The geometry of coexistence in large ecosystems (1507.05337v2)

Abstract: The role of species interactions in controlling the interplay between the stability of an ecosystem and its biodiversity is still not well understood. The ability of ecological communities to recover after a small perturbation of the species abundances (local asymptotic stability) has been well studied, whereas the likelihood of a community to persist when the interactions are altered (structural stability) has received much less attention. Our goal is to understand the effects of diversity, interaction strenghts and ecological network structure on the volume of parameter space leading to feasible equilibria, i.e., ones in which all populations have positive abundances. We develop a geometrical framework to study the range of conditions necessary for feasible coexistence in both mutualistic and consumer-resource systems. Using analytical and numerical methods, we show that feasibility is determined by just a handful of quantities describing the interactions, yielding a nontrivial complexity-feasibility relationship. Analyzing more than 100 empirical networks, we show that the range of coexistence conditions in mutualistic systems can be analytically predicted by means of a null model of random interactions, whereas food webs are characterized by smaller coexistence domains than those expected by chance. Finally, we characterize the geometric shape of the feasibility domain, thereby identifying the direction of perturbations that are more likely to cause extinctions. Interestingly, the structure of mutualistic interactions leads to very heterogeneous responses to perturbations, making those systems more fragile than expected by chance.