Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment

Takeuchi, Yasuhiro; Du, Nguyen Huu; Hieu, Nguyen Trong; Satō, Kazunori

doi:10.1016/j.jmaa.2005.11.009

Cited by 142 publications

(91 citation statements)

References 13 publications

(19 reference statements)

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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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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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show abstract

“…The Markov chain has important impacts on the population dynamics. Takeuchi et al [34] revealed the significant effect of Markovian switching on the population system: both its subsystems develop periodically but switching between them makes them become neither permanent nor dissipative. Let us take a more further step, in reality, due to some natural or man-made factors, the growth of species often undergoes some discrete changes of relatively short time interval at some fixed times, such as drought, flooding, hunting, planting, etc.…”

Section: Introductionmentioning

confidence: 99%

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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“…Therefore, many authors introduced stochastic perturbations into the deterministic models [16,19,29,33]. Takeuchi et al [28] considered the following predator-prey model with telegraph noise based on model (1.1):…”

Section: Introductionmentioning

confidence: 99%

“…In fact, two equilibrium states often do not coincide with each other. Takeuchi et al [28] discovered that the stochastic species system is neither permanent nor dissipative (see, e.g., [5]). This is an important result as it reveals a significant effect on the species system, i.e., both its subsystems evolved periodically, but the switching made them neither permanent nor dissipative.…”

Section: Introductionmentioning

confidence: 99%

The stochastic interactions between predator and prey under Markovian switching: competitive interaction between multiple prey

Li¹,

Zhang²

2017

J. Nonlinear Sci. Appl.

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In this paper, a class of predator-prey model with prey competition is proposed, in which the interactions of predation between predator and prey are randomised and subsequently evaluated under Markovian switching. By constructing appropriate Lyapunov functions and applying various analytical methods, sufficient conditions for the existence of unique global positive solution, stochastic permanence and mean extinction are established. In the permanence case, we also estimate the superior and inferior limits of the sample path in a time-averaged Markov decision. We conclude that the interactions between predator and two prey, two competitive prey themselves and the dynamical properties of switching subsystems are not only dependent on subsystem coefficients but also on the transition probability of the Markov chain (switching from one state to another). Specific examples and numerical simulations are provided to demonstrate our theoretical results.

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Evolution of predator–prey systems described by a Lotka–Volterra equation under random environment

Cited by 142 publications

References 13 publications

Dynamics of hybrid switching diffusions SIRS model

Dynamics of hybrid switching diffusions SIRS model

Dynamics of stochastic hybrid Gilpin-Ayala system with impulsive perturbations

The stochastic interactions between predator and prey under Markovian switching: competitive interaction between multiple prey

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