scholarly journals Asymptotic Behavior of Positive Solutions of a Competitive System Subject to Environmental Noise

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Huili Xiang ◽  
Zhuang Fang ◽  
Zuxiong Li ◽  
Zhijun Liu
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Numerical Simulations ◽  
Stochastic Differential Equations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Environmental Noise ◽  
Competitive System ◽  
Stochastic Boundedness

A competitive system subject to environmental noise is established. By using the theory of stochastic differential equations and Lyapunov function, sufficient conditions for the existence, uniqueness, stochastic boundedness, and global attraction of the positive solution of the above system are established, respectively. An example together with its corresponding numerical simulations is presented to confirm our analytical results.

Dynamic behavior of a stochastic SIRS model with two viruses

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiandong Zhao ◽  
Tonghua Zhang ◽  
Zhixia Han
Keyword(s):  
Growth Rate ◽  
Asymptotic Stability ◽  
Global Existence ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Environmental Noise ◽  
Sirs Model ◽  
The Moment ◽  
Disease Free

AbstractTo study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.

scholarly journals Asymptotic Behavior Analysis for a Three-Species Food Chain Stochastic Model with Regime Switching

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Junmei Liu ◽  
Yonggang Ma
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Function ◽  
Stochastic Model ◽  
Numerical Simulations ◽  
Behavior Analysis ◽  
Food Chain ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
Stationary Distributions ◽  
Theoretical Results

This paper discusses the asymptotic behavior of a class of three-species stochastic model with regime switching. Using the Lyapunov function, we first obtain sufficient conditions for extinction and average time persistence. Then, we prove sufficient conditions for the existence of stationary distributions of populations, and they are ergodic. Numerical simulations are carried out to support our theoretical results.

Dynamics of a stochastic rabies epidemic model with markovian switching

2021 ◽  
pp. 2150032
Author(s):  
Hao Peng ◽  
Xinhong Zhang ◽  
Daqing Jiang
Keyword(s):  
Lyapunov Function ◽  
White Noise ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Telegraph Noise ◽  
Global Positive Solution ◽  
Theoretical Results

In this paper, we analyze a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise. First, we prove the existence of the unique global positive solution. Second, by constructing an appropriate Lyapunov function, we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for the extinction of diseases. Finally, numerical simulations are introduced to illustrate our theoretical results.

scholarly journals Dynamical Behavior of a Stochastic SIRC Model for Influenza A

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 745 ◽  
Author(s):  
Tongqian Zhang ◽  
Tingting Ding ◽  
Ning Gao ◽  
Yi Song
Keyword(s):  
Lyapunov Function ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Epidemic Model ◽  
Influenza A ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Global Positive Solution ◽  
Theoretical Results ◽  
Stochastic Lyapunov Function

In this paper, a stochastic SIRC epidemic model for Influenza A is proposed and investigated. First, we prove that the system exists a unique global positive solution. Second, the extinction of the disease is explored and the sufficient conditions for extinction of the disease are derived. And then the existence of a unique ergodic stationary distribution of the positive solutions for the system is discussed by constructing stochastic Lyapunov function. Furthermore, numerical simulations are employed to illustrate the theoretical results. Finally, we give some further discussions about the system.

scholarly journals Extinction in Two-Species Nonlinear Discrete Competitive System

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Liqiong Pu ◽  
Xiangdong Xie ◽  
Fengde Chen ◽  
Zhanshuai Miao
Keyword(s):  
Numerical Simulations ◽  
Positive Solution ◽  
Discrete System ◽  
Sufficient Conditions ◽  
The Other ◽  
Discrete Equation ◽  
Toxic Substances ◽  
Competitive System ◽  
Globally Attractive ◽  
Nonlinear Discrete System

We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results.

scholarly journals Stochastically Ultimate Boundedness and Global Attraction of Positive Solution for a Stochastic Competitive System

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Shengliang Guo ◽  
Zhijun Liu ◽  
Huili Xiang
Keyword(s):  
Lyapunov Function ◽  
Positive Solution ◽  
Finite Time ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Ultimate Boundedness ◽  
Competitive System ◽  
Global Attraction ◽  
Stochastically Ultimate Boundedness ◽  
Theoretical Results

A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.

Extinction in a discrete Lotka–Volterra competitive system with the effect of toxic substances and feedback controls

2015 ◽  
Vol 08 (01) ◽  
pp. 1550012 ◽  
Author(s):  
Lijuan Chen ◽  
Fengde Chen
Keyword(s):  
Feedback Control ◽  
Lyapunov Function ◽  
Numerical Simulations ◽  
Sufficient Conditions ◽  
The Other ◽  
Toxic Substances ◽  
Two Dimensional ◽  
Competitive System ◽  
Feedback Controls ◽  
Analysis Technique

In this paper, we consider a discrete Lotka–Volterra competitive system with the effect of toxic substances and feedback controls. By using the method of discrete Lyapunov function and by developing a new analysis technique, we obtain the sufficient conditions which guarantee that one of the two species will be driven to extinction while the other will be permanent. We improve the corresponding results of Li and Chen [Extinction in two-dimensional discrete Lotka–Volterra competitive system with the effect of toxic substances, Dynam. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 15 (2008) 165–178]. Also, an example together with their numerical simulations shows the feasibility of our main results. It is shown that toxic substances and feedback control variables play an important role in the dynamics of the system.

scholarly journals Dynamics of a Stochastic Three-Species Food Web Model with Omnivory and Ratio-Dependent Functional Response

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-19
Author(s):  
Guirong Liu ◽  
Rong Liu
Keyword(s):  
Lyapunov Function ◽  
Numerical Simulations ◽  
Food Web ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Stochastic Comparison ◽  
Top Predator ◽  
Global Positive Solution ◽  
Persistent In Mean ◽  
Food Web Model

This paper is concerned with a stochastic three-species food web model with omnivory which is defined as feeding on more than one trophic level. The model involves a prey, an intermediate predator, and an omnivorous top predator. First, by the stochastic comparison theorem, we show that there is a unique global positive solution to the model. Next, we investigate the asymptotic pathwise behavior of the model. Then, we conclude that the model is persistent in mean and extinct and discuss the stochastic persistence of the model. Further, by constructing a suitable Lyapunov function, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Then, we present the application of the main results in some special models. Finally, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.

scholarly journals A Note on Stability of Stochastic Logistic Model by Incorporating the Ornstein-Uhlenbeck Process

2020 ◽  
Vol 12 (2) ◽  
pp. 52
Author(s):  
Tawfiqullah Ayoubi ◽  
Abdul Munir Khirzada
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Logistic Model ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Stochastic Logistic Model

In this research, we first prove that the stochastic logistic model (10) has a positive global solution. Subsequently, we introduce the sufficient conditions for the stochastically stability of the general form of stochastic differential equations (SDEs) in terms of equation (1), for zero solution by using the Lyapunov function. This result is verified via several examples in Appendix A. Besides; we prove that the stochastic logistic model, by incorporating the Ornstein-Uhlenbeck process is stable in zero solution. Furthermore, the simulated results are displayed via the 4-stage stochastic Runge-Kutta (SRK4) numerical method.

Analysis of a mutualism model with time-related coefficients in a stochastic environment

2020 ◽  
pp. 2050073
Author(s):  
Jun Wei Luo ◽  
Mei Li ◽  
Kai Liu ◽  
Rui Guan
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Existence And Uniqueness ◽  
Uniform Continuity ◽  
Stochastic Environment ◽  
Stochastic Perturbations ◽  
Stochastic Permanence ◽  
Stochastic Boundedness ◽  
Mutualism Model

In this paper, a mutualism model with stochastic perturbations is considered and some of its coefficients are related to time. Under some assumptions, we make efforts to prove the existence and uniqueness of a positive solution, and the asymptotic behavior to the problem is discussed. Furthermore, we also prove the properties of stochastic boundedness, uniform continuity and stochastic permanence of this system. At last, some numerical simulations are introduced to illustrate our main results.

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