  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.
