# Self-similar spreading in a merging-splitting model of animal group size

## TL;DR

This paper demonstrates that in a merging-splitting model of animal groups with initial distributions having power-law tails, the group size distribution evolves towards a self-similar spreading form characterized by monotone densities and distinct power-law behaviors.

## Contribution

It introduces a family of self-similar solutions for the model, revealing detailed asymptotic behavior of group size distributions with power-law tails.

## Key findings

- Solutions approach a self-similar form over time
- Existence of a one-parameter family of solutions
- Densities exhibit power-law behavior at small and large sizes

## Abstract

In a recent study of certain merging-splitting models of animal-group size (Degond et al., J. Nonl. Sci. 27 (2017) 379), it was shown that an initial size distribution with infinite first moment leads to convergence to zero in weak sense, corresponding to unbounded growth of group size. In the present paper we show that for any such initial distribution with a power-law tail, the solution approaches a self-similar spreading form. A one-parameter family of such self-similar solutions exists, with densities that are completely monotone, having power-law behavior in both small and large size regimes, with different exponents.