Feasibility of sparse large Lotka-Volterra ecosystems

Consider a large ecosystem (foodweb) with n species, where the abundances follow a Lotka–Volterra system of coupled differential equations. We assume that each species interacts with d=d_n other species and that their interaction coefficients are independent random variables. This parameter d reflects the connectance of the foodweb and the sparsity of its interactions especially if d is much smaller that n . We address the question of feasibility of the foodweb, that is the existence of an equilibrium solution of the Lotka-Volterra system with no vanishing species. We establish that for a given range of d , namely d∝ n or d≥log (n) with an extra condition on the sparsity structure, there exists an explicit threshold depending on n and d and reflecting the strength of the interactions, which guarantees the existence of a positive equilibrium as the number of species n gets large. From a mathematical point of view, the study of feasibility is equivalent to the existence of a positive solution x_n (component-wise) to the equilibrium linear equation: x_n = 1_n + M_n x_n , where 1_n is the n× 1 vector with components 1 and M_n is a large sparse random matrix, accounting for the interactions between species. The analysis of such positive solutions essentially relies on large random matrix theory for sparse matrices and Gaussian concentration of measure. The stability of the equilibrium is established. The results in this article extend to a sparse setting the results obtained by Bizeul and Najim in Bizeul and Najim ( 2021 ).