According to the competitive-exclusion principle, the number n of regulating variables describing a given community dynamics is an upper bound on the number of species (or types or morphs) that can coexist at equilibrium. On occasion, it is possible to reformulate a model with a lower number of regulating variables than appeared in the initial specification. We call the smallest number of such variables the dimension of the environmental feedback, or environmental dimension for short. For studying which species can invade a community, it is enough to know the sign of each species' long-term growth rate, i.e., invasion fitness. Therefore, different indicators of population growth - so-called fitness proxies, such as the basic reproduction number - are sometimes preferred. However, as we show, different fitness proxies may have different dimensions. Fundamental characteristics such as the environmental dimension should not depend on such arbitrary choices. Here, we resolve this difficulty by introducing a refined definition of environmental dimension that focuses on neutral fitness contours. On this basis, we show that this definition of environmental dimension is not only unambiguous, i.e., independent of the choice of fitness proxy, but also constructive, i.e., applicable without needing to assess an infinite number of possible fitness proxies. We then investigate how to determine environmental dimensions by analysing the two components of the environmental feedback: the impact map describing how a community's resident species affect the regulating variables and the sensitivity map describing how population growth depends on the regulating variables. The dimension of the impact map is lower than n when the set of feasible environments is of lower dimension than n, and the dimension of the sensitivity map is lower than n when not all n regulating variables affect the sign of population growth independently. While the minimum of the dimensions of the impact and sensitivity maps provides an upper bound on the environmental dimension, the combined effect of the two maps can result in an even lower environmental dimension, which happens when the sensitivity map is insensitive to some aspects of the impact map's image. To facilitate the applications of the framework introduced here, we illustrate all key concepts with detailed worked examples. In view of these results, we claim that the environmental dimension is the ultimate generalization of the traditional and widely used notions of the "number of regulating variables" or the "number of limiting factors", and is thus the sharpest generally applicable upper bound on the number of species that can robustly coexist in a community.

中文翻译：

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环境维度决定物种共存。

根据竞争排斥原则，描述给定群落动态的调节变量的数量 n 是可以平衡共存的物种（或类型或形态）数量的上限。有时，可以使用比初始规范中出现的更少的调节变量来重新构建模型。我们将最小数量的此类变量称为环境反馈维度，或简称为环境维度。对于研究哪些物种可以入侵一个群落，知道每个物种的长期增长率的标志，即入侵适应度就足够了。因此，人口增长的不同指标——所谓的适应度代理，例如基本繁殖数——有时是首选。然而，正如我们所展示的，不同的适应度代理可能有不同的维度。环境维度等基本特征不应依赖于此类任意选择。在这里，我们通过引入专注于中性健身轮廓的环境维度的精细定义来解决这个难题。在此基础上，我们表明环境维度的这种定义不仅是明确的，即独立于适应度代理的选择，而且是建设性的，即适用而无需评估无限数量的可能的适应度代理。然后，我们通过分析环境反馈的两个组成部分来研究如何确定环境维度：描述社区如何 s 常驻物种影响调节变量和描述人口增长如何取决于调节变量的敏感性图。当可行环境集的维数低于n时，影响图的维数低于n；当n个调节变量并非都独立影响人口增长的符号时，敏感性图的维数低于n。虽然影响图和敏感度图的最小维度提供了环境维度的上限，但两个地图的组合效应可能导致更低的环境维度，当敏感度图对环境维度的某些方面不敏感时就会发生这种情况。影响地图的图像。为了促进这里介绍的框架的应用，我们用详细的工作示例说明了所有关键概念。