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

一些具有时间延迟和/或空间差异的演化方程的动力学

• 批准号: 227048-2010
• 负责人: Zou, Xingfu
• 金额: $ 2.55万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=523889
• 关键词: Dynamics; some; evolution; equations; temporal

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.

在这项研究中，我建议研究一些进化方程的动力学，这些方程源于对结构化群体、传染病传播、免疫系统中病原体和细胞的相互作用、神经网络激活和其他相关领域的建模。这种模型方程的一个重要且新颖的特征是存在时间延迟和/或空间扩散/色散，以及反映空间和时间双重方面的空间非局部性。这种非局域性可能是由于种群动态中不成熟个体的分散造成的，或者是在对潜伏在空间异质环境中传播的传染病进行建模时潜伏个体流动性的结果。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，但也提出了更具挑战性和有趣的数学问题。主要关注的是时间延迟、空间扩散/色散和空间非局部非线性对此类方程的全局动力学和模式的单独和/或联合影响。研究主题包括（i）行波解决方案，可以解释群体生物学中的域入侵、稳态之间的过渡过程以及模式的传播； (ii) 稳态及其稳定性； (iii) 此类方程的全局动态和模式。我们建议开发一些新的工具和技术，这些工具和技术可能有助于为这类新方程建立新的理论，从而丰富现有的演化方程理论。 我们期望理论结果将在相关领域有潜在的应用。例如，通过将获得的结果应用到由此类方程表达的一些流行病模型中，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且有价值的指导和建议。