# Stationary distribution of stochastic population systems

Research output: Contribution to journalArticle

74 Citations (Scopus)

### Abstract

In this paper we consider the stochastic differential equation (SDE) population model dx(t) = diag(x1(t), . . . , xn(t))[(b + Ax(t))dt + σdB(t)] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004)  and Mao (2005) ) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.
Language English 398-405 8 Systems and Control Letters 60 6 7 Apr 2011 10.1016/j.sysconle.2011.02.013 Published - Jun 2011

### Fingerprint

Differential equations
Computer simulation

### Keywords

• Brownian motion
• stochastic differential equation
• Ito's formula
• stationary distribution

### Cite this

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title = "Stationary distribution of stochastic population systems",
abstract = "In this paper we consider the stochastic differential equation (SDE) population model dx(t) = diag(x1(t), . . . , xn(t))[(b + Ax(t))dt + σdB(t)] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004)  and Mao (2005) ) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.",
keywords = "Brownian motion, stochastic differential equation, Ito's formula, stationary distribution",
author = "Xuerong Mao",
year = "2011",
month = "6",
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In: Systems and Control Letters, Vol. 60, No. 6, 06.2011, p. 398-405.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Stationary distribution of stochastic population systems

AU - Mao, Xuerong

PY - 2011/6

Y1 - 2011/6

N2 - In this paper we consider the stochastic differential equation (SDE) population model dx(t) = diag(x1(t), . . . , xn(t))[(b + Ax(t))dt + σdB(t)] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004)  and Mao (2005) ) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.

AB - In this paper we consider the stochastic differential equation (SDE) population model dx(t) = diag(x1(t), . . . , xn(t))[(b + Ax(t))dt + σdB(t)] for n interacting species. The main aim is to study the stationary distribution of the solution. It is known (see e.g. Bahar and Mao (2004)  and Mao (2005) ) if the noise intensity is sufficiently large then the population may become extinct with probability one. Our main aim here is to find out what happens if the noise is relatively small. In this paper we will show the existence of a unique stationary distribution. We will then develop a useful method to compute the mean and variance of the stationary distribution. Computer simulations will be used to illustrate our theory.

KW - Brownian motion

KW - stochastic differential equation

KW - Ito's formula

KW - stationary distribution

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JO - Systems and Control Letters

T2 - Systems and Control Letters

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