## Abstract

It is well documented for many taxa that the frequency distribution of species’ body sizes is highly skewed towards large size (Blackburn & Gaston 1994a). Species of terrestrial mammals, for example, range more than a million-fold in body mass, from shrews weighing little more than 1 g to African elephants weighing several thousand kilograms. Between those limits, about 75% of terrestrial mammal species have a body mass of less than 1 kg (Blackburn & Gaston 1998). The same pattern is observed in marine and aerial mammals (Blackburn & Gaston 1998), in birds (Blackburn & Gaston 1994b) and in many other taxa (Blackburn & Gaston 1994a; Maurer 1998a). A highly skewed interspecific frequency distribution of body sizes seems a general characteristic of taxa. Of the hypotheses to explain this pattern of skew in body size frequency distributions, a model proposed by Brown, Marquet & Taper (1993) has perhaps received most recent attention (Blackburn & Gaston 1996; Brown, Taper & Marquet 1996; Kozlowski 1996; Chown & Gaston 1997; Jones & Purvis 1997; Maurer 1998a,b; Perrin 1998). The model describes a relation between body size and a quantity called ‘reproductive power’, which is a measure of the power of organisms to acquire resources from the environment and to transform acquired resources into offspring. The reproductive power of an individual is determined by two rates: (i) the rate at which the individual acquires resources from the environment and (ii) the rate at which it transforms those resources into offspring. The model assumes that the magnitudes of those rates scale allometrically, i.e. that the magnitudes relate to body mass as y = cmb where y is a rate, m is body size, and c and b are constants. Under that assumption reproductive power (dW/dt) is related to body mass according to a simple equation: where parameters c0 and b0 describe the rate of energy acquisition in excess of maintenance needs (rate i) and c1 and b1 scale the conversion of energy to reproductive work (rate ii). (For the derivation of equation 1a see Brown et al. 1993; Brown et al. 1996; Perrin 1998.) Coefficients c will be called ‘body mass coefficients’, and exponents b will be called ‘body mass exponents’, or in short ‘coefficients’ and ‘exponents’. Several students have discussed the logic and assumptions of the model. First, Kozlowski (1996) drew attention to the difficulty of using reproductive power as a measure of fitness while evolution in general precludes maximization of reproductive capacity. Then, Blackburn & Gaston (1996) discussed the assumption that there should be a single optimum body size for a higher taxon. Chown & Gaston (1997) discussed difficulties in the estimation of the body mass coefficients and exponents. And, finally, Perrin (1998) identified several difficulties inherent to the model that remained unclear even after Brown et al.’s (1996) reply to Kozlowski (1996). Especially, Perrin’s (1998) demonstration that rate ii (c1mb1) is not mass-specific and that b1 cannot be negative raises significant doubts about the validity of the model. Here, in addition to Kozlowski’s (1996) and Perrin’s (1998) analysis of the model, I show that the model is internally inconsistent, regardless of which parameters are substituted and regardless of what exactly is their meaning. To illustrate this, I elaborate on Brown et al.’s (1993) argument that size frequency distributions of taxa are quantitatively different because of taxon-specific constraints (c0, c1), but are qualitatively nearly similar because of identical body mass exponents (b0, b1). Brown et al. (1993) exemplified this by asserting that the avian order Strigiformes (owls) and family Tyrannidae (New World flycatchers) both exhibit skewed size frequency distributions, but differ in average size due to taxon-specific constraints. (Neither Brown et al. (1993) nor Maurer (1998b) tested this statement.) I investigate that assertion that taxon-specific constraints (i.e. c0, c1) correspond with the differences in the average size of the taxa Strigiformes and Tyrannidae. Failure of the hypothesis to withstand this straightforward test leads me to dispute the suggestion that reproductive power plays an important role in determining the frequency distribution of species body size. The simplest way to test whether reproductive power is the dominant determinant of the shape of the body size frequency distributions of two taxa would be to substitute taxon-specific estimates of c0 and c1 into equation 1a, and to see whether or not the shapes of the body size–reproductive power relations correspond with the shapes of the species body size frequency distributions of the taxa. However, the two fundamental processes in the model, the ‘rate of acquisition of energy from the environment’ and the ‘rate of conversion of energy into offspring’, are poorly defined. Consequently, much debate is possible on the question of what are appropriate relationships to describe the allometric scaling of these rates (Chown & Gaston 1997). Since model predictions strongly depend on the choice of parameters c0 and c1, such uncertainty prevents a solid test of the model (Blackburn & Gaston 1998). Furthermore, comparable, taxon-specific values of c1 are not available. Therefore, I investigate analytically what parameters are required for the model to describe size frequency distributions of different avian orders and families correctly. This more general analytical approach leaves less to be discussed. Given that body size relates to reproductive power as in equation 1a, the maximum reproductive power dW*/dt (with respect to body size) is: This maximum reproductive power is obtained by organisms that have the ‘optimal’ body size, and it can be shown (by setting to zero the derivative of equation 1a with respect to m) that the body size m* that maximizes reproductive power is: Now consider a taxon, for example the order Strigiformes and the higher taxon to which it belongs (the class Aves). Those two taxa have different modal body sizes (Brown et al. 1993; Maurer 1998a). If it is assumed that different taxa share the same body size exponents (which seems plausible considering the origin of allometric exponents; West, Brown & Enquist 1997), then the difference between modal body sizes of taxa must, as is understood from equation 3, be due to differences in the body mass coefficients. Thus, the reproductive power of a bird that belongs to the order Strigiformes as well as to the class Aves is: where c0A and c1A are body size coefficients specific of the class Aves, as well as: where c0S and c1S are body size coefficients specific of the order Strigiformes. Now it follows from equations 4a and 4b: that there is only a single realistic body size for which equations 4a and 4b predict the same reproductive power (Fig. 1). Thus species have different reproductive powers depending in which taxon they are regarded as occurring, or c0 and c1 are the same for all taxa. In either case the model contradicts itself. Prediction of reproductive power from body mass for avian taxa after a model by Brown et al. (1993). The heavy line is the class-specific reproductive power curve. Body mass coefficients c0A = 6·31 W c1A = 0·07 W (from Maurer 1998b) were substituted in equations 7a and 7b. These parameter estimates make maximum reproductive power of an avian taxon equals to the reproductive power of the class Aves at the same body size (the reproductive power curves intercept the class-specific curve at their maximum values). Further incomprehensibility of the model is revealed if it is investigated which values the body size coefficients and exponents have to take in order for the model to fit empirical data. It was argued by Brown et al. (1993) and subsequently by Maurer (1998b) that b0 = 0·75 and b1 = −0·25. (I assume for the moment that those values are realistic, because with those values the model has been shown to yield predictions in line with empirical findings, although Kozlowski (1996) and Perrin (1998) have already made clear that a negative value for b1 is quite incomprehensible.) Then, the relation between reproductive power and body mass for the class Aves becomes (by substitution of b0 = 0·75 and b1 = −0·25 in equation 1a): and reproductive power at optimal body mass is for any taxon: where . Following the requirement expressed in equation 5, that a species can have only one reproductive power (although it has already been shown the model contradicts itself at this point), we can equate equations 1b and 2b from which it follows that: Subsequent substitution of equation 2b in equations 6a and 6b yields: Equations 7a and 7b describe the relation between body size, the body size coefficients of the class Aves and the body size coefficients of any taxon contained within the class Aves (Fig. 1). If empirical estimates of the body size coefficients of the class Aves are substituted in equations 7a and 7b it is understood that the model requires unrealistically large differences in body size coefficients between avian orders (Fig. 2). Substituting c0A with 6·31 W and c1A with 0·07 W (Maurer 1998b) in equation 7 allows calculation of c0 and c1 for avian orders of which a modal body size is known. It can thus be shown that the largest and smallest of avian orders (Struthioniformes & Trochiliformes, Maurer 1998a) should differ about 900-fold in coefficient c0 (modal sizes estimated from Maurer 1998a) and the Tyrannidae and Strigiformes from Brown et al.’s (1993) paper should differ about 6·8-fold in coefficient c1. The relation between optimal body mass m* of an avian taxon and the body mass coefficients c0 and c1 of that taxon, as required by the model by Brown et al. (1993) to correctly predict the taxon’s reproductive power at m*. To obtain the relations, it was assumed that c0A = 6·31 W c1A = 0·07 W (Maurer 1998b) in equations 7a and 7b. It is evident that Struthioniform rate of ‘energy acquisition in excess of maintenance needs’ is not 917 times smaller than Trochiliform acquisition rate. There are large differences in energy requirements between organisms of different sizes, but these result from scaling principles that are expressed in the body mass exponent of allometric equations (West et al. 1997). Between-taxon differences as expressed in scaling coefficients (taxon-specific constraints sensuBrown et al. 1993) are typically small for related taxa. For instance, the metabolic rate of an ostrich is more than three thousand times as large as the metabolic rate of a hummingbird, but the body mass coefficients of the allometric equations for metabolic rate of different avian orders appear to differ less than three-fold (Zar 1968). Likewise, ostrich egg mass is 2800 times the egg mass of a hummingbird, but allometric coefficients relating egg to body mass again seem to differ less than three-fold between orders (Rahn, Paganelli & Ar 1975). The model thus requires unrealistically large differences in body mass coefficients between taxa to predict optimal body masses correctly. The above test of the model dismisses the argument that size–frequency distributions of taxa are quantitatively different because of taxon-specific constraints (c0, c1), while qualitatively nearly similar because of identical body mass exponents (b0, b1) (Brown et al. 1993; Maurer 1998b). It is tempting to think that, instead, differences between modal sizes of taxa are caused by competition-driven body size displacement (Brown et al. 1993; 1996). Kelt (1997) simulated the effect of a selection–competition interaction on assembly of body size in local communities. In his simulations interspecific competition drives species body sizes away from each other while the body size–fitness relation from the model by Brown et al. (1993) pulls body sizes towards a single optimum. The body size composition of simulated communities agreed with features of real North American mammal communities. However, considering Perrin’s (1998) and this paper’s comments on the model by Brown et al. (1993) there is little reason to assume a body size–fitness relation as Kelt (1997) uses in his simulations. Certainly those simulations do not explain size distributions of clades of which the members occur at different trophic levels in diverse communities all over the world. And Kelt’s model would predict body size distributions in lineages containing only species smaller than the optimum body size to be skewed to the left, which they are not (Maurer 1998a). It is concluded that the model by Brown et al. (1993) is logically inconsistent: either reproductive power is not the primary determinant of species richness, or/and body mass is not the principal determinant of reproductive power. That means that the model does not provide an explanation for the shape of interspecific body size frequency distributions of evolutionary lineages. That dismisses the model as such, but certainly not the idea behind it. The search for an ‘exchange rate’ for the biological currencies energy and fitness remains a formidable challenge, that deserves at least as much attention as its applicability to body size distributions has received. Body size frequency distributions offer perhaps not the best, and certainly not the only way to test energetic fitness definitions. As Kozlowski (1996), Perrin (1998) and the present study show it is necessary to search for alternative hypotheses to explain skew in interspecific body size frequency distributions. The lively discussion about Brown et al.’s (1993) energetic definition of fitness should not have concealed that pronounced skew in interspecific body size frequency distributions of taxa inevitably develops in the course of evolution, even if evolution is not biased with respect to body size (Stanley 1973; Maurer, Brown & Rusler 1992; McShea 1994). A general model predicts that unbiased evolution of body size would lead to a log-normal distribution of body sizes in a taxon (Maurer et al. 1992; McShea 1994). There is evidence that some empirically observed body size frequency distributions are skewed even after logarithmic transformation of body sizes (Blackburn & Gaston 1994a, 1998). It is yet unknown if the deviations from log-normality are statistically significant, but the large sample sizes and pronounced skew in for instance mammals (Blackburn & Gaston 1998) certainly suggest so. On the other hand there are statistical analyses that indicate little or no bias of body size on evolution (Nee, Mooers & Harvey 1992; Gittleman & Purvis 1998; Owens, Bennett & Harvey 1999). Nevertheless, much attention has already been paid to guesses why selection might favour small body size, why small species would face less risk of extinction, or why small species would be more prone to speciate. The energetic definition of fitness is one of those. At present however, it would seem more appropriate to investigate whether there is any evidence of biased body size evolution rather than to discuss what the bias is. I thank Brian Maurer for kindly explaining to me some analyses in his 1998b paper, Jan Kozlowski for drawing my attention to Perrin’s 1998 paper, and James H. Brown, Jan Kozlowski, Mikko Mönkkönen and three anonymous reviewers for helpful criticism.

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- Body Mass Exponents
- Body Size Frequency Distributions
- Body Size
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