On a nonlocal model of biological aggregation

Dejan Slepcev
Carnegie-Mellon University

I will discuss some features of a continuum model of biological aggregation introduced by Topaz, Bertozzi and Lewis. Individuals, described by the model, are attracted to each other at a distance, but avoid overcrowding via a local dispersal mechanism. In the continuum model for the population density the attraction is described via a nonlocal operator, while the repulsion is modeled by a differential operator.
The equation is a gradient flow in Wasserstein metric.

We show that the density profile develops interfaces between a near-constant-density aggregate state and the empty space.
The interfaces evolve under surface-tension-like ``forces''. More precisely, using the gradient flow structure and asymptotic analysis, we show that the sharp interface limit for the interfacial motion is the Hele-Shaw flow.

On long time scales the interfacial motion leads to coarsening of length scales present in the system. The rate of coarsening can be investigated using the Kohn-Otto framework.
We will describe a geometric viewpoint,
which unites the coarsening results in a variety of interfacial models.

The talk is in part based on joint work with Andrea Bertozzi.

Presentation (PDF File)

Back to Workshop I: Aspects of Optimal Transport in Geometry and Calculus of Variations