Non-Gaussian random matrices determine the stability of Lotka-Volterra communities

Abstract

The eigenvalue spectrum of a random matrix often only depends on the first and second moments of its elements, but not on the specific distribution from which they are drawn. The validity of this universality principle is often assumed without proof in applications. In this letter, we offer a pertinent counterexample in the context of the generalised Lotka-Volterra equations. Using dynamic mean-field theory, we derive the statistics of the interactions between species in an evolved ecological community. We then show that the full statistics of these interactions, beyond those of a Gaussian ensemble, are required to correctly predict the eigenvalue spectrum and therefore stability. Consequently, the universality principle fails in this system. Our findings connect two previously disparate ways of modelling complex ecosystems: Robert May's random matrix approach and the generalised Lotka-Volterra equations. We show that the eigenvalue spectra of random matrices can be used to deduce the stability of dynamically constructed (or 'feasible') communities, but only if the emergent non-Gaussian statistics of the interactions between species are taken into account.

Quenched complexity of equilibria for asymmetric generalized Lotka–Volterra equations

“…The motivation for introducing the rescaled parameters (49) is that the equations expressed in terms of these variables can be partially decoupled. Inspecting the equations one notices indeed that the convolutions on the right-hand side of equations ( 53)-( 55) do not depend on y explicitly; on the other hand, in (50) the order parameters appear multiplied by the suitable power of y which appears also to the left-hand side of the equations ( 53)- (55). Therefore, the value of y can be fixed at the end of the calculation using equation (50a), (where the product my is given in (53) and does not depend on y itself), once the values of the other conjugate variables are determined.…”

“…In turn, the order parameters appearing in (50) are themselves functions of the conjugate parameters x 1 , x 2 , β 1 , β 2 , r, φ, given by the following convolutions:…”

“…Finally, the divergence of the dynamics might be on yet another different line. The transition to the unbounded phase can also be characterized in terms of an isolated eigenvalue in the spectrum of the stability matrix: it is expected to occur at those values of parameters for which the isolated eigenvalue crosses zero (in the unique equilibrium phase, this has been argued in [50]).…”

“…This implies that the variable y can be fixed at the end of the calculation, via the identity κy = −x 2 + µ my. The remaining equations are those given in (50), with the expressions multiplying factors of y given by the integral convolutions in the above section (with β 3 = β 2 ). The derivation of the annealed equation ( 69) is completely analogous.…”

Complex systems in ecology: a guided tour with large Lotka–Volterra models and random matrices

“…Formulas (3.10) are recovered in this context. It is possible to study how these correlations impact spectral properties of the restricted matrix [69,70]. Rigorous proofs remain out of reach.…”