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March  2010, 13(2): 455-487. doi: 10.3934/dcdsb.2010.13.455

A non-linear degenerate equation for direct aggregation and traveling wave dynamics

Faustino Sánchez-Garduño 1, , Philip K. Maini 2, and  Judith Pérez-Velázquez 3,

1. 

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, México, 04510, D.F., Mexico
2. 

Centre for Mathematical Biology, Mathematical Institute, University of Oxford, OX1 3LB Oxford
3. 

Institute of Biomathematics and Biometry, Helmholtz Zentrum München, German Research Center for Enviromental Health, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany

Received  April 2009 Revised  October 2009 Published  December 2009

The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form $u_{t}=[D(u)u_{x}]_{x}+g(u)$. The model involves a density-dependent non-linear diffusion coefficient $D$ whose sign changes as the population density $u$ increases. For negative values of $D$ aggregation occurs, while dispersion occurs for positive values of $D$. We deal with a family of degenerate negative diffusion equations with logistic-like growth rate $g$. We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included.
Keywords: Direct aggregation, negative diffusion., traveling waves, degenerate diffusion, ill-posed problems.
Mathematics Subject Classification: Primary: 35K65, 35C07; Secondary: 47J0.
Citation: Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
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