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Coexistence holes characterize the assembly and disassembly of multispecies systems


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.

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Fig. 1: Coexistence holes characterize discontinuities in assembly and disassembly processes.
Fig. 2: Assembly and disassembly holes in classic ecological models.
Fig. 3: Homology reveals the essential structure of assembly and disassembly hypergraphs in terms of their skeletons.
Fig. 4: A structuralist approach to understanding the emergence of coexistence holes in the Lotka–Volterra model.
Fig. 5: Assembly and disassembly holes in empirical ecological systems.

Data availability

All of the data analysed in this study are publicly available.

Code availability

The code supporting the results is archived in the GitHub repository at


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We gratefully acknowledge financial support from CONACyT grant number A1-S-13909 (M.T.A.) and NSF grant number DEB-2024349 (S.S.).

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M.T.A. conceived of the idea of coexistence holes. M.T.A., C.S. and S.S. designed and realized the study. M.T.A., L.M. and A.K. performed the theoretical analysis. M.T.A., C.S. and S.S. wrote the manuscript. A.K. and M.T.A. wrote the software package to identify coexistence holes. All authors revised the manuscript.

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Correspondence to Marco Tulio Angulo or Serguei Saavedra.

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Peer review information Nature Ecology & Evolution thanks Stefano Allesina, Andrew Letten and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Angulo, M.T., Kelley, A., Montejano, L. et al. Coexistence holes characterize the assembly and disassembly of multispecies systems. Nat Ecol Evol 5, 1091–1101 (2021).

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