Emperor penguins (Aptenodytes forsteri
) are known to huddle to survive long periods of fast in the severe conditions of Antartcic winters. They are able to form huddles because they are not tied to a fixed nest. Huddles are discontinuous events that last for relatively short durations (on the order of a few hours) 
corresponding to storm events 
. The number density in a huddle at a colony may be as high as
. Researchers have observed directly penguins huddling at their colonies 
, and there is evidence indicating that emperor penguins also huddle during foraging trips 
Emperor penguins huddle to conserve energy, which is particularly important since they must fast for periods of 105 to 115 days 
. Emperor penguins benefit from huddles because of the reduction of body surface area exposed to the cold and owing to the warm temperature inside the huddle 
. The ambient temperature in the huddle is at least 20°C and may reach as high as 37.5°C 
. Despite the fact that huddles achieve these high ambient temperatures, emperor penguins benefit most from the huddle through the reduction of cold-exposed body surfaces 
Measurements of the body temperature of penguins in various environments and their relation to weather conditions have led to significant insight into huddle formation. 
Huddles occur more frequently at lower ambient temperatures and in higher wind speed, but the intensity of the huddle (i.e.
the number density of the huddle) depends on lower ambient temperatures only 
. Few theoretical models of huddling have been presented. Some modeling effort has assumed that penguins move from the windward to the leeward side of the huddle, but without providing much justification 
. Canals and Bozinovic 
modeled huddle formation in mice (Mus musculus) as a self-organizing event. In particular, they modeled huddling as a second-order phase transition triggered by cold temperatures.
An important feature of huddles is that each penguin has approximately equal opportunity to the warmth of the huddle. How each penguin obtains this equal access is thought to be the result of a complex phenomenon in which penguins reorganize themselves within the huddle 
. Gilbert et al
attribute heterogeneity of the huddle shape to ensuring this equal access, but without providing details as to how equality is achieved. On the other hand, Zitterbart et al
use ideas from condensed matter physics to explain how penguins within a huddle reorganize themselves. An important set of observations regarding penguin movement within huddles reported by Le Maho 
The huddles are not motionless; movement is extremely slow, but continuous. The huddle is urged along by the wind, the rear-flank birds (those most exposed to the wind) advancing slowly along the sides of the huddle in order to be protected from the wind. Thus, birds that at first are in the center of the huddle become members of the rear flank and move, in their turn, up the sidelines.
Further evidence showed that huddles move back and forth under the influence of the dominant winds 
We introduce here a systematic and quantitative mathematical model for penguin huddles. This mathematical model is aligned with the qualitative observations by Le Maho stated above. Moreover, it is consistent with the idea that penguins huddle tightly to reduce their cold-exposed body surfaces, and increase the ambient temperature. The key assumption of our mathematical model is that each individual penguin seeks to reduce its own heat loss. Thus, a penguin on the boundary of the huddle exposed to the wind will move downwind along the huddle boundary. In contrast, the penguins in the interior of the huddle neither have space to move nor experience significant heat loss, so they remain stationary. While penguins inside the huddle have been observed to make multiple small displacements 
, we consider here that these motions are small compared to the motion observed on the edge of the huddle. The accumulation of individual penguin movements along the boundary leads to coherent and robust huddle dynamics that we identify, describe, and quantify. In particular, we find from our simulation results that the number of penguins in the huddle, the wind strength quantified by the Péclet number, and the amount of uncertainty in the penguin movement are the key factors governing the dynamics of a huddle. Our simulation results show that the features of this model are sufficient to result in each penguin having equal access to the benefits of the huddle.
To avoid prohibitively large computations and to allow for concise analysis, our model does not account for all possible details and scenarios. Rather, our goal is to provide a simple model, based on reasonable and well defined assumptions on the geometry of the huddle and the fluid mechanics of the wind. Our model recovers important features of actual huddles such as their overall shape, downwind motion, and an equal distribution of access to the benefits of the huddle among penguins. We describe the overall framework of our model in Method, including our assumptions. In Results and Discussion, we show simulation results, and identify, describe, and quantify the dynamics of the huddle. We discuss how these results compare to field observations and outline how one may extend this mathematical model to include a number of particular effects in Conclusion.