Dispersal towards food: the singular limit of an Allen–Cahn equation

Danielle Hilhorst, Yong Jung Kim, Dohyun Kwon, Thanh Nam Nguyen

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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.

Original languageEnglish
Pages (from-to)531-565
Number of pages35
JournalJournal of Mathematical Biology
Issue number3
StatePublished - 1 Feb 2018


  • Fokker–Planck type diffusion
  • Food metric
  • Generation and propagation of interface
  • Perturbed motion by mean curvature
  • Singular limit


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