The usual theoretical condition for coexistence is that each species in a community can increase when it is rare (mutual invasibility). Traditional coexistence theory implicitly assumes that the invading species is common enough that we can ignore demographic stochasticity but rare enough that it does not compete with itself, even after it has reached a stationary spatial distribution. However, short-distance dispersal of discrete individuals leads to locally dense population clusters, and existing theory breaks down. We have an intuition that when we account for invader–invader competition, shorter-range dispersal should reduce the invader's ability to escape competition, but exactly how does this translate into lower population growth? And how will invader discreteness affect outcomes? We need a way of partitioning the contributions to coexistence, but current modern coexistence theory (MCT) does not apply under these conditions. Here we present a computationally based partitioning method to quantify the contributions to coexistence from different mechanisms, as in MCT. We also build up an intuition for how invader clumping and discreteness will affect these contributions by analyzing a case study, a lattice-based spatial lottery model. We first consider fluctuation-dependent coexistence, partitioning the contributions of variable environment, variable competition, demographic stochasticity, and their correlations and interactions. Our second example examines fluctuation-independent coexistence maintained by a fecundity–survival trade-off, and partitions the contributions to coexistence from interspecific differences in fecundity, in mortality, and in dispersal. We find that demographic stochasticity harms an invader, but only slightly. Localized invader dispersal, on the other hand, can have a strong effect. When invaders are more clumped, they compete with each other more intensely when rare, so they too become limited by environment-competition covariance. More invader clumping also means that variation in competition changes from helping the invader to harming it. More broadly, invader clumping is likely to weaken any coexistence mechanism that relies on the invader escaping competition from the resident, because invader clumping means that the resident is no longer the only source of competition.

中文翻译：

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走向离散和空间的“现代共存理论”

共存的通常理论条件是群落中的每个物种在稀有时可以增加（相互入侵）。传统的共存理论隐含地假设入侵物种足够普遍，我们可以忽略人口统计的随机性，但又足够罕见，以至于它不会与自身竞争，即使它已经达到了固定的空间分布。然而，离散个体的短距离分散导致局部密集的人口集群，现有理论失效。我们有一种直觉，当我们考虑入侵者与入侵者的竞争时，短程扩散应该会降低入侵者逃避竞争的能力，但这究竟如何转化为较低的人口增长？入侵者的离散性将如何影响结果？我们需要一种划分对共存贡献的方法，但当前的现代共存理论 (MCT) 不适用于这些条件。在这里，我们提出了一种基于计算的分区方法来量化不同机制对共存的贡献，如在 MCT 中。我们还通过分析案例研究（一种基于格的空间彩票模型）来建立对入侵者聚集和离散性将如何影响这些贡献的直觉。我们首先考虑依赖波动的共存，划分变量环境、变量竞争、人口随机性及其相关性和相互作用的贡献。我们的第二个例子检查了由繁殖力 - 生存权衡维持的与波动无关的共存，并将对共存的贡献与繁殖力、死亡率和扩散的种间差异分开。我们发现人口随机性对入侵者的伤害很小，但只是轻微的。另一方面，局部入侵者的分散可以产生强烈的影响。当入侵者更加聚集时，它们在稀有时相互竞争更加激烈，因此它们也受到环境竞争协方差的限制。更多的入侵者聚集也意味着竞争的变化从帮助入侵者变为伤害它。更广泛地说，入侵者聚集可能会削弱任何依赖入侵者逃避居民竞争的共存机制，