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The authors have declared that no competing interests exist.

Analyzed the data: AW FB ADK. Wrote the paper: AW FB ADK. Developing simulations: FB AW.

We present a systematic and quantitative model of huddling penguins. In this mathematical model, each individual penguin in the huddle seeks only to reduce its own heat loss. Consequently, penguins on the boundary of the huddle that are most exposed to the wind move downwind to more sheltered locations along the boundary. In contrast, penguins in the interior of the huddle neither have the space to move nor experience a significant heat loss, and they therefore remain stationary. Through these individual movements, the entire huddle experiences a robust cumulative effect that we identify, describe, and quantify. This mathematical model requires a calculation of the wind flowing around the huddle and of the resulting temperature distribution. Both of these must be recomputed each time an individual penguin moves since the huddle shape changes. Using our simulation results, we find that the key parameters affecting the huddle dynamics are the number of penguins in the huddle, the wind strength, and the amount of uncertainty in the movement of the penguins. Moreover, we find that the lone assumption of individual penguins minimizing their own heat loss results in all penguins having approximately equal access to the warmth of the huddle.

Emperor penguins (^{2}

Emperor penguins huddle to conserve energy, which is particularly important since they must fast for periods of 105 to 115 days

Measurements of the body temperature of penguins in various environments and their relation to weather conditions have led to significant insight into huddle formation.

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

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

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.

In our mathematical model, individual heat preservation is the goal of each and every penguin in the huddle, irrespective of the heat preservation of the huddle as a whole. In other words, each penguin will move or stay stationary to minimize its own heat loss. This mathematical model therefore requires the determination of the local rate of heat loss for each penguin in the huddle. This local rate of heat loss depends on the temperature distribution outside the huddle which, in turn, depends on the wind flow around the huddle.

Specifically, we focus on the dynamics of a single huddle, where all the penguins present are part of the huddle. This huddle is assumed to be situated on a flat plane, so there are no obstacles impeding penguin movement. Penguins in this huddle have uniform size and shape. Our model does not account for all heat exchanges between penguins and their environments. In particular, penguins are known to lose a large quantity of heat through their feet and eyes

Generate a huddle and determine the huddle boundary.

Compute the wind flow around the huddle.

Compute the temperature profile around the huddle.

Compute the local rate of heat loss for each penguin.

Add random variations to the rate of heat loss (optional).

Identify the penguin with the highest rate of heat loss (the “mover”) and move it to a location on the boundary where heat loss is minimal.

Determine the new huddle boundary.

Repeat over the desired number of iterations by going back to Step 2.

In what follows, we describe in detail each step of this procedure. Included in this discussion are any simplifying assumptions, and their justification.

From observing videos of huddling penguins

To determine the wind flow around the huddle, we need only to consider a two dimensional flow around a polygon. Moreover, we assume that this flow is inviscid and irrotational. These assumptions imply that we do not resolve turbulent wind flows, which would obfuscate the computation of the temperature profile needed to compute the local rate of heat loss. Rather, we find a smooth, regular wind flow around the huddle. Nonetheless, the key relationships in this mathematical model between the wind, the temperature, and individual heat loss are not compromised by these assumptions.

Because the wind flow is significantly faster that the movement of a penguin, we assume that this flow is steady. In other words, the wind flow does not depend on the time elapsed since a penguin has relocated, and is only dependent on the huddle shape. Consequently, we are able to use the mathematics of complex variables and the physical theory of potential flow to describe the flow around the huddle.

Let

Mathematically, the wind flow around the polygon is therefore given by

The temperature far outside the huddle, denoted by

Using the wind velocity around the huddle,

We compute the temperature distribution exterior to the unit circle in the canonical domain by discretizing the annulus between the unit circle and the circle of radius

Temperature distribution around a huddle of 100 penguins, for

When the flow around the huddle is turbulent, as is likely to be the case for all but the mildest winds

Penguins at the huddle boundary experience the most significant heat loss compared to those within the huddle interior since the temperature profile changes most abruptly outside the huddle boundary. We seek to find the penguin on the huddle’s edge with the largest rate of heat loss. Therefore, we compute the local rate of heat loss only for penguins on the huddle boundary.

The local rate of heat loss at a boundary is proportional to the derivative of the temperature in the direction normal to the boundary

The heat loss computation described above is idealized because everything is assumed to be known with absolute certainty. To account for variations that can occur in real huddles, we add uncertainty to this model through a random perturbation to the heat loss associated to each penguin on the boundary. We let

The parameter

Among all penguins located at the huddle boundary, we call the penguin with the highest effective heat loss rate, corresponding to the largest value of

In particular, the mover is relocated to a new position on the huddle boundary with at least two neighbors so that the huddle shape remains a polygon. The first neighbor is chosen as the penguin with the smallest heat loss rate corresponding to the smallest value of

Once the mover has been relocated, we may recompute the boundary of the huddle by connecting penguins with fewer than six neighbors. We then iterate the process outlined here by returning to step 2.

We begin by looking at the progression of a single huddle in the absence of random perturbations,

Dynamics of a huddle according to our deterministic model, with parameters

After only 10 steps, a more streamlined huddle starts to form as penguins on the windward side begin to relocate to the leeward side. Notice that the penguins located in thinner structures are the first to relocate, causing the initial indentations of the huddle to smooth out. As penguins continues to relocate, the huddle takes a somewhat elliptic shape. By the 50th time step, the outline of the final huddle formation is visible. During this phase, only penguins on the windward, upper, and lower edges of the huddle relocate. After 100 time steps, the huddle has reached its final shape, with flat sides and rounded windward and leeward ends. From here on, the thickness of the huddle remains constant. The huddle thus retains its basic structure while slowly traveling downwind. Moreover, regardless of their original position, every penguin eventually spends some time exposed to the wind, as layers of penguins on the windward side peel away and expose previously sheltered penguins.

We analyzed how the dynamics of the huddle are influenced by variations in the radius of a circle in the canonical domain where the temperature is assumed to match the temperature at infinity,

The huddle dynamics show a stronger dependence on variations in the Péclet number.

Logarithm of the width,

To explain the observed dependence of the huddle thickness on

For a turbulent Péclet number of order 100, as can be expected in actual conditions, we find that the huddle formed by our model is fairly elongated, with a length-to-thickness ratio of approximately 8 (the huddle shown on the right of

We also considered the effect of allowing the number of penguins,

Aspect ratios of huddles as a function of the number of penguins forming the huddle for fixed Péclet numbers,

We investigated the effects of varying

a) Aspect ratio of huddles of

The deterministic system therefore appears to capture qualitative features of real penguin huddles, but it is also sensitive to random variations likely to occur through wind variations, uneven terrain, short-range motions within the huddle, and individual penguin perceptions. Incorporating even a modest level of random perturbations,

Finally, we investigated how evenly the burden of facing the wind was shared among penguins, and how sensitive the sharing of heat was to random perturbations. As a measure of how evenly heat losses were distributed, we consider the time interval required for each individual penguin to become the most exposed and therefore to relocate. In a huddle of

By design, the model presented here contains only a minimal number of assumptions. Our model penguins are subject to heat loss in windy conditions, and they are simply trying to remain as warm as possible without resorting to displacing any of their fellow penguins. Our results were found to be robust to variations in initial conditions, and to the choice of the location where the temperature is assumed to match the temperature at infinity (i.e. the variable

By incorporating general observations, such as the hexagonal packing of huddles, and the usually long-range displacements of relocating penguins, our model reproduces several key features of the dynamics of huddles. By including a moderate random variation level of

The even sharing of heat loss is of particular interest, as we find that even by modeling penguins that are only intent on minimizing their own exposure to the elements, a nearly uniform heat loss distribution can be achieved for the entire huddle. Our model does not rule out that individual penguins may at times sacrifice for the benefit of the huddle as a whole. Rather it emphasizes that such behavior is not essential to achieving an even distribution of the heat loss among penguins.

It is worth noting that the huddles obtained via our model for realistic values of

There are several possible extensions that could make our model more realistic and that require further investigation. Firstly, the wind was assumed to remain constant, which is an idealization that could be relaxed by allowing the wind direction to vary over time. In addition, the wind velocity, captured by

Secondly, the manner in which penguins relocate could be made more realistic by considering that penguins move downwind along the huddle boundary until they find a position deemed sufficiently sheltered, rather than find the absolute best available location. This would require the addition of a new parameter, a shelter threshold, which would determine at what point penguins stop looking for a better location. Another aspect that could be incorporated in future models is the presence of natural obstacles, such as cracks or icebergs, which could be simulated by preventing relocating penguins from moving to certain fixed locations. Tthe criterion used to determine which penguin has to relocate is currently based on instantaneous exposure rather than cumulative exposure. Using cumulative heat loss to determine which penguin relocates may be more appropriate than using instantaneous heat loss, as it would take into account the duration of the exposure. At present, our model computes only relative exposure, making an individual cumulative heat budget impractical. However, if one were willing to solve the governing equations in a manner that allows for the cumulative heat loss to be tracked, which would require significantly more computational effort, then the penguin that has lost the most heat cumulatively could be chosen as the “mover”.

Finally, it is possible to model a huddle in a continuous rather than discrete fashion, with its boundary

The authors would like to acknowledge Prof. Eric Brown for several useful conversations.

^{th}edition.