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2010, Volume 13, Issue 2: 455-487. Doi: 10.3934/dcdsb.2010.13.455
This issue Previous Article Well posedness of a time-difference scheme for a degenerate fast diffusion problem Next Article On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer

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

  • 1.

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

  • 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

Received:  April 2009
Revised:  October 2009
Published:  September 2010
Abstract Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: Primary: 35K65, 35C07; Secondary: 47J06.

    Citation:
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