Stochastic differential delay equations of population dynamics

Abstract

In this paper we stochastically perturb the delay Lotka–Volterra model x ˙ ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ A ( x ( t ) − x ¯ ) + B ( x ( t − τ ) − x ¯ ) ] into the stochastic delay differential equation (SDDE) d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) { [ A ( x ( t ) − x ¯ ) + B ( x ( t − τ ) − x ¯ ) ] d t + σ ( x ( t ) − x ¯ ) d w ( t ) } . The main aim is to reveal the effects of environmental noise on the delay Lotka–Volterra model. Our results can essentially be divided into two categories: (i) If the delay Lotka–Volterra model already has some nice properties, e.g., nonexplosion, persistence, and asymptotic stability, then the SDDE will preserve these nice properties provided the noise is sufficiently small. (ii) When the delay Lotka–Volterra model does not have some desired properties, e.g., nonexplosion and boundedness, the noise might make the SDDE achieve these desired properties.