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Talk Abstract
Spike-Layer Steady States in Diffusion Systems

Michel Rascle, University of Nice

We consider a few models of chemotactism, inspired from the classical Keller-Segel model, namely

\begin{equation}
\left \{ \begin{array}{lll}
\partial_t u+ \text{div}(u \chi(s)\nabla s) & = & \; \mu\Delta u,\\
\partial_t s - \nu \Delta s         & = & \pm \; k(s,u)  
\end{array}  \right. \label{1} 
\end{equation}

where $\nabla$ denotes the gradient, u the concentration in predators, s the concentration in substrate. I will discuss a few different examples in which the population of predators concentrates in finite time to a delta-function : aggregation. I will mainly focus on cartoons, where the diffusion is neglected. In the unstable case (the + case), aggregation is linked with a severe pathology in the structure of the underlying system of conservation laws, which is typically mixed type. However, I will show how one can mathematically solve the problem, and even describe such a singular solution after aggregation. Such solutions seem to be numerically very stable!

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1998-1999 Mathematics in Biology

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