The Phillip Island penguin parade (a mathematical treatment)

TL;DR

This paper develops a mathematical model using differential equations to describe the collective movement and behaviors of penguins during their parade on Phillip Island, capturing phenomena like group formation and panic.

Contribution

It introduces a novel differential equation framework incorporating natural parameters to analyze penguin parade behaviors, including group formation and panic responses.

Findings

Model predicts successful group return scenarios

Illustrates how panic can cause penguins to freeze

Simulations match observed penguin movements

Abstract

Penguins are flightless, so they are forced to walk while on land. In particular, they show rather specific behaviors in their homecoming, which are interesting to observe and to describe analytically. In this paper, we present a simple mathematical formulation to describe the little penguins parade in Phillip Island. We observed that penguins have the tendency to waddle back and forth on the shore to create a sufficiently large group and then walk home compactly together. The mathematical framework that we introduce describes this phenomenon, by taking into account "natural parameters" such as the eye-sight of the penguins, their cruising speed and the possible "fear" of animals. On the one hand, this favors the formation of conglomerates of penguins that gather together, but, on the other hand, this may lead to the "panic" of isolated and exposed individuals. The model that we propose is based on a set of ordinary differential equations. Due to the discontinuous behavior of the speed of the penguins, the mathematical treatment (to get existence and uniqueness of the solution) is based on a "stop-and-go" procedure. We use this setting to provide rigorous examples in which at least some penguins manage to safely return home (there are also cases in which some penguins freeze due to panic). To facilitate the intuition of the model, we also present some simple numerical simulations that can be compared with the actual movement of the penguins parade.