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

TL;DR

This work develops 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, and proves that the possibility of stable coexistence in general ecologies is determined by similarly rare “irreducible ecologies”.

Abstract

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.

Figures (9)

• Figure 1: Lotka--Volterra ecological dynamics and impossible ecologies. (a) Example of an ecological topology on $N=5$ species, defined below. (b) Mathematical definition of the Lotka--Volterra model on $N$ species: the dynamics of the vector $\boldsymbol{x}$ of the population abundances of the $N$ species are determined by a vector $\boldsymbol{A}$ of growth rates and a matrix $\mathbfsfit{B}$ of interactions strengths. (c) The ecological topology is defined by the signs of $A_i\gtrless0$ and $B_{ij}\gtrless 0$ for $j\neq i$, defining competitive, mutualistic and directed predator-prey interactions between species. (d) List of the six non-trivial ecological topologies on $N=2$ species: one topology ("obligate mutualism", highlighted) is an impossible ecology in which stable and feasible coexistence is non-trivially impossible. (e) List of the four trivial ecologies on $N=2$ species: the highlighted species are dying on their own and have only deleterious interactions, prohibiting stable and feasible coexistence of all species.
• Figure 1: List of all 70 non-trivial ecologies on three species. Impossible and irreducible ecologies are highlighted by dashed and solid boxes, respectively.
• Figure 2: Three-species ecologies: irreducible ecologies. (a) Swarm plot of the probability $\mathbb{P}$ of stable and feasible coexistence for all 70 non-trivial three-species ecologies. Impossible ecologies have $\log_{10}{\mathbb{P}}=-\infty$. (b) List of the four impossible ecologies on three species. (c) Example of an irreducible ecology, decomposed into its three trivial or impossible two-species subecologies. (d) Cumulative distribution function of $\mathbb{P}$ for all 70 non-trivial three-species ecologies and restricted to the six irreducible ecologies. (e) List of the six irreducible ecologies on three species.
• Figure 2: List of the 29 non-extension ecologies that are found among the 2340 non-trivial ecologies of four species . (a) List of 11 ecologies asserted to be impossible; as discussed in the main text, this is an upper bound based on numerical calculations, and we do not have an analytical proof of impossibility for any of these ecologies. (b) List of the remaining 18 non-extension ecologies of four species, shown to be irreducible by direct non-uniform sampling of feasible equilibria yielding parameter values allowing stable and feasible coexistence.
• Figure 3: Enumeration of ecologies. (a) Example of a five-species ecology as a complete graph with two-coloured nodes and four-coloured edges. (b) Image of this ecology under a bijection onto directed graphs with allowed self-loops. See text for definition of the bijection. (c) Plot of the number of all and non-trivial ecologies against the number $N$ of species, showing a combinatorial explosion. Inset: fraction of trivial ecologies plotted against $N$.

Theorems & Definitions (2)

• Theorem
• Theorem