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DCDS-B

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.

DCDS

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.

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