Permanence and Extinction of Stochastic Logistic System with Feedback Control under Regime Switching

We study a stochastic logistic system with feedback control under regime switching. Sufficient conditions for extinction, non-persistence in the mean, weak persistence, and persistence in the mean are established. A very important fact is found in our results; that is, the feedback control is harmless to the permanence of species even under the regime switching and stochastic perturbation environments. Finally, some examples are introduced to illustrate the main results.

Dynamics of a stochastic predator–prey model with mutual interference

10.1142/s1793524514500260 ◽

2014 ◽

Vol 07 (03) ◽

pp. 1450026 ◽

Cited By ~ 2

Author(s):

Kai Wang ◽

Yanling Zhu

Keyword(s):

Numerical Simulation ◽

Positive Solution ◽

Sufficient Conditions ◽

Stochastic Perturbation ◽

Mutual Interference ◽

Predator Prey ◽

Predator Prey Model ◽

Persistence In The Mean ◽

The Mean ◽

Globally Positive Solution

In this paper, a stochastic predator–prey (PP) model with mutual interference is considered. Some sufficient conditions for the existence of globally positive solution, non-persistence in the mean, weak persistence in the mean, strong persistence in the mean and almost surely extinction of the the model are established. Moreover, the threshold between weak persistence in the mean and almost surely extinction of the prey is obtained. Some examples are given to show the feasibility of the results by numerical simulation. It is significant that such a model is firstly proposed with stochastic perturbation.

Dynamics of a stochastic SIRS epidemic model with standard incidence under regime switching

10.1142/s1793524521500741 ◽

2021 ◽

pp. 2150074

Author(s):

Jiang Xu ◽

Yinong Wang ◽

Zhongwei Cao

Keyword(s):

Lyapunov Function ◽

Stochastic System ◽

Epidemic Model ◽

Regime Switching ◽

Sufficient Conditions ◽

Sirs Epidemic Model ◽

Persistence In The Mean ◽

The Mean ◽

Stochastic Sirs Epidemic Model ◽

Stochastic Lyapunov Function

The goal of this paper is to introduce and initiate a study of a stochastic SIRS epidemic model with standard incidence which is perturbed by both white and telegraph noises. We first show persistence in the mean and then establish the sufficient conditions for extinction of the disease. Moreover, in the case of persistence, we obtain sufficient conditions for the existence of positive recurrence of the solutions by means of structuring suitable stochastic Lyapunov function with regime switching. Meanwhile, the threshold between persistence in the mean and extinction of the stochastic system is also obtained. Finally, we test our theory conclusion by simulations.

Asymptotic Behavior of a Chemostat Model with Stochastic Perturbation on the Dilution Rate

Abstract and Applied Analysis ◽

10.1155/2013/423154 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 10

Author(s):

Chaoqun Xu ◽

Sanling Yuan ◽

Tonghua Zhang

Keyword(s):

Dilution Rate ◽

Sufficient Conditions ◽

Stochastic Perturbation ◽

Positive Equilibrium ◽

Deterministic System ◽

Time Behavior ◽

Chemostat Model ◽

Persistence In The Mean ◽

Long Time ◽

We present a stochastic simple chemostat model in which the dilution rate was influenced by white noise. The long time behavior of the system is studied. Mainly, we show how the solution spirals around the washout equilibrium and the positive equilibrium of deterministic system under different conditions. Furthermore, the sufficient conditions for persistence in the mean of the stochastic system and washout of the microorganism are obtained. Numerical simulations are carried out to support our results.

Dynamics of a single predator multiple prey model with stochastic perturbation and seasonal variation

Filomat ◽

10.2298/fil2102535h ◽

2021 ◽

Vol 35 (2) ◽

pp. 535-549

Author(s):

Hong-Wen Hui ◽

Lin-Fei Nie

Keyword(s):

Seasonal Variation ◽

Sufficient Conditions ◽

Stochastic Perturbation ◽

Analysis Method ◽

Ultimate Boundedness ◽

Persistence In The Mean ◽

The Mean ◽

Stochastically Ultimate Boundedness ◽

Theoretical Results

Considering various factors are stochastic rather than deterministic in the evolution of populations growth, in this paper, we propose a single predator multiple prey stochastic model with seasonal variation. By using the method of solving an explicit solution, the existence of global positive solution of this model are obtained. The method is more convenient than Lyapunov analysis method for some population models. Moreover, the stochastically ultimate boundedness are considered by using the comparison theorem of stochastic differential equation. Further, some sufficient conditions for the extinction and strong persistence in the mean of populations are discussed, respectively. In addition, by constructing some suitable Lyapunov functions, we show that this model admits at least one periodic solution. Finally, numerical simulations clearly illustrate the main theoretical results and the effects of white noise and seasonal variation for the persistence and extinction of populations.

Dynamics of a stochastic phytoplankton-toxic phytoplankton-zooplankton system under regime switching

10.22541/au.161315119.96131469/v1 ◽

2021 ◽

Author(s):

He Liu ◽

Chuanjun Dai ◽

Hengguo Yu ◽

Qing Guo ◽

Jianbing Li ◽

...

Keyword(s):

White Noise ◽

Regime Switching ◽

Existence And Uniqueness ◽

Sufficient Conditions ◽

Aquatic Environments ◽

Combined Effects ◽

Toxic Phytoplankton ◽

Persistence In The Mean ◽

In this paper, a stochastic phytoplankton-toxic phytoplankton-zooplankton system with Beddington-DeAngelis functional response, where both the white noise and regime switching are taken into account, is studied analytically and numerically. The aim of this research is to study the combined effects of the white noise, regime switching and toxin-producing phytoplankton (TPP) on the dynamics of the system. Firstly, the existence and uniqueness of global positive solution of the system is investigated. Then some sufficient conditions for the extinction, persistence in the mean and the existence of a unique ergidoc stationary distribution of the system are derived. Significantly, some numerical simulations are carried to verify our analytical results, and show that high intensity of white noise is harmful to the survival of plankton populations, but regime switching can balance the different survival states of plankton populations and decrease the risk of extinction. Additionally, it is found that an increase in the toxin liberation rate produced by TPP will increase the survival change of phytoplankton, while it will reduce the biomass of zooplankton. All these results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton system in randomly disturbed aquatic environments.

The global dynamics of stochastic Holling type II predator-prey models with non constant mortality rate

Filomat ◽

10.2298/fil1718811z ◽

2017 ◽

Vol 31 (18) ◽

pp. 5811-5825

Author(s):

Xinhong Zhang

Keyword(s):

Mortality Rate ◽

Stochastic Model ◽

Sufficient Conditions ◽

Positive Periodic Solution ◽

Global Dynamics ◽

Type Ii ◽

Predator Prey ◽

Persistence In The Mean ◽

The Mean ◽

Predator Prey Models

In this paper we study the global dynamics of stochastic predator-prey models with non constant mortality rate and Holling type II response. Concretely, we establish sufficient conditions for the extinction and persistence in the mean of autonomous stochastic model and obtain a critical value between them. Then by constructing appropriate Lyapunov functions, we prove that there is a nontrivial positive periodic solution to the non-autonomous stochastic model. Finally, numerical examples are introduced to illustrate the results developed.

Dynamical Analysis of a Class of Prey-Predator Model with Beddington-DeAngelis Functional Response, Stochastic Perturbation, and Impulsive Toxicant Input

Complexity ◽

10.1155/2017/3742197 ◽

2017 ◽

Vol 2017 ◽

pp. 1-18 ◽

Cited By ~ 32

Author(s):

Feifei Bian ◽

Wencai Zhao ◽

Yi Song ◽

Rong Yue

Keyword(s):

Periodic Solutions ◽

Functional Response ◽

Regime Switching ◽

Sufficient Conditions ◽

Stochastic Perturbation ◽

Dynamical Analysis ◽

Positive Periodic Solutions ◽

Noise Perturbation ◽

Predator Model ◽

Predator System

A stochastic prey-predator system in a polluted environment with Beddington-DeAngelis functional response is proposed and analyzed. Firstly, for the system with white noise perturbation, by analyzing the limit system, the existence of boundary periodic solutions and positive periodic solutions is proved and the sufficient conditions for the existence of boundary periodic solutions and positive periodic solutions are derived. And then for the stochastic system, by introducing Markov regime switching, the sufficient conditions for extinction or persistence of such system are obtained. Furthermore, we proved that the system is ergodic and has a stationary distribution when the concentration of toxicant is a positive constant. Finally, two examples with numerical simulations are carried out in order to illustrate the theoretical results.

Stochastic dynamics of the transmission of Dengue fever virus between mosquitoes and humans

10.1142/s1793524521500625 ◽

2021 ◽

pp. 2150062

Author(s):

Manjing Guo ◽

Lin Hu ◽

Lin-Fei Nie

Keyword(s):

Dengue Fever ◽

Stochastic Dynamics ◽

Sufficient Conditions ◽

Lyapunov Methods ◽

Differential Model ◽

Persistence In The Mean ◽

The Mean ◽

And Behavior ◽

The Impact

Considering the impact of environmental white noise on the quantity and behavior of vector of disease, a stochastic differential model describing the transmission of Dengue fever between mosquitoes and humans, in this paper, is proposed. By using Lyapunov methods and Itô’s formula, we first prove the existence and uniqueness of a global positive solution for this model. Further, some sufficient conditions for the extinction and persistence in the mean of this stochastic model are obtained by using the techniques of a series of stochastic inequalities. In addition, we also discuss the existence of a unique stationary distribution which leads to the stochastic persistence of this disease. Finally, several numerical simulations are carried to illustrate the main results of this contribution.

Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching

10.1142/s1793524521500807 ◽

2021 ◽

pp. 2150080

Author(s):

Qun Liu ◽

Daqing Jiang

Keyword(s):

Markov Process ◽

Lyapunov Function ◽

Stochastic System ◽

Epidemic Model ◽

Regime Switching ◽

Dynamical Behavior ◽

Piecewise Deterministic Markov Process ◽

Positive Recurrence ◽

Persistence In The Mean ◽

In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching.

Study on a susceptible–infected–vaccinated model with delay and proportional vaccination

10.1142/s1793524518501024 ◽

2018 ◽

Vol 11 (08) ◽

pp. 1850102 ◽

Cited By ~ 1

Author(s):

Shuqi Gan ◽

Fengying Wei

Keyword(s):

Positive Solution ◽

Epidemic Model ◽

Sufficient Conditions ◽

Sufficient Condition ◽

Stochastic Epidemic ◽

Persistence In The Mean ◽

The Mean ◽

Stochastic Epidemic Model

A susceptible–infected–vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper [Formula: see text]-function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator [Formula: see text]. Further, if [Formula: see text], then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator [Formula: see text].