Spatio-temporal dynamics of general diffusive processes in heterogeneous and random environments

一般的扩散过程，如优势基因的传播、物种的入侵、森林类型的迁移、火焰的传播和传染病的传播，包括经典扩散过程和非局部扩散过程，并出现在许多科学领域，如化学、化学工程、燃烧理论、生态学、流行病学和种群生物学。演化方程已被广泛地用于模拟和研究一般扩散过程的演化。经典的进化方程研究假设环境在空间和时间上都是均匀的。然而，现实世界中许多一般扩散过程由于环境条件的变化而遇到时空波动，这种波动对这些过程的持续和传播有很大的影响。这激发了异质和随机环境下进化方程的研究。******提出的研究计划的主要目标是研究进化方程在异质和随机环境中的时空动力学。主要目标包括：******1。研究了异质或随机环境下几种重要进化方程的持续理论、全局动力学、传播解的扩展性质以及广义行进前沿的存在性、稳定性和定性性质，包括异质环境下的反应-扩散方程、踏脚石模型或随机Fisher-Kolmogorov-Petrowsky-Piscunov方程；和随机系数的积分差分方程；******2。通过研究一类受气候变化影响的积分差分方程来研究气候变化下的种群持久性，这些方程由在周围不利环境中移动的有限有利栖息地的位置、速度和几何形状所描述。******提出的研究计划有望推进和丰富异质和随机环境下的进化方程理论，引入新的数学思想和方法来研究更现实的模型进化方程，并为科学和工程的许多领域的应用提供数学框架和工具。

General diffusive processes, such as the spread of advantageous genes, the invasion of species, the migration of forest types, the propagation of flames, and the spread of infectious diseases, include both classical diffusive processes and nonlocal dispersal processes, and arise in many scientific areas such as chemistry, chemical engineering, combustion theory, ecology, epidemiology and population biology. Evolutionary equations have been widely used to model and study the evolution of general diffusive processes. Classical studies of evolutionary equations assume the environment is homogeneous in both space and time. However, many general diffusive processes in the real world encounter spatial and temporal fluctuations due to the change of environmental conditions, and such fluctuations have great influences on the persistence and spread of these processes. This motivates the study of evolutionary equations in heterogeneous and random environments. ******The primary goal of the proposed research program is to investigate spatio-temporal dynamics of evolutionary equations in heterogeneous and random environments. The main objectives include:******1. the investigation of the persistence theory, the global dynamics, the spreading properties of propagating solutions, and the existence, stability and qualitative properties of generalized travelling fronts for several important classes of evolutionary equations in heterogeneous or random environments, including reaction-diffusion equations in heterogeneous environments, stepping stone models or stochastic Fisher-Kolmogorov-Petrowsky-Piscunov equations, and integrodifference equations with random coefficients;******2. the study of population persistence under climate change by investigating a class of integrodifference equations subject to climate change described by the location, the velocity and the geometry of a finite favourable habitat, that moves inside the surrounding unfavourable environment.******The proposed research program is expected to advance and enrich the theory of evolutionary equations in heterogeneous and random environments, to introduce new mathematical ideas and methods to study evolutionary equations for more realistic models, and to provide mathematical frameworks and tools for applications in many areas of science and engineering.