# American Institute of Mathematical Sciences

July  2019, 24(7): 3089-3114. doi: 10.3934/dcdsb.2018302

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

 1 Department of Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan 2 Departamento de Matemática and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal

* Corresponding author: Teresa Faria

Received  April 2018 Revised  June 2018 Published  October 2018

Fund Project: Professor Yoshiaki Muroya passed away in October 2015, while the research for this paper was being conducted. The second author wishes to dedicate this paper to his memory.

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.

Citation: Yoshiaki Muroya, Teresa Faria. Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3089-3114. doi: 10.3934/dcdsb.2018302
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