本项目计划研究几乎周期介质中扩散系统传播现象和动力学。包括以下三个方面的内容：.1. 研究空间几乎周期介质中非局部扩散KPP方程传播速度。具体地，研究非局部扩散方程的广义主特征值理论、Harnack不等式和均质化方法。进一步，通过把问题转化环面旋转半群的轨道动力学来证明非局部扩散核的遍历性质可以保证传播速度的存在性。.2.研究空间几乎周期介质中随机扩散和非局部扩散KPP方程Cauchy问题的行波解。具体地，借鉴薛定谔算子谱理论研究方法研究随机扩散和非局部扩散KPP方程的主特征值问题。进一步主特征函数和自由边界问题已知的行波解构造上下解得到Cauchy问题行波解。.3.探索时空不均衡（几乎周期）介质中扩散系统传播现象。探索研究相应广义主特征值理论，均质化方法。研究相应KPP方程传播速度和行波解。

We plan to study the dynamics and spreading phenomena of diffusion systems in almost periodic media. Three problems will be considered:.1. Spreading speed of nonlocal KPP equations in almost periodic media. To solve this problem, we will first study the theory of generalized principal eigenvalues, Harnack inequality and homogenization of the nonlocal diffusion equations. Then we will prove the ergodic property of the diffusion kernel can guarantee the existence of the spreading speeds through transfer the problem to the dynamics of orbits of some subgroups of rotation on the torus..2. Traveling waves of KPP equations with random diffusion or nonlocal diffusion in almost periodic media. First, we study the principal eigenvalue problems of Schrodinger operators by the method in the spectral theory of Schrodinger operators. Then by construct the upper and lower solutions by the eigenfunction and the known traveling waves of free boundary problems to prove the existence of Cauchy problems of KPP equations. .3. Exploring the spreading phenomena and dynamics of diffusion system space-time almost periodic media. We will study the respective theory of generalized principal eigenvalues and homogenization. Then we will study the spreading speeds and traveling waves of respective KPP equations