Dynamics of some evolution equations with temporal delay and/or spatial diff

在这项研究中，我建议研究一些进化方程的动力学，这些方程来自于对结构化种群、传染病的传播、免疫系统中病原体与细胞的相互作用、神经网络的激活等相关领域的建模。这种模型方程的一个重要而新颖的特征是存在时间延迟和/或空间扩散/弥散，以及反映空间和时间双重方面的空间非局部性。这种非局部性可能是种群动力学中未成熟个体分散的结果，也可能是潜在个体在模拟具有潜伏期的传染病时的流动性的结果，该传染病在空间上不同的环境中传播。这样的模型方程更真实地描述了进化过程，可能有助于更好地理解现实世界中的过程，但也会提出更具挑战性和更有趣的数学问题。主要关注的是时间延迟、空间扩散/弥散和空间非局部非线性对这类方程的全局动力学和模式的单独和/或联合影响。研究的主题包括(I)可以解释种群生物学中区域入侵的行波解、稳态之间的转换过程和模式的传播；(Ii)稳态及其稳定性；(Iii)全球动力学和此类方程的模式。我们建议开发一些新的工具和技术，以帮助建立这类新的方程的新理论，从而丰富现有的发展方程理论。我们期望理论结果在相关领域具有潜在的应用价值。例如，通过将所得结果应用于一些由此类方程表示的传染病模型，我们期望能够更好地理解疾病的传播机制，从而为如何控制流行病提供一些有用的指导和建议。

In this research, I propose to investigate the dynamics of some evolution equations arising from modeling structured population, spread of infectious diseases, interaction of pathogens and cells in immune system, activation of neural networks and other related areas. One important and novel feature of such model equations is the presence of time delay and/or spatial diffusion/dispersion, as well as the spatial non-locality reflecting the dual aspect of the space and time. Such a non-locality could be a result of the dispersal of the immature individuals in population dynamics, or a consequence of mobility of latent individuals in modeling an infectious disease with latency spreading over a spatially heterogeneous environment. Such model equations more realistically describe the evolution processes and may help better understand the processes in the real world, and yet raise more challenging and interesting mathematical problems. The main concern is the individual and/or joint impact of the temporal delay, spatial diffusion/dispersion, and the spatial non-local nonlinearities on the global dynamics of and patterns in such equations. Among the topics of investigation are (i) traveling wave solutions which could account for the domain invasion in population biology, transition process between steady states, and propagation of patterns; (ii) steady states and their stability; (iii) global dynamics and patterns of such equations. We propose to develop some new tools and techniques that may help establish a new theory for this new class of equations, and thereby enrich the existing theory for evolution equations. We expect that the theoretic results will have potential applications in related areas. For example, by applying the obtained results to some epidemic disease models expressed by such equations, we expect to be able to better understand the mechanism of spread of the diseases, and thereby, provide some useful and valuable guidance and suggestions on how to control the epidemic diseases.