- Email: [email protected]
Optimum body sizes at different ambient temperatures: an energetics explanation of bergmann's rule
J. theor. Biol. (1980) 83,579-593
Optimum Body Sizes at Different tures: an Energetics Explanation WILLIAM Department
Ambient Temperaof Bergmann’s Rule
A. SEARCY~.
of Zoology, University of Washington, Seattle, Washington 98195, U.S.A. (Received 20 November 1978)
A model ispresentedof optimumbody sizesat different ambienttemperatures. Selectionis consideredto optimize the difference betweenforaging profit per time and maintenanceenergy costsper time, scaledto body size. Two different scalingfunctions are considered:one maximizesthe amount of activity that an individual can perform, and the other maximizesthe length of time an individual can survive on the profit gained in a given period of foraging. With both scalingfunctions, optimum sizeincreasesas ambient temperature decreases.
1. Introduction Bergmann’s Rule states that “races of warm blooded vertebrates from cooler climates tend to be larger than races of the same species from warmer climates” (Mayr, 1956). There is much evidence for the validity of this rule (Rensch, 1936,196O; Snow, 1954; Rosenzweig, 1968; Brown & Lee, 1969; James, 1970), but its adaptive significance is still much in question. The classical evolutionary explanatfon of the rule is that larger animals have an advantage in cooler climates because their smaller surface to volume ratios lead to lower rates of heat loss and metabolism per gram body weight (Mayr, 1963). The usual counter to this argument is that the main adaptation to lower temperatures is increased insulation, compared to which small increases in body size would have little effect (Scholander, 1954, 1956). As Mayr (1956) notes, even if increased insulation is the main adaptation to cold, increased size may also be an adaptation, and selection can cause the evolutionary enhancement of characteristics even if their benefit is small. A more fundamental criticism of the classical physiological explanation is that even though increased size leads to smaller per gram metabolic costs, it + Present address: Rockefeller University Field Research Center. Millbrook, 12545, U.S.A.
New York
579
0022-5193/80/080579+
15 $02.00/O
@ 1980 Academic Press Inc. (London) Ltd.
580
W.
increases metabolic
A.
SEARCY
costs per individual
(McNab,
1971). For example:
“a doubling of weight would increase the basal rate of metabolism by only 68%. However, I know of no way by which an animal can measure the 32% “saving” in the maintenance of the additional weight; it can only measure a 68% increase in energy expenditure (by an increase in the amount of food required)” (McNab, 1971). It is certainly true that there is no obvious selective advantage to minimizing metabolic rates per gram at the price of increasing total metabolism. Instead, selection should favor maximizing the difference between energy intake and maintenance energy costs per time per individual. The greater this difference, the more energy an individual can inveSt in activity, reproduction, or storage. Let us define F as the foraging profit (energy/time) of an individual,that is, as the energy harvested by an individual per time minus the additional energy cost it incurs by foraging rather than doing nothing. Let A4RT be maintenance energy expenditure, or metabolic rate (energy/time), at temperature T. Both F and lkfRT should vary with the size (w) of an animal. Optimal size will be a function of the difference between F and MRT. A small animal can do more with a certain amount of energy per time than can a large one so optimal size will also be a function of the cost of activity, reproduction, or thermoregulation to animals of different sizes. Suppose that C is the cost of some activity (or activities) crucial to the fitness of an animal, with units of energy cost per unit of activity. C should also vary with size, and optimal size would be that size where F-MRT C
is maximized.
(expression 1)
The units of the expression are energy/time -energy/time = activity/time. energy/unit activity
Alternatively, C (which I will call a scaling function) might represent the cost in energy of one unit of reproduction (e.g. kcal/egg), in which case expression 1 would reduce to reproduction/time. To determine how optimum size changes as temperature changes, we must determine: (1) how foraging profit per time varies with size, (2) how metabolic rate varies with size and temperature, and (3) what expressions for C are appropriate. 2. Energy Intake vs. Size
Let us assume we are considering an animal constrained by morphological and behavioral adaptations evolved over the history of its species to feed on
OPTIMUM
SIZES
AT DIFFERENT
TEMPERATURES
581
only a certain range of items which it obtains and handles in only certain ways. The evolution of major shifts of diet and foraging modes is prevented by interspecific competition. Then it is logical to assume that there will be some size below which the animal would become increasingly less efficient as its size decreases. For example, a hawk’s efficiency in capturing small mammals must decrease as the hawk becomes too small to subdue its prey. A rodent’s efficiency in husking seeds must decrease as its mouth becomes too small to husk the largest seeds in its diet. Conversely, above a certain size, further size increase should not increase efficiency and may even lower it. For example, once the hawk is large enough to have no trouble subduing its prey, further size increases may decrease its maneuverability in flight and so decrease its capture rate. Above a certain body size, a rodent must have increasing difficulty in manipulating small seeds in its teeth. We can assume a general relationship between size and foraging efficiency such as is shown in Fig. 1, with energy profit per time increasing with increasing consumer size over some range of sizes and then levelling off and perhaps declining with further size increase. I have assumed a unimodal shape for this F function; bimodal shapes are also possible (Schoener, 1969) though perhaps not as likely. In general, we can assume such size-foraging efficiency relationships to be independent of ambient temperature.
FIG. 1. Relationships between size and foraging efficiency and size and maintenance energy costs. F is energy profit per time foraging. BMR is energy cost of maintenance at a temperature within the thermal neutral zone; this curve is described by the equation a(~~) where w is in units of grams and b is approximately 0.75. MRT gives the energy cost of maintenance at temperature T, where T is less than the lower critical temperature for individuals of size w < w’.
582
W.
A.
SEAR(‘Y
3. Energy Costs vs. Size at Different
Temperatures
Metabolic costs at thermal neutrality increase as a function of weight to approximately the 0.75 power in a variety of vertebrate groups (Bartholomew, 1977). For example, in passerines, BMR = 46.63( Mu”“‘), where BMR is basal metabolic rate in mW and MI is weight in g (Lasiewski & Dawson, 1967; Calder, 1974). In rodents, BMR= 21.21(~“‘~) (Hart, 1971). Fourier’s law of heat flow predicts that MR, = h( Tb - r,) where !L is conductance (mW/‘C), Th is body temperature, and T, is ambient temperature (Calder, 1974). Below the lower critical temperature (T),, the lower limit of the thermal neutral zone), h is a constant for an animal of given weight. This constant increases with increasing size; for example, in passerines, h = 5.29( wO.~~) (Calder & King, 1974; Calder, 1974). If we set T, equal to T,,, then (for passerines) 46.63(w0”‘) = 5.29(w o’46) ( Tb - T,,), which reduces to 8.81(~~“~~) = Tb- TIC.Assuming Tb to be approximately independent of weight, lower critical temperature should decrease as w(“~~ increases (Calder, 1974). Similarly, Hart (1971) found that T,, in rodents should decrease as w”‘~ increases. Data giving T), and weight for different species of rodents fit the predicted relationship though with a good deal of variability (Hart, 1971). In Fig. 1, I have graphed BMR increasing with weight to approximately the 0.75 power. Imagine a small animal whose metabolic rate we measure at a series of temperatures below its thermal neutral zone. Metabolic rate should be higher at each successively lower temperature. However, for each lower temperature, we can imagine an animal just big enough so that the ambient temperature is within its thermal neutral zone. For example, we can find the approximate weight of a passerine just large enough to have its lower critical temperature at 20°C by solving 8.81(wn”h) = Tb-20. Assuming Tb is 40”, w = 23.4 g. Thus the line relating metabolic rate to weight at 20” for passerines must be above the BMR curve for small sizes and converge with it at about 23 g. For all vertebrate groups, there will be a family of such curves, one for each temperature below the T,,. for the smallest animal considered. In Fig. 1 is graphed one such curve: MRT where T < T,, for animals of w < w’. Empirically, there would be a great deal of variability about such curves, but if other temperature related adaptations, such as quality and amount of insulation, are held constant, such curves should be correct. 4. Two Scaling Functions
One quantity that selection may tend to maximize is the amount of activity an animal can perform. The energy cost of many activities seems to be scaled
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
583
to body size with approximately the same relationship as BMR so the cost of activity can be expressed as a constant multiple of BMR. This argument can be applied, for example, to cost of flight in passerines (King, 1974). The energy expenditure of birds in flight, in mW, is approximately 341*7( w”‘~) (Berger & Hart, 1972; King, 1974). If BMR is taken to be 46*63(w’.“) for passerines, then the cost of flight is about 7.3 x BMR for passerines of any size. King (1974), using a dif&ent expression for BMR, obtained the figure 9.3 x BMR. Hemmingsen (1960) found that for a large range of animals, the maximal rate of energy expenditure in activity is an approximately constant multiple of BMR. Maximizing the amount of activity an animal existing at temperature T can perform is equivalent to maximizing: F-MRT BMR
F =--BMR
MRT BMR
(expression 2).
The greater this difference, the more activity an individual can perform using the energy gained in a given period of foraging. This same expression can be used in some cases where selection is thought to favor the body size at which reproductive output per time is maximal. The cost of reproduction may often be a constant multiple of BMR for animals of different weights so maximizing the difference in expression 2 would maximize relative reproductive output. For example, egg weight, and presumably the cost of producing eggs, is a function of body weight of the mother to the 0.75 power in passerines (Ricklefs, 1974). Thus maximizing expression 2 for a passerine will maximize the number of eggs it can produce per time. Expression 2 is also appropriate in cases where most of the energy invested in reproduction is used in activity, as in male birds that invest heavily in territory defense and epigamic displays. Rosenzweig & Sterner (1970) give data on F/BMR for seven species of heteromyids eating four species of seeds. These authors measured caloric content of each seed type and the amount of time required by each heteromyid species to eat the seed. They then calculated F/BMR for each combination of seed and animal species. For two species of seeds (spinach and pumpkin), F/BMR decreased more or less uniformly with increasing consumer size. For the other two seeds (sunflower and squash), there was an intermediate sized consumer for which F/BMR was maximal, with values falling away with both increasing and decreasing consumer sizes. These data are compatible with the shape of F/BMR hypothesized in Fig. 2 (but see below). A curve of this shape is produced by dividing F in Fig. 1 by BMR. Kendeigh (1969) presents the lower critical temperatures of 15 species of passerines and the equations for calculating their standard metabolic rates at
584
W.
small
A.
SEARCY
Size
large
2. Optimum size where selection maximizes the difference between energy intake per time and maintenance energy costs scaled to energy costs at thermal neutrality. F is energy profit per time foraging. BMR is maintenance energy costs per time at a temperature within the thermal neutral zone. MRT is energy costs per time at temperature T, where T is below the lower critical temperature for individuals of w < w’. The maximum difference between the intake curve and the cost curve at each temperature occurs where the slope of the intake curve equals the slope of the cost curve. For all w < w’, the optimum size is higher at T than at r,,. FIG.
temperatures below thermal neutrality. From these data, A4RT can be calculated for any T < TI,. Table 1 gives the weights of these species and MRo*/BMR. This quantity decreases as size increases until it reaches a value of 1. Figure 2 shows the general shape that MR=/BMR should take. Values are high for small animals, decline as size increases up to the size where T = Tic, and thereafter are 1. With the model presented in expression 2, how does optimal body size change as ambient temperature is lowered? As an example, let us consider optimal body size at Tth(a temperature within the thermal neutral zone) and at some T such that T < T,, for animals of w < w’. The optimal body size at Tth is that size where the difference between F/BMR and BMR/BMR is greatest, that is, at the size corresponding to the peak of the F/BMR curve (see Fig. 2). The optimal body size at. T is at that w where the difference between F/BMR and MR=/BMR is maximal. This maximum occurs at the weight where the slopes of these two curves are equal. MR=/BMR has a negative slope for all w < w’. F/BMR has a negative slope only at weights to the right of its peak. Thus the two slopes will be equal at a size greater than the optimum size at Tfh as long as the peak of F/BMR occurs at a w < w’. In general we can say that if an animal is at its optimal size at a temperaure
OPTIMUM
SIZES
AT DIFFERENT TABLE
585
TEMPERATURES
1
The ratio of metabolic rate at 0°C to basal metabolic rate for passerinesof different sizes. Data calculated from equations in Kendeigh (1969, Table 1) Species Black-rumped waxbill (Esrrilda troglodytes) Black-rumped waxbill (Estrilda rroglodytes) House wren (Troglodyres aedon) Zebra finch (Taeniopygia casmnoris) Zebra finch (Taeniopygiu castunofis) Ortolan bunting (Emberiza horfuluna) White-throated sparrow (Zonotrichia albicollis) House sparrow (Passer domesticus) House sparrow (Passer domesticus) House sparrow (Passer domesticus) Yellow bunting (Emberiza citrinella) White-crowned sparrow (Zonorrichia leucophyrys) Red crossbill (Loxia curvirostru) White-winged crossbill (Loxia leucoprera) Cardinal (Richmondem cardinalis) Evening grosbeak (Hesperiphona vespertina) Evening grosbeak (Hesperiphona vespertina) Blue jay (Cyanocirta crisfuta) White-necked raven (Corvus cryptoleucus) Common raven (Corvus corax) Common raven (Corvus corax)
Weight Id 6.4 6.5
9.7 11.5 11.7 22.0 22,5 25.1 25.5 25.3 26.4 28.6 29.4 29.8 41 54.5 55.3 80.8 640 850 866
MR,./BMR 1.80 1.76 1.59 1.71 1.76 1.62 1.59 1.59 1.55 1.58 1.45 1.54 1.37 1.36 1.40 1.47 1.39 1.30 1.00 1.00 1.00
within its thermal neutral zone, and the ambient temperature is lowered below that animal’s T,,, the optimal size will increase. What happens if an animal is already existing at a temperature less than its T,, and the ambient temperature is lowered further? Figure 3 shows hypothetical MRT/BMR curves for TI and T2, with T,, > TI > T2. Since T1 > T2, the MR=,/BMR curve joins the BMR/BMR line at a smaller size than does the MRrJBMR curve. It follows from the graphical model (Fig. 3) that the optimum size at T2 will be greater than the optimum at T1. Thus any lowering of temperature causes an increase in optimal size. In Figs 2 and 3, I have assumed that the F/BMR curve is unimodal. If this assumption is relaxed, and more than one peak in the curve is allowed, then it is possible to get sudden jumps in optimal size as ambient temperature is lowered. For example, imagine that there is a second, lower peak in the F/BMR curve in Fig. 2 to the right of the first peak. Then as ambient temperaure is lowered, the optimal size might jump from the region of the first peak to the region of the second.
586
W. A. SFARCY
Size
Smdl
large
FIG. 3. Optimum size at two temperatures below the thermal neutral zone. Selection is again considered to maximize the difference between energy intake per time and maintenance energy costs per time scaled to energy costs at thermal neutrality. Optimum size is larger at the lower temperature (T,) than at the higher (T, ).
I stated above that Rosenzweig and Sterner’s data on F/BMR were consistent with the model. Note, however, that the model in expression 2 predicts that all species should be either at the peak of F/BMR or to the right of the peak. This prediction would be fulfilled by all seven heteromyid species if their diets were composed of spinach and pumpkin seeds; however, if the diets were only sunflower and squash seeds, the smallest species would have sizes to the left of the peak, thus violating the prediction. Whether or not these smallest species fit the model will depend on the proportions of seeds similar to spinach and pumpkin or sunflower and squash in their actual diets. A second possible scaling function is metabolic rate at ambient temperature (MZ?r-). This function is appropriate when the dominant selection pressure influencing size is selection for ability to survive long periods of energy stress. For instance, individuals living through a northern winter may have difficulty in finding food while at the same time they experience greater than normal metabolic costs because of lower temperatures. In this case selection would maximize: F-MRTa MRT*
z--l
F
(expression 3).
MRT~
The greater this difference, the longer an individual
can survive at T, on the
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
587
energy gained during a foraging period of given length. If this difference is negative over a certain period, the animal will have smaller energy reserves at the end of the period than at the start, and if it is negative long enough, the animal will die of starvation. What is the effect on optimum size of lowering ambient temperature using this model? The optimal body size at Tth will occur at the maximum of F/BMR and the optimum body size at T at the maximum of F/MR=. The maximum of any quotient x/y occurs where In (x)-In (y) is maximal. Thus if we graph F and BMR on a semilog plot (Fig. 4), the optimum size at Tth
/
opt Y 0’ Tth small
I
optw ot T Size
/ /
W’ large
FIG. 4. Optimum size where selection maximizes the difference between energy intake per time and maintenance energy costs per time scaled to energy costs at the ambient temperature (T,). Selection maximizes F/MR, - 1, where F is energy profit per time foraging and MR, is maintenance costs per time at temperature T,. F/MR, is maximal where In (F) - In (MR,) is maximal. T is a temperature below the lower critical temperature for individuals of w < w’. Using a semilog plot, optimum size at each temperature occurs where the slope of the intake curve and the slope of the cost curve are equal. For all WI< IV’, optimum size is higher at T than at thermal neutrality.
occurs at the weight where the difference between the two curves is greatest, that is, where their slopes are equal. Also graphed in Fig. 4 is MRT, where T < Tth for all individuals of w < w’. It can be seen that the optimum size at T will be greater than the optimum size at Tth as long as the optimum at Tth is less than w’. This can be easily proved. The slope of In (MR=) is dln (MR,) dw
1 =- MR,
588
W.
and the slope of In (BMR)
A.
SEARCY
is
d In (BMR) -L(“(y~). dw For the region where w < w’, l/MR,< l/BMR and d (MR=)/dw < d (BMR)/dw so the slope of In (MR,) is less than the slope of In (BMR). Since the slope of In (F) is decreasing as w increases, the point at which the slope of In (MRT) equals the slope of In (F) will be to the right of the point at which the slope of In (BMR) equals the slope of In (F). Again, we can say that the optimal weight at T < Tl, is greater than the optimal weight at T,,. The conclusions of this second model can be extended to time minimizers (Schoener, 1971), animals under selection to minimize time spent foraging. Minimizing foraging time can be important, for example, to animals that are particularly vulnerable to predation while foraging. Such animals can be considered to have a set maintenance energy requirement which they seek to meet in the shortest time possible. If MR, is in units of energy required per day and F in units of energy profit per hour foraging, then clearly a time minimizer is selected to minimize MRTJF = hours foraging/day. Minimizing MRTn/F is equivalent to maximizing F/MRTO so the conclusions of the model in Fig. 4 apply to time minimizers. 5. Some Assumptions
of the Models
One assumption of these optimal size models is that F does not change with T,. This assumption may not hold true for some homeothermic vertebrates because heat generated in foraging activity (or in digestion) may be used to substitute for heat generated for thermoregulation at low T,. Thus some of the costs of foraging would be absorbed in MRT,, raising the value of F. Heat generated in activity is used for thermoregulation in some birds and mammals but not others. For example, budgerigars (Melupsitfucus undulutus) must increase their metabolic rate over BMR when resting at 2O”C, but when flying they expend the same amount of energy per time at 20” as at 37” (Greenwald et al., 1967). In chaffinches (Fringilla coekbs), on the other hand, when T, is lowered from 20°C to Y, energy consumption increases by approximately the same amount at a high level of activity as at rest (Pohl, 1969). In cold acclimated rats, thermogenesis due to exercise is added to thermogenesis due to cold down to -lYC, but below -15” thermogenesis due to exercise can substitute for thermogenesis due to cold (Hart & Jansky, 1963). For animals in which energy expended in foraging activity is used for thermoregulation at low T,, the models of optimal size must be modified
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
589
somewhat. As stated above, the effect of using energy expended in foraging for thermoregulation is to raise F, the net profit of foraging, at low T,. Since small animals have a higher T,, than large ones, small animals are further below their T,, at low T,, and thus we can expect that at low T, the F curve will be shifted upwards more for small animals than large. The effect of shifting F upwards more for small animals than large would be to lower optimal size. However, the effect of shifting F upwards at low T, should not be as important as the shifting upwards of resting metabolic rate (MRTu) because F is raised at most only by the amount that MR, exceeds BMR during the period an animal is actually foraging while MRTa is raised for the entire period of low T,. Thus for animals that can substitute thermogenesis due to activity for thermogenesis due to cold there should be only a quantitative shift in the predictions of the models and not a qualitative one. Another assumption of the models is that the relationship between size and BMR found in interspecific comparisons also applies within species. McMahon (1973) has argued that BMR is proportional to weight to the 0.75 pdwer because of physical relationships between the diameter of objects and the length at which they buckle, between the length and diameter of objects and their weight, and between the diameter of muscles and their power output. All of these relationships should hold within species as well as between species, so theoretically BMR should be a function of the O-75 power of weight within species. Empirically found exponents for metabolism versus weight within various species of rodents range from 0.20 to almost 1.0 (Hart, 1971). Exponents within this range will still give the same general shapc
590
W. A. SEARCt
foraging rates should be important in the evolution of body size in these food-limited populations at least, and perhaps in others as well. A final assumption of these models is that the curves of F vs. body size have a hump, or at least an asymptote (Fig. l), and that the actual size of most birds and mammals is somewhere in the region where their F curves approach their maxima. The shapes of F curves can vary widely without effect on the models as long as the maxima exist. For example, the outcome of the models is not affected by the existence of regions where F is getting steeper with increasing size (positive second derivative) because optimum size will not occur in these regions. Similarly, F can be a linear function of size over some range of size without affecting the models because, again, optimum size will not occur within that range. I have already given arguments why all F curves should have a maximum at some size; these arguments can be summarized by saying that it is difficult to imagine an animal whose foraging rate would always increase as its size increases without any limit. A more important question is whether the actual size of most birds and mammals is in the size range where their F curves are approaching their maxima. In animals whose populations are food limited, we would expect selection to set size within this range. Whether the size of animals which are not food limited usually falls within this range is a question that can only be answered empirically. The advantage of these optimal size models is that they take into account changes in both energy intake and energy expenditure with size, whereas previous discussions of Bergmann’s Rule have been limited to one or the other. Although the direct applicability of these models may be limited by their assumptions, especially by assumptions about the shape of F in the region of an animal’s actual size, the models nevertheless illustrate a useful method of analysis. If a particular species of bird or mammal could be shown to have a radically different F curve in the region of its actual body size, then the conclusions of these models on how its optimum size would change with T, might not apply; nevertheless, one should still use this or a similar method of analysis to determine the animal’s energetically optimum sizes at different temperatures. 6. Discussion
The model presented here defines the optimum size of an individual as that size at which the difference between energy intake and maintenance energy costs per time, scaled to body size, is greatest. The model predicts that this optimum size should increase as ambient temperature is lowered and thus provides an explanation for Bergmann’s Rule. Note that optimum
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
591
size is not necessarily that size (w’) where the individual’s 7’,, is as low as the ambient temperature; in fact, in the examples graphed in Figs 2, 3, and 4, optimum size is well below w’. In other words, the model does not claim that an animal should be large enough so that it is within its thermal neutral zone at the reigning ambient temperature. If selection for maximizing the energy available for activity, reproduction, or thermoregulation were the only selection pressure influencing body size, then the explanation provided by the model for Bergmann’s Rule would stand alone. Actually, since there are manifestly other selection pressures acting on body size, other explanations may be compatible with this one. Furthermore, there may be cases where other selection pressures obscure or even reverse the trend described by Bergmann’s Rule and predicted by the model. James (1968, 1970) has shown that for many birds, size correlates more closely with wet bulb temperatures than with dry bulb temperatures (see also Power, 1971 and Hamilton, 1961). James interprets this as showing that Bergmann’s Rule is explained by selection for efficiency in dumping heat in warm, humid environments. Thus northern populations may be near their optimum size as given by my energetics model while southern populations are moved below their optimum size by selection for high surface to volume ratios to facilitate heat dumping. It has also been suggested that northern populations are under selection for large size because large indivuduals can withstand longer periods of starvation in winter storms (Calder, 1974; Ketterson & Nolan, 1976; Downhower, 1976). The ability to withstand starvation increases with size because maximum fat stores increase faster with size than do metabolic costs. Thus periods of starvation may select for sizes above the energetic optimum in the north, where periods of intense bad weather are most likely. It is also possible that geographical trends in body size may be due to changes in the energy intake versus size curve. McNab (1971) proposed for carnivorous mammals that the number of competing species declines towards the north so that larger food items are available there. If one accepts this reasoning, it is clear that the availability of larger prey might shift the peaks of the intake curves of the remaining predators towards larger sizes which would shift the energetic optimum size upwards. Deciding which of the hypothesized selective forces is most important in causing clines of body size in particular species will be very difficult, especially as the hypotheses are not mutually exclusive. One approach is to test whether the correlations predicted by one hypothesis are significantly stronger than the correlations predicted by another hypothesis. James (1970) took this approach when she showed that the correlations between
592
W. A. SEARCY
size and wet bulb temperatures (predicted by the heat dumping hypothesis) are higher than the correlations between size and dry bulb temperatures (predicted by the energetics model) in a significantly greater proportion of tests on eight species of birds. Another approach is to test predictions about what species should show the most pronounced clines. For example, the heat dumping hypothesis predicts that clines of size in temperature gradients should be more pronounced in species that encounter little drinking water than in species that encounter abundant water. The fat storage hypothesis predicts that size clines should be less pronounced in large species than in small because large species are less likely to encounter periods of starvation long enough to threaten survival. The physiological relationships between size and metabolic rate and size and conductance have long been known, but these relationships alone do not provide a complete, energetics explanation of Bergmann’s Rule. Before a complete explanation can be made, it is necessary to specify what an “energetically optimum size” is, that is, exactly what selection is acting to optimize in an energetics model. Secondly, it is necessary to specify some broad relationship between size and foraging efficiency. All of these elements are needed to form an energetics model of optimum size at different temperatures and so to provide an energetics explanation of Bergmann’s Rule. I would like to thank the many people who helped me in writing this paper, among them W. James Erckmann, James Quinn, Thomas Schoener, Gordon Orians, James Wittenberger, James Kenagy, Robert Stevenson, Sievert Rohwer, William Tschumy, Nelson Hairston Jr., Ken Yasukawa, Robert Seyfarth and Margaret McVey. I also thank Margaret Searcy for drawing the figures. REFERENCES G. A. (1977). In Animal Physiology: Principles and Adaptations (M. S. Gordon, ed.), 3rd Edn. New York: Macmillan. BERGER, M. & HART, J. S. (1972). J. Comp. Physiol. 81, 363. BROWN, J. H. & LEE, A. K. (1969).Evolution 23,329. CALDER, W. A. III (1974).In Avian Energetics (R. A. Payntered.).Nuttall Ornithological BARTHOLOMEW,
Club Publication No. 15. CALDER, W. A. III & KING, J. R. (1974).In Avian Biology vol. 4 (D. S. Farner & J. R. King eds). New York: Academic Press. DOWNHOWER, J. R. (1976). Nature 263,558. GREENWALD, L., STONE, W. B. & CADE, T. J. (1967). Comp. Biochem. Physiol. 22,91. HAMILTON, T. H. (1961). Evolution 15,180. HART, J. S. (1971).In ComparativePhysiology of Thermoregulation (G. C. Whittow, ed.). New York: AcademicPress. HART, J. S. & JANSKY, L. (1963). Can. J. Biochem. Physioi. 41, 629. HEMMIGNSEN, A. M. (1960).Reports Steno Memorial Hospital 9, 1.
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
593
JAMES, F. C. (1968). Am. Zool.8,815. JAMES, F. C. (1970). Ec&gy 51,365. KENDEIGH, S. C. (1969). Auk 86,13. KE~TERSON, E. D. & NOLAN, V. Jr. (1976). Ecology 57,679. KING, J. R. (1974). In Asian Energerics (R. A. Paynter, ed.). Nuttal Ornithological Club Publication No. 15. KONTOGIANNIS, J. E. (1968). Whys. 2001. 41, 54. LACK, D. (1954). The Natural Regulation of Animal Numbers. Oxford: Clarendon Press. LACK, D. (1966). Population Studies of Birds. Oxford: Clarendon Press. LASIEWSKI, R. C. & DAWSON, W. R. (1967). Condor 69,13. MAYR, E. (1956). Eoolution 10, 105. MAYR, E. (1963). Populations, Species, and Evolution. Cambridge, Mass.: Belknap Press. MCMAHON, T. (1973). Science 179, 1201. MCNAB, B. K. (1971). Ecology 52,845. POHL, H. (1969). Fed. Proc., Fed. Am. Sot. exp. Biol. 28, 1059. POWER, D. M. (1971). Sysl. 2001. 18, 363. RENSCH, B. (1936). Arch. Naturg. 5, 317. RENSCH, B. (1960). Evolution Aboue the Species Leuel. New York: Columbia University Press. RICKLEFS, R. E. (1974). In Aoian Energeks (R. A. Paynter, ed.). Nuttall Ornithological Club Publication No. 15. ROSENZWEIG, M. L. (1968). Am. Midl. Narur. 80,299. ROSENZWEIG, M. L., & STERNER, P. (1970). Ecology 51,217. SCHOENER, T. W. (1969). Am. Narur. 103,277. SCHOENER, T. W. (1971). Ann. Reu. Ecol. Sysr. 2, 369. SCHOLANDER, P. F. (1954). Evolution 9, 15. SCHOLANDER, P. F. (1956). Euolurion 10,339. SNOW, D. W. (1954). Evolution 8, 19.
Optimum Body Sizes at Different tures: an Energetics Explanation WILLIAM Department
Ambient Temperaof Bergmann’s Rule
A. SEARCY~.
of Zoology, University of Washington, Seattle, Washington 98195, U.S.A. (Received 20 November 1978)
A model ispresentedof optimumbody sizesat different ambienttemperatures. Selectionis consideredto optimize the difference betweenforaging profit per time and maintenanceenergy costsper time, scaledto body size. Two different scalingfunctions are considered:one maximizesthe amount of activity that an individual can perform, and the other maximizesthe length of time an individual can survive on the profit gained in a given period of foraging. With both scalingfunctions, optimum sizeincreasesas ambient temperature decreases.
1. Introduction Bergmann’s Rule states that “races of warm blooded vertebrates from cooler climates tend to be larger than races of the same species from warmer climates” (Mayr, 1956). There is much evidence for the validity of this rule (Rensch, 1936,196O; Snow, 1954; Rosenzweig, 1968; Brown & Lee, 1969; James, 1970), but its adaptive significance is still much in question. The classical evolutionary explanatfon of the rule is that larger animals have an advantage in cooler climates because their smaller surface to volume ratios lead to lower rates of heat loss and metabolism per gram body weight (Mayr, 1963). The usual counter to this argument is that the main adaptation to lower temperatures is increased insulation, compared to which small increases in body size would have little effect (Scholander, 1954, 1956). As Mayr (1956) notes, even if increased insulation is the main adaptation to cold, increased size may also be an adaptation, and selection can cause the evolutionary enhancement of characteristics even if their benefit is small. A more fundamental criticism of the classical physiological explanation is that even though increased size leads to smaller per gram metabolic costs, it + Present address: Rockefeller University Field Research Center. Millbrook, 12545, U.S.A.
New York
579
0022-5193/80/080579+
15 $02.00/O
@ 1980 Academic Press Inc. (London) Ltd.
580
W.
increases metabolic
A.
SEARCY
costs per individual
(McNab,
1971). For example:
“a doubling of weight would increase the basal rate of metabolism by only 68%. However, I know of no way by which an animal can measure the 32% “saving” in the maintenance of the additional weight; it can only measure a 68% increase in energy expenditure (by an increase in the amount of food required)” (McNab, 1971). It is certainly true that there is no obvious selective advantage to minimizing metabolic rates per gram at the price of increasing total metabolism. Instead, selection should favor maximizing the difference between energy intake and maintenance energy costs per time per individual. The greater this difference, the more energy an individual can inveSt in activity, reproduction, or storage. Let us define F as the foraging profit (energy/time) of an individual,that is, as the energy harvested by an individual per time minus the additional energy cost it incurs by foraging rather than doing nothing. Let A4RT be maintenance energy expenditure, or metabolic rate (energy/time), at temperature T. Both F and lkfRT should vary with the size (w) of an animal. Optimal size will be a function of the difference between F and MRT. A small animal can do more with a certain amount of energy per time than can a large one so optimal size will also be a function of the cost of activity, reproduction, or thermoregulation to animals of different sizes. Suppose that C is the cost of some activity (or activities) crucial to the fitness of an animal, with units of energy cost per unit of activity. C should also vary with size, and optimal size would be that size where F-MRT C
is maximized.
(expression 1)
The units of the expression are energy/time -energy/time = activity/time. energy/unit activity
Alternatively, C (which I will call a scaling function) might represent the cost in energy of one unit of reproduction (e.g. kcal/egg), in which case expression 1 would reduce to reproduction/time. To determine how optimum size changes as temperature changes, we must determine: (1) how foraging profit per time varies with size, (2) how metabolic rate varies with size and temperature, and (3) what expressions for C are appropriate. 2. Energy Intake vs. Size
Let us assume we are considering an animal constrained by morphological and behavioral adaptations evolved over the history of its species to feed on
OPTIMUM
SIZES
AT DIFFERENT
TEMPERATURES
581
only a certain range of items which it obtains and handles in only certain ways. The evolution of major shifts of diet and foraging modes is prevented by interspecific competition. Then it is logical to assume that there will be some size below which the animal would become increasingly less efficient as its size decreases. For example, a hawk’s efficiency in capturing small mammals must decrease as the hawk becomes too small to subdue its prey. A rodent’s efficiency in husking seeds must decrease as its mouth becomes too small to husk the largest seeds in its diet. Conversely, above a certain size, further size increase should not increase efficiency and may even lower it. For example, once the hawk is large enough to have no trouble subduing its prey, further size increases may decrease its maneuverability in flight and so decrease its capture rate. Above a certain body size, a rodent must have increasing difficulty in manipulating small seeds in its teeth. We can assume a general relationship between size and foraging efficiency such as is shown in Fig. 1, with energy profit per time increasing with increasing consumer size over some range of sizes and then levelling off and perhaps declining with further size increase. I have assumed a unimodal shape for this F function; bimodal shapes are also possible (Schoener, 1969) though perhaps not as likely. In general, we can assume such size-foraging efficiency relationships to be independent of ambient temperature.
FIG. 1. Relationships between size and foraging efficiency and size and maintenance energy costs. F is energy profit per time foraging. BMR is energy cost of maintenance at a temperature within the thermal neutral zone; this curve is described by the equation a(~~) where w is in units of grams and b is approximately 0.75. MRT gives the energy cost of maintenance at temperature T, where T is less than the lower critical temperature for individuals of size w < w’.
582
W.
A.
SEAR(‘Y
3. Energy Costs vs. Size at Different
Temperatures
Metabolic costs at thermal neutrality increase as a function of weight to approximately the 0.75 power in a variety of vertebrate groups (Bartholomew, 1977). For example, in passerines, BMR = 46.63( Mu”“‘), where BMR is basal metabolic rate in mW and MI is weight in g (Lasiewski & Dawson, 1967; Calder, 1974). In rodents, BMR= 21.21(~“‘~) (Hart, 1971). Fourier’s law of heat flow predicts that MR, = h( Tb - r,) where !L is conductance (mW/‘C), Th is body temperature, and T, is ambient temperature (Calder, 1974). Below the lower critical temperature (T),, the lower limit of the thermal neutral zone), h is a constant for an animal of given weight. This constant increases with increasing size; for example, in passerines, h = 5.29( wO.~~) (Calder & King, 1974; Calder, 1974). If we set T, equal to T,,, then (for passerines) 46.63(w0”‘) = 5.29(w o’46) ( Tb - T,,), which reduces to 8.81(~~“~~) = Tb- TIC.Assuming Tb to be approximately independent of weight, lower critical temperature should decrease as w(“~~ increases (Calder, 1974). Similarly, Hart (1971) found that T,, in rodents should decrease as w”‘~ increases. Data giving T), and weight for different species of rodents fit the predicted relationship though with a good deal of variability (Hart, 1971). In Fig. 1, I have graphed BMR increasing with weight to approximately the 0.75 power. Imagine a small animal whose metabolic rate we measure at a series of temperatures below its thermal neutral zone. Metabolic rate should be higher at each successively lower temperature. However, for each lower temperature, we can imagine an animal just big enough so that the ambient temperature is within its thermal neutral zone. For example, we can find the approximate weight of a passerine just large enough to have its lower critical temperature at 20°C by solving 8.81(wn”h) = Tb-20. Assuming Tb is 40”, w = 23.4 g. Thus the line relating metabolic rate to weight at 20” for passerines must be above the BMR curve for small sizes and converge with it at about 23 g. For all vertebrate groups, there will be a family of such curves, one for each temperature below the T,,. for the smallest animal considered. In Fig. 1 is graphed one such curve: MRT where T < T,, for animals of w < w’. Empirically, there would be a great deal of variability about such curves, but if other temperature related adaptations, such as quality and amount of insulation, are held constant, such curves should be correct. 4. Two Scaling Functions
One quantity that selection may tend to maximize is the amount of activity an animal can perform. The energy cost of many activities seems to be scaled
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
583
to body size with approximately the same relationship as BMR so the cost of activity can be expressed as a constant multiple of BMR. This argument can be applied, for example, to cost of flight in passerines (King, 1974). The energy expenditure of birds in flight, in mW, is approximately 341*7( w”‘~) (Berger & Hart, 1972; King, 1974). If BMR is taken to be 46*63(w’.“) for passerines, then the cost of flight is about 7.3 x BMR for passerines of any size. King (1974), using a dif&ent expression for BMR, obtained the figure 9.3 x BMR. Hemmingsen (1960) found that for a large range of animals, the maximal rate of energy expenditure in activity is an approximately constant multiple of BMR. Maximizing the amount of activity an animal existing at temperature T can perform is equivalent to maximizing: F-MRT BMR
F =--BMR
MRT BMR
(expression 2).
The greater this difference, the more activity an individual can perform using the energy gained in a given period of foraging. This same expression can be used in some cases where selection is thought to favor the body size at which reproductive output per time is maximal. The cost of reproduction may often be a constant multiple of BMR for animals of different weights so maximizing the difference in expression 2 would maximize relative reproductive output. For example, egg weight, and presumably the cost of producing eggs, is a function of body weight of the mother to the 0.75 power in passerines (Ricklefs, 1974). Thus maximizing expression 2 for a passerine will maximize the number of eggs it can produce per time. Expression 2 is also appropriate in cases where most of the energy invested in reproduction is used in activity, as in male birds that invest heavily in territory defense and epigamic displays. Rosenzweig & Sterner (1970) give data on F/BMR for seven species of heteromyids eating four species of seeds. These authors measured caloric content of each seed type and the amount of time required by each heteromyid species to eat the seed. They then calculated F/BMR for each combination of seed and animal species. For two species of seeds (spinach and pumpkin), F/BMR decreased more or less uniformly with increasing consumer size. For the other two seeds (sunflower and squash), there was an intermediate sized consumer for which F/BMR was maximal, with values falling away with both increasing and decreasing consumer sizes. These data are compatible with the shape of F/BMR hypothesized in Fig. 2 (but see below). A curve of this shape is produced by dividing F in Fig. 1 by BMR. Kendeigh (1969) presents the lower critical temperatures of 15 species of passerines and the equations for calculating their standard metabolic rates at
584
W.
small
A.
SEARCY
Size
large
2. Optimum size where selection maximizes the difference between energy intake per time and maintenance energy costs scaled to energy costs at thermal neutrality. F is energy profit per time foraging. BMR is maintenance energy costs per time at a temperature within the thermal neutral zone. MRT is energy costs per time at temperature T, where T is below the lower critical temperature for individuals of w < w’. The maximum difference between the intake curve and the cost curve at each temperature occurs where the slope of the intake curve equals the slope of the cost curve. For all w < w’, the optimum size is higher at T than at r,,. FIG.
temperatures below thermal neutrality. From these data, A4RT can be calculated for any T < TI,. Table 1 gives the weights of these species and MRo*/BMR. This quantity decreases as size increases until it reaches a value of 1. Figure 2 shows the general shape that MR=/BMR should take. Values are high for small animals, decline as size increases up to the size where T = Tic, and thereafter are 1. With the model presented in expression 2, how does optimal body size change as ambient temperature is lowered? As an example, let us consider optimal body size at Tth(a temperature within the thermal neutral zone) and at some T such that T < T,, for animals of w < w’. The optimal body size at Tth is that size where the difference between F/BMR and BMR/BMR is greatest, that is, at the size corresponding to the peak of the F/BMR curve (see Fig. 2). The optimal body size at. T is at that w where the difference between F/BMR and MR=/BMR is maximal. This maximum occurs at the weight where the slopes of these two curves are equal. MR=/BMR has a negative slope for all w < w’. F/BMR has a negative slope only at weights to the right of its peak. Thus the two slopes will be equal at a size greater than the optimum size at Tfh as long as the peak of F/BMR occurs at a w < w’. In general we can say that if an animal is at its optimal size at a temperaure
OPTIMUM
SIZES
AT DIFFERENT TABLE
585
TEMPERATURES
1
The ratio of metabolic rate at 0°C to basal metabolic rate for passerinesof different sizes. Data calculated from equations in Kendeigh (1969, Table 1) Species Black-rumped waxbill (Esrrilda troglodytes) Black-rumped waxbill (Estrilda rroglodytes) House wren (Troglodyres aedon) Zebra finch (Taeniopygia casmnoris) Zebra finch (Taeniopygiu castunofis) Ortolan bunting (Emberiza horfuluna) White-throated sparrow (Zonotrichia albicollis) House sparrow (Passer domesticus) House sparrow (Passer domesticus) House sparrow (Passer domesticus) Yellow bunting (Emberiza citrinella) White-crowned sparrow (Zonorrichia leucophyrys) Red crossbill (Loxia curvirostru) White-winged crossbill (Loxia leucoprera) Cardinal (Richmondem cardinalis) Evening grosbeak (Hesperiphona vespertina) Evening grosbeak (Hesperiphona vespertina) Blue jay (Cyanocirta crisfuta) White-necked raven (Corvus cryptoleucus) Common raven (Corvus corax) Common raven (Corvus corax)
Weight Id 6.4 6.5
9.7 11.5 11.7 22.0 22,5 25.1 25.5 25.3 26.4 28.6 29.4 29.8 41 54.5 55.3 80.8 640 850 866
MR,./BMR 1.80 1.76 1.59 1.71 1.76 1.62 1.59 1.59 1.55 1.58 1.45 1.54 1.37 1.36 1.40 1.47 1.39 1.30 1.00 1.00 1.00
within its thermal neutral zone, and the ambient temperature is lowered below that animal’s T,,, the optimal size will increase. What happens if an animal is already existing at a temperature less than its T,, and the ambient temperature is lowered further? Figure 3 shows hypothetical MRT/BMR curves for TI and T2, with T,, > TI > T2. Since T1 > T2, the MR=,/BMR curve joins the BMR/BMR line at a smaller size than does the MRrJBMR curve. It follows from the graphical model (Fig. 3) that the optimum size at T2 will be greater than the optimum at T1. Thus any lowering of temperature causes an increase in optimal size. In Figs 2 and 3, I have assumed that the F/BMR curve is unimodal. If this assumption is relaxed, and more than one peak in the curve is allowed, then it is possible to get sudden jumps in optimal size as ambient temperature is lowered. For example, imagine that there is a second, lower peak in the F/BMR curve in Fig. 2 to the right of the first peak. Then as ambient temperaure is lowered, the optimal size might jump from the region of the first peak to the region of the second.
586
W. A. SFARCY
Size
Smdl
large
FIG. 3. Optimum size at two temperatures below the thermal neutral zone. Selection is again considered to maximize the difference between energy intake per time and maintenance energy costs per time scaled to energy costs at thermal neutrality. Optimum size is larger at the lower temperature (T,) than at the higher (T, ).
I stated above that Rosenzweig and Sterner’s data on F/BMR were consistent with the model. Note, however, that the model in expression 2 predicts that all species should be either at the peak of F/BMR or to the right of the peak. This prediction would be fulfilled by all seven heteromyid species if their diets were composed of spinach and pumpkin seeds; however, if the diets were only sunflower and squash seeds, the smallest species would have sizes to the left of the peak, thus violating the prediction. Whether or not these smallest species fit the model will depend on the proportions of seeds similar to spinach and pumpkin or sunflower and squash in their actual diets. A second possible scaling function is metabolic rate at ambient temperature (MZ?r-). This function is appropriate when the dominant selection pressure influencing size is selection for ability to survive long periods of energy stress. For instance, individuals living through a northern winter may have difficulty in finding food while at the same time they experience greater than normal metabolic costs because of lower temperatures. In this case selection would maximize: F-MRTa MRT*
z--l
F
(expression 3).
MRT~
The greater this difference, the longer an individual
can survive at T, on the
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
587
energy gained during a foraging period of given length. If this difference is negative over a certain period, the animal will have smaller energy reserves at the end of the period than at the start, and if it is negative long enough, the animal will die of starvation. What is the effect on optimum size of lowering ambient temperature using this model? The optimal body size at Tth will occur at the maximum of F/BMR and the optimum body size at T at the maximum of F/MR=. The maximum of any quotient x/y occurs where In (x)-In (y) is maximal. Thus if we graph F and BMR on a semilog plot (Fig. 4), the optimum size at Tth
/
opt Y 0’ Tth small
I
optw ot T Size
/ /
W’ large
FIG. 4. Optimum size where selection maximizes the difference between energy intake per time and maintenance energy costs per time scaled to energy costs at the ambient temperature (T,). Selection maximizes F/MR, - 1, where F is energy profit per time foraging and MR, is maintenance costs per time at temperature T,. F/MR, is maximal where In (F) - In (MR,) is maximal. T is a temperature below the lower critical temperature for individuals of w < w’. Using a semilog plot, optimum size at each temperature occurs where the slope of the intake curve and the slope of the cost curve are equal. For all WI< IV’, optimum size is higher at T than at thermal neutrality.
occurs at the weight where the difference between the two curves is greatest, that is, where their slopes are equal. Also graphed in Fig. 4 is MRT, where T < Tth for all individuals of w < w’. It can be seen that the optimum size at T will be greater than the optimum size at Tth as long as the optimum at Tth is less than w’. This can be easily proved. The slope of In (MR=) is dln (MR,) dw
1 =- MR,
588
W.
and the slope of In (BMR)
A.
SEARCY
is
d In (BMR) -L(“(y~). dw For the region where w < w’, l/MR,< l/BMR and d (MR=)/dw < d (BMR)/dw so the slope of In (MR,) is less than the slope of In (BMR). Since the slope of In (F) is decreasing as w increases, the point at which the slope of In (MRT) equals the slope of In (F) will be to the right of the point at which the slope of In (BMR) equals the slope of In (F). Again, we can say that the optimal weight at T < Tl, is greater than the optimal weight at T,,. The conclusions of this second model can be extended to time minimizers (Schoener, 1971), animals under selection to minimize time spent foraging. Minimizing foraging time can be important, for example, to animals that are particularly vulnerable to predation while foraging. Such animals can be considered to have a set maintenance energy requirement which they seek to meet in the shortest time possible. If MR, is in units of energy required per day and F in units of energy profit per hour foraging, then clearly a time minimizer is selected to minimize MRTJF = hours foraging/day. Minimizing MRTn/F is equivalent to maximizing F/MRTO so the conclusions of the model in Fig. 4 apply to time minimizers. 5. Some Assumptions
of the Models
One assumption of these optimal size models is that F does not change with T,. This assumption may not hold true for some homeothermic vertebrates because heat generated in foraging activity (or in digestion) may be used to substitute for heat generated for thermoregulation at low T,. Thus some of the costs of foraging would be absorbed in MRT,, raising the value of F. Heat generated in activity is used for thermoregulation in some birds and mammals but not others. For example, budgerigars (Melupsitfucus undulutus) must increase their metabolic rate over BMR when resting at 2O”C, but when flying they expend the same amount of energy per time at 20” as at 37” (Greenwald et al., 1967). In chaffinches (Fringilla coekbs), on the other hand, when T, is lowered from 20°C to Y, energy consumption increases by approximately the same amount at a high level of activity as at rest (Pohl, 1969). In cold acclimated rats, thermogenesis due to exercise is added to thermogenesis due to cold down to -lYC, but below -15” thermogenesis due to exercise can substitute for thermogenesis due to cold (Hart & Jansky, 1963). For animals in which energy expended in foraging activity is used for thermoregulation at low T,, the models of optimal size must be modified
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
589
somewhat. As stated above, the effect of using energy expended in foraging for thermoregulation is to raise F, the net profit of foraging, at low T,. Since small animals have a higher T,, than large ones, small animals are further below their T,, at low T,, and thus we can expect that at low T, the F curve will be shifted upwards more for small animals than large. The effect of shifting F upwards more for small animals than large would be to lower optimal size. However, the effect of shifting F upwards at low T, should not be as important as the shifting upwards of resting metabolic rate (MRTu) because F is raised at most only by the amount that MR, exceeds BMR during the period an animal is actually foraging while MRTa is raised for the entire period of low T,. Thus for animals that can substitute thermogenesis due to activity for thermogenesis due to cold there should be only a quantitative shift in the predictions of the models and not a qualitative one. Another assumption of the models is that the relationship between size and BMR found in interspecific comparisons also applies within species. McMahon (1973) has argued that BMR is proportional to weight to the 0.75 pdwer because of physical relationships between the diameter of objects and the length at which they buckle, between the length and diameter of objects and their weight, and between the diameter of muscles and their power output. All of these relationships should hold within species as well as between species, so theoretically BMR should be a function of the O-75 power of weight within species. Empirically found exponents for metabolism versus weight within various species of rodents range from 0.20 to almost 1.0 (Hart, 1971). Exponents within this range will still give the same general shapc
590
W. A. SEARCt
foraging rates should be important in the evolution of body size in these food-limited populations at least, and perhaps in others as well. A final assumption of these models is that the curves of F vs. body size have a hump, or at least an asymptote (Fig. l), and that the actual size of most birds and mammals is somewhere in the region where their F curves approach their maxima. The shapes of F curves can vary widely without effect on the models as long as the maxima exist. For example, the outcome of the models is not affected by the existence of regions where F is getting steeper with increasing size (positive second derivative) because optimum size will not occur in these regions. Similarly, F can be a linear function of size over some range of size without affecting the models because, again, optimum size will not occur within that range. I have already given arguments why all F curves should have a maximum at some size; these arguments can be summarized by saying that it is difficult to imagine an animal whose foraging rate would always increase as its size increases without any limit. A more important question is whether the actual size of most birds and mammals is in the size range where their F curves are approaching their maxima. In animals whose populations are food limited, we would expect selection to set size within this range. Whether the size of animals which are not food limited usually falls within this range is a question that can only be answered empirically. The advantage of these optimal size models is that they take into account changes in both energy intake and energy expenditure with size, whereas previous discussions of Bergmann’s Rule have been limited to one or the other. Although the direct applicability of these models may be limited by their assumptions, especially by assumptions about the shape of F in the region of an animal’s actual size, the models nevertheless illustrate a useful method of analysis. If a particular species of bird or mammal could be shown to have a radically different F curve in the region of its actual body size, then the conclusions of these models on how its optimum size would change with T, might not apply; nevertheless, one should still use this or a similar method of analysis to determine the animal’s energetically optimum sizes at different temperatures. 6. Discussion
The model presented here defines the optimum size of an individual as that size at which the difference between energy intake and maintenance energy costs per time, scaled to body size, is greatest. The model predicts that this optimum size should increase as ambient temperature is lowered and thus provides an explanation for Bergmann’s Rule. Note that optimum
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
591
size is not necessarily that size (w’) where the individual’s 7’,, is as low as the ambient temperature; in fact, in the examples graphed in Figs 2, 3, and 4, optimum size is well below w’. In other words, the model does not claim that an animal should be large enough so that it is within its thermal neutral zone at the reigning ambient temperature. If selection for maximizing the energy available for activity, reproduction, or thermoregulation were the only selection pressure influencing body size, then the explanation provided by the model for Bergmann’s Rule would stand alone. Actually, since there are manifestly other selection pressures acting on body size, other explanations may be compatible with this one. Furthermore, there may be cases where other selection pressures obscure or even reverse the trend described by Bergmann’s Rule and predicted by the model. James (1968, 1970) has shown that for many birds, size correlates more closely with wet bulb temperatures than with dry bulb temperatures (see also Power, 1971 and Hamilton, 1961). James interprets this as showing that Bergmann’s Rule is explained by selection for efficiency in dumping heat in warm, humid environments. Thus northern populations may be near their optimum size as given by my energetics model while southern populations are moved below their optimum size by selection for high surface to volume ratios to facilitate heat dumping. It has also been suggested that northern populations are under selection for large size because large indivuduals can withstand longer periods of starvation in winter storms (Calder, 1974; Ketterson & Nolan, 1976; Downhower, 1976). The ability to withstand starvation increases with size because maximum fat stores increase faster with size than do metabolic costs. Thus periods of starvation may select for sizes above the energetic optimum in the north, where periods of intense bad weather are most likely. It is also possible that geographical trends in body size may be due to changes in the energy intake versus size curve. McNab (1971) proposed for carnivorous mammals that the number of competing species declines towards the north so that larger food items are available there. If one accepts this reasoning, it is clear that the availability of larger prey might shift the peaks of the intake curves of the remaining predators towards larger sizes which would shift the energetic optimum size upwards. Deciding which of the hypothesized selective forces is most important in causing clines of body size in particular species will be very difficult, especially as the hypotheses are not mutually exclusive. One approach is to test whether the correlations predicted by one hypothesis are significantly stronger than the correlations predicted by another hypothesis. James (1970) took this approach when she showed that the correlations between
592
W. A. SEARCY
size and wet bulb temperatures (predicted by the heat dumping hypothesis) are higher than the correlations between size and dry bulb temperatures (predicted by the energetics model) in a significantly greater proportion of tests on eight species of birds. Another approach is to test predictions about what species should show the most pronounced clines. For example, the heat dumping hypothesis predicts that clines of size in temperature gradients should be more pronounced in species that encounter little drinking water than in species that encounter abundant water. The fat storage hypothesis predicts that size clines should be less pronounced in large species than in small because large species are less likely to encounter periods of starvation long enough to threaten survival. The physiological relationships between size and metabolic rate and size and conductance have long been known, but these relationships alone do not provide a complete, energetics explanation of Bergmann’s Rule. Before a complete explanation can be made, it is necessary to specify what an “energetically optimum size” is, that is, exactly what selection is acting to optimize in an energetics model. Secondly, it is necessary to specify some broad relationship between size and foraging efficiency. All of these elements are needed to form an energetics model of optimum size at different temperatures and so to provide an energetics explanation of Bergmann’s Rule. I would like to thank the many people who helped me in writing this paper, among them W. James Erckmann, James Quinn, Thomas Schoener, Gordon Orians, James Wittenberger, James Kenagy, Robert Stevenson, Sievert Rohwer, William Tschumy, Nelson Hairston Jr., Ken Yasukawa, Robert Seyfarth and Margaret McVey. I also thank Margaret Searcy for drawing the figures. REFERENCES G. A. (1977). In Animal Physiology: Principles and Adaptations (M. S. Gordon, ed.), 3rd Edn. New York: Macmillan. BERGER, M. & HART, J. S. (1972). J. Comp. Physiol. 81, 363. BROWN, J. H. & LEE, A. K. (1969).Evolution 23,329. CALDER, W. A. III (1974).In Avian Energetics (R. A. Payntered.).Nuttall Ornithological BARTHOLOMEW,
Club Publication No. 15. CALDER, W. A. III & KING, J. R. (1974).In Avian Biology vol. 4 (D. S. Farner & J. R. King eds). New York: Academic Press. DOWNHOWER, J. R. (1976). Nature 263,558. GREENWALD, L., STONE, W. B. & CADE, T. J. (1967). Comp. Biochem. Physiol. 22,91. HAMILTON, T. H. (1961). Evolution 15,180. HART, J. S. (1971).In ComparativePhysiology of Thermoregulation (G. C. Whittow, ed.). New York: AcademicPress. HART, J. S. & JANSKY, L. (1963). Can. J. Biochem. Physioi. 41, 629. HEMMIGNSEN, A. M. (1960).Reports Steno Memorial Hospital 9, 1.
OPTIMUM
SIZES
AT
DIFFERENT
TEMPERATURES
593
JAMES, F. C. (1968). Am. Zool.8,815. JAMES, F. C. (1970). Ec&gy 51,365. KENDEIGH, S. C. (1969). Auk 86,13. KE~TERSON, E. D. & NOLAN, V. Jr. (1976). Ecology 57,679. KING, J. R. (1974). In Asian Energerics (R. A. Paynter, ed.). Nuttal Ornithological Club Publication No. 15. KONTOGIANNIS, J. E. (1968). Whys. 2001. 41, 54. LACK, D. (1954). The Natural Regulation of Animal Numbers. Oxford: Clarendon Press. LACK, D. (1966). Population Studies of Birds. Oxford: Clarendon Press. LASIEWSKI, R. C. & DAWSON, W. R. (1967). Condor 69,13. MAYR, E. (1956). Eoolution 10, 105. MAYR, E. (1963). Populations, Species, and Evolution. Cambridge, Mass.: Belknap Press. MCMAHON, T. (1973). Science 179, 1201. MCNAB, B. K. (1971). Ecology 52,845. POHL, H. (1969). Fed. Proc., Fed. Am. Sot. exp. Biol. 28, 1059. POWER, D. M. (1971). Sysl. 2001. 18, 363. RENSCH, B. (1936). Arch. Naturg. 5, 317. RENSCH, B. (1960). Evolution Aboue the Species Leuel. New York: Columbia University Press. RICKLEFS, R. E. (1974). In Aoian Energeks (R. A. Paynter, ed.). Nuttall Ornithological Club Publication No. 15. ROSENZWEIG, M. L. (1968). Am. Midl. Narur. 80,299. ROSENZWEIG, M. L., & STERNER, P. (1970). Ecology 51,217. SCHOENER, T. W. (1969). Am. Narur. 103,277. SCHOENER, T. W. (1971). Ann. Reu. Ecol. Sysr. 2, 369. SCHOLANDER, P. F. (1954). Evolution 9, 15. SCHOLANDER, P. F. (1956). Euolurion 10,339. SNOW, D. W. (1954). Evolution 8, 19.