Self-Organized Thermoregulation of Honeybee Clusters

J. theor. Biol. (1995) 176, 391 402 Self-Organized Thermoregulation of Honeybee Clusters JAMES WATMOUGH AND SCOTT CAMAZINE Department of Mathematics, University of British Columbia, Vancouver, B.C. V6T 1Z2 Pennsylvania State University, Department of Entomology, University Park, Pennsylvania, 16802, U.S.A. (Received 2 August 1994, Accepted in revised form on 19 May 1995) A simple model is used to examine the role of movement and metabolism in the thermoregulation of a cluster of honeybees (A. mellifera mellifera). The thermoregulation is thought to be a result of individual bees attempting to regulate their own body temperatures between allowable limits. The bees respond to decreases in their immediate, or local, temperature by a combination of increased metabolic activity, and movements towards their neighbours. This is modelled by coupling a thermotaxis-diffusion equation for the cluster density to a heat equation with a temperature and density dependent source. It is assumed that the heat transfer within the cluster is dominated by conduction. The temperature profiles predicted by the model agree qualitatively with those observed. The model also predicts the density profiles within the cluster. These agree with qualitative observations in the literature. 1. Introduction Precise thermoregulation by aggregations of honeybees occurs in two distinct situations. The first is reproductive swarming. Swarms ranging from 2000 to 20000 bees may remain exposed on a tree branch for several days (Heinrich, 1981a, b, 1985). At ambient temperatures below 15 C, the swarm regulates its core temperatures within a few degrees of 35 C, and its surface (mantle) temperature near 15 C. For warmer ambient temperatures, the core temperature remains regulated near 35 C, whereas the mantle temperature will stay a few degrees above the ambient temperature. The second situation involves a winter cluster of 10000 to 30000 bees (Seeley, 1978, 1985). Over an ambient temperature range of 15 C to 10 C, the core temperature of the cluster is maintained between 18 C and 32 C, and the mantle temperature between 9 C and 15 C (Heinrich, 1985; Simpson, 1961; Southwick & Mugaas, 1971). In contrast to homeothermic animals, honey bee colonies do not possess a centralized physiological mechanism capable of monitoring and adjusting E-mail: watmough math.ubc.ca E-mail: Scott Camazine agcs.cas.psu.edu temperatures within the cluster (Brooks & Koizumi, 1974; Southwick, 1983). In fact, it has been demonstrated that the thermoregulation of the cluster does not rely on any communication between bees on the mantle and those in the core (Heinrich, 1981a). This evidence supports the assumption that thermoregulation is a result of the collective actions of the individual bees. That is, it is a result of each bee responding to the local temperature according to a simple behavioural algorithm. Bees on the mantle are not aware of the conditions of the bees in the core, and those in the core do not produce excess heat in order to warm the bees on the mantle. The regulation of the core temperature near a set point results from each bee attempting to remain at a comfortable temperature within the confines of the behaviours permitted it by its own individual physiology. Several attempts have been made to model the temperature distributions within the cluster (Lemke & Lamprecht, 1990; Omholt & Lo nvik, 1986; Omholt, 1987; Myerscough, 1993). The models of Omholt, Omholt and Lo nvik, and Lemke and Lamprecht unfortunately all begin with assumptions about the global behaviour of the cluster that should, ideally, be predictions of the model. Specifically, each of these 0022 5193/95/190391+12 $12.00/0 391 1995 Academic Press Limited

392 J. WATMOUGH AND S. CAMAZINE models assumes the metabolic heat production of the bees to be a function of the bee s position within the swarm. In addition, the swarm radius is specified from experimental data, and not predicted by the model. In contrast, we base our assumptions solely on the behaviours of the individual bees, and not on the observed global behaviour of the cluster. In this aspect, our model is similar to that of Myerscough (1993). Myerscough s model uses the local energetics of the individual bees to predict the global temperature profile within the cluster. It assumes, however, that the density at any point within the cluster can be determined a priori from the local temperature. We take this model one stage further by removing this assumption and using the observed local motions of the bees to predict the global density profile. With our model we are able to capture several features of swarm and cluster thermoregulation that have eluded previous models. These include the formation of the core and mantle as regions of different densities, and the inverse relationship between the core temperature and the ambient temperature (Heinrich, 1981a, b; Southwick & Mugaas, 1971). In contrast to previous models, this model incorporates the motions as well as the metabolism of the bees. In addition, we base the model solely on the responses of individual bees to their environment. We do not use observed global features and responses of the cluster as inputs, but rather derive these as outputs of our model. 2. Derivation of the Model Our model is based on the following assumptions: 1. Each bee bases her behaviour exclusively on the temperature she experiences in her local environment. 2. If a bee is too warm, she will move in the direction of decreasing temperature. If she is too cool, she will move in the direction of increasing temperature. 3. The heat production of an individual bee is based on her metabolic rate. Below a threshold temperature a bee increases her metabolism above the resting level by shivering with her flight muscles. Above this threshold temperature she remains at her resting metabolism. The resting metabolism is an increasing function of temperature as given by Kammer & Heinrich (1974). 4. The cluster is spherically symmetric. 5. Heat transfer through the cluster is due to conduction. Assumption 1 is based on the experiments of Heinrich (1981a), which fail to show that the thermoregulation is coordinated by any form of communication among the bees. In contrast, each bee appears to rely on information taken from her immediate surroundings. The key mechanisms leading to thermoregulation are divided into movements (Assumption 2) and heat production (Assumption 3). The assumed movements are based on the observation that the bees on the mantle will huddle closer together as the ambient temperature falls (Heinrich, 1981a; Ribbands, 1953; Southwick, 1983). Unfortunately, these observations were qualitative rather than quantitative. In contrast, the heat production data are based on several quantitative experiments. Assumptions and 4 and 5 are physical. The assumption of spherical symmetry appears valid according to the observations of Heinrich (1981a), at least at lower ambient temperatures. At higher ambient temperatures (above 15 C) the cluster tends to elongate. The assumption that heat transfer within the cluster is dominated by conduction may be valid at lower ambient temperatures. Below ambient temperatures of 15 C the bees on the mantle are observed to huddle close together, restricting the flow of air through the cluster. However, at higher ambient temperatures visible channels through the cluster are opened which serve to rapidly convect heat from its centre (Heinrich, 1981a, b). Thus, at ambient temperatures above 15 C convection will become more important than conduction as a heat transfer mechanism, and Assumption 5 will no longer be valid. MODELLING THE TEMPERATURE PROFILE Let T(r, t) and (r, t) be the temperature and density of bees at position r and time t. With the assumption of spherical symmetry, r will be the radial distance from the centre of the cluster. Assumptions 3 and 5 allow us to model the temperature distribution within the cluster using a classical heat equation with a source term. c T t =1 r r 2 r2 ( ) T + f(t); (1) r where c is the heat capacity of the cluster; ( ) is the coefficient of heat conduction; and f(t) is the metabolic output per bee. The heat capacity of the cluster is assumed to be one Joule per Kelvin per cubic centimeter (1 J K 1 cm 3 ). This is based on the assumption that the heat capacity of the bees is due to their water content, and that the heat capacity of the airspaces between the bees is negligible. This estimate is in fact the average heat

SELF-ORGANIZED THERMOREGULATION 393 capacity over the entire swarm. Assuming that the average mass of a honeybee is 150 mg, the heat capacity should be approximated as 0.15 J K 1 per bee, and should increase linearly with the local density of the cluster. In addition, the movement of the bees should induce an additional flux into the above equation. It is important to note that these omissions will not affect the steady state solutions of the equation. The first term on the right-hand side of eqn (1) is the heat energy per unit time gained by the bees a distance r from the centre of the cluster due to conduction from their neighbours. The second term is the change in the local temperature due to the heat produced by the bees metabolism. The function f(t) is derived from Assumption 3. A steady state temperature profile is reached when the heat produced by the bees is equal to the heat lost to their neighbours. Heat transfer within the cluster will occur by conduction through the bees and convection through the airspaces between the bees. Due to the lack of experimental data on the conductivity within the cluster we will assume only that the local conductivity depends on the local density of the bees in the cluster. As the bees move closer together, the density increases and the conductivity decreases (Fig. 1). Southwick (1985) suggests that the conductivity may drop as low as 3 10 4 W cm 1 K 1 in the cluster mantle since the bees are able to trap air in their body hair forming an insulating layer. This value of is comparable to the conductivity of dry air and would seem an optimistic lower limit. Heat transfer due to convection within the cluster is ignored since the rate of airflow through the cluster is assumed to be low. The function f(t) represents the production of heat both passively, by the normal metabolic output of the bees, and actively, by shivering the flight muscles. The passive heating is assumed to follow the Q 10 rule similar to bumblebees (Kammer & Heinrich, 1974). Lemke & Lamprecht (1990) suggest that the resting metabolic output of an individual bee increases by a factor of 2.4 for every 10 C increase in the local temperature. This factor seems in agreement with the data referenced by Seeley & Heinrich (1981). They suggest a resting metabolism of one milliwatt (1 mw) at 18 C ( f 18 ). As the local temperature drops below a setpoint, the bees begin to shiver. Their metabolic heat production is assumed to increase exponentially to a maximum of 50 milliwatts, the metabolism of a bee in flight (Seeley & Heinrich, 1981; Heinrich, 1985). The resulting form of f(t) is sketched in Fig. 2. FIG. 1. The following form for the conductivity, ( ), is used: ( )= air ( air bees)( /10) 2/3. If air bees then the mantle of the cluster, where the density is higher, will have a lower conductivity and act as an insulating layer. This model does not take into account the effect of the density of the mantle layer on the rate of heat loss by convection from the interior of the cluster. Nor does it attempt to model convection through the core of the cluster.

394 J. WATMOUGH AND S. CAMAZINE FIG. 2. Metabolic heat production of a single bee as a function of the local temperature. 50f 18 T T m; f(t)= f 1850e 2.044(T T m) T m T T s; f 18(2.4) (T 18)/10 T T s. Above T s heat is produced according to the Q 10 law for the resting metabolism of insects. Below this threshold the bees will increase their heat production by shivering their flight muscles. We assume a 50-fold increase in heat production as the temperature drops from T s to T m. We have used the value f 18=0.001 Watts/bee based on the values referenced by Seeley & Heinrich (1981). MODELLING THE BEE DENSITY PROFILE Assumptions 1 and 2 can be modelled using a thermotaxis-diffusion equation. t =1 r 2 r r2 ( ) r 1 r 2 r r2 (T) T r ; (2) where ( ) is the motility of the bees, and (T) is the thermotactic velocity. The reader is referred to the texts of Edelstein-Keshet (1988) and Murray (1989) for definitions of these terms and an introduction to similar models for chemotaxis. The second term on the right-hand side of eqn (2) represents the changes in the local density of the bees due to movements aligned with the temperature gradient as per Assumption 2. The thermotactic velocity (T) is a measure of the net motion of the bees in the direction of increasing temperature. This will depend on the speed at which the bees move up the temperature gradient as well as the fraction of the bees responding at the given temperature. The minimal requirements of this function are that it be positive if the local temperature is below the huddling temperature T h, and negative if the local temperature is above this level. This indicates that the bees move up the temperature gradient if they are cold, and down the temperature gradient if they are too warm (Fig. 3). The diffusive term, the first term on the right hand side of eqn (2), sums all movements which are uncorrelated with the temperature gradient. These movements will be assumed to be essentially random, and result in a net flux of bees in the direction of decreasing densities. The magnitude of these motions is specified by the motility ( ). By introducing a dependence of the motility on the local bee density it is possible to prevent both over and under crowding within the cluster. We assume that the random motions are several orders of magnitude faster in regions of very high or very low density. Behaviourally, this corresponds to bees avoiding regions where the density is either too high or too low. Heinrich (1981b) notes that the average density within a cluster is roughly 2 bees cm 3 at higher ambient temperatures. As the temperature falls to 0 C the average density increases to a maximum of 8 bees cm 3. To obtain average densities in this range we allow the local densities within the cluster to range from 2 to 10 bees cm 3. The motility function used in the calculations is plotted in Fig. 4. THE BOUNDARY CONDITIONS Spherical symmetry leads to the boundary conditions (0, t)=0, t 0, (3) r T (0, t)=0, t 0, (4) r at the centre of the cluster. Balancing the heat energy generated by the bees with the heat lost from the cluster leads to the boundary condition T =h c (T(, t) T a ) (5) ( ) r r= at the edge of the cluster. The rate of heat loss is proportional to the difference between the temperature at the surface of the cluster and the ambient temperature T a. The rate of heat loss will depend on the properties of the cluster surface as well as on the moisture content and speed of the surrounding air, and will vary over the surface of the cluster and with time. The coefficient h c represents the average rate of heat loss from the cluster surface due to convection. For heat transfer by free convection of air, the convective heat transfer coefficient (h c ) will be on the order of 6 10 4 to 3 10 3 W cm 2 K 1 (Kreith & Bohn, 1986, table 1.4). The temperature profiles given by

SELF-ORGANIZED THERMOREGULATION 395 FIG. 3. For simplicity we assume that the thermotactic velocity as a function of temperature takes the form of a hyperbolic tangent, (T)= o tanh Th T 4. This function is plotted above with o=1 and T h=25 to show the shape of the curve. The essential features are that (T) be decreasing, and that it pass through zero at the huddle temperature T h. Heinrich (1981a, figure 1) can be used in conjunction with (5) to estimate that the ratio T r T T a r= h c = ( (, t)) (6) is approximately between 1 and 3 cm 1. Thus, the above range for h c suggests that the conductivity within the cluster, ( ), is in the range of 6 10 4 to 9 10 3 W cm 1 K 1. These values are comparable to heat conduction coefficients of 3 10 4 W cm 1 K 1 for dry air, 5 10 3 for dry sand, and 10 2 for moist sand. In addition we impose the constraint of constant density (, t)= R (7) at the edge of the cluster. To our knowledge there has been no research into the actual conditions at the boundary of the cluster. It is likely that additional mechanisms are responsible for the cohesion of the cluster. We have also experimented with boundary conditions where the mantle density depends on the ambient temperature, and the cluster radius. Neither refinement appears to have a significant effect on the results. CHANGES IN CLUSTER SIZE The radius of the cluster will vary over time. Let R o be the initial radius, and be the radius at time t. An equation for the rate of change of the radius R can be derived from a constraint on the total number of bees in the cluster. If the number of bees in the cluster remains constant then d dt 0 r 2 (r, t)dr=0. (8) Note that the integral is the total number of bees in the cluster at time t. Expanding this derivative yields 0 r 2 t (r, t)dr+(r2 (r, t)) dr =0. (9) dt 0 Using eqn (2) we can evaluate the remaining integral r2 ( ) T r r2 (T) r +r2 (r, t) dr dt 0 =0. (10)

396 J. WATMOUGH AND S. CAMAZINE FIG. 4. The exact form of ( ) used for the computations is: ( )= 0(100) 2 0 0(10) 10 1000 0 2; 2 10; 10 13; 13 This function is plotted above with 0=1 to show the shape of the function. This function was used to ensure that the bees remained at densities between 1 and 12 bees cm 3. Finally, the boundary conditions (3), (4) and (7), lead to the equation for the rate of change of the cluster radius: R dr dt T = ( ) + (T) r r r= (11) Again, this relation results from assuming that only a very small fraction of the bees enter or leave the cluster. The notations used in the above equations are summarized in the following table. The capitals in the rightmost column represent the dimensions of the quantities. We use T and L for time and length, K for temperature (Kelvin), N for a number, and E for energy. r Distance from centre of cluster [L] t Time [T] T(r, t) Temperature at a point within the cluster [K] (r, t) Density of the bees at a point within the cluster [NL 3 ] Cluster radius at time t [L] c Heat capacity of a honey bee [EK 1 L 3 ] ( ) Heat conduction coefficient [ET 1 K 1 L 1 ] f(t) Metabolic heat production of bees [EN 1 ] ( ) Motility coefficient [L 2 T 1 ] (T) Thermotaxis coefficient [LK 1 T 1 ] h c Rate of heat convection from the cluster surface [ET 1 K 1 L 2 ] T a Ambient temperature [K] R Density of bees at edge of cluster [NL 3 ] The model can be best understood by the following scenario. Consider a cluster of bees at equilibrium at an ambient temperature above the huddling temperature T h. As the ambient temperature falls, the rate of heat loss from the surface of the cluster [see eqn (5)] will increase, causing the temperature on the surface of the cluster to decrease. As the surface temperature falls below T h the bees on the surface will begin to move closer to their neighbours, increasing the local density of the cluster, and decreasing the radius R. If the ambient temperature continues to fall, so that the surface temperature falls below the shiver temperature T s, the bees on the cluster surface will begin to shiver, increasing their metabolic heat production.

SELF-ORGANIZED THERMOREGULATION 397 3. Results CORE AND MANTLE TEMPERATURES Figure 5 shows the solutions of the equations subjected to a forced ambient temperature. The ambient temperature oscillates between 15 and +15 C over a 24-hr period. Figure 6 shows these same results as functions of the ambient temperature rather than the time. The results of several cluster sizes are shown in the data. These are comparable at least qualitatively to the experimental data of Heinrich (1981a, figure 2). The two key features of these results are (i) the increase in core temperature with an increase in the number of bees in the cluster, and (ii) the increase in core temperature with a decrease in ambient temperature. These solutions are very close to the steady-state solutions obtained by running the simulation at a steady ambient temperature until the solution appears to converge. For this reason we do not expect that they will be significantly altered if we relax the assumption that the heat capacity is uniform throughout the swarm. METABOLIC HEAT PRODUCTION Figure 7 shows the dependence of the total metabolic heat output of the cluster on both the ambient temperature and the cluster size. For T a 0 C, these results are comparable with the experimental data of Heinrich (1981a, figure 9) and Southwick (1983, 1988). For ambient temperatures above 0 C the data of Heinrich (1981a) continue to decrease linearly with ambient temperature. However, Southwick s (1988) data show a broad minimum of metabolic output similar to the one predicted by our model. TEMPERATURE AND DENSITY PROFILES Typical temperature and density profiles are shown in Fig. 8. There is a distinctive boundary between the higher densities of the mantle and the lower densities of the core. Note that only a relatively small increase in the radius of the cluster is required for a significant reduction in the thickness of the mantle. Since the cluster is spherical, the decrease in the thickness of the mantle does not result in a significant decrease in the number of bees in the mantle. In fact, the model predicts that the majority of the bees will be found in the mantle even at the higher ambient temperatures. Figure 9 shows the metabolic output of the bees as a function of their position within the swarm. These figures show that even at higher ambient temperatures, the bees on the mantle are the primary heat producers. These results agree with those of Omholt (1987). FIG. 5. Numerical computations of the core and mantle temperatures using an oscillating ambient temperature. The results are shown for a medium sized cluster and a large cluster. In each graph, the upper, dashed line is the core temperature, the middle, dashed line the mantle temperature and the solid line the ambient temperature. The circles below the time axis represent the cluster size. This changes very little below 0 C, but increases rapidly with ambient temperatures above 0 C. The parameter values used were c=1 J K 1 cm 3, h c=2.5 mw cm 2 K 1, bees=5 mw cm 1 K 1, air=10 mw cm 1 K 1, o=8 10 4 cm K 1 s 1, T h=25 C, o=8 10 4 cm 2 s 1. The number of gridpoints used was 360. THE ROLE OF SHIVERING Figure 10 shows the core and mantle temperatures resulting from variations in the shivering response. If the temperature below which the bees begin to shiver is lowered, the equilibrium mantle temperature decreases. The core temperature, however, increases, as a result of the increased mantle thickness. This shows that the movement of the bees is not sufficient for thermoregulation, but that the shivering at low temperatures is also necessary. 4. Discussion The conductivities and heat transfer coefficients used in the numerical results shown in this paper were

398 J. WATMOUGH AND S. CAMAZINE FIG. 6. Numerical computations of the core and mantle temperatures versus the ambient temperature. This data was computed using the same parameter values as for Fig. 5. Note that the core temperature shows a slight negative correlation with the ambient temperature, whereas the mantle shows a positive correlation. Both the core and the mantle temperatures remain nearly constant for temperatures below five degrees celsius. There is also a definite increase in core temperature with increased cluster size. at the high end of those anticipated. Thermoregulation did not occur at lower conductivities due to a rapid overheating of the core. It is possible that a more detailed model incorporating the density dependence of the heat capacity would allow for lower conditivities. However, initial simulations using such a model have not shown any significant differences. This result also suggests that the convective heat transfer within the swarm may be more important than we have assumed. Recall that we have assumed that convection FIG. 7. Numerical computations of the metabolic output of the entire cluster for several cluster sizes. These computations are taken from the same simulations as the temperatures of Fig. 6. Note that at temperatures below 0 C, the metabolic output of the cluster increases linearly with a decrease in ambient temperature. Also, the metabolic output per bee is relatively independent of cluster size.

SELF-ORGANIZED THERMOREGULATION 399 FIG. 8. Numerical computations of the temperature and density profiles for a cluster of 13109 bees. Note that the thickness of the mantle decreases, but that the density in the mantle remains in the upper limit set by the motility function. For higher temperatures the mantle begins to collapse. The decrease in thickness of the mantle, and the ensuing collapses does not imply that there are fewer bees which can be designated as mantle bees. The number of bees in the mantle will increase as the square of the cluster radius. The temperature profiles are in a rough qualitative agreement with the data of Heinrich (1981a). We are unable to reproduce the stepped temperature profile he observes. within the cluster would be negligible at ambient temperatures below 15 C. Unfortunately, a model incorporating convection would require a far more complicated geometry than the model presented here. Southwick & Moritz (1987) observe that the air within the cluster is purged periodically by organized fanning. This is followed by a slower influx of fresh air. They speculate that the fanning activity of the bees is triggered by high levels of carbon dioxide. This suggests that convective heat exchange may also occur FIG. 9. These data are taken from the same calculations as Fig. 8. Here we show the cumulative fraction of metabolic output with increasing distance from the centre of the cluster. Note that even at higher ambient temperatures, at least 80% of the heat is produced by the outer third of the swarm.

400 J. WATMOUGH AND S. CAMAZINE FIG. 10. Dependence of the temperature and density profiles on the shivering temperature T s. The solid line represents computations performed with T s=13 and T m=10; the dashed line computations performed with T s=3 and T m=0 (see definitions in Fig. 2). Observe that the mantle is thicker in the latter case. Thus, even though the surface temperature is lower, the core temperature is higher. in a periodic manner when the increased density of the mantle reduces the air flow through the cluster, and causes a buildup of carbon dioxide. Our model shows a definite improvement in thermoregulation over Myerscough s model. Her model did not produce the observed inverse relationship between the core and ambient temperatures. Our model, which includes the motions of the bees in addition to their energetics, is able to reproduce this feature. Both models produce broader temperature profiles in the core region than those observed in live clusters. The data of Heinrich (1981a) consistently show a two step temperature profile. If the heat transfer within the cluster is dominated by conduction, then this stepped temperature profile implies the existence of a second band of higher density and increased heat production in the interior of the cluster. Equation (1) can be manipulated to show that the slope of the temperature profile a distance r from the core of the cluster is proportional to the ratio of the metabolic heat produced by the bees inside the sphere of radius r and the conductivity of the cluster at the point r. Thus, if there is a region inside the cluster where the temperature profile is steep, then this region must have an increased heat production and a decreased conductivity. It is possible that this observation could be explained by introducing two classes of bees, a first class which behaves as described above, and a second class which has a preference for higher temperatures. The thermotactic mechanism may provide a means to separate the second class of bees to the core, and the first class to the surface of the cluster. Each group of bees could produce a band of higher density and higher heat production. The older bees creating the mantle, and the younger bees the proposed inner band. There does appear to be evidence for the segregation of young and old bees in this manner. Younger bees, who are less able to regulate their own temperatures remain near the core, while the older bees compose the mantle (Heinrich, 1981a). The data of Southwick (1988) do not show this stepped temperature profile. It is not known at this time whether this difference could be due to the absence of younger bees in his clusters. However, since his experiments were performed in the winter, this may be the case. Our model predicts the formation of the mantle and the core previously observed. These predictions are grounded on the motions of the individual bees. Much experimental work is now necessary to measure the parameters introduced in the model, and to examine the density profiles of thermoregulating clusters in more detail. In addition, further numerical and analytical work is necessary to examine the hypothesis that simple differences in behaviour between old and young bees can lead to their positional separation, and the formation of an inner mantle layer. An inverse relationship between the core temperature and the ambient temperature results from the

SELF-ORGANIZED THERMOREGULATION 401 mechanics and energetics chosen in the model. This relationship has been observed by several experimentalists, but has not been explained by any previous models. These results support the speculation (Moritz & Southwick, 1992) that the active thermoregulatory behaviour is performed by the bees on the mantle. As the ambient temperature decreases, these bees attempt to keep their local temperature above a minimum setpoint. This results in an increase in the temperature at the cluster core. This heating of the core has been noted to approach lethal temperatures in some isolated cases (Heinrich, 1981a). The fact that the bees on the mantle will attempt to keep warm even at the expense of overheating the core of the cluster strongly suggests that the thermoregulation is self-organized. Such an overheating should not arise if either the heating is performed by the core bees in order to keep the bees on the mantle above a certain set point, or if the heating and huddling are performed by the bees on the mantle in order to keep the bees in the core at a set point. The overheating of the core suggests that the bees on the mantle are huddling and shivering in an attempt to keep themselves within a certain temperature range, and that this desire is sometimes in conflict with the attempts of the core bees to remain at a set point of 36 C. We gratefully acknowledge support from the National Sciences and Engineering Research Council of Canada under grant number OGPIN 021. We also wish to thank John Stockie and Brian Wetton for their helpful hints on the numerical methods, and Mary Myerscough and Leah Edelstein-Keshet for their many constructive comments. REFERENCES BROOKS, C. & KOIZUMI, K. (1974). The hypothalamus and control of integrative processes. 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Appendix Numerical Methods The above equations were solved by first transforming them to the dimensionless coordinate system x(r, t)= r t=t u(x, t)=t(r, t), v(x, t)= (r t), (A.1) (A.2) (A.3) (A.4) In these coordinates, eqns (2) and (1) of the main text become cu t cxr'(t) u x = ( u) +vf(u) () 2 v t xr'(t) v x = ( (v) v (u)v u) () 2 (A.5) (A.6) for 0 x 1, t 0. The boundary conditions become u x (0, t)=0 v x (0, t)=0 u x (1, t)= h c (u(1, t) T a ) v(1, t)= R (A.7) (A.8) (A.9) (A.10)

402 J. WATMOUGH AND S. CAMAZINE Equation (11) of the main text becomes dr R dt = ( ) R x=1 + (T) T R x=1 (A.11) These equations were discretized in both space and time by first freezing the coefficients, and then using a centered difference scheme for the diffusive terms, and an upwind difference scheme for the convective terms. The resulting method is first order in both space and time. By solving eqns (A.5), (A.6) and (A.11) separately, the method requires the inversion of two tridiagonal matrices. We first calculate the change in the cluster radius by freezing the coefficients (v) and (u) in eqn (A.11) at the current values of u and v. Next, the coefficients (v) and f(u) of eqn (A.5) are frozen at their current values, and the new temperature profile is computed using the new cluster radius. Equation (A.6) is solved by freezing the coefficients (v) and (u), the current density profile v, and the new temperature profile u. Finally, the total number of bees in the cluster is recalculated, and the density profile is adjusted to maintain the population at its initial level.