Non-Gaussian random matrices predict the stability of feasible Lotka-Volterra communities

50 years ago Robert May sparked the `diversity-stability debate' in ecology. He assumed that the "community matrix" describing the interactions in an ecosystem has random entries. May concluded from the analysis of the eigenvalues of these matrices that large ecosystems would be less stable than smaller ones. A decade-long debate ensued, including a number of recent high-profile papers extending May’s work to matrices with more structure. Much of the work in this area relies on random matrix theory, that is, techniques for the calculation of the eigenvalue spectrum of ensembles of random matrices. One major criticism of May’s approach is that it does not address the question how the ecological community arises from a dynamics, and whether the equilibrium is `feasible' or not.

Here, I will first discuss recent work in which we calculate the bulk and outlier eigenvalues of the most general Gaussian ensemble of random matrices, which does not systematically give preference to any species over another. I will then show how May’s approach can be used for feasible communities arising from the survivors in a dynamic Lotka-Volterra model with random interactions. The ensemble of interactions among extant species turns out to be non-Gaussian, even if the original interaction matrix among all species is Gaussian. I will then demonstrate that random-matrix universality does not apply, i.e. a Gaussian calculation fails to predict the leading eigenvalue correctly. I will show how tools from the theory of disordered systems can be used to account for non-Gaussian features of the interactions, and to obtain the spectra of the community matrix of survivors. The stability criteria from these eigenvalue spectra agree with those obtained from the Lotka-Volterra equations. Hence, we have demonstrated how May’s random-matrix approach can be used to characterise the stability of feasible equilibria. Feasibility is encoded in the higher-order non-Gaussian statistics of the community matrices arising from the survivors in Lotka-Volterra systems.