Sydney Mathematical Research Institute, University of Sydney, L4.42, Quadrangle A14, Sydney, NSW, Australia.
Department of Mathematics, Tufts University, Bromfield-Pearson Hall, 503 Boston Avenue, Medford, MA, 02155, USA.
J Math Biol. 2020 Apr;80(5):1323-1351. doi: 10.1007/s00285-019-01464-y. Epub 2020 Jan 10.
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.
从最简单的形式来看，竞争排斥原理指出，在争夺较少资源的情况下，许多物种不能共存。然而，从经验上观察到，在某些情况下共存是可能的。一个例子是 Hutchinson 的“浮游生物悖论”。这是一个大量浮游植物物种共存的例子，同时它们在争夺非常有限的资源。实验和理论研究表明，环境的时间波动可以促进竞争物种的共存。Hutchinson 推测，一种可以共存，因为非平衡条件将使不同的物种在不同的时间被环境所青睐。在本文中，我们在各种情况下展示了可变（随机）环境如何使一组受资源或其他密度相关因素限制的竞争物种共存。如果环境波动通过白噪声来建模，并且竞争者的人均增长率线性依赖于资源，我们证明存在竞争排斥。然而，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，我们表明在比物种数量更少的资源上共存是可能的。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，即使增长率线性依赖于资源，所有物种也都有可能共存。我们在一个例子中（Benaim 和 Lobry '16 中的一个变体）表明，与 Hutchinson 的解释相反，可以在两个环境之间切换，其中相同的物种受到青睐，仍然可以共存。
