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‘Traveling wave solutions’ and their bifurcations in some mathematical models of population with taxis
by
Faina Berezovskaya
Mathematics Department, Howard University, Washington DC
Coauthors: Artem Novozhilov, Georgy Karev,
National Institute of Health,Bethesda
An analysis of traveling wave solutions of some kinds of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of models describing certain forms of taxis, the movement of populations due to attraction or repulsion characteristics of one of counteragents. The Keller-Segel model describing chemotaxis processes as well as some models of tumor encapsulation belong to the considering class. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front-front, front-impulse and train traveling wave solutions are formulated. We also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. In particular, the simplest mathematical model that has a family of impulse-impulse solution is presented. The latter family of solutions corresponds to an elliptic sector in the wave system.
The results can be used for construction and analysis of different mathematical models describing systems with taxis.
Date received: February 15, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caws-47.