Dynamics of the logistic equation with delay
type: Item
instance of: scholarly article
title: Dynamics of the logistic equation with delay
zbMATH Open document ID: 1335.34110
DOI: 10.1134/S0001434615070093
published in: Mathematical Notes
publication date: 2015-10-26T00:00:00Z
review text: This paper is about the dynamics of the equation \[ u'(t) = \lambda[1-\alpha u(t) -(1-\alpha)u(t-1)]u(t) \] for \(1 < \alpha <1\). It is the generalisation of the author's work on Hutchinson equation. It is known from the roots of the characteristic equation \(\mu = -\lambda[\alpha +(1-\alpha)e^{-\mu}]\) that all positive solutions tend to \(u_0 = 1\) as \(t\to\infty\) if \(\alpha > 1/2\). But if \(0< \alpha <1/2\), there is a bifurcation point \(\lambda = \lambda_0(\alpha)\). Under some conditions, it is shown that the equation has an asymptotically orbitally stable periodic solution close to \(u_0\). Further, global dynamics is investigated and asymptotics of the relaxation cycle for large \(\lambda\) is discussed.
reviewed by: Zhanyuan Hou
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K17
Mathematics Subject Classification ID: 34K26
zbMATH DE Number: 6498243
zbMATH Keywords: logistic equation with delay
zbMATH Keywords: Andronov-Hopf bifurcation
zbMATH Keywords: Hutchinson equation
zbMATH Keywords: relaxation cycle
zbMATH Keywords: periodic solution
author: S. A. Kashchenko
MaRDI profile type: Publication
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full work available at URL: s0001434615070093
OpenAlex ID: W1542299752
description: scientific article; zbMATH DE number 6498243