Equilibrium in a large Lotka–Volterra system with pairwise correlated interactions

Abstract

Consider a Lotka–Volterra (LV) system of coupled differential equations: ẋk=xk(rk−xk+(Bx)k),x=(xk),1≤k≤n,where r=(rk) is a n×1 vector and B a n×n matrix. Assume that the interaction matrix B is random and follows the elliptic model: B=1αnA+μn1n1nT,where A=(Aij) is a n×n matrix with N(0,1) entries satisfying the following dependence structure (i) the entries Aij on and above the diagonal are i.i.d., (ii) for i<j each vector (Aij,Aji) is standard Gaussian with covariance ρ, and independent from the other entries; vector 1n stands for the n×1 vector of ones. Parameters α,μ are deterministic and may depend on n.Leveraging on Random Matrix Theory, we analyze this LV system as n→∞ and study the existence of a positive equilibrium. This question boils down to study the existence of a (componentwise) positive solution to the linear equation: x=r+Bx,depending on B’s parameters (α,μ,ρ), a problem of independent interest in linear algebra.In the case where no positive equilibrium exists, we provide sufficient conditions for the existence of a unique stable equilibrium (with vanishing components), and following Bunin (2017), present a heuristics estimating the number of positive components of the equilibrium and their distribution.The existence of positive equilibria for large Lotka–Volterra systems has been raised in Dougoud et al. (2018), and addressed in various contexts by Bizeul and Najim (2021) and Akjouj and Najim (2022).Such LV systems are widely used in mathematical biology to model populations with interactions, and the existence of a positive equilibrium known as a feasible equilibrium corresponds to the survival of all the species xk within the system.