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Thermoregulation in the winter cluster of the honeybee, Apis Mellifera
J. theor. BioL (1987) 128, 219-231
Thermoregulation in the Winter Cluster of the Honeybee,
Apis
Mellifera S-rIG W. OMHOLT N-6530 Bruhagen, Norway
(Received 15 December 1986, and in revised form 7 April 1987) A model is presented based on the assumption that the apparently coordinated thermoregulatory responses of the broodless winter cluster of the honeybee, Apis mellifera, are results of the individuals acting behaviourally and physiologically to regulate their own body temperature. The predictive potential of the model is elucidated by a set of mathematically formulated relationships. The results include predictions concerning radial distributions of metabolic rate, and temperature profiles, as a function of ambient temperature. Furthermore, the apparently equal mass specific metabolic rate observed from weak and strong winter clusters is shown to be consistent with the proposed model. Consideration is also given to the generally accepted view of how the thermoregulation is accomplished, which is shown to give very unrealistic predictions.
I. Introduction There is still no general consensus as to which behavioural and physiological mechanisms are responsible for the thermoregulatory responses of the honeybee colony. Assuming that the individual bees are in some way or another aware of the needs o f the colony per se and directly responding to this in an a p p r o p r i a t e way, the accepted view seems to be that supercoordinating principles are operative in colonial thermoregulation (Farrar, 1943; Free & Spencer-Booth, 1958; Himmer, 1932; Milner & Demuth, 1921; Owens, 1971; Phillips & Demuth, 1914; Ribbands, 1953; Savitskii, 1980; Southwick, 1982, 1983; Southwick & Mugaas, 1971; Wedmore, 1947; Wilson, 1971). There is a widespread belief that the bees in the core region are responsible for the main part of the heat production and that their production is tuned to the survival o f the bees at the mantle of the cluster. Such principles, however, have been shown to be non-existent in swarm thermoregulation (Heinrich, 1981). In such a situation, Occam's razor is a convenient tool to apply. The basic aim of the present work is, in the case of the broodless winter cluster, to elucidate the consequences o f the assumption that the apparently coordinated thermoregulatory responses o f a bee society result from individuals acting behaviourally and physiologically to regulate their own body temperature in the context o f proximal and evolutionary constraints and possibilities afforded by their social life (Heinrich, 1985). Based on such a hypothesis, a set of simple mathematical relationships describing different characteristics o f the winter cluster are developed. From these are derived several predictions which are in agreement with empirical observations. 219
0022-5193/87/180219+ 13 $03.00/0
© 1987 Academic Press Ltd
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It is believed that the conceptual apparatus introduced in this paper represents an essential component of the theoretical framework needed for a comprehensive and rational experimental and theoretical approach to the study of the winter cluster.
2. Description of the Model (A) T H E R M O R E G U L A T O R Y
MECHANISMS
The thermoregulatory response pattern of the broodless winter cluster is assumed to be due to very simple behavioural rules followed by the individual bees: When a bee experiences a fall in temperature by its thermoreceptors on the antenna, it responds by increasing its metabolism to a maximum level dictated by its thorax temperature, and by moving towards a positive temperature gradient. When it experiences a rise in temperature, it responds by remaining at a resting metabolic level defined by its thorax temperature, and by moving towards a negative temperature gradient. This model may be called the COFY (Care Only For Yourself) model. While it cannot generate all the different behavioural patterns associated with the winter cluster (see Discussion), it is assumed to be responsible for the major thermoregulatory response characteristics of the whole colony. The main underlying premises for the model are: (1) The observed volume changes and heterogenous structures of winter clusters are due to changes in ambient temperature (Owens, 1971; Ribbands, 1953; Simpson, 1961). (2) The bees are equipped with thermoreceptors on the antenna which react to a sudden decrease in temperature by increasing impu' ~ cy, whereas an increasing temperature causes a decrease (Lacher, 196,,). l nese receptors respond to temperature changes down to 0.25°C, and it is reasonable to assume that they play a significant role in the monitoring of the bees' temperature environment. (3) Individual bees gain heat through rapid small-amplitude contraction-relaxation cycles of their thoracic wing muscles. Heat production is a result of the low mechanical efficiency of this musculature which makes up the bulk of the thorax, and is among the most metabolically active tissues known (Esch, 1964; Esch & Bastian, 1968; Heinrich, 1980a, b, 1981; Kronenberg & Craig-Heller, 1982; Southwick, 1983). (4) The thorax contains the ganglia that regulate the contraction of the indirect flight muscles. The neural output from these ganglia is highly temperature-dependent (Bastian & Esch, 1970; Esch & Bastian, 1968; Hanegan, 1973; Heinrich, 1980a; Machin et al., 1962). This suggests a close relationship between the thorax temperature and heat production capability. By using thermocouples placed in individual bees within a group, sharp and irregular fluctuations of thoracic temperature may be observed, indicating that the individual bee has an on/off response pattern (Bastian & Esch, 1970; Esch, 1960; Kammer & Heinrich, 1974; Kronenberg & Craig Heller, 1982). These observations suggest that for a specific surrounding air temperature (Ten), the metabolic limits
THERMOREGULATION
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of an individual bee are defined by the resting metabolism (q(Ten)res,), and the maximum limit (q(Ten)max), which is the heat generated when the bursts are at the maximum attainable intensity, and when the direct flight muscles are not activated. The COFY model predicts that the macroscopic behaviour of the winter cluster is the result of a "force de majeur" principle, and a very intricate pattern of stabilizing and destabilizing individual behaviour defines the dynamics. A quasi-steady state condition is supposed to exist when the stabilizing forces are stronger than the destabilizing. This may happen when the ambient temperature (T~) has not changed significantly for several hours and the colony is not moving honey into the clustering area (Owens, 1971). In this state there may be only minor fluctuations about an average value of the different macroscopic parameters by which the cluster is defined. Thus, for a given To, the cluster may be characterized by a specific cluster radius (R(T~)) (assuming spherical geometry), a specific radial bee density distribution (D(r, To)), a specific radial distribution of metabolic rate per individual (q(r, T~)), and a specific temperature profile ( T(r, To)). The resting metabolism is known with some certainty (Heinrich, 1981; Kammer & Heinrich, t974; Omhoit and L0nvik, 1986). Due to the paucity of experimental data, the maximum metabolism is assumed to be proportional to the resting metabolism in this study, so that q(T~,)max=q(Ten)restX 1~, where 1~ is the proportionality constant (see Discussion for further validation). Under steady state conditions it is assumed that only a minor proportion of the bees are not producing at a maximum or a minimum level. Those will be situated at a layer in the cluster where their cooling rate is greater than that compensated by their resting metabolism, but small enough to be compensated by a metabolic rate less than or-t.I~Up.b'!i a~eir maximum production.
(B) DESCRIPTION OF THE QUASI STEADY STATE CONDITION
( a ) Radial distributions of metabolic rate per individual bee The heat flux from a cluster at a specific ambient temperature (Q(Ta)) may be given by
Q( Ta) = k( T~)4rrR( To)2( T p - Ta)
(1)
where k ( T ~ ) = coefficient of heat transfer at Ta, Tp = temperature at the periphery of the cluster. The cluster structure can be approximately characterized by an outer shell where the bee density is considerably greater than in the inner core, and where the thickness of the shell increases with decreasing Ta (Owens, 1971; Ribbands, 1953; Simpson, t961; Wedmore, 1947). It is therefore natural to introduce two density levels, D .... and Dshetl, SO that
Nh = Jo
"° 4rrr2D .... d r + JnlT-.}.... 4"rrr2Dsheil dr
(2)
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where Nb = n u m b e r o f bees in the cluster; R(TQ) .... = the core radius at a specific
r~. Given an ambient temperature range (Tmi., Tmax), and by quantifying Q(Tmax), Q(T~in), Tp, D¢ .... D~h¢ll, R(Tmin) ..... R(Tm~)core, one may calculate the radial limits, R ( T ~ . ) and R(T~,~), and the heat transfer coefficients k(Tmi.) and k(Tm~) from eqns (1) and (2). According to the comprehensive observations made by Owens (1971), the volume of a normally strong cluster decreases linearly between the Ta-limits - 2 0 ° C and 5°C. This implies that within this temperature range (see section 2(d)) R(T.~) = [ ( ( T m ~ - T~)R(T.~i.)3+(Ta - Tmin)R(Tmax)3)/(Tmax -
Tmin)]1/3.
(3)
Q(Ta) between the To-limits may then be quantified by applying a specific distribution of k(T~) (throughout this study assumed to be linearly decreasing between Tm~. and Tm~). The radial borderline between bees remaining at a resting metabolism and those at an enhanced metabolism may be calculated from the relationship ~lor ¢
Q(T~) = E k=!
n
VkD .... qk +
i
n
~" k = ncor¢+
VkDshe,,q k + VpD~,~,,qp + I
E
VkD~h~,,qkl~ (4)
k=p+l
where n = number of concentric spherical shells into which the cluster is divided; n .... = greatest shell n u m b e r where the bee density is D¢o~; n ~ , = greatest shell n u m b e r where the bees are staying at a resting metabolism; Vk = volume of the kth shell (when k < nco~ or T~ = Tm~, the wideness of the shell is R( Tmax)/n, otherwise it is ( R ( T ~ ) - R ( T , , ) .... ) / ( n - n .... )); q k = r e s t i n g metabolism per bee in the kth shell; p = the shell where the metabolic level is somewhere between minimum and m a x i m u m metabolism. The radial distribution of metabolic rate per individual when all the bees are at a resting level, may be a p p r o x i m a t e d by an error function (cf. Omholt & Le~nvik, 1986), so that q ( r ) ~ = q ( 0 ) ~ t exp - ar 2.
(5)
By the simplifying assumption that the radial temperature profile is rather invariable with changes in T~ (see Discussion), the discrete version of this function when it is calibrated against R(T,,,ox) and Q(Tm~,), m a y be applied in eqn (4) for every T~. The model predicts that there should be an agreement between n .... and n ~ , . However, n .... and n .... are not necessarily identical, as some of the bees may be forced to stay at a high density even if they are at resting metabolism. This may happen because the bees farther out might press inwards and the bees farther in might push outwards (cf. Results). Such a situation is more likely to exist at low rather than at high To's, indicating that primarily there should be correspondence between n,¢~, a n d n ..... but that the model is not violated if n,~t > n .... when To is in the lower domain. However, according to the model, n~¢~,cannot be less than n .... This relationship m a y be used as a calibration criterium in order to determine an appropriate magnitude of I,..
THERMOREGULATION IN THE WINTER CLUSTER
223
( b ) Temperature distributions These may be calculated from the relationship
T(r, To) = Tp+
f
mL~
j(r)/A(r) dr
(6)
dO
where j(r) is the energy current density at radial distance r
j(r) =
Io
4~r2D(r, To)q(r, TO) dr/4~r 2
(7)
and A(r) is the coefficient o f heat conduction. The coefficient of heat conduction is assumed to be related to the bee density (Simpson, 1961; Wedmore, 1947), so that A.... = A(r, D .... ) and A~h~H= A(r, D~h~t) are constants. By assuming a specific proportionality constant between A.... and A~h~, the coefficients of heat conduction may be calculated by application o f eqns (2) and (5), even if one takes into consideration that the outermost bees have started to form a thin shell at Tm,x.
(c) Mass specific metabolic rate Some workers seem to be surprised at not being able to observe an increasing mean metabolic rate per individual with a decreasing number of bees in the cluster, considering the increasing surface to volume ratio (Free & Simpson, 1963; Southwick, 1983). The C O F Y model may offer an explanation:
In weak colonies there is a relatively greater radial contraction than in stronger ones, because a greater part of the cluster will experience a greater heat loss than can be compensated by resting metabolism. Testable predictions from this hypothesis may be obtained in the following way. Let N b ( m i n ) < N b < Nb(max), and let Mmr(TO) define a specific mean metabolic rate per individual for Nb(max) as a function of ambient temperature. By specifying the different cluster radii (R(Ta, Nb(max))) between Tmaxand Train, and the limits and form of k(TO), M,.r(TO) is obtained from the relationship
MInt(TO) =4~rk( TO)R( TO, N b ( m a x ) ) 2 ( T p - Ta)/ Nb(max).
(8)
From the assumption that M,.r(TO) is identical for every Nb, it follows that the radius for a specific cluster as a function of ambient temperature may be expressed as
R( TO, Nb) = [NbM,.r( TO)/4-rrk( TO)( Tp - T~)] ~/2.
(9)
However, if R(TO, Nb) implies a mean bee density greater than a specified maximum Dmax (corresponding to the minimum radius R(Nb)ml.), the actual mean metabolic rate M(TO, Nb) has to be greater than MInt(TO), and is given by
M( TO, Nb) =47rR( Nb)~.k( TO)( Tp - TO)/ Nb.
(10)
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Furthermore, the cluster's degree of contraction (Ca) may be expressed as
(11)
Ca = R(Nh)min/ R( Ta, Nh). ( d) Quantification of parameters
The ambient temperature in this study varies between -20°C (Tm~,) and 5°C (Tmax), which is a temperature range wide enough to disclose the different patterns predicted by the model, but narrow enough to avoid dealing with the special cases arising at the upper and lower existence limits o f the cluster. D .... and Dsh~, are normally in the range o f 1-2 and 4.5-6 bees/ml respectively (Heinrich, 1985; Owens, 1971; Wedmore, 1947), and are given the values 1.25 and 5-5 in this study. From a normally strong winter colony of 15 000 individuals, which corresponds to the basic Nb-value to be used throughout this study, Q(T,,~x) and Q(Tmi°) are assumed to be 80 and 350 cal/min respectively (Free & Simpson, 1963; Southwick, 1983). The minimum and maximum size of the cluster core, R(Tmi,) .... and R( Tm~x)¢.... are presumed to have values of 6.0 and 12.5 cm respectively. The mantle temperature Tp, normally lies within the range 8-12°C (Free & Spencer-Booth, 1960; Owens, 1971), and is a s s u m e d t o stay constantly at 10°C irrespective of the ambient temperature. The core temperature Tc at T~.~ is set to be 24°C (Himmer, 1932; Southwick & Mugaas, 1971). The resting metabolic rate o f a bee staying at the mantle (q(R(T,,)~es~), is assumed to be 0-0015 c a l / b e e / m i n , which is in agreement with the results obtained by Heinrich (1981), Kammer & Heinrich (1974) and Omholt & Letnvik (1986).
3. Results (A) D I S T R I B U T I O N S
OF
METABOLIC
RATE
When Ta is high (or extremely low) the metabolic rate per individual increases rapidly from the periphery to the centre. This is in accordance with the general view of how the heat is produced (cf. Omholt & Le~nvik, 1986). However, when Ta is in the medium range, the metabolic rate will decrease at a resting level from the centre to a point R(Ta)~est, where it increases to a level somewhat greater than q(0)re~, and then decreases outwards. Even at -20°C, only the bees outside a radial distance o f 0.8 x R ( - 2 0 ) are in an enhanced metabolic state (Fig. 1). This implies that the active thermoregulation, under normal temperature conditions, may be located in the outer part o f the cluster, as proposed by Heinrich (1985), M~bus (1980b) and Steiner (1947). Furthermore, the model predicts that the share o f the total heat production from the different parts o f the cluster is dependent on Ta, with a decreasing part of the heat being produced in the core region with decreasing Ta, until To becomes so low that almost all the bees are staying at a maximum metabolism. However, even then the share of the core bees is not likely to exceed 50% (Fig. 2).
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225
I-0 0"95 0"90 ~: 0.85 0-80
I
-
20
I
I
I
I
I
-15 -I0 -5 0 Ambient temperoture(*C)
5
FIG. 1. T h e relative radial proportion of the cluster remaining at a resting metabolism as a function of ambient temperature, when Nb = 15 000 bees; R( Tmi.) . . . . = 6.0 cm; R(Tm,~) .... = 12.5 cm; Dco,c = 1-25 b e e s / m l ; D, hell= 5"5 b e e s / m l ; Tp = 10°C; q(R(T,)),,,st=O.O015 c a l / b e e / m i n ; q(0),c~,=0.02395 cal/bee/min; I~ = 8-75; Q(Tmi .) = 3 5 0 . 0 c a l / m i n ; Q ( T m , ~ ) = 80-0 c a l / m i n ; k ( T ~ ) =0.0076 c a l / c m 2 / m i n / ° C ; k(Train) = 0"0106 c a l / c m Z / m i n / ° C .
I00
8O
=5"C g 60 S
t20.C
-6 40
g E < 20
0.2
0.4 0.6 0,8 Rodiol distonce
1
PO
FIG. 2. The percentage o f the total a m o u n t o f heat produced up to a given radial distance, when T, = - 2 0 ° C a n d 5°C. All parameter values as listed for Fig. 1. (B) T E M P E R A T U R E
PROFILES
The model is able to predict an inverse relationship between Tc and T~, in agreement with the observations of Corkins (1932), Himmer (1932), Owens (1971), Southwick & Mugaas (1971), Steiner (1947) and Wilson & Milum (1927). However, there is no significant increase of Tc until T~ is below -10°C, and the model predicts even a slight decrease in Tc below 24°C when T~ =0°C (Fig. 3). This may explain why there has been so much discussion on the existence of a long-term inverse relationship, as long as some investigators have not measured the core temperature below an ambient of -10°C.
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38
36
34
oo ®
52
50
X
x 24 k L,,,,I -2(3
I -15
+ I -I0
~ 'I ~ X ~ 4-5
0
l 5
Ambient temperature ( ° C )
FIG. 3. Predicted core temperatures as a function of ambient temperature, when N~ = 15 000 bees;
R ( T , . i . ) ..... =6-0 cm; R(T,.,,.~) ..... = 12"5 cm; D .... =1-25 bees/ml; D~h~.=5-5 bees/ml; T r = I 0 = C ; q( R( 7",,))re., = 0"0015 cal/bee/min; q(0)rc~, = 0-0295 cat/bee/rain; l,. = 8.75; Q ( T , ~ . ) = 350.0 cal/min; Q( Tm,,~} = 80.0 cal/min; k(Tm.~) =0"0076 c a l / c m ' / m i n / ° C ; k(T..,.) =0.0106 cal/cm2/min/°C; A.... = A.h¢. =0"036 c a l / c m / m i n / ° C . The (+) and (x) signs show respectively the effect from a 25% increase and decrease of the coefficient of heat conduction in the outer shell compared to that of the core.
The magnitude of the coefficient of heat conduction seems to be rather sharply defined in order to obtain an inverse relationship between T~ and Ta (Fig. 3). If A is inversely related to the bee density, Tc increases to unreasonably high levels at low T,,'s. As long as a constant A gives a bit too high estimate, this implies that it may increase slightly with increasing bee density. The predicted temperature profiles imply that the distance between the isotherms shortens in the outer part o f the cluster with decreasing T,, (Fig. 4). This seems to be in agreement with observations on swarm clusters (Heinrich, 1981, 1985), as welt as winter clusters (Owens, 1971). The rather constant radial distance to the 15.6°C isotherm observed by Owens (1971) during December and January, is predicted by the model. Owens' data show a variation of approximately 0.35 cm about the mean value, while the model predicts a variation of approximately 0.5 cm.
(C) MASS S P E C I F I C MEAN METABOLIC RATE
From Fig. 5 it follows that the lack of increase of mean metabolic rate with decreasing Nb may be explained by a higher degree of contraction by the smaller
THERMOREGULATION
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227
Fa= 5 ° C
m
=_zo°c!!!!ll
nl,l I
i
I
i
I
~
i
I
I
I
i 1~=_5oc
I I I I I I
~:15°C FIG. 4. P r e d i c t e d i s o t h e r m a l d i s t r i b u t i o n at d i f f e r e n t a m b i e n t t e m p e r a t u r e s . T h e o u t m o s t m a r k s r e p r e s e n t the 10°C i s o t h e r m s , a n d t h e i s o t h e r m a l d i f f e r e n c e is 2.5°C. T h e p a r a m e t e r v a l u e s a r e as listed f o r Fig. 3, e x c e p t t h a t A~h~~= t - 2 5 X A. . . . -
colonies. The contraction potential of even weak clusters are considerable, and if used to the full extent, Nb has to be less than 5000 and T, below -10°C, before an increase in mean metabolic rate is expected. These results are in agreement with the view that small colonies do not freeze to death unless Ta is extremely low, but actually starve to death at low T,, by not being able to move honey into the clustering area (Owens, 1971).
4. Discussion (A) SHORTCOMINGS
OF THE
MODEL
The model is not able to account for the cluster behaviour in connection with movements of honey into the clustering area. This is, however, short-term behaviour (Owens, 1971), and does not interfere with the predictions given above. On the other hand, it would have been very interesting to know whether an individualistic or a super-coordinated approach is the most convenient to apply in this connection. Presence of brood in the winter cluster is a rather frequent p h e n o m e n o n (Avitabile, 1978; Free & Simpson, 1963; Jeffree, 1956; M~bus, 1980a). It is not clear whether the bees actively regulate the brood temperature or not (Heinrich, 1985; Kronenberg & Craig-Heller, 1982; Ritter & Koeniger, 1977), but if they do, other thermoregulatory mechanisms are likely to be involved. Especially at low ambient temperatures there may be a build-up of carbon dioxide within the cluster to such a level that the core bees respond by fanning (Seeley, 1974). Such behaviour will have some impact on the production and temperature profiles, but does not contradict the main premises behind the COFY model.
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+ -+
+
+ E_ .t:]
I
-20
I
I
-15
-10
[
I
I
-5
0
5
Ambient temperature (°C)
i,o! _~ 0-9 ~G §
r~
0"8
0.7
0"6
5
I0
15
20
25
Cluster size (/V~)/1000
FIG. 5. (a) Calculated mean metabolic rates per bee from clusters of different strengths as a function of ambient temperature, when the clusters are allowed to make use of their contraction potential in order to keep a mean metabolic rate identical to that of the strongest cluster which is depicted by the graph. The plus signs express the metabolic rate from the cluster sizes not being able to cope with Nh(max), where those at -20°C represent clusters of 2500, 5000 and 7500 bees respectively, and that at -15°C a cluster of 2500 bees. The applied parameter values are: N h ( m a x ) = 2 5 000 bees; Dm~x = 5.5 bees/ml; k(T, nl,) = 0-0106 cal/cm2/min/°C; k(T,,,~) = 0-0076 cal/cm2/min/°C; Tp = 10°(2; R ( - 2 0 , Nh(max)) = 12.08 cm; R ( - 1 5 , N h ( m a x ) ) = 1 3 . 3 4 cm; R ( - 1 0 , N h ( m a x ) ) = 1 4 . 4 8 cm; R ( - 5 , N h ( m a x ) ) = 1 5 - 3 2 cm; R(0, Nb(max)) = 16.14 cm; R(5, Nb(max)) = 16.89 cm. (b) The calculated contraction degree of clusters of different strength at Tu = -20°C, -10°C and 5°C. The parameter values are as listed for (a).
(B) ROBUSTNESS OF THE MODEL
From the outset, the calculations are based on rather idealized conditions in assuming spherical geometry, neglecting the effects from combs, and by representing the bees as homogenous energy-producing units uniformly distributed in thin layers. In order to be sure that the model has captured the more significant features, the predictions should be rather insensitive to changes in the model parameters from a qualitative point of view. This is especially important in connection with parameters that have a very scarce empirical backing, such as the form of k(Ta) and q(Ten) . . . . A linear k(T~) seems to represent a rather crude approximation, because it gives a somewhat small difference between Q ( - 2 0 ) and Q ( - 1 5 ) . However, a k(T~) that
THERMOREGULATION
1N
THE
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CLUSTER
229
increases more rapidly towards T.,in has only a minor effect on the results, as long as the calculated limits k ( T m J and k(Tma~) agree very closely with empirical data (Biidel & Herold, 1960; Savitskii, 1980). The difference between k(Tmax) and k(Tmin) is likely to be caused by an increase in the water loss from the cluster surface, as long as the temperature difference is too small to effect a significant increase of the conduction and radiation components of k( Ta). Metabolic measurements o f individual bees exposed to different environmental temperatures indicate that the heat production capability at 15-20°C may be somewhat greater than what is assumed above (Allen, 1959; Cahill & Lustick, 1976; Free & Spencer-Booth, 1958; Heussney & Roth, 1963; Heussner & Stussi, 1964). Also, the length o f the action potentials from the dorsoventral thorax muscles o f bees individually exposed to different environmental temperatures shows a rather moderate difference between 18-30°C (Esch & Bastian, 1968). These two conditions may imply that q( Te,)ma, increases a bit more rapidly up to 15-20°C, and then increases more slowly towards To, than what is assumed by q( Te,)man = le x q( Te,)rest. However, a change o f q(T~,)ma~ along these lines will have very little quantitative and no qualitative influence on the results. In this connection it is nice to observe that the maximum heat production capability predicted by the model by applying the calibration criterium nre~t> n .... for Ta > - 1 5 ° C , is in agreement with production levels o f unisolated colonies at -40°C (Free & Simpson, 1963). This is a temperature level rather close to the lowest T~ at which a normally strong winter cluster is able to compensate its heat loss. The effects from an increasing T~ with a decreasing T~, and hence a steeper temperature gradient from the periphery inwards, are not accounted for in the calculations behind Figs 1 and 2. The former condition will increase the resting metabolism in the core region, and the latter will, according to the C O F Y model, enhance the maximum heat production capability of the bees in the outer parts. In the present context, both conditions have an insignificant impact on the results. The above considerations imply that the model generates predictions that are rather robust from a qualitative and semi-quantitative point of view. (C) C O N S I D E R A T I O N S
OF THE
MANTLE
TEMPERATURE
The model cannot disclose why the mantle bees stay at a temperature level of approximately 10°C. This is about 5°C below the mantle temperature of swarm clusters (Heinrich, 1981, 1985). It is reasonable to assume that this difference is due to the acclimatisation experienced by the bees under prolonged exposure to low temperatures (Free & Spencer-Booth, 1960). This is further validated by the fact that the total metabolic rate has a minimum when To is 10°C for winter clusters, and 15°C for swarm clusters (Free & Simpson, 1963; Heinrich, 1981; Southwick, 1983). From an evolutionary point of view there seems to be two major reasons why the mantle bees should have a temperature just a little above their chill-coma temperature. One is that the reduction of the mantle temperature implies a reduction
230
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o f h o n e y c o n s u m p t i o n d u r i n g t h e winter. T h e o t h e r is t h a t a low m a n t l e t e m p e r a t u r e r e d u c e s t h e loss o f w a t e r f r o m all t h e b e e s in t h e cluster. T h i s m a y b e e s p e c i a l l y i m p o r t a n t in t h e core r e g i o n , b e c a u s e the loss o f w a t e r i n c r e a s e s d r a m a t i c a l l y a b o v e 30°C ( B e a m e n t , 1961; F r e e & S p e n c e r - B o o t h , 1958; O m h o l t , 1987; W i g g l e s w o r t h , 1945). (D) COORDINATED
OR
INDIVIDUALISTIC
BEHAVIOUR?
It has to b e n o t i c e d t h a t the C O F Y m o d e l i m p l i e s the a b o v e p r e d i c t i o n s , b u t n o t the o t h e r w a y r o u n d . T h a t is, the shell b e e s m a y still be r e s p o n s i b l e for the active t h e r m o r e g u l a t i o n even if the b e h a v i o u r a l a n d p h y s i o l o g i c a l m e c h a n i s m s p r o p o s e d in this p a p e r are not e x a c t l y the c o r r e c t ones. H e n c e , the a b o v e p r e d i c t i o n s c o r r o b o r ate the C O F Y m o d e l , b u t t h e r e m a y b e o t h e r m o d e l s t h a t m a k e a b e t t e r fit with reality. H o w e v e r , the a p p a r e n t l y s u p e r o r g a n i s m i c r o o t e d h y p o t h e s i s t h a t gives t h e c o r e bees the central role in c l u s t e r t h e r m o r e g u l a t i o n , is c e r t a i n l y not s u p p o r t e d by this t h e o r e t i c a l analysis. B e c a u s e by c r e a t i n g a m o d e l , w h e r e all the b e e s in the shell r e m a i n at a resting m e t a b o l i s m , a n d w h e r e the c o r e b e e s p r o d u c e an e q u a l a m o u n t o f e n e r g y ( i m p l y i n g t h a t t h e y a r e m o v i n g r e l a t i v e l y freely w i t h i n the c o r e ) w h i c h a d d s u p to the d i f f e r e n c e b e t w e e n Q(T,,) a n d t h a t p r o d u c e d in t h e shell, e x t r e m e l y high core t e m p e r a t u r e s at low Ta's are p r e d i c t e d (Fig. 6). In o t h e r w o r d s , it s e e m s to be h i g h l y u n l i k e l y that the shell bees are n o t p l a y i n g a p r i m a r y role in the t h e r m o r e g u l a t i o n o f the w i n t e r clusters The author wishes to thank Nils Chr. Stenseth, Egil Villumstad, Knut Lonvik, Arne Hagen and Anders Kummervold for their encouragement and criticism. I00
o v
t~
80
60
E
~, 4o 0
20 -20
1
I
-15
-I0
,
I
I
I
-5
0
5
Ambient temperature (°C)
FtG. 6. The core temperatures at different ambient temperatures predicted from a model presuming the core bees to be responsible for the increase in heat production at decreasing ambient temperatures. The core bees are supposed to move freely within the core and to produce an equal amount of heat which adds up to the difference between Q(T,~) and that produced by the shell bees staying at a resting metabolism. Nh = 15 000 bees; R(Tmi,).... = 6.0 cm; R(Tmax) 12'5 cm; D~,,~e = 1.25 bees/ml; D,h~H= 5-5 bees/ml; Tp = t0°C; q(R(T,,))rc,, = 0-0015 cal/bee/min; q(0)r~t = 0-0295 cal/bee/min; Q(Train) = 350"0 cal/min; Q(T,.~) =80.0 cal/min; k(T,,,~) =0.0076 cal/cm2/min/°C; k(Tmi,,) =0-0106 cal/cm2/min/°C; A.... = A~h,,= 0-036 cal/cm/min/°C. When A~hc,= 5"0 x A...... T, (-20) decreases to ....
68°C.
=
THERMOREGULATION
IN T H E W I N T E R
CLUSTER
231
REFERENCES ALLEN M. D. (1959). Z exp. Biol. 36, 92. AVITABILE, A. (1978). J. apic. Res. 17, 69. BAST1AN, J. & ESCH, H. (1970). Z. vergl. Physiol. 67, 307. BEAMENT, J. W. L. (1961). Biol. Rev. 36, 281. BUDEL, A. &. HEROLD, E. (eds) (1960). Biene und Bienensucht. Munchen: Ehrenwirth Verlag. CAHILL, K. & LUSTICK, S. (1976). Comp. Biochem. Physiol. 55a, 355. CORKINS, C. L. (1932). J. Econ. Entomol. 25, 820. ESCH, H. (1960). Z. vergl. Physiol. 43, 305. ESCH, H. (1964). Z. vergl. Physiol. 48, 547. ESCH, H. & BASTIAN, J. (1968). Z. vergl. Physiol. 58, 429. FARRAR, C. L. (1943). Glean. Bee Cult. 71,513. FREE, J. B. &. SIMPSON, J. (1963). Ent. exp. & appl. 6, 234. FREE, J. B. & SPENCER-BOOTH, Y. (1958). Z exp. Biol. 35, 930. FREE, J. B. & SPENCER-BOOTH, H. Y. (1960). Ent. exp.& appl. 3, 222. HANEGAN, J. (1973). J. exp. Biol. 59, 67. HEINRICH, B. (1980a). J. exp. Biol. 85, 61. HEINRICH, B. (1980b). J. exp. Biol. 85, 73. HEINRICH, B. (1981). J. exp. Biol. 91, 25. HEINR1CH, B. (1985). In: (Holldobler & Lindauer eds). Experimental Behavioral Ecology. New York: G. Fischer Verlag. HEUSSNER, A. & ROTH, M. (1963). C. R. Hebd. Seance Acad. Sci., Paris 256, 284. HEUSSNER, A. & STUSSl, T. (1964). lnsectes Soc. 1!, 239. HIMMER, A. (1932). Cambridge Phil. Soc. Biol. Rev. 7, 224. JEFFREE, E. P. (1956). lns'ectes Soc. 3, 417. KAMMER, A. E. & HEINRICH, B. (1974). J. exp. Biol. 61,219. KRONENBERG, F. & CRAIG HELLER, H. (1982). J. comp. Physiol. 148, 65. LACHER, V. (1964). Z. vergl. Physiol. 48, 587. MACHIN, K. E., PRINGLE, J. W. S. & TAMASIGE, M. (1962). Proc. R. Soc. Lond. B 155, 493. MILNER, R. D. & DEMUTH, G. S. (1921). U.S. Dept. Agr. Bull. No. 988. MOBus, R, (1980a). Proc. 27th Int. Congr. Apic. p. 282. Athens: Apimondia Publ. House. MOaus, R. (1980b). Proc. 27th Int. Congr. Apic. p. 286. Athens: Apimondia Publ. House. OMHOLT, S. W. (1987). J. theor. Biol. (in press). OMHOLT, S. W. & LONVlK, K. (1986). J. theor. Biol. 120, 447. OWENS, C. D. (1971). U.S. Dep. Agr. Bull. No. 1429. PHILLIPS, E. F. & DEMUTH, G. S. (1914). U.S. Dep. Agr. Bull. No. 93. RIBBANDS, C. R. (1953). The Behaviour and Social Life o f Honeybees. London: Bee Research Association Ltd. RITTER, W. & KOENIGER, N. (1977). Proc. 8th Int. Cong. Int. Union Stud. Soc. Insects (Wageningen) p. 283. SAVITSKII, V. E. (1980). Soviet Agricultural Sciences 4, 51. SEELEY, T. D. (1974). J. Insect Physiol. 20, 2301. SIMPSON, J. (1961). Science 133, 1327. SOUTHWlCK, E. E. (1982). Comp. Biochem. Physiol. 71A, 277. SOUTHWlCK, E. E. (1983). Comp. Biochem. Physiol. 75A, 641. SOUTHWlCK, E. E. & MUGAAS, J. N. (1971). Comp. Biochem. Physiol. 40A, 935. STEINER, A. (1947). Bhft. zur Schw. Bienenz. 2, 140. WEDMORE, E. B. (1947). The Ventilation of Bee-Hives. Petts Wood, Kent: Bee Craft. WIGGLESWORTH,V. B. (1945). J. exp. Biol. 21, 97. WILSON, E. O. (1971). The Insect Societies. London: Harvard University Press.
Thermoregulation in the Winter Cluster of the Honeybee,
Apis
Mellifera S-rIG W. OMHOLT N-6530 Bruhagen, Norway
(Received 15 December 1986, and in revised form 7 April 1987) A model is presented based on the assumption that the apparently coordinated thermoregulatory responses of the broodless winter cluster of the honeybee, Apis mellifera, are results of the individuals acting behaviourally and physiologically to regulate their own body temperature. The predictive potential of the model is elucidated by a set of mathematically formulated relationships. The results include predictions concerning radial distributions of metabolic rate, and temperature profiles, as a function of ambient temperature. Furthermore, the apparently equal mass specific metabolic rate observed from weak and strong winter clusters is shown to be consistent with the proposed model. Consideration is also given to the generally accepted view of how the thermoregulation is accomplished, which is shown to give very unrealistic predictions.
I. Introduction There is still no general consensus as to which behavioural and physiological mechanisms are responsible for the thermoregulatory responses of the honeybee colony. Assuming that the individual bees are in some way or another aware of the needs o f the colony per se and directly responding to this in an a p p r o p r i a t e way, the accepted view seems to be that supercoordinating principles are operative in colonial thermoregulation (Farrar, 1943; Free & Spencer-Booth, 1958; Himmer, 1932; Milner & Demuth, 1921; Owens, 1971; Phillips & Demuth, 1914; Ribbands, 1953; Savitskii, 1980; Southwick, 1982, 1983; Southwick & Mugaas, 1971; Wedmore, 1947; Wilson, 1971). There is a widespread belief that the bees in the core region are responsible for the main part of the heat production and that their production is tuned to the survival o f the bees at the mantle of the cluster. Such principles, however, have been shown to be non-existent in swarm thermoregulation (Heinrich, 1981). In such a situation, Occam's razor is a convenient tool to apply. The basic aim of the present work is, in the case of the broodless winter cluster, to elucidate the consequences o f the assumption that the apparently coordinated thermoregulatory responses o f a bee society result from individuals acting behaviourally and physiologically to regulate their own body temperature in the context o f proximal and evolutionary constraints and possibilities afforded by their social life (Heinrich, 1985). Based on such a hypothesis, a set of simple mathematical relationships describing different characteristics o f the winter cluster are developed. From these are derived several predictions which are in agreement with empirical observations. 219
0022-5193/87/180219+ 13 $03.00/0
© 1987 Academic Press Ltd
220
s . w . OMHOLT
It is believed that the conceptual apparatus introduced in this paper represents an essential component of the theoretical framework needed for a comprehensive and rational experimental and theoretical approach to the study of the winter cluster.
2. Description of the Model (A) T H E R M O R E G U L A T O R Y
MECHANISMS
The thermoregulatory response pattern of the broodless winter cluster is assumed to be due to very simple behavioural rules followed by the individual bees: When a bee experiences a fall in temperature by its thermoreceptors on the antenna, it responds by increasing its metabolism to a maximum level dictated by its thorax temperature, and by moving towards a positive temperature gradient. When it experiences a rise in temperature, it responds by remaining at a resting metabolic level defined by its thorax temperature, and by moving towards a negative temperature gradient. This model may be called the COFY (Care Only For Yourself) model. While it cannot generate all the different behavioural patterns associated with the winter cluster (see Discussion), it is assumed to be responsible for the major thermoregulatory response characteristics of the whole colony. The main underlying premises for the model are: (1) The observed volume changes and heterogenous structures of winter clusters are due to changes in ambient temperature (Owens, 1971; Ribbands, 1953; Simpson, 1961). (2) The bees are equipped with thermoreceptors on the antenna which react to a sudden decrease in temperature by increasing impu' ~ cy, whereas an increasing temperature causes a decrease (Lacher, 196,,). l nese receptors respond to temperature changes down to 0.25°C, and it is reasonable to assume that they play a significant role in the monitoring of the bees' temperature environment. (3) Individual bees gain heat through rapid small-amplitude contraction-relaxation cycles of their thoracic wing muscles. Heat production is a result of the low mechanical efficiency of this musculature which makes up the bulk of the thorax, and is among the most metabolically active tissues known (Esch, 1964; Esch & Bastian, 1968; Heinrich, 1980a, b, 1981; Kronenberg & Craig-Heller, 1982; Southwick, 1983). (4) The thorax contains the ganglia that regulate the contraction of the indirect flight muscles. The neural output from these ganglia is highly temperature-dependent (Bastian & Esch, 1970; Esch & Bastian, 1968; Hanegan, 1973; Heinrich, 1980a; Machin et al., 1962). This suggests a close relationship between the thorax temperature and heat production capability. By using thermocouples placed in individual bees within a group, sharp and irregular fluctuations of thoracic temperature may be observed, indicating that the individual bee has an on/off response pattern (Bastian & Esch, 1970; Esch, 1960; Kammer & Heinrich, 1974; Kronenberg & Craig Heller, 1982). These observations suggest that for a specific surrounding air temperature (Ten), the metabolic limits
THERMOREGULATION
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221
of an individual bee are defined by the resting metabolism (q(Ten)res,), and the maximum limit (q(Ten)max), which is the heat generated when the bursts are at the maximum attainable intensity, and when the direct flight muscles are not activated. The COFY model predicts that the macroscopic behaviour of the winter cluster is the result of a "force de majeur" principle, and a very intricate pattern of stabilizing and destabilizing individual behaviour defines the dynamics. A quasi-steady state condition is supposed to exist when the stabilizing forces are stronger than the destabilizing. This may happen when the ambient temperature (T~) has not changed significantly for several hours and the colony is not moving honey into the clustering area (Owens, 1971). In this state there may be only minor fluctuations about an average value of the different macroscopic parameters by which the cluster is defined. Thus, for a given To, the cluster may be characterized by a specific cluster radius (R(T~)) (assuming spherical geometry), a specific radial bee density distribution (D(r, To)), a specific radial distribution of metabolic rate per individual (q(r, T~)), and a specific temperature profile ( T(r, To)). The resting metabolism is known with some certainty (Heinrich, 1981; Kammer & Heinrich, t974; Omhoit and L0nvik, 1986). Due to the paucity of experimental data, the maximum metabolism is assumed to be proportional to the resting metabolism in this study, so that q(T~,)max=q(Ten)restX 1~, where 1~ is the proportionality constant (see Discussion for further validation). Under steady state conditions it is assumed that only a minor proportion of the bees are not producing at a maximum or a minimum level. Those will be situated at a layer in the cluster where their cooling rate is greater than that compensated by their resting metabolism, but small enough to be compensated by a metabolic rate less than or-t.I~Up.b'!i a~eir maximum production.
(B) DESCRIPTION OF THE QUASI STEADY STATE CONDITION
( a ) Radial distributions of metabolic rate per individual bee The heat flux from a cluster at a specific ambient temperature (Q(Ta)) may be given by
Q( Ta) = k( T~)4rrR( To)2( T p - Ta)
(1)
where k ( T ~ ) = coefficient of heat transfer at Ta, Tp = temperature at the periphery of the cluster. The cluster structure can be approximately characterized by an outer shell where the bee density is considerably greater than in the inner core, and where the thickness of the shell increases with decreasing Ta (Owens, 1971; Ribbands, 1953; Simpson, t961; Wedmore, 1947). It is therefore natural to introduce two density levels, D .... and Dshetl, SO that
Nh = Jo
"° 4rrr2D .... d r + JnlT-.}.... 4"rrr2Dsheil dr
(2)
222
s.w.
OMHOLT
where Nb = n u m b e r o f bees in the cluster; R(TQ) .... = the core radius at a specific
r~. Given an ambient temperature range (Tmi., Tmax), and by quantifying Q(Tmax), Q(T~in), Tp, D¢ .... D~h¢ll, R(Tmin) ..... R(Tm~)core, one may calculate the radial limits, R ( T ~ . ) and R(T~,~), and the heat transfer coefficients k(Tmi.) and k(Tm~) from eqns (1) and (2). According to the comprehensive observations made by Owens (1971), the volume of a normally strong cluster decreases linearly between the Ta-limits - 2 0 ° C and 5°C. This implies that within this temperature range (see section 2(d)) R(T.~) = [ ( ( T m ~ - T~)R(T.~i.)3+(Ta - Tmin)R(Tmax)3)/(Tmax -
Tmin)]1/3.
(3)
Q(Ta) between the To-limits may then be quantified by applying a specific distribution of k(T~) (throughout this study assumed to be linearly decreasing between Tm~. and Tm~). The radial borderline between bees remaining at a resting metabolism and those at an enhanced metabolism may be calculated from the relationship ~lor ¢
Q(T~) = E k=!
n
VkD .... qk +
i
n
~" k = ncor¢+
VkDshe,,q k + VpD~,~,,qp + I
E
VkD~h~,,qkl~ (4)
k=p+l
where n = number of concentric spherical shells into which the cluster is divided; n .... = greatest shell n u m b e r where the bee density is D¢o~; n ~ , = greatest shell n u m b e r where the bees are staying at a resting metabolism; Vk = volume of the kth shell (when k < nco~ or T~ = Tm~, the wideness of the shell is R( Tmax)/n, otherwise it is ( R ( T ~ ) - R ( T , , ) .... ) / ( n - n .... )); q k = r e s t i n g metabolism per bee in the kth shell; p = the shell where the metabolic level is somewhere between minimum and m a x i m u m metabolism. The radial distribution of metabolic rate per individual when all the bees are at a resting level, may be a p p r o x i m a t e d by an error function (cf. Omholt & Le~nvik, 1986), so that q ( r ) ~ = q ( 0 ) ~ t exp - ar 2.
(5)
By the simplifying assumption that the radial temperature profile is rather invariable with changes in T~ (see Discussion), the discrete version of this function when it is calibrated against R(T,,,ox) and Q(Tm~,), m a y be applied in eqn (4) for every T~. The model predicts that there should be an agreement between n .... and n ~ , . However, n .... and n .... are not necessarily identical, as some of the bees may be forced to stay at a high density even if they are at resting metabolism. This may happen because the bees farther out might press inwards and the bees farther in might push outwards (cf. Results). Such a situation is more likely to exist at low rather than at high To's, indicating that primarily there should be correspondence between n,¢~, a n d n ..... but that the model is not violated if n,~t > n .... when To is in the lower domain. However, according to the model, n~¢~,cannot be less than n .... This relationship m a y be used as a calibration criterium in order to determine an appropriate magnitude of I,..
THERMOREGULATION IN THE WINTER CLUSTER
223
( b ) Temperature distributions These may be calculated from the relationship
T(r, To) = Tp+
f
mL~
j(r)/A(r) dr
(6)
dO
where j(r) is the energy current density at radial distance r
j(r) =
Io
4~r2D(r, To)q(r, TO) dr/4~r 2
(7)
and A(r) is the coefficient o f heat conduction. The coefficient of heat conduction is assumed to be related to the bee density (Simpson, 1961; Wedmore, 1947), so that A.... = A(r, D .... ) and A~h~H= A(r, D~h~t) are constants. By assuming a specific proportionality constant between A.... and A~h~, the coefficients of heat conduction may be calculated by application o f eqns (2) and (5), even if one takes into consideration that the outermost bees have started to form a thin shell at Tm,x.
(c) Mass specific metabolic rate Some workers seem to be surprised at not being able to observe an increasing mean metabolic rate per individual with a decreasing number of bees in the cluster, considering the increasing surface to volume ratio (Free & Simpson, 1963; Southwick, 1983). The C O F Y model may offer an explanation:
In weak colonies there is a relatively greater radial contraction than in stronger ones, because a greater part of the cluster will experience a greater heat loss than can be compensated by resting metabolism. Testable predictions from this hypothesis may be obtained in the following way. Let N b ( m i n ) < N b < Nb(max), and let Mmr(TO) define a specific mean metabolic rate per individual for Nb(max) as a function of ambient temperature. By specifying the different cluster radii (R(Ta, Nb(max))) between Tmaxand Train, and the limits and form of k(TO), M,.r(TO) is obtained from the relationship
MInt(TO) =4~rk( TO)R( TO, N b ( m a x ) ) 2 ( T p - Ta)/ Nb(max).
(8)
From the assumption that M,.r(TO) is identical for every Nb, it follows that the radius for a specific cluster as a function of ambient temperature may be expressed as
R( TO, Nb) = [NbM,.r( TO)/4-rrk( TO)( Tp - T~)] ~/2.
(9)
However, if R(TO, Nb) implies a mean bee density greater than a specified maximum Dmax (corresponding to the minimum radius R(Nb)ml.), the actual mean metabolic rate M(TO, Nb) has to be greater than MInt(TO), and is given by
M( TO, Nb) =47rR( Nb)~.k( TO)( Tp - TO)/ Nb.
(10)
224
s . W . OMHOLT
Furthermore, the cluster's degree of contraction (Ca) may be expressed as
(11)
Ca = R(Nh)min/ R( Ta, Nh). ( d) Quantification of parameters
The ambient temperature in this study varies between -20°C (Tm~,) and 5°C (Tmax), which is a temperature range wide enough to disclose the different patterns predicted by the model, but narrow enough to avoid dealing with the special cases arising at the upper and lower existence limits o f the cluster. D .... and Dsh~, are normally in the range o f 1-2 and 4.5-6 bees/ml respectively (Heinrich, 1985; Owens, 1971; Wedmore, 1947), and are given the values 1.25 and 5-5 in this study. From a normally strong winter colony of 15 000 individuals, which corresponds to the basic Nb-value to be used throughout this study, Q(T,,~x) and Q(Tmi°) are assumed to be 80 and 350 cal/min respectively (Free & Simpson, 1963; Southwick, 1983). The minimum and maximum size of the cluster core, R(Tmi,) .... and R( Tm~x)¢.... are presumed to have values of 6.0 and 12.5 cm respectively. The mantle temperature Tp, normally lies within the range 8-12°C (Free & Spencer-Booth, 1960; Owens, 1971), and is a s s u m e d t o stay constantly at 10°C irrespective of the ambient temperature. The core temperature Tc at T~.~ is set to be 24°C (Himmer, 1932; Southwick & Mugaas, 1971). The resting metabolic rate o f a bee staying at the mantle (q(R(T,,)~es~), is assumed to be 0-0015 c a l / b e e / m i n , which is in agreement with the results obtained by Heinrich (1981), Kammer & Heinrich (1974) and Omholt & Letnvik (1986).
3. Results (A) D I S T R I B U T I O N S
OF
METABOLIC
RATE
When Ta is high (or extremely low) the metabolic rate per individual increases rapidly from the periphery to the centre. This is in accordance with the general view of how the heat is produced (cf. Omholt & Le~nvik, 1986). However, when Ta is in the medium range, the metabolic rate will decrease at a resting level from the centre to a point R(Ta)~est, where it increases to a level somewhat greater than q(0)re~, and then decreases outwards. Even at -20°C, only the bees outside a radial distance o f 0.8 x R ( - 2 0 ) are in an enhanced metabolic state (Fig. 1). This implies that the active thermoregulation, under normal temperature conditions, may be located in the outer part o f the cluster, as proposed by Heinrich (1985), M~bus (1980b) and Steiner (1947). Furthermore, the model predicts that the share o f the total heat production from the different parts o f the cluster is dependent on Ta, with a decreasing part of the heat being produced in the core region with decreasing Ta, until To becomes so low that almost all the bees are staying at a maximum metabolism. However, even then the share of the core bees is not likely to exceed 50% (Fig. 2).
THERMOREGULATION
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225
I-0 0"95 0"90 ~: 0.85 0-80
I
-
20
I
I
I
I
I
-15 -I0 -5 0 Ambient temperoture(*C)
5
FIG. 1. T h e relative radial proportion of the cluster remaining at a resting metabolism as a function of ambient temperature, when Nb = 15 000 bees; R( Tmi.) . . . . = 6.0 cm; R(Tm,~) .... = 12.5 cm; Dco,c = 1-25 b e e s / m l ; D, hell= 5"5 b e e s / m l ; Tp = 10°C; q(R(T,)),,,st=O.O015 c a l / b e e / m i n ; q(0),c~,=0.02395 cal/bee/min; I~ = 8-75; Q(Tmi .) = 3 5 0 . 0 c a l / m i n ; Q ( T m , ~ ) = 80-0 c a l / m i n ; k ( T ~ ) =0.0076 c a l / c m 2 / m i n / ° C ; k(Train) = 0"0106 c a l / c m Z / m i n / ° C .
I00
8O
=5"C g 60 S
t20.C
-6 40
g E < 20
0.2
0.4 0.6 0,8 Rodiol distonce
1
PO
FIG. 2. The percentage o f the total a m o u n t o f heat produced up to a given radial distance, when T, = - 2 0 ° C a n d 5°C. All parameter values as listed for Fig. 1. (B) T E M P E R A T U R E
PROFILES
The model is able to predict an inverse relationship between Tc and T~, in agreement with the observations of Corkins (1932), Himmer (1932), Owens (1971), Southwick & Mugaas (1971), Steiner (1947) and Wilson & Milum (1927). However, there is no significant increase of Tc until T~ is below -10°C, and the model predicts even a slight decrease in Tc below 24°C when T~ =0°C (Fig. 3). This may explain why there has been so much discussion on the existence of a long-term inverse relationship, as long as some investigators have not measured the core temperature below an ambient of -10°C.
226
s.w.
OMHOLT
38
36
34
oo ®
52
50
X
x 24 k L,,,,I -2(3
I -15
+ I -I0
~ 'I ~ X ~ 4-5
0
l 5
Ambient temperature ( ° C )
FIG. 3. Predicted core temperatures as a function of ambient temperature, when N~ = 15 000 bees;
R ( T , . i . ) ..... =6-0 cm; R(T,.,,.~) ..... = 12"5 cm; D .... =1-25 bees/ml; D~h~.=5-5 bees/ml; T r = I 0 = C ; q( R( 7",,))re., = 0"0015 cal/bee/min; q(0)rc~, = 0-0295 cat/bee/rain; l,. = 8.75; Q ( T , ~ . ) = 350.0 cal/min; Q( Tm,,~} = 80.0 cal/min; k(Tm.~) =0"0076 c a l / c m ' / m i n / ° C ; k(T..,.) =0.0106 cal/cm2/min/°C; A.... = A.h¢. =0"036 c a l / c m / m i n / ° C . The (+) and (x) signs show respectively the effect from a 25% increase and decrease of the coefficient of heat conduction in the outer shell compared to that of the core.
The magnitude of the coefficient of heat conduction seems to be rather sharply defined in order to obtain an inverse relationship between T~ and Ta (Fig. 3). If A is inversely related to the bee density, Tc increases to unreasonably high levels at low T,,'s. As long as a constant A gives a bit too high estimate, this implies that it may increase slightly with increasing bee density. The predicted temperature profiles imply that the distance between the isotherms shortens in the outer part o f the cluster with decreasing T,, (Fig. 4). This seems to be in agreement with observations on swarm clusters (Heinrich, 1981, 1985), as welt as winter clusters (Owens, 1971). The rather constant radial distance to the 15.6°C isotherm observed by Owens (1971) during December and January, is predicted by the model. Owens' data show a variation of approximately 0.35 cm about the mean value, while the model predicts a variation of approximately 0.5 cm.
(C) MASS S P E C I F I C MEAN METABOLIC RATE
From Fig. 5 it follows that the lack of increase of mean metabolic rate with decreasing Nb may be explained by a higher degree of contraction by the smaller
THERMOREGULATION
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CLUSTER
227
Fa= 5 ° C
m
=_zo°c!!!!ll
nl,l I
i
I
i
I
~
i
I
I
I
i 1~=_5oc
I I I I I I
~:15°C FIG. 4. P r e d i c t e d i s o t h e r m a l d i s t r i b u t i o n at d i f f e r e n t a m b i e n t t e m p e r a t u r e s . T h e o u t m o s t m a r k s r e p r e s e n t the 10°C i s o t h e r m s , a n d t h e i s o t h e r m a l d i f f e r e n c e is 2.5°C. T h e p a r a m e t e r v a l u e s a r e as listed f o r Fig. 3, e x c e p t t h a t A~h~~= t - 2 5 X A. . . . -
colonies. The contraction potential of even weak clusters are considerable, and if used to the full extent, Nb has to be less than 5000 and T, below -10°C, before an increase in mean metabolic rate is expected. These results are in agreement with the view that small colonies do not freeze to death unless Ta is extremely low, but actually starve to death at low T,, by not being able to move honey into the clustering area (Owens, 1971).
4. Discussion (A) SHORTCOMINGS
OF THE
MODEL
The model is not able to account for the cluster behaviour in connection with movements of honey into the clustering area. This is, however, short-term behaviour (Owens, 1971), and does not interfere with the predictions given above. On the other hand, it would have been very interesting to know whether an individualistic or a super-coordinated approach is the most convenient to apply in this connection. Presence of brood in the winter cluster is a rather frequent p h e n o m e n o n (Avitabile, 1978; Free & Simpson, 1963; Jeffree, 1956; M~bus, 1980a). It is not clear whether the bees actively regulate the brood temperature or not (Heinrich, 1985; Kronenberg & Craig-Heller, 1982; Ritter & Koeniger, 1977), but if they do, other thermoregulatory mechanisms are likely to be involved. Especially at low ambient temperatures there may be a build-up of carbon dioxide within the cluster to such a level that the core bees respond by fanning (Seeley, 1974). Such behaviour will have some impact on the production and temperature profiles, but does not contradict the main premises behind the COFY model.
228
s.w.
OMHOLT
+ -+
+
+ E_ .t:]
I
-20
I
I
-15
-10
[
I
I
-5
0
5
Ambient temperature (°C)
i,o! _~ 0-9 ~G §
r~
0"8
0.7
0"6
5
I0
15
20
25
Cluster size (/V~)/1000
FIG. 5. (a) Calculated mean metabolic rates per bee from clusters of different strengths as a function of ambient temperature, when the clusters are allowed to make use of their contraction potential in order to keep a mean metabolic rate identical to that of the strongest cluster which is depicted by the graph. The plus signs express the metabolic rate from the cluster sizes not being able to cope with Nh(max), where those at -20°C represent clusters of 2500, 5000 and 7500 bees respectively, and that at -15°C a cluster of 2500 bees. The applied parameter values are: N h ( m a x ) = 2 5 000 bees; Dm~x = 5.5 bees/ml; k(T, nl,) = 0-0106 cal/cm2/min/°C; k(T,,,~) = 0-0076 cal/cm2/min/°C; Tp = 10°(2; R ( - 2 0 , Nh(max)) = 12.08 cm; R ( - 1 5 , N h ( m a x ) ) = 1 3 . 3 4 cm; R ( - 1 0 , N h ( m a x ) ) = 1 4 . 4 8 cm; R ( - 5 , N h ( m a x ) ) = 1 5 - 3 2 cm; R(0, Nb(max)) = 16.14 cm; R(5, Nb(max)) = 16.89 cm. (b) The calculated contraction degree of clusters of different strength at Tu = -20°C, -10°C and 5°C. The parameter values are as listed for (a).
(B) ROBUSTNESS OF THE MODEL
From the outset, the calculations are based on rather idealized conditions in assuming spherical geometry, neglecting the effects from combs, and by representing the bees as homogenous energy-producing units uniformly distributed in thin layers. In order to be sure that the model has captured the more significant features, the predictions should be rather insensitive to changes in the model parameters from a qualitative point of view. This is especially important in connection with parameters that have a very scarce empirical backing, such as the form of k(Ta) and q(Ten) . . . . A linear k(T~) seems to represent a rather crude approximation, because it gives a somewhat small difference between Q ( - 2 0 ) and Q ( - 1 5 ) . However, a k(T~) that
THERMOREGULATION
1N
THE
WINTER
CLUSTER
229
increases more rapidly towards T.,in has only a minor effect on the results, as long as the calculated limits k ( T m J and k(Tma~) agree very closely with empirical data (Biidel & Herold, 1960; Savitskii, 1980). The difference between k(Tmax) and k(Tmin) is likely to be caused by an increase in the water loss from the cluster surface, as long as the temperature difference is too small to effect a significant increase of the conduction and radiation components of k( Ta). Metabolic measurements o f individual bees exposed to different environmental temperatures indicate that the heat production capability at 15-20°C may be somewhat greater than what is assumed above (Allen, 1959; Cahill & Lustick, 1976; Free & Spencer-Booth, 1958; Heussney & Roth, 1963; Heussner & Stussi, 1964). Also, the length o f the action potentials from the dorsoventral thorax muscles o f bees individually exposed to different environmental temperatures shows a rather moderate difference between 18-30°C (Esch & Bastian, 1968). These two conditions may imply that q( Te,)ma, increases a bit more rapidly up to 15-20°C, and then increases more slowly towards To, than what is assumed by q( Te,)man = le x q( Te,)rest. However, a change o f q(T~,)ma~ along these lines will have very little quantitative and no qualitative influence on the results. In this connection it is nice to observe that the maximum heat production capability predicted by the model by applying the calibration criterium nre~t> n .... for Ta > - 1 5 ° C , is in agreement with production levels o f unisolated colonies at -40°C (Free & Simpson, 1963). This is a temperature level rather close to the lowest T~ at which a normally strong winter cluster is able to compensate its heat loss. The effects from an increasing T~ with a decreasing T~, and hence a steeper temperature gradient from the periphery inwards, are not accounted for in the calculations behind Figs 1 and 2. The former condition will increase the resting metabolism in the core region, and the latter will, according to the C O F Y model, enhance the maximum heat production capability of the bees in the outer parts. In the present context, both conditions have an insignificant impact on the results. The above considerations imply that the model generates predictions that are rather robust from a qualitative and semi-quantitative point of view. (C) C O N S I D E R A T I O N S
OF THE
MANTLE
TEMPERATURE
The model cannot disclose why the mantle bees stay at a temperature level of approximately 10°C. This is about 5°C below the mantle temperature of swarm clusters (Heinrich, 1981, 1985). It is reasonable to assume that this difference is due to the acclimatisation experienced by the bees under prolonged exposure to low temperatures (Free & Spencer-Booth, 1960). This is further validated by the fact that the total metabolic rate has a minimum when To is 10°C for winter clusters, and 15°C for swarm clusters (Free & Simpson, 1963; Heinrich, 1981; Southwick, 1983). From an evolutionary point of view there seems to be two major reasons why the mantle bees should have a temperature just a little above their chill-coma temperature. One is that the reduction of the mantle temperature implies a reduction
230
s.w.
OMHOLT
o f h o n e y c o n s u m p t i o n d u r i n g t h e winter. T h e o t h e r is t h a t a low m a n t l e t e m p e r a t u r e r e d u c e s t h e loss o f w a t e r f r o m all t h e b e e s in t h e cluster. T h i s m a y b e e s p e c i a l l y i m p o r t a n t in t h e core r e g i o n , b e c a u s e the loss o f w a t e r i n c r e a s e s d r a m a t i c a l l y a b o v e 30°C ( B e a m e n t , 1961; F r e e & S p e n c e r - B o o t h , 1958; O m h o l t , 1987; W i g g l e s w o r t h , 1945). (D) COORDINATED
OR
INDIVIDUALISTIC
BEHAVIOUR?
It has to b e n o t i c e d t h a t the C O F Y m o d e l i m p l i e s the a b o v e p r e d i c t i o n s , b u t n o t the o t h e r w a y r o u n d . T h a t is, the shell b e e s m a y still be r e s p o n s i b l e for the active t h e r m o r e g u l a t i o n even if the b e h a v i o u r a l a n d p h y s i o l o g i c a l m e c h a n i s m s p r o p o s e d in this p a p e r are not e x a c t l y the c o r r e c t ones. H e n c e , the a b o v e p r e d i c t i o n s c o r r o b o r ate the C O F Y m o d e l , b u t t h e r e m a y b e o t h e r m o d e l s t h a t m a k e a b e t t e r fit with reality. H o w e v e r , the a p p a r e n t l y s u p e r o r g a n i s m i c r o o t e d h y p o t h e s i s t h a t gives t h e c o r e bees the central role in c l u s t e r t h e r m o r e g u l a t i o n , is c e r t a i n l y not s u p p o r t e d by this t h e o r e t i c a l analysis. B e c a u s e by c r e a t i n g a m o d e l , w h e r e all the b e e s in the shell r e m a i n at a resting m e t a b o l i s m , a n d w h e r e the c o r e b e e s p r o d u c e an e q u a l a m o u n t o f e n e r g y ( i m p l y i n g t h a t t h e y a r e m o v i n g r e l a t i v e l y freely w i t h i n the c o r e ) w h i c h a d d s u p to the d i f f e r e n c e b e t w e e n Q(T,,) a n d t h a t p r o d u c e d in t h e shell, e x t r e m e l y high core t e m p e r a t u r e s at low Ta's are p r e d i c t e d (Fig. 6). In o t h e r w o r d s , it s e e m s to be h i g h l y u n l i k e l y that the shell bees are n o t p l a y i n g a p r i m a r y role in the t h e r m o r e g u l a t i o n o f the w i n t e r clusters The author wishes to thank Nils Chr. Stenseth, Egil Villumstad, Knut Lonvik, Arne Hagen and Anders Kummervold for their encouragement and criticism. I00
o v
t~
80
60
E
~, 4o 0
20 -20
1
I
-15
-I0
,
I
I
I
-5
0
5
Ambient temperature (°C)
FtG. 6. The core temperatures at different ambient temperatures predicted from a model presuming the core bees to be responsible for the increase in heat production at decreasing ambient temperatures. The core bees are supposed to move freely within the core and to produce an equal amount of heat which adds up to the difference between Q(T,~) and that produced by the shell bees staying at a resting metabolism. Nh = 15 000 bees; R(Tmi,).... = 6.0 cm; R(Tmax) 12'5 cm; D~,,~e = 1.25 bees/ml; D,h~H= 5-5 bees/ml; Tp = t0°C; q(R(T,,))rc,, = 0-0015 cal/bee/min; q(0)r~t = 0-0295 cal/bee/min; Q(Train) = 350"0 cal/min; Q(T,.~) =80.0 cal/min; k(T,,,~) =0.0076 cal/cm2/min/°C; k(Tmi,,) =0-0106 cal/cm2/min/°C; A.... = A~h,,= 0-036 cal/cm/min/°C. When A~hc,= 5"0 x A...... T, (-20) decreases to ....
68°C.
=
THERMOREGULATION
IN T H E W I N T E R
CLUSTER
231
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