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.

我们考虑一个标量延迟微分方程\（\ dot {x}（t）=-dx（t）+ f（（1- \ alpha）\ rho x（t- \ tau）+ \ alpha \ rho x（t- 2 \ tau））\）具有瞬时死亡率\（d> 0 \），非线性Rick生殖函数f，在所有发育阶段的存活率\（\ rho \）和比例常数\（\ alpha \在[0，1] \）中，种群经历了滞育发展。我们考虑通过Hopf分叉机制局部生成的周期解分支的全局连续性，并建立周期为\（（（3 \ tau，6 \ tau）\）内的周期解的存在性适用于各种参数值。我们证明了存在周期解的存在不仅在于当局部Hopf分叉发生在正平衡附近时，对于第一临界值\（\ tau ^ * \）附近的延迟\（\ tau \），而且对于所有\（\ tau > \ tau ^ * \）。我们使用基于等价度的全局Hopf分叉理论，结合Li-Muldowney技术的应用，排除周期为（（3 \ tau \）的周期解，从而获得了周期解的这种（全局）存在。。我们进行了一些数值模拟，以说明这种全局连续性完全是由于滞育延迟造成的，因为在给定的生物现实范围内，仅具有正常发育延迟的延迟微分方程的解全部收敛于正平衡。