标题： 时滞自调控的周期性是一种捕食者-猎物过程 显示英文标题

标题： Periodicity in delayed self-regulation is a predator-prey process

摘要： 我们统一了两种不同的周期性机制：时滞自调控和平面捕食者-猎物反馈。 我们考虑标量时滞微分方程 $\dot x(t) = rf(x(t), x(t - 1))$，其中$f$在时滞分量上是单调的。 由于单调时滞反馈系统的庞加莱-本迪克森定理，当时滞参数$r$增大时，典型的全局动力学表现为存在周期轨道。 本文中，我们证明了，随着时滞的变化，每个连通分支的周期轨道构成一个环面，其全局坐标由对应的周期解的时间和振幅给出。 在每个环面上，变量$x(t)$和$x(t - 1)$满足一个可积的常微分方程，并且该方程满足捕食者-猎物反馈关系。 此外，我们完全通过由底层捕食者-猎物系统生成的两个时间映射来刻画时滞微分方程的周期解集合。 显示英文摘要

摘要： We unify two different periodicity mechanisms: delayed self-regulation and planar predator-prey feedback. We consider scalar delay differential equations $\dot x(t) = rf(x(t), x(t - 1))$ where $f$ is monotone in the delayed component. Due to a Poincar\'{e}--Bendixson theorem for monotone delayed feedback systems, the typical global dynamics present periodic orbits as the delay parameter $r$ increases. In this article, we show that, as we vary the delay, each connected component of periodic orbits is an annulus with global coordinates given by the time and the amplitude of the corresponding periodic solutions. On each annulus, the variables $x(t)$ and $x(t - 1)$ solve an integrable ordinary differential equation that satisfies a predator-prey feedback relation. Moreover, we completely characterize the set of periodic solutions of the delay differential equation in terms of two time maps generated by the underlying predator-prey system.