IJPAM: Volume 11, No. 4 (2004)


Arifah Bahar$^1$, Xuerong Mao$^2$
$^{1,2}$Department of Statistics and Modelling Science
University of Strathclyde
Glasgow G1 1XH, U.K.
$^1$e-mail: bahar@stams.strath.ac.uk
$^2$e-mail: xuerong@stams.strath.ac.uk

Abstract.In this paper we stochastically perturb the delay Lotka-Volterra model $\dot x(t) = \diag(x_1(t), \cdots, x_n(t))[b + Ax(t)+ Bx(t-\T)]$ into the stochastic delay differential equation $dx(t) = \diag(x_1(t), \cdots, x_n(t))$ $[(b + Ax(t)+Bx(t-\T))dt +\be dw(t)]$, and show that under certain conditions,the original delay equation and the associated stochastic delay equation behave similarly in the sense that both have positive solutions which will not explode to infinity in a finite time and, in fact, will be ultimately bounded. In other words, we show that under certain condition the noise will not spoil these nice properties. However, we will also show, under some other conditions, that although the solution to the original delay equation may be persistent, the solution to the associated stochastic delay equation will become extinct with probability one. This reveals the important fact that the environmental noise may make the population extinct.

Received: December 9, 2003

AMS Subject Classification: 60J65, 60H10, 34K40

Key Words and Phrases: Brownian motion, stochastic differential delay equation, ultimate boundedness, extinction, Itô formula

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 11
Issue: 4