UC Santa Barbara

In this paper, we will build mathematical pelagic fish migration models and inspect simulations to discover the behavior for motion amongst a school. I focus on the Icelandic Capelin, and explore all possible migration patterns and their limitations as governed by mathematical equations and not biological observations. We develop a system of ordinary differential equations from a discrete system for the most general motion. First, we will perturb the system to develop a system of stochastic differential equations to study the unique behavior under naturally occurring external forces such as currents or reefs. Then, we will categorize the found transitory and long term behavior of these systems and compare them to the unperturbed solutions. Finally, we will prove that the system exhibits a subcritical pitchfork bifurcation.