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.