# Merging-splitting group dynamics via Bernstein function theory (or: How to count fish using mathematics)

#### Bob Pego (Carnegie Mellon University)

Wednesday, October 18, 2017 -
12:00pm to 1:00pm
Location:
Physics 119

We study coagulation-fragmentation equations inspired by a simple model derived in fisheries science to explain data on the size distribution of schools of pelagic fish. The equations lack detailed balance and admit no H-theorem, but we are anyway able to develop a rather complete description of equilibrium profiles and large-time behavior, based on complex function theory for Bernstein and Pick (Herglotz) functions. The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers that count all ternary trees with $n$ nodes. The structure of equilibrium profiles and other related sequences is explained through a new and elegant characterization of the generating functions of completely monotone sequences, as those Pick functions analytic and nonnegative on a half line. This is joint work with Jian-Guo Liu and Pierre Degond.

Last updated: 2017/11/18 - 10:21am