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

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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.