Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

Du, Nguyen Huu; Kon, Ryusuke; Satō, Kazunori; Takeuchi, Yasuhiro

doi:10.1016/j.cam.2004.02.001

Cited by 98 publications

(60 citation statements)

References 5 publications

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“…One of the distinct features of such systems is that the underlying dynamics are subject to changes with respect to certain configurations. It is recognized that using a continuous-time Markov chain is suitable to depict such kind of systems where they differ by factors such as rain falls or nutrition [5,33].…”

Section: Introductionmentioning

confidence: 99%

“…By now, there are no result related to the general stochastic Gilpin-Ayala system under regime switching with impulsive effects. Inspired by the above discussions, in this paper, we study the general stochastic hybrid system with impulsive effects, or system under regime switching: 5) where λ > 0, θ > 0. It is worth pointing out that many classical and important stochastic models are the special cases of (1.5).…”

Section: Introductionmentioning

confidence: 99%

“…Remark 3.11. Let us consider some subsystem 5) for the state i. Similarly, we can prove that ifh(i) := lim sup t→∞ t −1 0<t k <t ln(1 + b k ) + h(i) < 0, then species x(t) of (3.5) will go extinct, ifh(i) = 0, then species x(t) of (3.5) will be non-persistent in the mean, ifh(i) > 0, then species x(t) of (3.5) will be weakly persistent, see [28].…”

Section: Definition 32 ([13]mentioning

confidence: 99%

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Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations

Wu¹

2017

J. Nonlinear Sci. Appl.

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This paper is mainly concerned with the dynamics of the stochastic Gilpin-Ayala model under regime switching with impulsive perturbations. The goal is to analyze the effects of Markov chain and impulse on the dynamics. Some asymptotic properties are considered and sufficient criteria for stochastic permanence, extinction, non-persistence in the mean and weak persistence are obtained. The critical value among the extinction, non-persistence in the mean and weak persistence is explored. Our results demonstrate that the dynamics of the model have close relations with the impulse and the stationary distribution of the Markov chain.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Definition 32 ([13]mentioning

confidence: 99%

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Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations

Wu¹

2017

J. Nonlinear Sci. Appl.

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“…However, this is not the only way to introduce stochasticity into the deterministic model (2). There is another type of environmental noise, namely color noise, say telegraph noise [14][15][16][17][18]. Telegraph noise can be illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition, climatic characteristics or socio-cultural factorsfactors.…”

Section: S(t) = μ − μS(t) − β S(t)i (T) + γ R(t) Dt D I (T) = [−(μ +mentioning

confidence: 99%

Dynamics of hybrid switching diffusions SIRS model

Settati¹,

Lahrouz²,

Jarroudi³

et al. 2015

J. Appl. Math. Comput.

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The main aim of this paper is to study the effect of the environmental noises in the asymptotic properties of a stochastic version of the classical SIRS epidemic model. The model studied here include white noise and telegraph noise modeled by Markovian switching. We obtained conditions for extinction both in probability one and in pth moment. We also established the persistence of disease under different conditions on the intensities of noises, the parameters of the model and the stationary distribution of the Markov chain. The highlight point of our work is that our conditions are sufficient and almost necessary for extinction and persistence of the epidemic. The presented results are demonstrated by numerical simulations.

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“…[4, 12]) and the hybrid population systems are illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition or as rain falls [1,11].…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions of stochastic differential delay equations with Markovian switching

Mao

Shen

2010

Journal of Difference Equations and Applications

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This version is available at https://strathprints.strath.ac.uk/16869/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Abstract. Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching (SDDEwMSs) and to establish the convergence theory for the Euler-Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching.

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Dynamical behavior of Lotka–Volterra competition systems: non-autonomous bistable case and the effect of telegraph noise

Cited by 98 publications

References 5 publications

Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations

Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations

Dynamics of hybrid switching diffusions SIRS model

Approximate solutions of stochastic differential delay equations with Markovian switching

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