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

Nearly 50 years ago Robert May sparked the “diversity-stability debate” in ecology. Assuming that the so-called community matrix has random entries May claims that an increased number of species promotes instability. A decade-long debate has 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. I will explain what tools from disordered systems can deliver for the dynamics of random Lotka-Volterra models. In this context, I will also discuss recent work on the spectra of random matrices with generalised correlations, and show how May’s approach can be justified retrospectively, provided one focuses on the right ensemble of matrices. This ensemble turns out to be non-Gaussian, and I will demonstrate how tools from disordered systems can be used to calculate their spectra. Universality does not apply in this context, higher-order non Gaussian statistics need to be accounted for to properly predict the leading eigenvalue of community matrices arising from random Lotka-Volterra dynamics.

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