Abstract:
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.
Although the equations lack detailed balance and admit no H-theorem,
we are able to develop a rather complete description of equilibrium profiles
and large-time behavior, based on complex function theory for Bernstein and Pick functions.
The generating function for discrete equilibrium profiles also generates the Fuss-Catalan numbers
(derived by Lambert in 1758) 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 (-?,1).
This is joint work with Bob Pego and Pierre Degond. |