and extinction is obtained. For the second model, sufficient conditions for extinction, stochastic non-persistence in the mean, stochastic weak persistence, stochastic weak persistence in the mean, stochastic strong persistence in the mean and stochastic permanence are established. The threshold between stochastic weak persistence and extinction is derived. For the third system, the threshold between stochastic weak persistence and extinction is obtained.     

Persistence and extinction of a stochastic Gilpin–Ayala model with jumps

This paper considers a stochastic Gilpin–Ayala model with jumps. First, we show the model that has a unique global positive solution. Then we establish the sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of the solution. The threshold between weak persistence and extinction is obtained. Finally, we make simulations to conform our analytical results. The results show that the jump process can change the properties of the population model significantly. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics of a general non-autonomous stochastic Lotka-Volterra model with delays

In this paper, a general non-autonomous n-species Lotka-Volterra model with delays and stochastic perturbation is investigated. For this model, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The influences of the stochastic noises to the properties of the stochastic model are discussed. The property permanence for the model is preserved with the sufficiently small noise and sufficiently large noise may cause extinction of the model. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results.     

Persistence and extinction of a stochastic logistic model with delays and impulsive perturbation

A stochastic logistic model with delays and impulsive perturbation is proposed and investigated. Sufficient conditions for extinction are established as well as nonpersistence in the mean, weak persistence and stochastic permanence. The threshold between weak persistence and extinction is obtained. Furthermore, the theoretical analysis results are also derivated with the help of numerical simulations.     

PERSISTENCE AND EXTINCTION OF A STOCHASTIC LOGISTIC MODEL WITH DELAYS AND IMPULSIVE PERTURBATION

A stochastic logistic model with delays and impulsive perturbation is proposed and investigated. Sufficient conditions for extinction are established as well as nonpersistence in the mean, weak persistence and stochastic permanence. The threshold between weak persistence and extinction is obtained. Furthermore, the theoretical analysis results are also derivated with the help of numerical simulations.     

Persistence and extinction in stochastic non-autonomous logistic systems

This paper studies two widely used stochastic non-autonomous logistic models. For the first system, sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The critical number between weak persistence and extinction is obtained. For the second system, sufficient criteria for extinction, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and stochastic permanence are established. The critical number between weak persistence in the mean and extinction is obtained. It should be pointed out that this research is systematical and complete. In fact, the behaviors of the two models in every coefficient cases are cleared up by the results obtained in this paper.     

Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations

In this article, an impulsive stochastic tumor-immune model with regime switching is formulated and explored. Firstly, it is proven that the model has a unique global positive solution. Then sufficient criteria for extinction, non-persistence in the mean, weak persistence and stochastic permanence are provided. The threshold value between extinction and weak persistence is gained. In addition, the lower- and the upper-growth rates of tumor cells are estimated. The results demonstrate that the dynamics of the model are intimately associated with the random perturbations and impulsive perturbations. Finally, biological implications of the results are addressed with the help of real data and numerical simulations.     

Evolutionary analysis of adaptive dynamics model under variation of noise environment

This paper proposes a stochastic model for the evolutionary adaptive dynamics of species subject to trait-dependent intrinsic growth rates and the influence of environmental noise. The aim of this paper is twofold: (a) mathematically we make an attempt to investigate the evolutionary adaptive dynamics for models with noises; (b) biologically we investigate how the noises in environment affect the evolutionary stability. We first investigate the extinction and permanence of the population in the presence of environmental noises. Combining evolutionary adaptive dynamics with stochastic dynamics, we then establish a fitness function with stochastic disturbance and obtain the evolutionary conditions for continuously stable strategy and evolutionary branching. Our study finds that under intense competition among species, increasing stochastic disturbance can lead to rapidly stable evolution towards smaller trait values, but there is an opposite effect under weak competition among species. This yields an interesting evolutionary threshold, beyond which any increasing stochastic disturbance can go against evolutionary branching and promote evolutionary stability. We then carry out the evolutionary analysis and numerical simulations to illustrate our theoretical results. Finally, for demonstrating the emergence of high-level polymorphism we perform long-term simulation of evolutionary dynamics.     

Dynamics of a non-autonomous stochastic Gilpin-Ayala model

This paper is concerned with a stochastic non-autonomous Gilpin-Ayala model. First, it is shown that this model has a global positive solution. Then sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence of the solution are established. The critical number between weak persistence and extinction is obtained. Finally, the lower- and upper-growth rate of the solution are investigated. Several numerical figures are introduced to illustrate the results. Some recent results are improved and generalized.     

Dynamics analysis and numerical simulations of a stochastic non-autonomous predator–prey system with impulsive effects

In this paper, we propose a stochastic non-autonomous Lotka–Volterra predator–prey model with impulsive effects and investigate its stochastic dynamics. We first prove that the subsystem of the system has a unique periodic solution which is globally attractive. Furthermore, we obtain the threshold value in the mean which governs the stochastic persistence and the extinction of the prey–predator system. Our results show that the stochastic noises and impulsive perturbations have crucial effects on the persistence and extinction of each species. Finally, we use the different stochastic noises and impulsive effects parameters to provide a series of numerical simulations to illustrate the analytical results.     

Persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise

In this paper, we consider the persistence and extinction of a stochastic non-autonomous Gilpin–Ayala system driven by Lévy noise. Sufficient criteria for extinction, non-persistence in the mean and weak persistence of the system are established. The threshold between weak persistence and extinction is obtained. From the results we can see that both persistence and extinction have close relationships with Lévy noise. Some simulation figures are introduced to demonstrate the analytical findings.     

随机的改进Lotka-Volterra竞争模型

本文提出且讨论了一类两种群随机的改进Lotka Volterra竞争模型.白噪声及有色噪声都在本文中被考虑.我们得到了全局唯一正解、随机有界性、随机持久和随机灭绝等种群动力性质的充分条件.     

Dynamics of stochastic single-species models

A type of stochastic single-species model is proposed and studied. The sufficient conditions of the existence of a unique solution, the existence of its stationary distribution, and stochastic permanence are obtained. Besides, the threshold conditions for its strong stochastic persistence and extinction are found. Finally, some examples and numerical simulations are introduced to support our main results.     

Survival analysis of a cooperation system with random perturbations in a polluted environment

Taking white noises, Markovian switchings and Lévy jump noises into account, a stochastic cooperation system of two species in a polluted environment is developed and analyzed. Persistence–extinction thresholds are obtained for each population. The results reveal that white noises, Markovian switchings and Lévy jumps have sufficient effect to the persistence and extinction of the species.     

Persistence and extinction of a stochastic delay predator-prey model under regime switching

The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.     

Stochastic Lotka–Volterra systems with Lévy noise

This paper is concerned with stochastic Lotka–Volterra models perturbed by Lévy noise. Firstly, stochastic logistic models with Lévy noise are investigated. Sufficient and necessary conditions for stochastic permanence and extinction are obtained. Then three stochastic Lotka–Volterra models of two interacting species perturbed by Lévy noise (i.e., predator–prey system, competition system and cooperation system) are studied. For each system, sufficient and necessary conditions for persistence in the mean and extinction of each population are established. The results reveal that firstly, both persistence and extinction have close relationships with Lévy noise; Secondly, the interaction rates play very important roles in determining the persistence and extinction of the species.     

Stochastic logistic equation with infinite delay

This paper perturbs the famous logistic equation with infinite delay into the corresponding stochastic system This study shows that the above stochastic system has a global positive solution with probability 1 and gives the asymptotic pathwise estimation of this solution. In addition, the superior limit of the average in time of the sample path of the solution is estimated. This work also establishes the sufficient conditions for extinction, nonpersistence in the mean, and weak persistence of the solution. The critical value between weak persistence and extinction is obtained. Then these results are extended to n‐dimensional stochastic Lotka–Volterra competitive system with infinite delay. Finally, this paper provides some numerical figures to illustrate the results. The results reveal that, firstly, different types of environmental noises have different effects on the persistence and extinction of the population system; secondly, the delay has no effect on the persistence and extinction of the stochastic system.Copyright © 2012 John Wiley & Sons, Ltd.     

Global dynamics analysis of a nonlinear impulsive stochastic chemostat system in a polluted environment

This paper intends to develop a new method to obtain the threshold of an impulsive stochastic chemostat model with saturated growth rate in a polluted environment. By using the theory of impulsive differential equations and stochastic differential equations, we obtain conditions for the extinction and the permanence of the microorganisms of the deterministic chemostat model and the stochastic chemostat model. We develop a new numerical computation method for impulsive stochastic differential system to simulate and illustrate our theoretical conclusions. The biological results show that a small stochastic disturbance can cause the microorganism to die out, that is, a permanent deterministic system can go to extinction under the white noise stochastic disturbance. The theoretical method can also be used to explore the threshold of some impulsive stochastic differential equations.     

Persistence and extinction of a stochastic delay Logistic equation under regime switching

A stochastic delay Logistic equation under regime switching is proposed and studied. Sufficient conditions for extinction, non-persistence in the mean and weak persistence of the solutions are established. The critical value between weak persistence and extinction is obtained.