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An analysis of merging-splitting group dynamics by Bernstein function theory

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 (-\infty,1).