The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate

The consensus that complexity begets stability in ecosystems was challenged in the seventies, a result recently extended to ecologically-inspired networks. The approaches assume the existence of a feasible equilibrium, i.e. with positive abundances. However, this key assumption has not been tested. We provide analytical results complemented by simulations which show that equilibrium feasibility vanishes in species rich systems. This result leaves us in the uncomfortable situation in which the existence of a feasible equilibrium assumed in local stability criteria is far from granted. We extend our analyses by changing interaction structure and intensity, and find that feasibility and stability is warranted irrespective of species richness with weak interactions. Interestingly, we find that the dynamical behaviour of ecologically inspired architectures is very different and richer than that of unstructured systems. Our results suggest that a general understanding of ecosystem dynamics requires focusing on the interplay between interaction strength and network architecture.Author summary: Finding conditions leading to stable equilibria, i.e. equilibria robust to small perturbations, is a key feature in understanding how persistent are ecological systems. The number of species involved in the ecosystem as well as the number of relations between them has historically been pointed out to be the prominent ingredients determining stability. The latter is however always computed for a particular equilibrium, which needs and is usually assumed to contain only positive abundances. This assumption has until now only barely been tested in relation to system size. We provide here analytical results complemented by numerical simulations which show that it is almost impossible to find equilibria containing only positive abundances in species rich systems or, if so, the parameters are constrained to lead to ecologically-nonsensical abundances in unstructured systems. Interestingly, only ecologically-inspired architectures permit one to obtain feasible equilibria with a rich and sensible dynamical behavior.