几类非单调系统行波解的渐近性和稳定性研究

This project mainly studies asymptotics and stability and related problems of traveling wave solutions for three classes of non-monotone systems, including delayed continuous and discrete reaction diffusion system, integro-difference system and delayed integro-differntial system. By analyzing the spectrum of the linear operator perturbed with a small delay, we will give the conditions which the asymptotic spreading speed and the minimal wave speed of the delayed systems are determined by the linearized systems at the unstable equilibrium, and then establish the asymptotic spreading speed and the minimal wave speed. For non-monotone and non-cooperative system, by constructing a pair of appropriate weak monotone systems and a pair of non-monotone upper and lower solutions by making use of cross monotonicity of them, we will first establish the existence and asymptotic behavior of traveling wave solutions, then by using the asymptotic behavior of weak monotone system and comparison principle, we will obtain the existence of the entire solutions, the weighted energy method will be extended and applied to study the stability of traveling wave solutions by selecting the appropriate weighted function at last. By means of constructing a pair of suitable upper and lower solutions and using the asymptotic behavior of them, and combining comparison principle and the sliding method, we establish the directions of the unstable manifold and the stable manifold which the traveling wave solutions of competition system at infinity to flow out and flow into. To illustrate our results, we will give some examples to application and realize numerical simulation. The results of this project will improve and extend the previous.

本项目主要深入研究时滞连续与离散反应扩散系统、积分-差分系统和时滞积分-微分系统等三类非单调系统行波解的渐近性和稳定性及其相关问题. 通过分析无时滞系统在小时滞扰动下对应算子的谱, 给出时滞系统的渐近波速和最小波速由不稳定平衡态处的线性化系统所确定的条件, 并确立时滞系统的渐近波速和最小波速. 对于非单调且非合作系统, 通过构造一对恰当的弱单调系统, 并利用其交叉单调性, 构造一对非单调的上下解, 得到行波解的存在性; 利用弱单调系统行波解的渐近性和比较原理, 得到整体解的存在性; 通过选择合适的加权函数, 将加权能量法推广并应用到行波解的稳定性研究上. 通过构造合适的上下解, 利用其渐近行为并结合强比较原理和滑行方法, 确定竞争系统单稳行波解在无穷远处流出和进入不稳定流形和稳定流形的方向. 同时, 对每一类系统给出实例加以应用, 并实现数值模拟. 本项目的研究结果将改进和延拓已有结果.

本项目从以下四个方面研究了三类非单调系统的行波解及其渐近性态：一是研究了连续反应扩散系统行波解的存在性、渐近性和整体解的存在性，利用上下解方法和Schauder不动点定理得到行波解的存在性，分析了时滞对波速的影响，结合行波解的渐近性得到了整体解的存在性；二是研究了离散反应扩散系统和离散时间差分系统行波解的渐近性和唯一性，并分析了时滞对渐近性的影响，利用强比较原理和滑行方法得到了行波解的唯一性，三是研究了非局部扩散系统的行波解及其渐近性态，将滑行方法延拓到证明非局部微分系统行波解的唯一性上；四是研究了非稠定半线性随机发展方程的稳定性, 并将结果应用到随机年龄结构模型。本项目的研究结果改进或延拓了部分已有结果。