Global Continuation of Periodic Oscillations to a Diapause Rhythm

Abstract

We 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 population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within $$(3\tau , 6\tau )$$ for a wide range of parameter values. We show this existence of periodic solutions not only for the delay $$\tau $$ near the first critical value $$\tau ^*$$ when a local Hopf bifurcation takes place near the positive equilibrium, but for all $$\tau >\tau ^*$$. We 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 $$3\tau $$. We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium.