Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls

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.

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“…Generally, the dependency will be on a certain segment of prehistory of the process, and, in certain cases, on the whole of it, leading to infinite delays. This is the reason why stability of systems with a distributed delay has been intensively investigated, let us mention some recent publications [1,4,17,18,20,24,25,28,30,31], see also references therein. In biological models [21,22,32], a distributed delay sometimes is accepted by default.…”