Abstract
In this paper, we study pathwise estimation of the global positive solutions for the stochastic functional Kolmogorov-type systems with infinite delay. Under some conditions, the growth rate of the solutions for such systems with general noise structures is less than a polynomial rate in the almost sure sense. To illustrate the applications of our theory more clearly, this paper also discusses various stochastic Lotka-Volterra-type systems as special cases. MSC:34K50, 60H10, 92D25, 93E03.
Highlights
1 Introduction The stochastic functional Kolomogorov-type systems for n interacting species is described by the following stochastic functional differential equation: dx(t) = diag(x, . . . , xn) f dt + g(xt) dw(t), ( . )
Population systems are often subject to environmental noise
Since this paper mainly examines the pathwise estimation of the solution for stochastic functional Kolmogorov-type systems with infinite delay, we assume that there exists a unique global positive solution for all discussed equations
Summary
Our interest is to discuss pathwise estimation of the global positive solutions for stochastic functional Kolomogorov-type systems with infinite delay. Little is yet known about the pathwise property of the stochastic functional Kolmogorov-type systems with infinite delay they may be seen as a generalized stochastic functional Lotka-Volterra system. This paper will examine the pathwise estimation of the stochastic functional Kolmogorov-type systems under the general noise structures.
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