Abstract We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential system with delay-dependent parameters. By choosing the normal development time delay as a bifurcation parameter, we analyze the stability switches of the positive equilibrium, and examine the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium. Under some technical conditions, we show that global Hopf branches are bounded and connected by a pair of Hopf bifurcation values. This allows us to show that diapause can lead to the occurrence of multiple stability switches, coexistence of two stable limit cycles, among other rich dynamical behaviours.

中文翻译：

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延迟微分方程中的复杂动力学，具有滞育的蜱生长的两个延迟

摘要 我们考虑了具有滞育的蜱种群的延迟微分方程，该方程源自年龄结构的种群模型，由于正常和滞育介导的发育具有两个时间滞后。我们推导了生物学重要平衡的全局渐近稳定性的阈值条件，并给出了具有延迟相关参数的延迟微分系统中 Hopf 分岔出现的一般几何标准。通过选择正常的发展时间延迟作为分岔参数，我们分析了正平衡的稳定性转换，并从正平衡检查周期解的Hopf分岔的开始和终止。在某些技术条件下，我们证明了全局 Hopf 分支由一对 Hopf 分岔值有界和连接。