The Mathematical Institute, University of Oxford, Eprints Archive

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

Sánchez-Garduño , F and Maini, P. K. and Pérez-Velázquez, J (2010) A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems Series B, 13 (2). pp. 455-487.

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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.

Item Type:Article
Uncontrolled Keywords:Direct aggregation, degenerate diffusion, traveling waves, ill-posed problems, negative diffusion.
Subjects:A - C > Biology and other natural sciences
Research Groups:Centre for Mathematical Biology
ID Code:874
Deposited By:Philip Maini
Deposited On:19 Dec 2009 08:32
Last Modified:25 Oct 2010 16:39

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