Wave propagation study of abstract dynamical systems with applications

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The primary goal of this proposal is to investigate wave propagation dynamics of abstract dynamical systems arising from reaction diffusion systems, integrodifference systems, nonlocal dispersal systems.  These systems  can model important phenomena of moving patterns in spatial single or multiple species models occurred in the study of competition and collaboration, ecological invasion and disease infection. Although there are various models describing spatial evolution of species, a universal  approach  in our studies of the dynamics is to consider the solution semiflow of the modeling system. This semiflow is also called as the dynamical system of the given model. Traveling wave patterns are important biological phenomena observed in these models, with a movement of a fixed speed as well as an unchanged solution profile. Determinacy of the waves, especially the spreading speed, is a challenging job. For a monotone dynamical system with monostability, we will study whether the system has a single spreading speed or there are multiple spreading speeds. We want to deeply understand how and when the spreading speed is determined by a linear conjecture, and how and when multiple spreading speeds and stacked wavefronts can appear. In either of above cases (single or multiple spreading speeds) we will study linear selection (linear conjecture) or nonlinear selection of the spreading speeds.  For a monotone dynamical system with bistable nonlinearity, we will study how to determine the wave speed sign which indicates the outcome of competition between two bistable states. The main scope of my research is: 1). Study the selection mechanism of the spreading speed(s) of abstract dynamical systems with monostable nonlinearity in three cases: a) The dynamical system is time periodic. b) The dynamical system is in a periodic habitat. c) The dynamical system is time periodic and in a periodic habitat. 2). Study the uniqueness and sign of the traveling speed of the abstract dynamical system with bistable nonlinearity in the following cases: a) Continuous or discrete in time or space (lattice). b) Particularly in time periodic case. c) In periodic habitat and/or time periodic case. 3). Further extend our idea and investigation to nonmonotone systems and stochastic dynamical systems. We will apply our theory to infectious disease models, chemotaxis models as well as models in biological, physical, chemical and other science and engineering fields. We intend to improve research results in some important past contributions, solve or answer open problems or conjectures in recent related investigations. This is designed to lead to  an important breakthrough in the deep understanding of traveling waves to some complex partial differential equation models. Theoretically it develops the idea of linear conjecture of some mathematical biologists and physicists. Moreover, these methods can be applied to solve problems in the life sciences and engineering.