Spatio-temporal dynamics of general diffusive processes in heterogeneous and random environments

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General diffusive processes, such as the spread of advantageous genes, the invasion of species, the migration of forest types, the propagation of flames, and the spread of infectious diseases, include both classical diffusive processes and nonlocal dispersal processes, and arise in many scientific areas such as chemistry, chemical engineering, combustion theory, ecology, epidemiology and population biology. Evolutionary equations have been widely used to model and study the evolution of general diffusive processes. Classical studies of evolutionary equations assume the environment is homogeneous in both space and time. However, many general diffusive processes in the real world encounter spatial and temporal fluctuations due to the change of environmental conditions, and such fluctuations have great influences on the persistence and spread of these processes. This motivates the study of evolutionary equations in heterogeneous and random environments. The primary goal of the proposed research program is to investigate spatio-temporal dynamics of evolutionary equations in heterogeneous and random environments. The main objectives include: 1. the investigation of the persistence theory, the global dynamics, the spreading properties of propagating solutions, and the existence, stability and qualitative properties of generalized travelling fronts for several important classes of evolutionary equations in heterogeneous or random environments, including reaction-diffusion equations in heterogeneous environments, stepping stone models or stochastic Fisher-Kolmogorov-Petrowsky-Piscunov equations, and integrodifference equations with random coefficients; 2. the study of population persistence under climate change by investigating a class of integrodifference equations subject to climate change described by the location, the velocity and the geometry of a finite favourable habitat, that moves inside the surrounding unfavourable environment. The proposed research program is expected to advance and enrich the theory of evolutionary equations in heterogeneous and random environments, to introduce new mathematical ideas and methods to study evolutionary equations for more realistic models, and to provide mathematical frameworks and tools for applications in many areas of science and engineering.