Global continuation of periodic oscillations to a diapause rhythm
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.
scientific article; zbMATH DE number 7616286
scientific article