For a family of $n$-dimensional periodic delay differential equations which encompasses a broad set of models used in structured population dynamics, the existence of a positive periodic solution is obtained under very mild conditions. The proof uses the Schauder fixed point theorem and relies on the permanence of the system. A general criterion for the existence of a positive periodic solution for Nicholson's blowflies periodic systems (with both distributed and discrete time-varying delays) is derived as a simple application of our main result, generalizing the few existing results concerning multi-dimensional Nicholson models.
In the case of a Nicholson system with discrete delays all multiples of the period, the global attractivity of the positive periodic solution is further analyzed, improving results in recent literature.
Monotone traveling wavefronts of the KPP-Fisher delayed equation
在2000年代初，Gourley（2000），Wu et al.（2001），Ashwin et al.（2002）开始研究延迟Kolmogorov-Petrovskii-Piskunov-Fisher方程的正波前。此后，该模型成为单稳态时滞反应扩散方程行波解研究中最热门的研究对象之一。本文给出了KPP-Fisher方程中单调波的存在唯一性问题的完全解。我们表明，每个单调行波可以通过迭代过程中发现。所提出的方法是基于使用特殊的单调积分算子（这是不同于通常的吴邹算子）和适当的上下解与他们相关联。对无穷远处的最终行进锋的渐近展开的分析是我们方法的另一个关键组成部分。
