Nonexistence of periodic solutions in delayed Lotka--Volterra systems.
type: Item
instance of: scholarly article
title: Nonexistence of periodic solutions in delayed Lotka--Volterra systems.
zbMATH Open document ID: 1091.34037
DOI: 10.1016/S1468-1218(01)00017-7
author: E. Beretta
author: Ryusuke Kon
author: Yasuhiro Takeuchi
published in: Nonlinear Analysis. Real World Applications
publication date: 2003-05-25T00:00:00Z
review text: The authors consider Lotka-Volterra systems with distributed delays \[ {dx_i\over dt}= x_i\Biggl(e_i+ \sum^n_{j=1} a_{i_j} x_j+ \sum^n_{j=1} \gamma_{ij} \int^t_{-\infty} f_{ij}(t- \tau) x_j(\tau)\,d\tau\Biggr),i=1,2,\dots, n, \tag{*} \] where the delay kernels \(f_{ij}\) are such continuous nonnegative functions that \((*)\) can be transformed in an extended system of \(n+p\) autonomous ordinary differential equations. Exploiting the fact that if the extended system has no nonconstant periodic solution, then also system \((*)\) has no nonconstant periodic solution, the authors formulate conditions that the extended autonomous system has no nonconstant periodic solution. Hereby, extended negative criteria of Bendixson by \textit{M. Y. Li} and \textit{J. S. Muldowney} [J. Differ. Equations 106, No.~1, 27--39 (1993; Zbl 0786.34033)] play a crucial role. The case \(n= 2\), \(p\leq 2\) is investigated in detail.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 1915792
zbMATH Keywords: Li and Muldowney's criterion
zbMATH Keywords: Nonexistence of periodic solutions
zbMATH Keywords: Delayed Lotka--Volterra system
zbMATH Keywords: Butler--McGehee's lemma
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915792