随机Lotka-Volterra食物链
Stochastic Lotka-Volterra food chains
摘要：
本文研究了具有环境随机性的Lotka-Volterra系统所模拟的简单食物链中物种的持续生存和绝灭。文献中对确定性Lotka-Volterra系统已有较好的结果，但对随机Lotka-Volterra系统的研究却很少。我们分析的食物链由一个猎物和$n-1$捕食者组成。第一种捕食者吃掉第$j-1$种，然后被第$j+1$种捕食者吃掉;这样，每个物种最多只与另外两个物种相互作用--在营养链中，它们的位置在它的上面或下面。我们可以分类，根据一个明确的数量取决于系统的相互作用系数，物种灭绝，收敛到他们独特的不变概率测度。我们的工作可以看作是Gard和Hallam '79的确定性结果到随机环境的自然扩张。作为一个结果，我们表明，环境随机性使物种更有可能灭绝。然而，如果环境的波动很小，持久性的确定性设置是保存在随机系统。我们的分析还表明，正如预期的那样，新的顶级捕食者的加入使不同的物种更容易灭绝。我们的分析的另一个新颖之处是，我们可以描述的行为时，噪声是退化的系统。这是相关的，因为环境对不同物种的影响之间可能存在很强的相关性。
英文摘要：
We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $n-1$ predators. The $j$th predator eats the $j-1$th species and is eaten by the $j+1$th predator; this way each species only interacts with at most two other species - the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting. As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction. Another novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.
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