A central goal of ecological research has been to understand the limits on the maximum number of species that can coexist under given constraints. However, we know little about the assembly and disassembly processes under which a community can reach such a maximum number, or whether this number is in fact attainable in practice. This limitation is partly due to the challenge of performing experimental work and partly due to the lack of a formalism under which one can systematically study such processes. Here, we introduce a formalism based on algebraic topology and homology theory to study the space of species coexistence formed by a given pool of species. We show that this space is characterized by ubiquitous discontinuities that we call coexistence holes (that is, empty spaces surrounded by filled space). Using theoretical and experimental systems, we provide direct evidence showing that these coexistence holes do not occur arbitrarily—their diversity is constrained by the internal structure of species interactions and their frequency can be explained by the external factors acting on these systems. Our work suggests that the assembly and disassembly of ecological systems is a discontinuous process that tends to obey regularities.

生态研究的一个中心目标是了解在给定限制下可以共存的最大物种数量的限制。然而，我们对社区可以达到这样一个最大数量的组装和拆卸过程知之甚少，或者这个数量实际上是否可以在实践中达到。这种限制部分是由于进行实验工作的挑战，部分是由于缺乏一种可以系统地研究这些过程的形式主义。在这里，我们引入了一种基于代数拓扑和同调理论的形式主义来研究给定物种池形成的物种共存空间。我们表明，这个空间的特点是无处不在的不连续性，我们称之为共存孔（即，被填充空间包围的空白空间）。使用理论和实验系统，我们提供了直接证据表明这些共存漏洞不是任意发生的——它们的多样性受到物种相互作用的内部结构的限制，它们的频率可以通过作用于这些系统的外部因素来解释。我们的工作表明，生态系统的组装和拆卸是一个不连续的过程，往往服从规律。