Global Attractivity and Extinction for Lotka-Volterra systems with infinite delay and feedback controls

TL;DR

This paper investigates conditions for stability and extinction in complex multi-species Lotka-Volterra models with infinite delays and feedback controls, extending previous results to more general interaction signs without Lyapunov functionals.

Contribution

It provides new sufficient conditions for the existence and attractivity of equilibria in Lotka-Volterra systems with infinite delays, without restrictions on interaction signs or Lyapunov functionals.

Findings

Established criteria for equilibrium existence and stability.

Derived sharper extinction conditions for boundary equilibria.

Extended analysis to non-competitive systems with delays.

Abstract

The paper deals with a multiple species Lotka-Volterra model with infinite distributed delays and feedback controls, for which we assume a weak form of diagonal dominance of the instantaneous negative intra-specific terms over the infinite delay effect in both the population variables and controls. General sufficient conditions for the existence and attractivity of a saturated equilibrium are established. When the saturated equilibrium is on the boundary of $\R^n_+$, sharper criteria for the extinction of all or part of the populations are given. While the literature usually treats the case of competitive systems only, here no restrictions on the signs of the intra- and inter-specific delayed terms are imposed. Moreover, our technique does not require the construction of Lyapunov functionals.