In its simplest form, the competitive exclusion principle states that a number of species competing for a smaller number of resources cannot coexist. However, it has been observed empirically that in some settings it is possible to have coexistence. One example is Hutchinson's 'paradox of the plankton'. This is an instance where a large number of phytoplankton species coexist while competing for a very limited number of resources. Both experimental and theoretical studies have shown that temporal fluctuations of the environment can facilitate coexistence for competing species. Hutchinson conjectured that one can get coexistence because nonequilibrium conditions would make it possible for different species to be favored by the environment at different times. In this paper we show in various settings how a variable (stochastic) environment enables a set of competing species limited by a smaller number of resources or other density dependent factors to coexist. If the environmental fluctuations are modeled by white noise, and the per-capita growth rates of the competitors depend linearly on the resources, we prove that there is competitive exclusion. However, if either the dependence between the growth rates and the resources is not linear or the white noise term is nonlinear we show that coexistence on fewer resources than species is possible. Even more surprisingly, if the temporal environmental variation comes from switching the environment at random times between a finite number of possible states, it is possible for all species to coexist even if the growth rates depend linearly on the resources. We show in an example (a variant of which first appeared in Benaim and Lobry '16) that, contrary to Hutchinson's explanation, one can switch between two environments in which the same species is favored and still get coexistence.

中文翻译：

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随机环境中的竞争排斥原则。

竞争排斥原则以最简单的形式规定，争夺较少资源的许多物种无法共存。但是，从经验上观察到，在某些情况下，可能会共存。一个例子就是哈钦森的“浮游生物悖论”。在这种情况下，大量浮游植物物种共存，同时争夺数量非常有限的资源。实验和理论研究都表明，环境的时间波动可以促进竞争物种的共存。哈钦森推测，一个人可以共存，因为非平衡条件将使不同物种在不同时间受到环境的青睐。在本文中，我们展示了在各种环境中，可变（随机）环境如何使一组受较少资源或其他密度相关因素限制的竞争物种共存。如果环境波动以白噪声为模型，而竞争者的人均增长率与资源线性相关，则证明存在竞争排斥。但是，如果增长率和资源之间的依赖关系不是线性的，或者白噪声项是非线性的，则表明存在比物种更少的资源并存的可能性。更令人惊讶的是，如果时间环境变化来自于在有限数量的可能状态之间随机切换环境，那么即使增长率线性依赖于资源，所有物种也可能共存。