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

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 ut = [D(u)ux]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.