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

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_=[D(u)u_]_+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.