scholarly journals Stochastic Permanence, Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System with Random Perturbation

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Li Zu ◽  
Daqing Jiang ◽  
Donal O’Regan
Keyword(s):  
Numerical Simulation ◽  
Stationary Distribution ◽  
Nonlinear Diffusion ◽  
Logistic Model ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Single Species ◽  
Unique Positive Solution ◽  
Stochastic Perturbations ◽  
Stochastic Permanence

We analyze the influence of stochastic perturbations on a single-species logistic model with the population’s nonlinear diffusion amongnpatches. First, we show that this system has a unique positive solution. Then we obtain sufficient conditions for stochastic permanence and persistence in mean, stationary distribution, and extinction. Finally, we illustrate our conclusions through numerical simulation.

scholarly journals Existence, Stationary Distribution, and Extinction of Predator-Prey System of Prey Dispersal with Stochastic Perturbation

2012 ◽  
Vol 2012 ◽  
pp. 1-24 ◽  
Author(s):  
Li Zu ◽  
Daqing Jiang ◽  
Fuquan Jiang
Keyword(s):  
Numerical Simulation ◽  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Ergodic Property ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

We consider a predator-prey model in which the preys disperse amongnpatches (n≥2) with stochastic perturbation. We show that there is a unique positive solution and find out the sufficient conditions for the extinction to the system with any given positive initial value. In addition, we investigate that there exists a stationary distribution for the system and it has ergodic property. Finally, we illustrate the dynamic behavior of the system withn=2via numerical simulation.

scholarly journals Stationary Distribution and Extinction in a Stochastic SIQR Epidemic Model Incorporating Media Coverage and Markovian Switching

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1122
Author(s):  
Yanlin Ding ◽  
Jianjun Jiao ◽  
Qianhong Zhang ◽  
Yongxin Zhang ◽  
Xinzhi Ren
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Stationary Distribution ◽  
Dynamic Characteristics ◽  
Epidemic Model ◽  
Regime Switching ◽  
Media Coverage ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Theoretical Results

This paper is concerned with the dynamic characteristics of the SIQR model with media coverage and regime switching. Firstly, the existence of the unique positive solution of the proposed system is investigated. Secondly, by constructing a suitable random Lyapunov function, some sufficient conditions for the existence of a stationary distribution is obtained. Meanwhile, the conditions for extinction is also given. Finally, some numerical simulation examples are carried out to demonstrate the effectiveness of theoretical results.

Stationary distribution and periodic solution of stochastic chemostat models with single-species growth on two nutrients

2019 ◽  
Vol 12 (06) ◽  
pp. 1950063 ◽  
Author(s):  
Miaomiao Gao ◽  
Daqing Jiang ◽  
Tasawar Hayat
Keyword(s):  
Periodic Solution ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Markov Processes ◽  
Lyapunov Functions ◽  
Autonomous System ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Single Species

In this paper, we consider two chemostat models with random perturbation, in which single species depends on two perfectly substitutable resources for growth. For the autonomous system, we first prove that the solution of the system is positive and global. Then we establish sufficient conditions for the existence of an ergodic stationary distribution by constructing appropriate Lyapunov functions. For the non-autonomous system, by using Has’minskii theory on periodic Markov processes, we derive it admits a nontrivial positive periodic solution. Finally, numerical simulations are carried out to illustrate our main results.

Stochastic dynamics for the solutions of a modified Holling–Tanner model with random perturbation

2014 ◽  
Vol 25 (11) ◽  
pp. 1450105 ◽  
Author(s):  
Zhenjie Liu
Keyword(s):  
Stochastic System ◽  
Stochastic Dynamics ◽  
Deterministic Model ◽  
Random Perturbation ◽  
Sufficient Conditions ◽  
Unique Positive Solution ◽  
Initial Value ◽  
Environmental Forcing ◽  
Predator Prey ◽  
Predator Prey Model

In this paper, we consider a stochastic nonautonomous predator–prey model with modified Leslie–Gower and Holling II schemes in the presence of environmental forcing. The deterministic model is the modified Holling–Tanner model which is an extension of the classical Leslie–Gower model. We show that there is a unique positive solution to the stochastic system for any positive initial value. Sufficient conditions for strong persistence in mean and extinction to the stochastic system are established.

Dynamics behaviors of a delayed competitive system in a random environment

2015 ◽  
Vol 08 (05) ◽  
pp. 1550062 ◽  
Author(s):  
Ronghua Tan ◽  
Huili Xiang ◽  
Yiping Chen ◽  
Zhijun Liu
Keyword(s):  
Global Existence ◽  
Random Environment ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Unique Positive Solution ◽  
Competitive System ◽  
Stochastic Perturbations ◽  
Stochastically Ultimate Boundedness ◽  
White Noises

In the real world, the population systems are often subject to white noises and a system with such stochastic perturbations tends to be suitably modeled by stochastic differential equations. This paper is concerned with the dynamic behaviors of a delay stochastic competitive system. We first obtain the global existence of a unique positive solution of system. Later, we show that the solution of system will be stochastically ultimate boundedness. However, large noises may make the system extinct exponentially with probability one. Also, sufficient conditions for the global attractivity of system are established. Finally, illustrated examples are given to show the effectiveness of the proposed criteria.

scholarly journals Analysis of a mutualism model with stochastic perturbations

2015 ◽  
Vol 08 (06) ◽  
pp. 1550072 ◽  
Author(s):  
Mei Li ◽  
Hongjun Gao ◽  
Chenfeng Sun ◽  
Yuezheng Gong
Keyword(s):  
Ecological Model ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Stochastic Perturbations ◽  
Stochastic Permanence ◽  
Local Existence And Uniqueness ◽  
Mutualism Model ◽  
Stochastically Bounded ◽  
Uniformly Continuous

This paper is concerned with a mutualism ecological model with stochastic perturbations. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded, uniformly continuous and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistent are also established. At last, some figures are presented to illustrate our main results.

scholarly journals Stochastic Delay Logistic Model under Regime Switching

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Zheng Wu ◽  
Hao Huang ◽  
Lianglong Wang
Keyword(s):  
Regime Switching ◽  
Logistic Model ◽  
Sufficient Conditions ◽  
Function Technique ◽  
Stochastic Permanence ◽  
Stochastic Delay ◽  
Gronwall’S Inequality ◽  
Diffusion In Random Environment ◽  
Stochastically Ultimate Boundedness ◽  
Generalized Itô Formula

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue ofV-function technique,M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

scholarly journals The Bifurcation and Control of a Single-Species Fish Population Logistic Model with the Invasion of Alien Species

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yi Zhang ◽  
Qiaoling Zhang ◽  
Jinghao Li ◽  
Qingling Zhang
Keyword(s):  
Alien Species ◽  
Logistic Model ◽  
Singular System ◽  
Sufficient Conditions ◽  
Single Species ◽  
Fish Population ◽  
Point Of View ◽  
Practical Significance ◽  
Singularity Induced Bifurcation ◽  
And Control

The objective of this paper is to study systematically the bifurcation and control of a single-species fish population logistic model with the invasion of alien species based on the theory of singular system and bifurcation. It regardsSpartina anglicaas an invasive species, which invades the fisheries and aquaculture. Firstly, the stabilities of equilibria in this model are discussed. Moreover, the sufficient conditions for existence of the trans-critical bifurcation and the singularity induced bifurcation are obtained. Secondly, the state feedback controller is designed to eliminate the unexpected singularity induced bifurcation by combining harvested effort with the purification capacity. It obviously inhibits the switch of population and makes the system stable. Finally, the numerical simulation is proposed to show the practical significance of the bifurcation and control from the biological point of view.

scholarly journals Survival Analysis of a Nonautonomous Logistic Model with Stochastic Perturbation

2012 ◽  
Vol 2012 ◽  
pp. 1-21
Author(s):  
Chun Lu ◽  
Xiaohua Ding
Keyword(s):  
Survival Analysis ◽  
Asymptotic Stability ◽  
White Noise ◽  
Global Asymptotic Stability ◽  
Logistic Model ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Stochastic Permanence ◽  
The Mean

Taking white noise into account, a stochastic nonautonomous logistic model is proposed and investigated. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, stochastic permanence, and global asymptotic stability are established. Moreover, the threshold between weak persistence and extinction is obtained. Finally, we introduce some numerical simulink graphics to illustrate our main results.

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