Title：Lie symmetries to Degenerate Parabolic Systems
Speaker： Zhaosheng FengProfessor
Date：June 16, 2016 (Thursday).Time3:00-4:00 PM
Location：Science building E518

Brief sketch of speaker：Dr. Zhoasheng Feng, a Full-Professor of Mathematics in the School of Mathematical and Statistical Sciences at University of Texas-Rio Grande Valley. His research interests lie in analysis on differential equations (odes and pdes), dynamical systems, computational analysis and methods, mathematical physics and mathematical biology. So far he has edited five books, monographs and Proceedings, and published 148 research papers in international academic journals. He is currently serving in five editorial boards.

Abstract：The history of the theory of reaction-diffusion systems begins with the three famous works by Luther (1906), Fisher and Kolmogorov etc. (1937). Since theseseminal papers much research has been carried out in an attempt to extend the original results to more complicated systems which arise in several  fields. For example, inecology and biology the early systematic treatment of dispersion models of biological
populations [Skellam (1951)] assumed random movement. There the probability thatan individual which at time t = 0 is at the point x_1 moves to the point x_2 in theinterval of time ∆t is the same as that of moving from x_2 to x_1 during the same time interval. On this basis the diffusion coefficient in the classical models of populationdispersion appears as constant. In this talk, we introduce the Lie symmetry reduc-
tion method and apply it to study the case that some species migrate from denselypopulated areas into sparsely populated areas to avoid crowding. We consider a moregeneral parabolic system by considering density-dependent dispersion as a regulatorymechanism of the cyclic changes. Here the probabi- lity that an animal moves from thepoint x_1 to x_2 depends on the density at x_1. Under certain conditions, we apply the
higher terms in the Taylor series and the center manifold method to obtain the localbehavior around a non-hyperbolic point of codimension one in the phase plane, anduse the Lie symmetry reduction method to explore bounded traveling wave solutions.