# Louis Ryan

## Thesis

Analysis of Swarm Behavior in Two Dimensions

- Advisor
- Andrew J. Bernoff
- Second Reader(s)
- Chad M. Topaz

## Abstract

We investigate the steady state solutions that can exist for a two dimensional swarm of biological organisms, which have pairwise social interaction forces. The three steady states we investigate using a continuum model are a ribbon migrating swarm, a circular migrating swarm, and a milling swarm. We solve these numerically by reformulating the integral equation that arises from the continuum model as an energy minimization problem. For the ribbon migrating solution, we are able to determine an analytic solution from Carleman's equation which arises after an asymptotic expansion of the social interaction potential. Using this technique we are able to show the existence of a square root singularity that emerges at the boundary of the compactly supported swarm. The analytic solution agrees with the numerical solution for certain parameter values in the social interaction potential. We then demonstrate the existence of solutions for a migrating and milling circular swarm which contain a square root singularity. The milling swarm looks similar to the infinite ribbon, so we are able to use an asymptotic expansion of the potential to obtain an analytic solution in this case as well. The singularities in the density of the swarm suggest that the Morse potential should not be used for modeling biological swarming.

## Proposal

Modeling Predator–Prey Effects in Swarms