Impossible ecologies: Interaction networks and stability of coexistence in ecological communities

Does an ecological community allow stable coexistence? Identifying the general principles that determine the answer to this question is a central problem of theoretical ecology. Random matrix theory approaches have uncovered the general trends of the effect of competitive, mutualistic, and predator-prey interactions between species on stability of coexistence. However, an ecological community is determined not only by the counts of these different interaction types, but also by their network arrangement. This cannot be accounted for in a direct statistical description that would enable random matrix theory approaches. Here, we therefore develop a different approach, of exhaustive analysis of small ecological communities, to show that this arrangement of interactions can influence stability of coexistence more than these general trends. We analyse all interaction networks of $N\leqslant 5$ species with Lotka-Volterra dynamics by combining exact results for $N\leqslant 3$ species and numerical exploration. Surprisingly, we find that a very small subset of these networks are “impossible ecologies”, in which stable coexistence is non-trivially impossible. We prove that the possibility of stable coexistence in general ecologies is determined by similarly rare “irreducible ecologies”. By random sampling of interaction strengths, we then show that the probability of stable coexistence varies over many orders of magnitude even in ecologies that differ only in the network arrangement of identical ecological interactions. Finally, we demonstrate that our approach can reveal the effect of evolutionary or environmental perturbations of the interaction network. Overall, this work reveals the importance of the full structure of the network of interactions for stability of coexistence in ecological communities.

💡 Research Summary

This paper presents a groundbreaking analysis of how the detailed architecture of interaction networks, beyond just the proportions of interaction types, fundamentally determines the possibility of stable coexistence in ecological communities. Moving beyond the statistical approaches of random matrix theory, the authors employ an exhaustive analysis of small communities (N≤5 species) governed by Lotka-Volterra dynamics to uncover principles that are otherwise obscured in large-system limits.

The core methodology involves enumerating all possible “ecological topologies”—defined by the signs of species’ intrinsic growth rates and their pairwise interactions (competition, mutualism, predation)—and estimating the probability (P) of achieving a feasible (all species present) and stable coexistence equilibrium via random sampling of interaction strengths. For N=2 and N=3, analytical proofs complement numerical exploration.

The study yields several profound discoveries. First, it identifies “impossible ecologies”: rare network structures where stable, feasible coexistence is provably impossible for any choice of quantitative interaction strengths, despite each species having a potential source of growth (e.g., obligate mutualism between two species). Second, it introduces the concept of “irreducible ecologies”: networks that are themselves possible but contain no possible sub-ecology (obtained by removing one species). A key theorem proves that any non-trivial extension of a possible ecology (by adding a species) remains possible. This implies that the possibility of coexistence for any larger community is ultimately dictated by the set of these rare irreducible ecologies, which act as fundamental “building blocks” for stability.

For N=4 and N=5 species, large-scale numerical analysis reveals the staggering impact of network structure. The probability of stable coexistence varies over many orders of magnitude (exceeding a factor of 10^12) across different topologies. Crucially, this massive variation persists even among ecologies with identical counts of competitive, mutualistic, and predatory interactions, demonstrating that the specific arrangement of these links is a critical determinant of stability, independent of their gross statistics.

The findings suggest that the fraction of impossible and irreducible ecologies decays exponentially as the number of species increases, hinting that network structures enabling stable coexistence in large communities may be highly specific and non-generic. Overall, this work shifts the paradigm in theoretical ecology by establishing that the microscopic details of network wiring—which have been largely ignored in statistical approaches—are not just minor details but primary drivers of coexistence stability. It provides a new theoretical framework for understanding how evolutionary changes or environmental perturbations that rewire interaction networks could dramatically alter a community’s fate.