# Population dynamical behavior of Lotka-Volterra system under regime switching

Xiaoyue Li, Daqing Jiang, Xuerong Mao, National Natural Science Foundation of China (Funder), Royal Society of Edinburgh (Funder)

Research output: Contribution to journalArticle

111 Citations (Scopus)

### Abstract

In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.
Original language English 427-448 21 Journal of Computational and Applied Mathematics 232 2 https://doi.org/10.1016/j.cam.2009.06.021 Published - 2009

### Fingerprint

Regime Switching
Lotka-Volterra System
Permanence
Stationary Distribution
Dynamical Behavior
Extinction
Brownian movement
Sample Path
Markov processes
Probability distributions
Brownian motion
Positive Solution
Markov chain
Probability Distribution
Sufficient Conditions
Coefficient
Relationships
Standards

### Keywords

• brownian motion
• stochastic differential equation
• generalized It^o's formula
• markov chain
• stochastic permanence.

### Cite this

Li, X., Jiang, D., Mao, X., National Natural Science Foundation of China (Funder), & Royal Society of Edinburgh (Funder) (2009). Population dynamical behavior of Lotka-Volterra system under regime switching. Journal of Computational and Applied Mathematics, 232(2), 427-448. https://doi.org/10.1016/j.cam.2009.06.021
Li, Xiaoyue ; Jiang, Daqing ; Mao, Xuerong ; National Natural Science Foundation of China (Funder) ; Royal Society of Edinburgh (Funder). / Population dynamical behavior of Lotka-Volterra system under regime switching. In: Journal of Computational and Applied Mathematics. 2009 ; Vol. 232, No. 2. pp. 427-448.
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abstract = "In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.",
keywords = "brownian motion, stochastic differential equation, generalized It^o's formula, markov chain, stochastic permanence.",
author = "Xiaoyue Li and Daqing Jiang and Xuerong Mao and {National Natural Science Foundation of China (Funder)} and {Royal Society of Edinburgh (Funder)}",
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journal = "Journal of Computational and Applied Mathematics",
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Li, X, Jiang, D, Mao, X, National Natural Science Foundation of China (Funder) & Royal Society of Edinburgh (Funder) 2009, 'Population dynamical behavior of Lotka-Volterra system under regime switching', Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 427-448. https://doi.org/10.1016/j.cam.2009.06.021

Population dynamical behavior of Lotka-Volterra system under regime switching. / Li, Xiaoyue; Jiang, Daqing; Mao, Xuerong; National Natural Science Foundation of China (Funder); Royal Society of Edinburgh (Funder).

In: Journal of Computational and Applied Mathematics, Vol. 232, No. 2, 2009, p. 427-448.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Population dynamical behavior of Lotka-Volterra system under regime switching

AU - Li, Xiaoyue

AU - Jiang, Daqing

AU - Mao, Xuerong

AU - National Natural Science Foundation of China (Funder)

AU - Royal Society of Edinburgh (Funder)

PY - 2009

Y1 - 2009

N2 - In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.

AB - In this paper, we investigate a Lotka-Volterra system under regime switching dx(t) = diag(x1(t); : : : ; xn(t))[(b(r(t)) + A(r(t))x(t))dt + (r(t))dB(t)]; where B(t) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.

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KW - stochastic differential equation

KW - generalized It^o's formula

KW - markov chain

KW - stochastic permanence.

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Li X, Jiang D, Mao X, National Natural Science Foundation of China (Funder), Royal Society of Edinburgh (Funder). Population dynamical behavior of Lotka-Volterra system under regime switching. Journal of Computational and Applied Mathematics. 2009;232(2):427-448. https://doi.org/10.1016/j.cam.2009.06.021