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

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.

Singular limit of an Allen-Cahn equation with nonlinear diffusion

“…We introduce a useful cut off signed distance function d as follows. Recall the signed distance function d defined in (16), and interface Γ t satisfying (13). Choose d 0 > 0 small enough so that the signed distance function d is smooth in the set…”

“…We first prove (i). Recall that d(x, t) is the cut-off signed distance function to the interface Γ t moving according to equation (13), and d ε (x, t) is the signed distance function corresponding to the interface…”

“…It is well known that Problem (IP ) possesses locally in time a unique smooth solution. Fix T > 0 such that the solution of (IP ), in (13), exists in [0, T ] and denote the solution by Γ = ∪ 0≤t<T (Γ t × {t}). From Proposition 2.1 of [7] such a T > 0 exists, and one can deduce that Γ ∈ C 4+ν, 4+ν 2 in [0, T ], given that Γ 0 ∈ C 4+ν .…”

Global wellposedness of nutrient-taxis systems derived by a food metric

“…In [8], the authors derived a discrete velocity kinetic model corresponding to food metric diffusion and presented a sufficient condition for the existence of a traveling wave solution to the discrete velocity model. In [14], Hilhorst et al considered the singular limit of an Allen-Cahn equation with the food metric diffusion. They consider the food metric chemotaxis model to represent spatial heterogeneity of the food distribution.…”

“…The same authors in [7] derived a porous medium type equation from the food metric (1.1) and prove that its traveling wave solution has compact support with an interface that divides into zero and non-zero regions [9]. In spite of the aforementioned studies [7,8,9,14], there has been little work in the literature investigating the existence and boundedness results. As far as we know, none of them have obtained the general well-posedness result for (1.2), even in one dimension.…”