Spatial and temporal dynamics of evolution 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, almost periodic solutions and entire solutions, persistence and extinction, global convergence and attractors, the existence, uniqueness and stability of traveling waves, and spreading speeds. In this project we also propose to obtain general results on traveling waves for abstract monotone systems with ignition structure, propagation dynamics in space-time heterogeneous media and shifting environments, principal eigenvalues for periodic nonlocal dispersal systems, 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, positive operators, nonlinear analysis, and numerical simulations. The novelty of the research lies in the dynamical systems approach to evolution equations with spatial and temporal heterogeneities. The proposed research is expected to establish the mathematical theory of spatial-temporal dynamics and biological invasions, and to develop the theory of principal eigenvalues and basic reproduction ratios. Progress in this project 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 propagation.