Stochastic delay population systems

Abstract

In this article we stochastically perturb the classical non-autonomous delay Lotka–Volterra model

into the stochastic delay population system (SDPS)

Different from most of the existing papers [A. Bahar and X. Mao, Stochastic delay Lotka–Volterra model, J. Math. Anal. Appl. 292 (2004), 364–380, A. Bahar and X. Mao, Stochastic delay population dynamics, J. Pure Appl. Math. 11 (2004), 377–400, X. Mao, Delay population dynamics and environmental noise, Stochastics Dyn. 5(2) (2005), pp. 149–162], the system parameters in this article are time-dependent. We will give a sufficient condition under which the SDPS will have a unique global positive solution. We will then establish some new asymptotic properties for the moments of the solution. In particular, we will discuss two fundamental problems in population systems, namely ultimate boundedness and extinction.

Keywords:

• stochastic delay Lotka–Volterra model
• Itô's formula
• ultimate boundedness
• extinction

AMS Subject Classifications:

• 60J65
• 60H10
• 34K40

Acknowledgement

The author would like to thank the referees for their helpful comments.