Structural stability of invasion graphs for Lotka--Volterra systems

TL;DR

This paper analyzes the invasion graph structure in Lotka-Volterra systems with stable matrices, showing edges represent heteroclinic connections and exploring the robustness of this structure under parameter perturbations.

Contribution

It establishes that invasion graph edges correspond to heteroclinic connections and introduces a concept of structural stability in ecological models.

Findings

Edges of invasion graph represent all heteroclinic connections.

The invasion graph structure is stable under parameter perturbations.

A new definition of structural stability in ecology is proposed.

Abstract

In this paper, we study in detail the structure of the global attractor for the Lotka--Volterra system with a Volterra--Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in [19] and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics.