EQUILIBRIUM IN A LARGE LOTKA-VOLTERRA SYSTEM WITH PAIRWISE CORRELATED INTERACTIONS

Maxime Clenet (1, 2) , Hafedh El Ferchichi , Jamal Najim (3, 2)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 Université Gustave Eiffel
3 CNRS - Centre National de la Recherche Scientifique

We study the equilibria of a large Lokta-Volterra system of coupled differential equations in the case where the interaction coefficients form a large random matrix. In the case where this random matrix follows an elliptic model , we study the existence of a (componentwise) positive equilibrium and describe a phase transition for the matrix normalization. If there is no positive equilibrium, we provide conditions on the model parameters for the existence of a stable equilibrium (with vanishing components) and state heuristics to compute the number of positive components of the equilibrium. Lotka-Volterra systems are important in mathematical biology/ theoretical ecology.

Mots clés: Lotka-Volterra system; large random matrices; elliptic model

Domaines: Probabilités [math.PR]; Biodiversité et Ecologie; Ecosystèmes

Maxime Clenet, Hafedh El Ferchichi, Jamal Najim. EQUILIBRIUM IN A LARGE LOTKA-VOLTERRA SYSTEM WITH PAIRWISE CORRELATED INTERACTIONS. Stochastic Processes and their Applications, 2022, 153, pp.423-444. ⟨10.1016/j.spa.2022.08.004⟩. ⟨hal-03681747⟩