Analysis of nonlinear dynamics via asymptotic and perturbation methods

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My research will concentrate on the following rather related fields:     First, I will study the  spatial movement and evolution of   single or multiple species modeled by mathematical equations. The recent progress towards the combined effect of spatial diffusion and temporal delay leads to some interesting mathematical equations with time delay and spatial interaction. The qualitative and quantitative analyses of nonlinear dynamics (e.g. traveling wave, stability of their movement in an area, difficulty in prediction) of these mathematical equations are the main purpose of my research in the future.     Besides the investigation of spatial movement of species, I will also study the interaction between Ecology and infectious diseases. When infectious diseases are exhibited in a single species or multiple species, new evolution behaviors are formed. Throughout recorded history, non-indigenous vectors that arrive, establish and spread in new areas have fomented epidemics of human (or animal) disease such as malaria, yellow fever, typhus, rabies, rabbit hemorrhagic disease, plague and west Nile.  In order to better understand the transmission of the diseases in  species, we need to investigate the interaction between ecology  and the spread of the diseases. A systematic study of dynamical transmission of these diseases is another  subject of my  interest. For the short-term goal, I will investigate the spread of rabies in Europe caused by red foxes. A new mathematical model will be constructed by taking into account  the dispersal behavior of juvenile foxes.  I will extend our novel methods and research experiences to study the spatial spread of infectious disease such as west Nile, rabbit hemorrhagic disease and so on.     Mathematical modeling and analysis of real problems arising in the fields of signal transmission of neural networks and in the field of periodic oscillation of stem cell population in hematological disease will also be in my research scope.