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Generalized stochastic delay Lotka-Volterra systems

Mao, X. and Yin, J. and Wu, F. (2009) Generalized stochastic delay Lotka-Volterra systems. Stochastic Models, 25 (3). pp. 436-454. ISSN 1532-6349

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This article deals with a class of generalized stochastic delay Lotka-Volterra systems of the form dX(t) = diag(X1(t), X2(t),..., Xn(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 + )-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.