A non-linear degenerate equation for direct aggregation and traveling wave dynamics
Faustino Sánchez-Garduño Philip K. Maini Judith Pérez-Velázquez
Discrete & Continuous Dynamical Systems - B 2010, 13(2): 455-487 doi: 10.3934/dcdsb.2010.13.455
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

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