A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations
type: Item
instance of: scholarly article
title: A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations
zbMATH Open document ID: 1312.34087
published in: Publicacions Matemàtiques
publication date: 2014-12-19T00:00:00Z
review text: Volterra introduced systems of the form \[ \dot x_j = \varepsilon_j x_j + \sum_{k=1}^n a_{jk} x_j x_k, j = 1, \dots, n, \] in the context of the interaction of \(n\) biological species. Attention is confined to the restriction to \({\mathbb R}_+^n\). Such a system is ``stably dissipative'' if for it and any nearby system there exists a diagonal matrix \(D > 0\) such that \(A D \leqslant 0\), where \(A\) is the \(n \times n\) interaction matrix \(A = (a_{jk})\). This survey is organized into two parts. In the first one, the author discusses such systems and proves that if there is a singularity then the system has a global attractor on which the dynamics are Hamiltonian. In the second part systems with delays, also introduced by Volterra, are discussed. Examples are included that illustrate the complexity of the dynamics possible in such systems.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D45
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 6381941
zbMATH Keywords: Lotka-Volterra
zbMATH Keywords: stably dissipative system
author: Waldyr Muniz Oliva
MaRDI profile type: Publication
full work available at URL: publmat extra14 21
OpenAlex ID: W2022485483
DOI: 10.5565/PUBLMAT_EXTRA14_21
description: scientific article; zbMATH DE number 6381941