Attractivity of Saturated Equilibria for Lotka-Volterra Systems with Infinite Delays and Feedback Controls

TL;DR

This paper demonstrates that feedback controls do not affect the attractivity of saturated equilibria in Lotka-Volterra systems with infinite delays, improving understanding of their stability without additional control conditions.

Contribution

The authors show that feedback controls do not influence the attractivity of saturated equilibria in systems with infinite delays, extending previous stability results.

Findings

Feedback controls do not alter attractivity of saturated equilibria.

Reduced systems maintain global dynamics features.

Improved conditions for stability in delayed systems.

Abstract

In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original system is reduced to a lower dimensional one which maintains its global dynamics features.