Generalized Stochastic Delay Lotka–Volterra Systems

Abstract

This article deals with a class of generalized stochastic delay Lotka–Volterra systems of the form dX(t) = diag(X ₁(t), X ₂(t),…, X _n (t))[(f(X(t)) + g(X(t − τ)))dt + h(X(t))dB(t)]. Under some unrestrictive conditions on f, g, and h, we show that the unique solution of such a stochastic system is positive and does not explode in a finite time with probability one. We also establish some asymptotic boundedness results of the solution including the time average of its (β + α)-order moment, as well as its asymptotic pathwise estimation. As a by-product, a stochastic ultimate boundedness of the solution for this stochastic system is directly derived. Three examples are given to illustrate our conclusions.

Keywords:

• Asymptotic boundedness of moment
• Brownian motion
• Lotka–Volterra model
• Stochastic delay differential equation
• Stochastic ultimate boundedness

Mathematics Subject Classification:

• 60H10
• 60J65
• 34K40

ACKNOWLEDGMENTS

The first author thanks the Department of Statistics and Modelling Science, University of Strathclyde, for its hospitality during the conduct of this research. The authors are also grateful to Prof. Peter Taylor and an anonymous referee for their very careful comments and helpful suggestions.

This work was supported in part by the NSF of Guangdong Province (program no. 04300573).