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Dispersal towards food: the singular limit of an Allen-Cahn equation
Danielle Hilhorst1, Yong-Jung Kim2, Dohyun Kwon3
1CNRS and Laboratoire de Mathématiques, University of Paris-Sud Paris-Saclay, 91405, Orsay Cedex, France.
Abstract:
The effect of dispersal under heterogeneous environment is studied in terms of the singular limit of an Allen-Cahn equation. Since biological organisms often slow down their dispersal if food is abundant, a food metric diffusion is taken to include such a phenomenon. The migration effect of the problem is approximated by a mean curvature flow after taking the singular limit which now includes an advection term produced by the spatial heterogeneity of food distribution. It is shown that the interface moves towards a local maximum of the food distribution. In other words, the dispersal taken in the paper is not a trivialization process anymore, but an aggregation one towards food.
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