Stability of a delayed predator—prey model in a random environment**Project supported by the National Natural Science Foundation of China (Grant Nos. 11272051 and 11302172).

Abstract

The stability of the first-order and second-order solution moments for a Harrison-type predator–prey model with parametric Gaussian white noise is analyzed in this paper. The moment equations of the system solution are obtained under Itô interpretations. The delay-independent stable condition of the first-order moment is identical to that of the deterministic delayed system, and the delay-independent stable condition of the second-order moment depends on the noise intensities. The corresponding critical time delays are determined once the stabilities of moments lose. Further, when the time delays are greater than the critical time delays, the system solution becomes unstable with the increase of noise intensities. Finally, some numerical simulations are given to verify the theoretical results.