论文标题
人口增长的随机模型中的稳定性，持久性和灭绝
The stability, persistence and extinction in a stochastic model of the population growth
论文作者
论文摘要
在本文中，我们考虑了人口增长的随机扰动物流模型的全球定性特性。在此模型中，假定随机扰动为白噪声类型，与当前的人口大小成正比。使用直接的Lyapunov方法，我们建立了此随机微分方程的全局特性。特别是，我们发现方程的溶液围绕一个间隔振荡，并明确发现了此间隔的终点。此外，我们发现，如果噪声的幅度超过了一定的临界水平（也明确发现），则会发生零溶液的随机稳定（“稳定性”）。在这种情况下，（i）原点是间隔的下边界，（ii）在有限的时间内几乎确定（A.S.），由于随机性而灭绝。
In this paper we consider the global qualitative properties of a stochastically perturbed logistic model of population growth. In this model, the stochastic perturbations are assumed to be of the white noise type and are proportional to the current population size. Using the direct Lyapunov method, we established the global properties of this stochastic differential equation. In particular, we found that solutions of the equation oscillate around an interval, and explicitly found the end points of this interval. Moreover, we found that, if the magnitude of the noise exceeds a certain critical level (which is also explicitly found), then the stochastic stabilisation ("stabilisation by noise") of the zero solution occurs. In this case, (i) the origin is the lower boundary of the interval, and (ii) the extinction of the population due to stochasticity occurs almost sure (a.s.) for a finite time.
