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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Acknowledgements
This research was performed in the context of the CNRS GDRI ReaDiNet. Hilhorst would like to thank the Center for Mathematical Challenges for the support during her visits at KAIST where this work was completed. Kim was supported in part by National Research Foundation of Korea under contact no. 2017R1A2B2010398. The authors are grateful for the ideas included in the unpublished article “Motion of interfaces in a spatially inhomogeneous Allen–Cahn equation with equal well-depth potential” by Hilhorst, Matano, and Schätzle.
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Hilhorst, D., Kim, YJ., Kwon, D. et al. Dispersal towards food: the singular limit of an Allen–Cahn equation. J. Math. Biol. 76, 531–565 (2018). https://doi.org/10.1007/s00285-017-1150-5
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DOI: https://doi.org/10.1007/s00285-017-1150-5
Keywords
- Fokker–Planck type diffusion
- Food metric
- Singular limit
- Generation and propagation of interface
- Perturbed motion by mean curvature