时空周期单调半流的传播理论及应用研究

Periodic environment is common in the nature world, such as seasonal succession and ebb and flow. How does a periodic environment influence the species' propagation patterns? Traveling waves defined by one-variable function can be used to study the propagation problem in homogeneous environment. In time or space periodic environment it is not appropriate but pulsating waves defined by two-variable function is. In time and space periodic environment, pulsating wave is no longer appropriate because of double periodicity. Is there a more generalized wave so that it can characterize the influence of time-space periodicity on propagation patterns? This proposal intends to investigate such a problem from a viewpoint of monotone dynamical systems. It includes three main parts: (1) we refine the propagation characteristics as much as possible from several different aspects including numerics and use them to define a generalized wave; (2) we decompose the time-space periodicity into two addible periodicity from an evolution viewpoint and then establish the theory of such a generalized wave for monotone dynamical systems with monostable and bistable structures, respectively; (3) we apply the theoretical results to some concrete models, such as a delayed reaction-diffusion model for single species and a two-speices competition model. The study of this project needs new ideas and methods and it is helpful for the development of the propagation theory of monotone dynamical systems.

自然界中周期环境是常见的，如季节变化和潮起潮落，那么周期环境对种群的传播模式有什么影响呢？在齐性环境下，由一元函数所定义的行波可以用来研究传播问题；在时间或者空间周期的环境下，行波不再适用，但可以用由二元函数所定义的脉动波来研究此问题；在时空均是周期的环境下，双周期性使得脉动波不再适用。那么是否存在一种比脉动波更广义的波，且能刻画时空周期性对传播模式的影响呢？本项目拟从单调动力系统角度来探讨该问题。主要内容有三部分：（1）从数值等方面提炼足够多的传播特征，并用这些特征定义一个广义的波；（2）从演化的角度把时空周期性拆成可叠加的两个周期性，对具有单稳定和双稳定结构的两类单调半流分别建立关于该广义波的存在性理论；（3）把理论结果应用到一些具体的模型，如单种群时滞反应扩散模型和两种群竞争模型。该项目的研究需要新的思想和方法，有助于单调动力系统传播理论的发展。

自然界中周期环境是常见的，如季节变化和潮起潮落，那么周期环境对种群的传播模式有 什么影响呢?在齐性环境下，由一元函数所定义的行波可以用来研究传播问题;在时间或者空 间周期的环境下，行波不再适用，但可以用由二元函数所定义的脉动波来研究此问题;在时空 均是周期的环境下，双周期性使得脉动波不再适用。那么是否存在一种比脉动波更广义的波， 且能刻画时空周期性对传播模式的影响呢? 本项目找到了一个能够刻画该影响的波，称之为“几乎脉动波”，英文译为“almost pulsating wave”，并对单稳型单调半流建立了该波的存在性理论。结合其他方法和技巧，把相关理论了应用到几个周期环境下的种群模型。