Acta Biotheoretica 39: 91-106, 1991. © 1991 Kluwer Academic Publishers. Printed in the Netherlands,

SOME REFLECTIONS ON EVOLUTIONARY THEORIES,

WITH A CLASSIFICATION OF FITNESS

Klaus Henle

Zoologisches Institut, Univers i~ t Frankfurt , Siesmayerstr. 70, D-6000 F ra nk fu r t - l l , F.R. Germany

(Received 2-1-1990)

ABSTRACT

Using a classical life history model (the Smith & Fretwell model of the evolution of offspring size), it is demonstrated that even in the presence of overwhelming empirical support, the testability of predictions derived from evolutionary models can give no guarantee that the underlying fitness concept is sound. Non-awareness of this problem may cause considerable justified but avoidable criticism. To help understanding the variable use of fitness in evolutionary models and recognizing potentially problematic areas which need careful consideration, a hierarchical classification of definitions of fitness used in evolutionary models is presented. As a conclusion, it is advocated to use the term fitness more conscientiously than currently often practised and to think more about ways to develop fitness-free evolutionary theories compatible with Darwin's ideas.

KEYWORDS: Smith & Fretwell model; survival of the fittest tautology; fitness definitions; fitness-free evolutionary theories.

1. I N T R O D U C T I O N

My essay intends to contribute to a bet ter understanding of evolution and the model l ing of evolutionary ideas by drawing attention to some often over looked issues which easily can at tract avoidable criticism, by developing a classification of current approaches to the use of fitness in evolutionary models, and by pointing to potent ia l alternatives. To state it explicitly, my essay does not challenge evolutionary theory principally, and it should not be misconstrued as such. I t aims at identifying problems which need careful considerations to avoid controversy and to advance our unders tanding of evolution.

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When Darwin (1859) wrote his famous Origin of Species, he not only inspired an exceptionally broad research field for scientists who tried to confirm or refine his evolutionary ideas, but also attracted considerable criticism ever since. The strongest attacks were and are motivated by the conservative religious background of the opponents, who deny evolution completely. Ruse (1982) presented a very informative introduction to this controversy and adequately defended evolutionary ideas against such attacks using but a few examples of the overwhelming empirical evidence for natural selection. This controversy needs no further comment and will not be addressed in this essay. Another view - that evolutionary theories based on Survival of the fittest as causal explanation are tautological (a concept avoided by Darwin in the first four editions of his Origin) is shared by some scientists who do not deny completely the empirical evidence for natural selection (e.g. Popper, 1974; Peters, 1976; Witten, 1978; Tuljapurkar and Orzack, 1980; Jongeling, 1985; see also references in Endler [1986]). These scientists caution that the fitness concept as presently understood is at the very least inadequate. Such criticism usually is immediately countered by other scientists (e.g. Ferguson, 1976; Caplan, 1977; Castrodeza, 1977; Stebbins, 1977; Stearns and Schmid-Hempel, 1987), and most biologists and some philosophers strongly dismiss all similar accusations (e.g. Williams, 1970; Ruse, 1982).

Is the criticism completely unjustified, as its strong rejection seems to indicate? The rejection generally is presented by pointing to potential problems in the terminology, by adding aspects of evolutionary theories not discussed by the critics, and by citing papers which state that a general consensus on the correct use of fitness in evolutionary models now has been reached among biologists. With the noticeable exception of Endler (1986), the rejections usually lack a discussion of potential problems in the definition of fitness. If it is true that meanwhile a general consensus on the correct fitness definition exists, why does the criticism surfaces time and again? Is it because evolutionary theories are easily misunderstood, or does the criticism raise out of an unease with the contrast of a multitude of different fitness definitions available in the literature with the affirmation of a general consensus by many biologists (e.g., Partridge and Harvey, 1988)? These questions arose in frequent discussions of life history ideas with many biologists and philosophically interested scientists during the last five years and inspired the following essay.

To begin with, the criticism has to be posed more precisely as different meanings can be and have been given to the term tautology (see Caplan, 1977; Castrodeza, 1977; Sober, 1984). The criticism claims that evolutionary theories are circular because fitness is first defined by means of survival and then used to explain differential survival (they survive because they are fit, and they are fit because they survive). Thus one has to ask: Do evolutionary theories exist which do not include survival in the definition of fitness, or can they be developed?

2. THE SMITH & FRETWELL MODEL, AN ILLUSTRATIVE EXAMPLE

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The Smith & Fretwell model is a classic life history theory which makes no reference to survival in the definition of fitness. It is also an excellent example to illustrate a general problem which is frequently overlooked in evolutionary ideas, an avoidable problem which deserves and may attract heavy critique if neglected. For these reasons, it is of interest to discuss the model in some detail.

In their model, Smith & FretweU (1974) tried to define an optimum compromise between producing many small and a few large offspring. In brief, they assumed: (a) As the energy expended on individual offspring is increased, the number of offspring a parent can produce is decreased. (b) The fitness of individual offspring increases with the energy expended on it. (c) The parents' total contribution to future generations should be maximized. They defined parental fitness as the product of offspring number and offspring fitness. Under these assumptions, they developed a graphical and an analytical model which demonstrates how the finite amount of reproductive energy should be distributed into different sizes and numbers of offspring (fig. 1). Brockelman (1975) accepted the Smith & Fretwell model as a fundamental concept of life history theory and used it to analyze conditions and factors which cause the optimal resource allocation per offspring to be high or low. He compiled a fairly extensive literature documenting cases in which higher energy investment in juveniles increased their fitness. Wilbur (1977) noted that several functional relationships between offspring fitness and offspring weight can represent the assumptions of the Smith & Fretwell model and pointed out that the optimal offspring size is determined by the form of the function used. He further argued that a parameter for mortality unrelated to offspring size should be included in the model.

The Smith & FretweU model did not remain without critics. Janzen (1977) and Capinera (1979) both advocated that there is no optimal propagule size but a range of optimal sizes that varies with habitat and they dismissed the validity of the Smith & Fretwell model with regard to plants and insects. Variability in offspring size has been observed in many animal groups and has been viewed as adaptation to variable environments (e.g. Kaplan, 1980; Crump, 1981; Stamp and Lucas, 1983; Brody and Lawlor, 1984; Thompson, 1984). Kaplan and Cooper (1984) documented large between-year variability in the offspring size of amphibians and argued that the idea of an optimal egg size is hard to defend - and consequently, they argued, the Smith & FretweU model must be wrong. They developed their own model to explain between-year variability in offspring number and called their model the adaptive-coin- flipping-principle. They did not try to analyze the Smith & Fretwell model for a potential logical error nor examined which of their assumptions differed from those of Smith & Fretwell and may have caused the different conclusions. McGinley et al. (1987) defended the Smith & Fretwell model against these critics and pointed out that Kaplan & Cooper used a parabolic function between offspring fitness and offspring size while Smith & Fretwell (1974) used a convex function (see fig. 1). They demonstrated that the Smith & Fretwell model can be generalized for use in variable environments and that under the original assumption of a convex relationship between

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0 . 3

0 . 2

0 . 1

0.0

W = O f f s p r i n g F i t n e s s

0 5 10 15 2 0

I = Effort/Offspring y

I

2 5

I y

W = P a r e n t a l F i t n e s s p

1.o

0 . 5

o.0 I y

0 5 10 1 5 2 0 2 5

I = E f f o r t / O f f s p r i n g y

Fig. 1: Fitness set analysis of parental strategy, the curved line in the upper part of the figure is a graphical representation of assumption (b) of the Smith-Fretwell model. The straight lines are adaptive functions of two possible parental types. The intersection involving the adaptive function with the highest slope determines the optimal parental strategy defined as the product of offspring number and offspring fitness. Intersections of other adaptive functions determine the relationship between parental fitness and parental strategy (effort per offspring) plotted in the lower graph. After Smith and Fretwell (1974).

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offspring fitness and offspring size, variation in offspring size can be adaptive only under a limited range of environmental variability. They further presented a large body of literature on empirical studies supporting their arguments.

The original Smith & Fretwell model and the adaptive-coin-flipping-principle could be integrated into a generalized model combining the approaches of Wilbur (1977) and McGinley et al. (1987). The possibility to demonstrate limits to the applicability of two special theories and to unify them to a more general model, and a large body of data supporting a model usually is accepted by many evolutionary biologists as a demonstration of the power and reliability of the model. However, a large body of supporting data distracts from an important, often neglected problem: the support by data is only a necessary but never a sufficient condition for the acceptance of a theory as flawless. Remarkably, it became apparent in my discussions of this issue with other biologists that a surprisingly large number of them had difficulties to accept this principle of logique. Already Gould (1980, 1985) variously illustrated the same problem in his essays on racism, sexism, and evolutionary theories. Anyone still sceptical should keep in mind that given a flawed logic the predictions of the Smith & Fretwell model could be combined with any theory be it based on Darwin's, Lamarck's, Aristotl's, or creationists' ideas of evolution.

Within the Smith & Fretwell model, there are two serious logical problems overlooked not only by those who accepted and extended the model but also by those who rejected it. One of the problems can be identified by examining figure l more in detail. It may be assumed for a moment that the model works and all parents optimized their fitness in the way suggested by the model. It can be shown that this is not an evolutionary stable strategy. Consider the case that one mutant parent invests more resources in each offspring than is optimal. Consequently, it has fewer but fitter offspring. Their offspring grow up and become parents themselves. They are fitter than the offspring of non-mutants but according to the model they are selected against because as mutants they do not optimize their parental fitness. The apparent paradox "they are less fit (read: do not optimize their fitness) because they are fitter" has only one solution within the model: parents have to maximize the fitness of their offspring and cannot optimize their own fitness. (It may be argued against my criticism that I may have mixed two concepts of fitness: viability or adaptedness, and reproductive success. This argument points to another serious problem of the Smith & Fretwell model - the definition of fitness - which will be discussed further below.) A further logical consequence of the functional relationship depicted in figure l is a one-way-direction in offspring size selection which ends only once a single but maximally large offspring is produced. Irreversibility of reproductive life history traits has been suggested only rarely (e.g., Fitch, 1970; Vitt and Price, 1982), but the idea is worth pursuing. The consequence is a continuons reduction in the number of offspring in the evolution of animals, true at least as trend in vertebrates (compare data in Bertin, 1958; Duellman and Trueb, 1986; Fitch, 1970; Klomp, 1970; Eisenberg, 1981).

I admit that I myself would not have noticed the flaw in the Smith & Fretwell model had I not needed the above consequence and been tempted for a while to use it to explain this trend within vertebrates. However, a more serious fallacy of the model precludes its use. The proponents of the model (e.g., Smith and Fretwell, 1974;

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Brockelman, 1975; McGinley et al., 1987) "defined" parental fitness as the product of offspring number and offspring fitness. Defining fitness with fitness is no real definition. - It could be argued that parental and offspring fitness are two separate entities and that thus parental fitness is defined in the model. However, then we require a fitness-free definition of offspring fitness which has not been given for the model. (It should be noted that not the definition of one model parameter by a relation with other model parameters is criticized but the lack of a definition of one parameter: offspring fitness.) - Failure to define a central component of a theory is unacceptable (see below).

Above criticism of the Smith & Fretwell model sometimes met with strong sceptism. The lack of presenting a genetic or demographic discussion of the model has been objected, and the various citations and generalizations of the model in the literature have been taken as a guarantee for its soundness. However, these and any other objections are irrelevant as long as fitness remains undefined. (By the way, the Smith & Fretwell model is neither an explicit genetic nor demographic model.) To say it in the words of Box (1976): "Since all models are wrong, the scientist must be alert to what is importantly wrong. It is inappropriate to be concerned about mice when there are tigers abroad."

The papers dealing with the Smith & Fretwell model cited above all have been published in leading international journals apparently without any of the reviewers or editors noticing the flaws mentioned. I regard this as an indication for the proness of even simple evolutionary models to logical pitfalls and for the ease with which biologists may be distracted from the tedious work of carefully examining the assumptions and limitations of evolutionary models by data wlfich seem to support their already existing ideas. However, it could also be interpreted as a sign of a general lack of critical thought. The case of the Smith & Fretwell model is not an exception. Evolutionary ideas based on flawed racist of sexist prejudices frequently have been advocated even by many eminent scientists in the past (see Gould, 1980, 1985) - and supporting data distracting from the flawed logic have been accumulated for such ideas like for the Smith & Fretwell model - not necessarily intentionally, sometimes probably unconsciously. Neglect of this problem facilitates misuse of evolutionary ideas as in Singapore's marriage policies to battle a feared intellectual deterioration (see essay by Gould, 1985). Such misuse and the easy distraction from a logical analysis of a theory by data comforting one's general ideas and the fact that many biologists do not mind to be called and call themselves Darwinists or Neo- Darwinists (a terminology of theologies but not of other natural sciences - at least I never heard of Einsteil~ts, Galileists, or Bohrists) can easily cast general doubt into the soundness of evolutionary theories. It certai~fly contributcd to the rejection of evolutionary ideas by many of its critics (e.g., see references cited in the introduction and by Endler, 1986). The same facts easily can give the impression of dogmatism and of a lack of self-criticism and offer easy arguments for any opponents to evolutionary science. Such an exposure is unnccessary, and all that contributes to it should be avoided. It implies the necessity of undcrtaking the tedious task of analyzing particularly rigorously all parts of any evolutionary thcory bcfore accepting or refining it. When models are applied or refined, potential limitations and problems should bc named clearly rather than put aside or denied to improve awareness of critical issues

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which is a prerequisite for their solution. The definition of fitness is such a problem which needs particular attention as

it is a central component of evolutionary models. According to Tuomi and Haukioja (1979) and Stearns (1986), fitness is a technical tool to represent some models of natural selection. To this end, Endler (1986) argues that fitness can be used only as a description of natural selection and accepts the criticism of tautology if fitness is used causally. Regarding the definition of fitness, Partridge and Harvey (1988) pretend that a general consensus has been reached among biologists. In contrast, Stearns (1986) believes that different definitions are necessary to capture the entire process of natural selection, and that the different definitions also have to reflect the goals of the specialists which use them. Some of the different fitness concepts or thc confusion around them have been discussed by Stearns (1982) and Endler (1986). Above discussion of the Smith & Fretwell model shows that nevertheless examples still appear in recent literature which use fitness in a criticizable way. Also, the taxonomy of fitness remains still in its infancy (Stearns, 1986). The following classification of the potentially possible use of fitness in evolutionary models may help recognize the consequences of various approaches, to identify those for which criticism may be justified, and to select an appropriate one.

3. A CLASSIFICATION OF THE USE OF FITNESS IN E V O L U T I O N A R Y MODELS

Theoretically, three approaches are possible. Fitness may be used either as a description for natural selection or various parts of it like adaptedness, one may include fitness in an evolutionary model (as a parameter and, potentially, as a causal explanation), or one may avoid it.

I) Fitness as a description of natural selection

Fitness variously has been used as a description of natural selection. Several different concepts of fitness (e.g., adaptedness and population fitness) used in this sense have been published (see Endler, 1986; Krimbas, 1984). Various authors already discussed in detail the advantages and disadvantages of these concepts, and others analyzed the distinction between various concepts (e.g.; Brandon, 1978; Krimbas, 1984; Endler, 1986; Lcwontin, I978; Wallace, 1981). It is beyond the scope of this paper to further analyze the relative merits and problems of all these concepts of fitness, in which fitness is used as a description of natural selection. The interested reader may refer to the cited publications as an introduction. Two general comments, however, should be made. First, according to the view of Dunbar (1982), fitness used as a description for natural selection has no explanatory value; however, Endler (1986) opposed this opinion, and Krimbas (1984) took an intermediate position between the former two authors. Second, no matter which position one prefers to take, one should ask: Why is fitness used at all as a description of natural sclection given the confusion,

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potential misunderstandings, and different concepts it still has in the literature? Would it not be better to use other descriptors - descriptors without any of these disadvantages?

II) Fitness as a parameter of evolutionary models

Fitness has been a central component of Darwin's (1859) ideas of evolution. Almost all biologists who accepted his theory also include fitness in their models of evolution, but in various ways as shown in the following subcategories.

II.1) No definitions and pseudodefinitions

Williams (1970) probably was the first to express the view that fitness is a primitive term, meaning that it cannot be defined within the context of evolutionary theory and must be assumed as a given undefined term to get the theory going. Rosenberg (1983) and Rosenberg and Williams (1986) further advocated this idea. Such models have the same predictive power as a model about one's future wealth and social status, built on one's perspective employer's offer to pay a salary of $100,000.- per month if he refuses to define the currency (after all, he may pay in Karst dollars = old Yugoslav Dinars!)!

Pseudodefinitious have the appearance of a definition but do not really present a definition. The "definition" of parental fitness as the product of offspring number and offspring fitness in the Smith & Fretwell model is an excellent example of a pseudodefinition. (If offspring fitness were defined, parental fitness would be a proper definition, not a pseudodefinition.) Steams' (1976) definition of fitness as "something everyone understands but no one can define precisely" is another pseudodefinition. Pseudodefinitious have to be treated as if no definition was presented, but contain the danger to mislead because they can create the impression of a proper definition. The above discussion of the Smith & Fretwell model shows that pseudodefinitions can easily escape detection.

11.2) Parameter substituted for fitness, and fitness definitions

To substitute one or several parameters for fitness or to regard these parameters as components of fitness is an elegant tool employed in empirical studies. These studies may advance considerably our understanding of natural selection by analyzing how various substitute parameters correlate with differential survival or reproductive success. Cooper (1984) called them derived or indirect measures of fitness and discussed various aspects of them. He pointed out that they may be incomplete measures, and that they assume that everything else is being held constant. These conditions have to be accounted for in the development of evolutionary models.

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They can be presented in a mathematical form: A parameter (p) can be substituted for fitness and maximized instead of fitness if and only if the functional relationship between fitness (w) and p holds:

wl = f(Pt) < w2 = f(P2) for all Pl < P2 (1)

P in eq. (1) may be a composite parameter as in life time reproductive success if defined as the number of offspring reaching sexual maturity, i.e., p = N (l-q), where N is the number of offspring produced and q offspring mortality between birth and maturity.

II.2.A) Definitions not based on demographic parameters

These are indirect or derived fitness measures in the terminology of Cooper (1984). Net energy gain over a certain period of time is one example particularly favoured by physiological ecologists. Dominance status, territory size, mating ability, health, and a plethora of other definitions could be and have been used (see Krimbas, 1984). All such definitions of fitness make sense in an evolutionary context if and only if they influence the demographic parameters b (birth rate) and/or d (death rate). Maximizing such fitness definitions is equivalent to maximizing b under constant d, minimizing d under constant b, or, if b and d are simultaneously influenced, maximizing the Malthusian parameter r (variously the derived parameters ~., O, or m are used instead of r). Else, the fittest individuals will not be those best-represented in future generations. Consequently, in theoretical models such definitions can be used for fitness and maximized if these linear dependencies exist, but then models based on such definitions become equivalent to those directly based on b, d, or r.

II.2.B) Definitions based on demographic parameters

II.2.B.a) Definitions of fitness based only on reproductive parameters

If critics of evolutionary theories argue that all evolutionary models have to be circular, they argue that models falling into this category are impossible. The study by Sitch et al. (1988) on the gall-making wasp Cynips divisa is a counter-example of a successful study in which fitness is defined solely by reproductive parameters (the number of offspring produced by pathenogenetic females - these females carry a full complement of eggs already at their birth). For this and any similar study, the criticism of circularity is certainly never justified. Unfortunately, such models remain at most special cases. Independence of mortality from reproductive parameters is a necessary condition for these models. Given the frequent costs in form of decreased survival associated with reproduction (see reviews by Stearns, 1976, and Shine, 1980) and size-dependent differential survival of offspring (see data cited by Brockelman,

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1974; McGinley, 1987), it is unlikely that such models will have any wide apphcation.

II.2.B.b) Definitions of fitness based on survival

Definitions of fitness falling into this category are the prime target of the critique of circularity. Few would argue that the critique is justified in cases in which fitness is simply defined as survival and afterwards used to explain differential survival. Such clear cases rarely have been published, although one can find some in which a direct substitution of fitness by survival occurs (e.g. Wilbur, 1977). For other definitions, it may not be immediately obvious that they fall into this category. Such definitions have been suggested more frequently (e.g., Slobodkin, 1964; Holling, 1973; Slobodkin and Rapoport, 1974; Cooper, 1984; see also examples in Stearns, 1982 - a list certainly not exhaustive). They have in common that fitness is defined on various measures of persistence or probabilities of not becoming extinct. Persistence and extinction are like the two sides of the same coin, inseparable from one another, and persistence is just another term for survival. Critique of circularity is justified for models based on such definitions if they use fitness to explain differential survival. Willbur's (1977) comment on the smith & Fretwell model cited above is such a case because the model uses fitness as a causal explanation for the selection of offspring size and number. However, if fitness does not appear causally in a theory but as a description of natural selection as in Cooper (1984), criticism is not justified. For a more detailed discussion of this difference, see Endler (1986). A provocative question should follow but apparently never has been asked before: If fitness cannot be used as a cause but merely as a description of natural selection, why is it used at all in models of natural selection given the confusion, potential misunderstandings, and different concepts it still has in the literature? To use Cooper's (1984) preferred definition of fitness Expected Time to Extinction as an example: why not saying genes (or life history traits, etc.) A and B have an expected time to extinction of a and b? Fullstop. I think it is worth considering whether confusion could be avoided in this way.

II.2.B.c) Definitions of fitness based on the Malthusian parameter, r

This is the most frequently used definition of fitness and applied successfully in life history theories and population genetics (e.g., see Stearns, 1976; Charlesworth, 1980; Abugov, 1986). In a recent review in the journal "Science", Partridge and Harvey (1988) regarded this fitness definition as generally accepted (and the only acceptable one) in life history theory. [Note, that life time reproductive success usually is viewed as a direct derivative of r (Steams, 1986) - but this need not be correct (Henle, in prep.).] Advantages and disadvantages of the use of this parameter have been discussed variously, notably by Kempthorne and Pollak (1970), Stearns (1976, 1982), and Charlesworth (1980). The lack of accounting for the diploid genetics of most

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organisms has been regarded as the most serious weakness of models using r (see discussion by Stearns, 1982). However, the same argument could be applied to any evolutionary theory which does not incorporate diploid genetics. By no means is it peculiar to models using r. At least in principle, it would be possible to construct models using r as fitness definition and accounting for diploid genetics although such models may become intractable (Stearns, 1976). Thus the criticism is a question of whether the neglect of diploid genetics, and age-structure, is a too strongly simplifying abstraction of the real world rather than a special criticism against defining fitness with r.

However, to my knowledge two other problems have never been really addressed. Witten (1978) modeled mathematically the fate of clonal populations differing in r under various regimes of perturbances and showed that the populations with the highest r had the highest chance of extinction. Second, if evolution selects for optimal life histories, and if optimal life histories maximize r (Partridge and Harvey, 1988), why is there a propensity towards a reduction in rm~ in the evolution of animals (at least in vertebrates, see above)?

Some further critical points need to be mentioned. The correct use of the parameter r has been debated (Nur, 1987; Murray, 1988). The commonest usage seems to be the one which applies r only to that period of gene spread that is critical for fixation of a mutant or its maintenance as a stable polymorphism (Roughgarden, 1979; Charlesworth, 1980). Terminology may also be confusing: e.g., Stearns (in litt.) states that no evolutionary theorist ever uses r as the parameter of the rate of increase of a population. In contract, Partridge and Harvey (1988) explicitly define r in this way. Given that the first introduction of the Malthusian parameter to biological theories occurred in the modelling of populations, any other use of the parameter should be clearly labelled. Furthermore, defining r as absolute or relative fitness has different consequences (see below). Often, models lack sufficiently precise statements for an unequivocal understanding of the intended use of r.

I I . 2 . B . c . a ) R a s a d e f i n i t i o n o f a b s o l u t e f i t n e s s

Absolute fitness is the absolute contribution to the breeding population by a phenotype or a class of phenotypes while relative fitness describes contribution relative to the contribution of other phenotypes (Endler, 1986). The same definition can also be applied for genes, genotypes, or other units of interest in models on natural selection. To analyze the consequences of defining absolute fitness (w) as r, the relation of r and ~ and the definition of ~ is first needed (compare Pianka, 1988):

w == r = ln~. = In(N,.~/N~ (2)

where N t is the number of entity A (gene or any other level to which one may apply fitness) at time t. From Eq. (2) it follows:

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Nt÷ i = f i X i Nt i=l

(3)

and, if fitness is maximized as usually assumed in evolutionary aheories:

Nt+i > ~n Nt (4)

[Substituting ). for r and maximizing ~. instead of r is justified because eq. (2) fulfils the condition of eq. (1)]. For any J. > 1, which must have been true for all populations at least at some stage in their evolution unless we want to assume that all present day populations existed in at least their present size since the origin of life, we get:

(5) lim XnN, = oo

which is impossible in a limited universe. Thus the auxiliary hypothesis that catastrophies happen regularly during evolution and reduce all populations of all animals and all plants to the same percentage of their population size before the onset of the catastrophy has to be introduced. Or, natural selection does not maximize fitness. Else, ~. (and consequently r) is not acceptable as a definition of absolute fitness. Indeed, the empirical evidence against using r as a definition of absolute fitness is overwhelming. Comparing the reproductive potential of mammals (Eisenberg, 1981) or birds (Klomp, 1970) with those of reptiles (Fitch, 1970), amphibians (Duellman and Trueb, 1986), or fishes (Bertin, 1958) very strongly suggests that, at least in vertebrates, evolution ran exactly in the opposite direction, towards a reduction of rm~. A further logical consequence of accepting r as definition of absolute fitness is that life started with organisms which had a potential population growth at least as low as elephants or whales and evolved towards life forms with a higher and higher potential population growth, a consequence obviously against all evidence.

II.2.B.c.fl) R as a definit ion of relative fitness

To use r as a definition of relative fitness necessitates to compare the fitness of two entities to one another. One cannot examine r for one entity only irrespective of other entities; otherwise one models absolute fitness and runs into the consequences of eq. (5). Therefore, at least two rs have to compared. Relative fitness (w) cannot be defined as r of one entity; but it may be defined as the ratio of the two rs:

w = r / r b

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(6)

where r, and r b apply to the two entities a and b of the relevant level (e.g., gene or a particular life history trait). Defined in this way, the consequences outlined above for absolute fitness are avoided. W can be maximized without any N (number of copies of a gene, number of individuals carrying a life history trait, etc.) growing to infinity, because r b in eq. (6) may become zero. I have not seen any publication on life history theories which explicitly defined relative fitness as in eq. (6) although some may have done it implicitly (e.g., Witten, 1978). Usually, either eq. (2) is presented or it is stated that r is used as definition of fitness without specifying whether absolute or relative fitness is meant. (I assume that in most cases relative fitness is meant.) The use of definition (2) instead of definition (6) is only acceptable with a clear statement that one deals with relative fitness, and if one can provide evidence that the mathematically less cumbersome eq. (2) is a sufficient approximation for eq. (6) for the purpose of a suggested model (which is the case if the population size remains constant).

With an increased awareness of the points outlined above, the presentation of life history models using the Malthusian parameter to define relative fitness could become more precise and less prone to misunderstandings. Then, this use of fitness seems to be particularly promising (see Charlesworth, 1980; Partridge and Harvey, 1988). However, a final potentially problematic issue regarding r as fitness definition has to be addressed. R is a function of b and d. The question arises whether survival is incorporated in the definition through the back door. Those who rejected Peters' (1976) critique of circularity of evolutionary theories (e.g. Ferguson, 1976; Caplan, 1977; Castrodeza, 1977; Stebbins, 1977) avoided this question. The lack of re~erence to this issue in publications of evolutionary models defining relative or absolute fitness with r suggests that evolutionary theorists generally do not regard it as a problem (or has nobody thought about it before?). I regard it as unacceptable, unless one follows Endler (1986) and uses fitness only as a description and not as a cause of natural selection. However, at least in some life history models, fitness appeared as causal explanation (e.g., in the Smith & Fretwell model) and others may be misunderstood in this way as the many criticisms (see literature cited in the introduction and by Endler, 1986) demonstrate. Again, the question may be asked: could not the potential for confusion be diminished if fitness is dispensed of completely?

III.) Evolutionary models without fitness

The vast majority of biologists include fitness as a central component in their evolutionary theories. The suggestion to think about alternatives which dispose of fitness completely, frequently meets with astonished disbelief or even straight rejection. It is argued that exclusion of the term f i t n e s s from evolutionary models would be incompatible with Darwin's (1859) theory and therefore would be impossible, unless Darwin was wrong (Murray, in litt.). This need not be the case

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(Henle, in prep.) . I t is also demonst ra ted by a model published by Charnov (1986) which came to my at tent ion while rewriting an earl ier draft of this essay. In his paper , the word fitness appears exactly zero times. One may criticize his model as too idealistic and too restrictive (which is a great weakness as it requires a constant popula t ion size); however, it certainly offers no potential for confusion - quite in contrast to many models based on fitness.

Charnov apparent ly did not realize the significance of his approach; at least, he did not comment on it. By combining demographic and genetic models, it is possible to avoid fitness, yet remain in good accordance with Darwin's principle ideas. Analyzing various concepts unders tood under the term adaptation, Krimbas (1984) came to a similar conclusion. Using such an approach, one cou ld concentrate on analyzing how the well-defined demographic parameters change under natural selection, whether general pat terns emerge, and why the parameters changed in the way they did instead of relying on fitness as an undefined or ambiguously defined pa rame te r with all its associated pitfalls. To conclude, one ought to use the term fitness far more conscientiously, call certain parameters such as survival chance by their original names instead of equating fitness with them, and start thinking more about developing evolutionary theories along above lines.

ACKNOWLEDGEMENTS

I wish to thank the evolutionary biologists in Canberra, Australia, and Frankfur t and Stuttgart, Germany, for many hours of stimulating discussions. Comments from B. Murray, K. Sandau, S. Stearns, J. Tuomi, and an anonymous reviewer also contr ibuted to a clearer formulation of my ideas. Special thanks are due to R.E. Barwick, J. Caughley, B. Hammer, and D. McCorquodale for repeated discussions and reviews of ear l ier drafts of this essay.

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