The improved results on the stochastic Kolmogorov system with time-varying delay
Yangzi Hu Fuke Wu
This paper discusses the stochastic Kolmogorov system with time-varying delay. Under two classes of sufficient conditions, this paper solves the non-explosion, the moment boundedness and the polynomial pathwise growth simultaneously. This is an important improvement for the existing results, since the moment boundedness and the polynomial pathwise growth do not imply each in general. Moreover, these two classes of conditions only depends on the parameters of the system and are easier to be used. Finally, a two-dimensional Komogorov model is examined to illustrate the efficiency of our result.
keywords: non-explosion time-varying delay polynomial pathwise growth. moment boundedness Stochastic Kolmogorov system
Stochastic Lotka-Volterra system with unbounded distributed delay
Fuke Wu Yangzi Hu
In general, population systems are often subject to environmental noise. To examine whether the presence of such noise affects these systems significantly, this paper perturbs the Lotka--Volterra system

$\dot{x}(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))$

into the corresponding stochastic system

$dx(t)=\mbox{diag}(x_1(t), \cdots, x_n(t))[(r+Ax(t)+B\int_{-\infty}^0x(t+\theta)d\mu(\theta))dt+\beta dw(t)].$

This paper obtains one condition under which the above stochastic system has a global almost surely positive solution and gives the asymptotic pathwise estimation of this solution. This paper also shows that when the noise is sufficiently large, the solution of this stochastic system will converge to zero with probability one. This reveals that the sufficiently large noise may make the population extinct.

keywords: Stochastic Lotka-Volterra system; Global solutions; Pathwise estimation; Extinction.

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