A non-linear degenerate equation for direct aggregation and traveling wave dynamics
Faustino Sánchez-Garduño Philip K. Maini Judith Pérez-Velázquez
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
Singular travelling waves in a model for tumour encapsulation
John R. King Judith Pérez-Velázquez H.M. Byrne
In this paper we study an existing mathematical model of tumour encapsulation comprising two reaction-convection-diffusion equations for tumour-cell and connective-tissue densities. The existence of travelling-wave solutions has previously been shown in certain parameter regimes, corresponding to a connective tissue wave which moves in concert with an advancing front of the tumour cells. We extend these results by constructing novel classes of travelling waves for parameter regimes not previously treated asymptotically; we term these singular because they do not correspond to regular trajectories of the corresponding ODE system. Associated with this singularity is a number of further (inner) asymptotic regions in which the dynamics is not governed by the travelling-wave formulation, but which we also characterise.
keywords: reaction-convection-diffusion systems. singular travelling waves

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