Global continuation of periodic oscillations to a diapause rhythm
type: Item
instance of: scholarly article
title: Global continuation of periodic oscillations to a diapause rhythm
author: Xue Zhang
author: Francesca Scarabel
author: Xiang-Sheng Wang
author: Jianhong Wu
published in: Journal of Dynamics and Differential Equations
publication date: 2022-11-14T00:00:00Z
review text: In this work, the authors consider a scalar delay differential equation \[ \dot{x}(t)=-dx(t)+f((1-\alpha)\rho x(t-\tau)+ \alpha\rho x(t-2\tau)) \] with an instant mortality rate \(d>0\), the nonlinear Rick reproductive function \(f\), a survival rate during all development stages \(\rho\), and a proportion constant \(\alpha\in[0,1]\) with which a population undergoes a diapause development. The global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism is discussed and the existence of periodic solutions with periods within \((3\tau,6\tau)\) for a wide range of parameter values is explored. This paper shows the existence of periodic solutions not only for the delay \(\tau\) near the first critical value \(\tau^\ast\) when a local Hopf bifurcation takes place near the positive equilibrium, but for all \(\tau>\tau^\ast\). The authors obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li-Muldowney technique to rule out periodic solutions with period \(3\tau\). Up to now, there are few works on this topic. The acquired outcomes and the exploration idea enrich the bifurcation theory of delayed differential equation.
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 34K18
Mathematics Subject Classification ID: 34K60
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 7616286
zbMATH Keywords: delay differential equation
zbMATH Keywords: ticks
zbMATH Keywords: diapause
zbMATH Keywords: periodic solution
zbMATH Keywords: global Hopf bifurcation
reviewed by: Changjin Xu
describes a project that uses: MATCONT
MaRDI profile type: Publication
full work available at URL: s10884-020-09856-1
OpenAlex ID: W3034824007
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DOI: 10.1007/S10884-020-09856-1
description: scientific article; zbMATH DE number 7616286