The metabolic rate of an animal is maintained through the steady consumption of fuel and oxygen. From the viewpoint of scaling, much attention has been given to the supply of oxygen, and relatively little to the supply of fuel. In this and the next two chapters we shall focus on the gas transport system, especially as it pertains to oxygen.

First we shall consider the effects of scale on gas-exchange organs: lungs and gills. These organs must have a size and diffusion capacity adequately scaled to the need for oxygen. In the next chapter, the role of blood in gas transport will be considered, and because information is available mostly for vertebrates, in particular for mammals, the discussion will be limited to these. Following that we shall examine the circulatory system, which consists of a pump (the heart) and conduits through which blood is pumped.

When we analyze the function of the respiratory system from the viewpoint of scale, we should keep in mind that most of the available information has been obtained for animals at rest. The normal animal, if not asleep, spends much of its time being active and moving about, and the supply system for oxygen must be scaled to meet these extra demands (except during brief periods of non-steady-state conditions, such as a burst of maximal activity).

The regular decrease in specific metabolic rate with increasing body size must somehow be reflected in the metabolic rates of the various organs that make up the whole animal. To extend this reasoning, the observed differences should also be reflected in the metabolic rates of the cells that make up these organs. We could therefore ask if it is fruitful to study the scaling problem from the viewpoint of cell metabolism.

Tissue metabolism and cell size

The peculiar situation is that large and small animals have cells that are roughly of the same size, within an order of magnitude of 10 µm (Teissier, 1939). For example, a microscopist would be hard put to recognize differences between microscopic sections of a horse muscle and a mouse muscle, except that the mitochondrial density is higher in the muscle from the smaller animal.

Because cell size in various animals is much the same, independent of body size, a large organism is not made up of larger cells, but of a larger number of cells of roughly the same size. It could therefore be expected that, as more cells of the same size are added to make up a larger organism, the metabolic rate should increase in proportion to the increased number of cells. As we have seen, this is not so, and we must therefore explore alternatives.

Biological significance and statistical significance

Properly calculated allometric equations (or regression lines) will be accompanied by statistics that give information about significance and confidence limits. Statistics are necessary because we cannot rely on subjective evaluations of whether or not data and numbers are significant.

Allometric equations, y = axb (or corresponding linear-regression lines), have two important numerical terms: the proportionality coefficient a (the intercept at unity) and the exponent b (the slope of the regression line). These two terms have different meanings and can answer different questions. An example will help.

The proportionality coefficient can be used to answer questions such as this: Do marsupials, in general, have lower metabolic rates than eutherian mammals (see p. 64)? The equations for the metabolic rates of these two groups have the same exponent, and we can therefore directly compare the proportionality coefficients, which are lower for marsupials. This tells us that marsupials, in general, have lower metabolic rates than eutherian mammals. The exponent, on the other hand, tells us that the metabolic rate changes with changing body size in the same way in marsupials and eutherian mammals. This suggests that the same principles determine the scaling of metabolic rates in the two groups (although the coefficient told us that the levels of their metabolic rates differ systematically).

The preceding chapters in this book have dealt with structures and functions and how they are related to body size. We have discussed bones and muscles, energy metabolism and oxygen supply, why time has a different meaning for a mouse and an elephant, and so on. We have considered animals that move about, running and jumping, swimming and flying, and how body size affects the energy cost of locomotion. This is important, for real animals do not sit around doing nothing; they spend much of their time moving about and being active. One important fact is evident: Although comparing animals at rest can provide a great deal of information, it is in the active animal that we are apt to find the limits and constraints on the various functions that make up the whole animal.

Wherever we look at the functions of living organisms, we find that size is important and that a change in size has consequences that require appropriate adjustments or changes. Various functions must be appropriately adjusted; they must be modified as dictated by a change in scale.

Some variables remain size-independent; for example, physical and chemical constants cannot be changed. That is, animals must find the best possible solutions within the existing limitations or rules as determined by the realities of the physical world. When animals meet constraints that set limits to further change in scale, discontinuities in design may solve the problem.

Ecologists generally place more importance on science as explanation than science as prediction. In this book, that emphasis was reversed by stressing the predictive and descriptive roles of theory. Nevertheless, one cannot but wonder why the power formula, in general, and the mass exponents of ¾, ¼, and −¼, in particular, are so effective in describing biological phenomena. This chapter addresses such questions by examining some of the explanations proposed for the ¾ law. I do not consider the problem solved and so offer only a review without conclusion.

Two basic components of allometric explanations

Similitude

One of the striking features of allometry is that very different processes show parallel responses to variations in body size. This parallelism is usually referred to as the principle of similitude (Thompson 1961) or similarity(Kleiber 1961). This may be stated in a restricted form: For example, the size relations of biological rates can be described by a power formula in which the exponent of mass is approximately ¾. A wider formulation could be used instead: Over the longer term, the ratio of physiological processes is constant. The second formulation is the more testable for it applies wherever a range of rates are encountered. The first is, however, a more accurate description of the empiricisms on which the principle is founded. Either statement is empirically based and should be viewed as a general theory.

The body of this book is broadly restricted to an allometric analysis of the balanced growth equation and its extensions. This equation provides a convenient framework upon which many diverse relations may be organized. However, the validity of the individual relations is not dependent on the concept of energetic balance, and not all allometric relations can be arranged in that context. For example, larger animals are less sensitive to high-frequency sounds (Heffner & Heffner 1980). I am prepared to believe that this has ecological implications and should be mentioned but I see no place for it in the balanced growth equation. The interest an equation holds is in no way diminished (and is arguably increased), because it bears little relation to the central theme of the balanced growth equation.

This chapter briefly examines the role of body size in three areas: animal behavior, ecological economics, and evolution. In general, these areas rarely use body size as an independent variable in interspecific comparisons, and a consideration of these topics carries me further from my area of expertise. Consequently, the discussions below are briefer and more speculative than many earlier arguments. I hope points raised here will be sufficiently interesting that more capable workers will pursue them, if only to offer falsifications.

Animal behavior

All patterns of animal behavior exist within the range of possibilities defined by physiology and ecology. Since these scale with size, the range of behavior must also do so.

One of the central tenets in Thomas Kuhn's (1970) famous book on scientific growth and revolution is that normal, vigorous fields of science present many unsolved, but soluble, problems and questions. Pursuit of this premise leads to the paradox that if a scientific monograph is to be a success, it must also be a failure. When all holes and doubts are filled, when the approach has been prodded and viewed from all angles, when it has been pulled apart and put together in every possible way, when the definitive review has been written, then the field is closed to future investigation and withers as a scientific pursuit. The information represented by the science may still be used by technologists in various applications, but scientists show little interest. They have moved on to the next challenge.

This book does not endanger allometry as a scientific field. Instead the book should be a sufficiently successful failure that it encourages others to study and advance our knowledge of biological scaling. There is certainly room for improvement. Even as a review, this book is incomplete because it ignores the rich Soviet literature. Other gaps in our knowledge are indicated throughout the text. These should appear as interesting opportunities for further research.

Although testing proposed relations is an essential part of science, surprisingly few tests are encountered in allometry. This is a major flaw.

Only a small number of numerical skills are required to deal with body size relationships, and most of these are straightforward. They include a basic understanding of simple algebra, an ability to manipulate numbers expressed as powers and logarithms, a grasp of the implications of power formulas like Equation 1.1, and an appreciation of the strengths and weaknesses of regression analysis. This chapter is intended as a crutch for those who are nervous with the algebra of body size. Many readers will find this chapter a tedious exercise in the obvious and should pass over it.

Basic tools

Logarithms

Most analyses of body size relations begin by converting or “transforming” observed values to their logarithms. Logarithmic transformation is a simple device to ease and improve diagrammatic and statistical descriptions of the effect of body size on other attributes. This primer, therefore, begins by recalling the basic characteristics of logarithms.

Like any numbers, logarithms can be added, subtracted, multiplied, and divided, provided all logarithms in the calculation are converted to the same base. However, addition, subtraction, multiplication, and division performed with logarithms do not correspond to the same operations in “normal” arithmetic based on the antilogarithms. Because logarithms represent the power to which some base must be raised, a change of one logarithmic unit corresponds to an order of magnitude change in the antilogarithm.

Ingestion rate is the largest term in the balanced growth equation and sets an upper limit to all other variables. Allometric relations describing the components of the energy budget can seem reasonable and coherent only if ingestion is greater than any other single rate and equal to the sum of all other rates. Such checks on coherence are particularly important, since all allometric relations have a large residual variation and always contain unevaluated sources of error.

Ingestion rate is also the basis for any calculation of the efficiency with which an animal converts food to new tissue or metabolic power. Such efficiencies, calculated as the ratios of the energy used in growth or respiration to that eaten, normalize differences in absolute rates among animals of different size or metabolic grouping. In this and in subsequent chapters, energetic efficiencies highlight important differences and similarities within the animal kingdom. For example, such calculations will show if homeothermy has resulted in more efficient use of food for growth or if this strategy has simply increased heat loss through respiration.

Ingestion is far more than a tool for ecological bookkeepers. It also represents predation, certainly the most apparent and probably the most significant interaction between an animal and its community. As a measure of a predator's demands, ingestion rate estimates the impact of a given animal on its ecosystem.

Introduction

The allometric approach is often unsatisfying, because it draws far more from physiology than ecology. Thus, bare scaling relationships (Table 12.1) suggest that all animals of the same size will behave in the same way, despite differences in their biotic and physical environments. Clearly, this is not the case; but empirical relations rarely exist that would permit descriptions of the habitat's modifications of allometric equations. Just as we cannot tell what effect a particular environment will have on a given organism, we also have no way of predicting what effects the organism will have on its ecosystem.

Simulation models address this problem, because the models permit the investigator a larger creative role than do empirical relations alone. If a necessary piece of information is unknown, the modeler can supply a “reasonable assumption” (a good guess) instead. Because such models treat whole, if hypothetical, ecosystems, they allow a further extension of body size relations to include the effect of size on community level processes, like succession and material flow. Most of this chapter discusses the implications of putting a single allometric organism or population into the biotic environment provided by a community of such organisms. This is a simple demonstration of the role of imagination in extending and connecting empirical relationships.

For most animals, body size is that universal characteristic that is most easily measured. As a result, many researchers include estimates of size in descriptions of their experimental animal, and empirical theories can be built from published data. Other variables, for example, protein content or metabolic rate, may be equally universal and, perhaps, even better predictors. However, because their determination is more difficult, they are far less frequently reported. Body size is also attractive because, as a continuous variable, it may be more easily treated mathematically and the power curve's repeated success makes the researcher's choice of statistical models clear. In part, body size is a good independent variable, because it is practical and convenient.

Only body temperature is as practical and convenient an independent variable as body size. Every organism has some body temperature, and, as a rule, this is easily measured. Long experience has shown that body temperature influences physiological rates, and temperature is, therefore, reported frequently. Like body mass, temperature is a continuous variable and is, therefore, amenable to regression analysis. Although no one mathematical function dominates the description of thermal response, the choice of statistical models is small (Bottrell 1975; McLaren 1963), and so statistical effort is reduced. Unlike body mass, there is a strong physicochemical rationale for the effect of temperature on chemical and biochemical rates.

Introduction

The most general allometric equations describe metabolic rate as a linear function of body mass raised to an exponent of approximately ¾ (Hemmingsen 1960; Appendix III). This regularity holds a special fascination for biologists, because its wide applicability suggests that this may be a rare example of a general biological law (Wilkie 1977). Before this claim is accepted, it should be examined as closely as possible.

One could seek confirmation by examining those allometric equations that relate metabolic rate to body size in particular taxa. The equations assembled in Appendix III suggest that the generality holds at these more restricted levels. The mode of the frequency distribution of the slopes (Figure 4.1) lies between 0.725 and 0.750; the median of the distribution is 0.735 and the mean is 0.738 (SD = 0.11; N = 146). There is no great advantage in promoting 0.74 over 0.75, for the two values are not significantly different and the latter is widely accepted and slightly easier to compute (Kleiber 1961). Figure 4.1, therefore, confirms the ¾ law or Kleiber's rule as a valid statistical generalization. This does not imply that ¾ is the “true” value of the slope for all equations, only that ¾ is a reasonable approximation.

Figure 4.1 may overestimate the dispersion in the slopes, because the slope in body size relations is not completely independent of the intercept.

Production is the last component of the balanced growth equation that we can examine in depth and probably the most important. Production determines the amount of exploitation by man or natural predators that a population can withstand, the capacity of a population to recover from depredation, and its ability to resist control. Conversely, production defines the role of an animal population as a continuing resource for other members of its community, and, therefore, it determines much of the population's role in directing mass and energy to other parts of the community. Finally, the relationships of production to ingestion and respiration describe efficiencies of resource use around which animal communities must be organized and upon which human utilization of both wild and domestic stocks depends (Ames 1980).

The primary goal of this chapter is an equation or set of equations that predicts average rates of total production from animal size. These equations are essential for the allometric definition of the balanced growth equation, which is an underlying theme of this book, and for further analysis of the implications of body size in animal ecology. However, relations that predict average individual rates of production are rare and often imprecise. Were this chapter to limit itself to predicting individual production rates, it would seriously underestimate our knowledge of the scaling of the production process. Body size–production relations for use in the balanced growth equation will be treated at the end of the chapter.

This book is an exercise in predictive ecology. It draws together a widely dispersed body of empirical relations that relate biological form and process to body size. These relationships are then applied to ecological problems; that is to say, to problems whose solutions require some knowledge about temporal and spatial patterns in the characteristics of organisms and in their abundance.

Although the book is largely based on the work of autecologists and environmental physiologists and is applied principally to the interests of ecologists, it should appeal to anyone interested in the development of a broad, quantitative science of organism function. The book was written for senior undergraduates, who should have some feeling for the goals and jargon of ecology. Nevertheless, the approach and information are original enough that many established scientists will find it interesting too. I hope the book is sufficiently clear that it is also open to the educated (if perseverant) layman.

In writing this book, I have used a simple design. I begin with a sketch of one typical body size relationship (Chapter 1) to show why the subject of this book deserves better recognition and wider use. In brief, these body size relations represent scientific theories and hypotheses, which include many examples of general, quantitative biological laws.