Spatial and temporal dynamics of nonlinear systems with applications

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This research program is aimed at the global dynamics of nonlinear evolution systems with spatial and temporal *heterogeneities, which are inspired by population biology and epidemiology. We will study the existence, *uniqueness, bifurcations, and global stability of steady states and periodic solutions, persistence and extinction, *global convergence and attractors, the existence, uniqueness and stability of traveling waves, and spreading speeds. *In this program, we also propose to obtain general results on spreading speeds and traveling waves for abstract *evolution systems with weak compactness and basic reproduction ratios for non-autonomous population models.**The investigation methods involve dynamical systems, Liapunov functions, upper-lower solutions, perturbations *and bifurcations, comparison principle, skew-product semiflows, linearization, nonlinear analysis, and numerical *simulations. The novelty of the research lies in the dynamical systems approach to evolution equations with *spatial and temporal heterogeneity.**The proposed research is expected to establish the mathematical theory of spatial-temporal dynamics and biological *invasions, and to develop the theory of non-autonomous dynamical systems. Progress in this research can also help *to answer important questions such as whether and how a community of interacting populations can persist (avoid *extinction) or whether an infectious disease becomes endemic in a population, to estimate the speeds of biological *invasion and epidemic spreading, and to understand biological phenomena such as spatial-temporal patterns and *wave propagations.