Phenotypic plasticity and the handicap principle

Z theoa Biot (1984) 110, 275-297

Phenotypic Plasticity and the Handicap Principle NADAV N U R t AND OREN HASSON§

Department of Zoology, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel (Received I I April 1983, and in revised form 9 April 1984) Two quantitative models of the handicap principle are presented which incorporate phenotypic plasticity, i.e. flexibility in the expression of a handicapping trait. The first model ("multiplicative model") applies to instances in which greater development of the trait improves male mating success while lowering survival. Two examples treated are antler size in deer and tail length in birds. The second model ("additive model") applies to instances in which an individual performs a display which, though potentially costly, may nevertheless deter an opponent (or predator). Stotting in gazelles and roaring among stags illustrate application of the second model. The conclusions of the models are that optimal development of the handicap will correlate positively with the "condition" (e.g. the nutritional state) of the individual. This allows a female (or predator) to benefit by choosing (or avoiding) a mate (or prey) of better condition on the basis of the size of the handicap alone. The benefit to the choosing individual is discussed. The multiplicative model appears quite robust. Most importantly, in nearly all plausible situations, the model indicates that those individuals displaying the greatest handicap will at the same time survive best, and vice versa. On the other hand, the additive model appears valid only under a narrow range of conditions; therefore caution is advised in applying this model. The models suggest conditions which favor the evolution of phenotypic plasticity and indicate why, in nature, survival and fecundity (e.g. mating success due to intrasexual or intersexual selection) are often positively correlated. 1. Introduction

The tails of peacocks, massive antlers of some deer species, and the gaudy coloration of birds o f paradise are just some of a host o f traits which have perplexed evolutionary biologists, since they a p p e a r to be deleterious to those individuals possessing them (e.g. Selander, 1965). Zahavi (1975, 1977a) suggested that m a n y such traits have evolved and are maintained, t Present address: Biology Department, University of Rochester, Rochester, New York 14627, U.S.A. § Present address: Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, Arizona 85721, U.S.A. 775 0022-5193/84/180275 +23 $03.00/0 O 1984 Academic Press Inc. (London) Ltd.

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not despite their being a "handicap" to their bearers (as Fisher, 1958, had suggested), but because they are a "handicap". Zahavi's "handicap principle" has elicited much controversy (e.g. Davis & O'Donald, 1976, Dawkins, 1976; Maynard Smith, 1976, 1978a, b; Bell, 1978; Eshel, 1978; Halliday, 1978; Harvey & Arnold, 1982; see also Andersson, 1982a; Dominey, 1983), but much of this may reflect misinterpretation of the concept. Confusion evidently remained even following Zahavi's response (1977a) to previous criticisms (cf. Halliday, 1978; Maynard Smith, 1978a). In his response, Zahavi (1977a) introduced what we feel to be an essential ingredient of the handicap principle: phenotypic plasticity. That is, the idea that "the phenotypic manifestation of the handicap is adjusted to correlate to the phenotypic quality of the individual" (Zahavi, 1977a, pp. 603-604). Subsequent investigators have often failed to appreciate the significance of such flexibility of phenotypic expression and its implications for the handicap principle (notable exceptions are Andersson, 1982a; Dominey, 1983; Kodric-Brown & Brown, 1984). In this paper we demonstrate the significance of phenotypic plasticity with regard to the handicap principle and indicate conditions which would favor the evolution of such plasticity. We also propose two different quantitative models which apply the handicap principle. We hope this will clarify the assumptions, limitations and generalizability of the handicap principle. The first model (termed the "multiplicative model") applies to instances in which the actions of sexual selection and natural selection (in the narrow, Darwinian sense (Mayr, 1972)) oppose each other. Antler size in deer (Cervidae) is the first paradigm presented (section 2). Epigamic characters, e.g. extreme tail length in birds, are treated in section 3. We then present a complementary model (termed the "additive model") which applies to a wide range of other situations, not necessarily involving sexual selection. Stotting behavior of gazelles, an example of predator-prey communication, is considered in section 4 and roaring among stags, an example of intraspecific communication, is examined in section 5. 2. Antler Size in Deer: A Multiplicative Model

A "handicap" is, for our purposes, a trait in which greater expression of that trait decreases survival, holding all other factors constant. The horns of rhinoceros beetles, mandibles of scarabid beetles and antlers of deer (Cervidae) are likely to be "handicaps" by our criterion. The antlers of deer represent considerable investment: a new pair, which may weigh 20 kg or more, is grown every year. In growing a new pair, the deer must draw on his stored calcium reserves (Grzimek, 1972). In addition to the nutrient

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investment, large antlers may encumber the animal whilst escaping from predators. Antler size (x) and the annual survival rate (S) are presumably inversely related (Fig. 1). S(x) is depicted, for simplicity, as a linear function, but this is by no means essential; as shown below, convex or concave functions yield the same qualitative conclusions. (b)

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The presumed deleterious effect of antlers on male survival must be counter-balanced by another selective force, otherwise males could be expected to lose their antlers with evolutionary time. Clutton-Brock et aL (1979, 1982) present evidence that fighting ability increases with antler size and that mating success, in turn, depends on fighting ability. Since male mating success (F) depends on competition with other males, we assume F to be proportional to a male's ranking relative to other males in the population; that is, a male's mating success reflects the cumulative distribution function of antler size. If the frequency distribution of antler size is normal, then the mating success function F(x) will be sigmoidal as shown in Fig. 1. As with S(x), F(x) can be of various shapes without affecting the qualitative conclusions of the model. The expected annual reproductive output of an individual will be related to the product of F(x) and S(x), which we term V(x). This can be seen as follows: for an organism breeding once a year (as in the Cervidae), the probability of surviving from the previous year until the time of breeding is S, by definition. A male's expected fecundity if he survives equals Fa, where a = the expected number of offspring produced given that a male has acquired a mate. Thus a male's expected annual reproductive output equals SFa; that is, his output is directly proportional to SF = V, if a can be assumed to be independent of x. Expected annual reproductive output,

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in turn, bears a clear relationship to Darwinian fitness, one which is particularly close where the population is stationary (Schaffer, 1983), as is likely for avian and large-mammal populations. The product S(x)F(x) has been portrayed in Fig. 1. For a diverse array o f F and S functions, V(x) will peak at an intermediate value o f x, which we denote ~, the optimum antler size. That ~ should occur at an intermediate value (i.e. greater than the minimum value of x and less than the maximum value of x) requires only that F(x)-~ 0 when x is sufficiently small and S(x)=O when x is sufficiently large. Since in this model fitness is related to the product of two functions, we refer to this as a multiplicative model, in contrast with an additive model (presented in sections 3 and 4). A population of males will not be phenotypically (or genotypically) homogeneous: males can be expected to differ in their "intrinsic viability". We are concerned with those viability differentials which may be attributed to differences in "condition". For a given antler size x, the probability o f survival o f a male in " p o o r " condition will be lower than a male in average condition; while the survival of a male in " g o o d ' condition will be greater than that of an average male. By "condition" we refer to a general, long-term property o f the organism. For example, those animals with a better nutritional history, who are better foragers, or with a better territory, will likely be in better condition. The condition of a male is actually a continuous variable, but for simplicity we assume only three states. Each phenotypic class, " p o o r " , "average" and " g o o d " , will adhere to its own S(x) curve: So(x), S~(x) and S2(x), respectively (Fig. 2). The phenotypic differences 03 F

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may be thought to arise either purely for non-genetic reasons (case 1) or from a combination of environmental influences and genetic factors which are themselves independent of antler size (case 2). To simplify the model, we assume that mating success, and hence fecundity, depend on antler size, but are independent of the condition of the male except through the effect of condition on antler size itself. As can be seen in Fig. 2, the optimum antler size for each male phenotypic class differs: "poor" males would maximize fitness if possessing the smallest antlers (~o) while "good" males would maximize fitness if they possessed the largest antlers (~2). This would favor the evolution of phenotypic plasticity. Where there is little plasticity in the expression of the genotype, i.e. antler size is canalized, the one best antler size would be the optimum for the average-condition male, ~t. However, a genotype (genotype P) in which antler size is adjusted to the condition of the male, such that £0 for a "poor" male, ~ for an "average" male, and ~: for a "good" male, would be at a selective advantage relative to a non-plastic genotype (genotype N), hence leading to the fixation of the plastic genotype. No other "strategy" could do better than P since by definition the strategy P is: adopt antler size ~ which maximizes fitness for an/-condition male. An important point is that, despite the fact that mating success is determined by antler size, it would never pay for a male of condition i to "cheat", i.e. by growing antlers larger than ~, since ~ is by definition a male's optimum antler size. As a result, antler size is a reliable indicator of condition. This should be of great importance to females: they can reliably choose "better" males by choosing males with large antlers. There are two circumstances under which females could benefit by choosing "better" males, depending on the genetic basis of differences in condition. First, where males provide parental care, even if differences in male condition are purely non-genetic (case 1 above), females would benefit by choosing "better" males (Motro, 1982). We assume that a male in better condition will be able to provide more (or better) parental care. The parental care need not include provisioning food, it may be merely supplying an adequate feeding territory for the mother or defending the mother or young. Second, even in the absence of any parental care, if differences in male condition have a genetic basis (case 2 above), then females can benefit by choosing a male in better condition. As a result their offspring will be in better condition as well, and therefore more viable. This argument assumes that there exists sufficient additive variance for characteristics pertaining to fitness such that "choosy" females would enjoy a selective advantage over non-choosy females (Maynard Smith, 1978a); for an example of a mechanism generating sufficient genetic variance see Hamilton & Zuk (1982). Whether natural selection would favor females

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which prefer more viable males may depend also on the frequency of the preference allele among females (see Kirkpatrick, 1981, for discussion). In Fig. 2, it is evident that So(~) is less than S~(~) which is less than $2(~). That is, better-condition males, despite adopting a larger handicap than their competitors, nevertheless survive better. Is this result specific to the types of functions portrayed in Fig. 2? To investigate this question, we considered three simple types of S-curves, convex, linear and concave, and the same three types of F-curves, and then calculated x2, x~, Xo and $2(x), S~(~) and So(~) for each pairwise combination (Fig. 3). For this analysis, convex and concave functions were represented by parabolic functions, i.e. of the form F=ax2+bx (i.e. F = 0 when x = 0 ) and S=dx2+ex+c~ ( i = 0 ,

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FiG. 3. Nine cases (depicted in (a)-(i) in figure) for the multiplicative model.For each case ((a)-(i)) solid lines depict mating success, F, dashed lines, survival for 0-, I- and 2-condition males, So, S I and $2, respectively. In (a), (b) and (c), F(x) is convex, in (d), (e) and (f), F(x) is linear and in (g), (h) and (i), F(x) is concave. In (a), (d), (g), Si(x) are a family o f convex curves, in (b), (e) and (h), Si(x ) are linear functions and in (c), (f) and (i), Si(x) are concave functions. All concave and convex curves are parabolic (i.e. of the form y = ax 2 + bx + c, a ~ 0). F and S vary from 0 to I, x is measured on an arbitrary scale. So, S~ and S 2 differ by a constant interval, here 0.25. Points indicate ,~i, JDi (i = 0 , l, 2), where Yi = S~(.~). Note that in every case, -Xo< -xl < x2 and Yo < )3t < -v2.

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1, 2). The c; differ among the /-condition males (such that c2 > cl > Co) but a, b, d and e are common to all males. Thus in each case ((a)-(i) in Fig. 3), three parallel S-curves were generated. In Fig. 3, the difference between curves (c2-c~ and Cl-CO) was set at 0.25. Note that, for the sake of computational simplicity, F(x) is no longer being treated as a sigmoid curve (which has both convex and concave components) but rather as a curve that is convex or concave or neither. The results, presented in Fig. 3, are that in every case -~2> ~ > Xo and $2(~)> S l ( ~ ) > So(~). This ordering is of great significance for females: by choosing males with the largest handicaps, females are actually choosing the males with the best survival prospects, provided that males are indeed optimizing the expression of their handicap. Figure 3 demonstrates the generality of the result, that those in the best condition would sport the largest handicap and yet stilVsurvive the best, but not its universality. While we know of no plausible S- or F-curves which refute the above result, less-than-plausible functions which do can be suggested. One example of the latter sort of function, mentioned by Parker (1983) and an anonymous referee, is the negative exponential function, S = e "x (a <0). We consider this function implausible because S drops to zero only where x is infinitely large. In this case a 200 kg deer with 200 kg (or even 400 kg) antlers would have a positive survival probability.

3. Epigamic Characters: A Further Example We have argued that given a character beneficial in intrasexual competition for mates but deleterious for survival, females would benefit by choosing males possessing the greatest development of the character (greatest "handicap"). The extent to which female deer exercise choice is in dispute (Clutton-Brock, 1982), but in other species where males engage in male-male combat, females have been shown to exercise choice (e.g. Cox & Le Boeuf, 1977). We now consider epigamic characters, that is, characters (such as nuptial coloration and long tails among breeding males) which may serve to attract females but are not likely to be useful in direct male-to-male competition. For example, in the guppy Poecilia reticulata, brightly-colored males are the most susceptible to predation (i.e. S decreases with an increase in the degree of conspicuousness, x), yet females prefer these males (i.e. F increases with an increase in x) (Haskins et al., 1961). The existence of female choice itself is not in dispute (see Brown (1975) and Searcy (1982) for references; for a particularly good example, see Andersson (1982b)), but what is controversial is the mechanism by which female preference for deleterious characters has evolved. Several different

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routes can be suggested by which female preference for such a character could arise. Fisher's (1958) scenario, applied to a character such as excessively long tails in males, was as follows: in the past when male tails were short, tail length may have been positively correlated with survival. Females which chose males with somewhat longer tails were favored since their offspring inherited longer tails and thus survived better. Fisher suggested that once female choice for long-tailed males is established in the population, male tails may become so long as to jeopardize male survival. Females nevertheless continue to choose long-tailed males because their sons accrue an advantage through sexual selection. This is but one possible route suggested for the evolution of female preference for deleterious characters. Lande (1981) and Kirkpatrick (1982) discuss the evolution of female choice under conditions where the preferred character is initially deleterious. Both Kirkpatrick and Lande, in their treatments explicitly exclude the possibility of direct selection acting on female choice. Under these circumstances, in which genetic drift and pleiotropy play a large role in the evolution of female choice, preference for a deleterious trait can arise without preference itself ever having been beneficial to the female. We suggest an additional mechanism by which female preference for a deleterious character may become established. Let us assume that mating with larger males is beneficial for females. The potential advantages for the female of mating with a larger-than-average male are manifold; for example: (i) large body size may directly enhance survival (Boag & Grant, 1981; Thornhill, 1983), (ii) large body size may reflect the superior foraging ability and/or ability to compete for food resources among such individuals, and (iii) larger-bodied males may prove superior at competing for mates (e.g. Clutton-Brock et al., 1982; Thornhill, 1983). Females mating with larger than average males would pass on the males' superior traits to their offspring. Whether such a preference would evolve through natural selection depends also on the distribution of female preference in the population (Kirkpatrick, 1981). Females may benefit in a more direct fashion as a result of mating with large males, for example: (iv) copulation with larger-bodied males, who are usually the dominant males, may less likely be interrupted by rivals, and thus mating with large males is more efficient, and (v) larger-bodied males may be able to provide superior parental care or other material benefits (Thornhill, 1983). These direct benefits accrue to females irrespective of the behavior of other females in the population. To estimate body size, however, females may actually rely on criteria such as tail length, because differences in a one-dimensional character may be easier to assess than differences in a three-dimensional character, i.e. volume (A. Zahavi,

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personal communication). Once females begin choosing males on the basis of tail length, this will select for an increase in male tail length. However, any increase in tail length beyond the point where male viability is maximal will prove costly to the male. The balance between cost and benefit will vary depending on the condition of the male as we have discussed above. We consider, finally, a fourth mechanism by which female preference for a deleterious character may become established, using bright coloration as our paradigmatic trait. Brightly-colored males, since they are conspicuous to fertile females, will be more quickly identified as suitable mates. In contrast, a female may have to observe an inconspicuous male for some time to identify him and may even overlook him. Brightly-colored males may thus secure a mate faster and thus secure more mates than dull mates but at the same time will be more vulnerable to predation. This mechanism seems most credible where visibility is poor, e.g. in turbid waters (Haas, 1976) or where vegetation is thick (e.g. woodland, tall grass; see Andersson, 1982b). The multiplicative model outlined above (section 2) may be applied once female preference for long-tailed (or brightly-colored) males has evolved (by whatever mechanism). The optimal tail-length will be proportional to a male's condition (which will be related to his ability to evade predators). At the same time, we expect natural selection to act to reduce the heritability of the deleterious trait by favoring plasticity in the expression of the handicap. As a result, males will develop the "handicap" only to the extent they can afford it. The extent of development of the handicapping trait is controlled by the severity of natural selection. Where S drops steeply with an increase in x, e.g. where predation pressure is intense or where individuals with long tails are at a severe disadvantage in their attempts to capture highly mobile prey, Xo, x~ and x2 will be small. Though the tails of males do not appear extravagant to the human observer and though the differences in expressed tail-length may appear slight, these differences may prove significant to females. A change in the S(x) function, e.g. due to the relaxation of predation pressure, will result in an accompanying shift in the optimal expressions of the trait. During a period of small, successive changes, the change in the expression of the character will be slight and the system will remain at near-equilibrium. Amelioration of the environment (the family of S(x) curves shifting upwards and/or a decrease in the rate at which the S(x) curves drop) will also cause £ to shift toward the right. Only then will the development of trait appear extravagant and only then will most evolutionary biologists suspect that female choice is at work, though in fact, female choice may be ubiquitous (Thornhill, 1983).

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We note that, in amplifying Zahavi ( 1977 a), we have treated the handicap principle as a logical outcome of the Fisherian model, not necessarily a substitute for it (contra Halliday, 1978; Maynard Smith, 1978a). That is, the Fisherian scenario provides initial conditions (which are sufficient though not necessary) so as to bring about the evolution of plasticity in the expression of a handicapping trait coupled with female preference for those individuals with the largest handicap. Dominey (1983) has recently discussed this point in some detail. 4. Stotting in Gazelles: An Additive Model

In this section, we consider the stotting behavior of gazelles (Gazella sp.), which serves as an example of the application of the handicap principle outside the context of sexual selection and at the same time exemplifies a second model, which we term, because of its mathematical form, an additive model. The multiplicative model, in which the cost is expressed as a decrease in annual survival and the benefit expressed as a gain in fecundity, is not appropriate for many behavioral traits which may affect fecundity or survival, but not both. "Stotting" (also termed "spronking", see Walther (1969) for review) involves the jumping of a gazelle in a vertical direction, often while fleeing from a hunting predator. Such jumping is, on the face of it, disadvantageous since the prey is expending valuable time and energy without appreciably increasing the distance between itself and the predator (Estes & Goddard, 1967). If we measure the expression of the behavior by its intensity, x, where an intensity of 0 corresponds to the absence of stotting, then we posit that the intensity of stotting and the probability of the prey successfully evading its pursuer if pursued, P(x), are negatively correlated (Fig. 4). "Intensity" will include several factors such as the height, number of jumps per encounter and number of jumps per unit time. The advantage of stotting for the stotter may be that the gazelle thus deters the pursuit of the predator (Zahavi, in Dawkins, 1976). The gazelle's ability to jump high will be closely related to its ability to outrun the predator. A feeble animal would be incapable of stotting at all or of stotting in the same manner as a normal, healthy animal. We expect predators to be selective in their choice of potential prey and choose those animals least likely to escape (the weak, lame, etc.). Imagine one predator which chooses to pursue one among N available prey. Assume that in the absence of stotting, a predator has little information by which to select prey: the probability of pursuit for each prey is then approximately I/N. If we introduce one stotter into the herd, then the predator will presumably not

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choose the stotter, provided that the stotter has shown himself to be in healthy condition. In this case, the probability of pursuit of the stotter is close to 0 while the probability of pursuit of the other gazelles will now be approximately 1/(N-1). Thus, under the pressure of predator-choice, stotting, when a gazelle is in good shape, will be favored by natural selection, provided that a healthy stotter can nevertheless escape from a predator more easily than a non-healthy non-stotter. Where the majority of the population stots, with average intensity x, predator-selection will favor individuals which stot more intensely than average (intensity x + Ax). Thus the intensity (height, rate of jumping, etc.) will be positively related to the probability of avoiding the initiation of pursuit (Fig. 4). We assume that the probability of deterring pursuit is related to a gazelle's stotting relative to the stotting of the other members of the population, i.e. deterrence is frequency-dependent. The deterrence function, D(x), is related to the cumulative distribution function for stotting intensity, though not identical to it (Fig. 4). In the example shown, D(x) arbitrarily varies from 0.25 to 0.75. That is, no gazelle can achieve total immunity from pursuit in which case

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D = 1.0, merely by stotting at extremely high levels. Stotting acts only to lower the probability of pursuit. It is an essential feature of the model that the maximum value of D is less than l (see below) ; i.e. pursuit is a stochastic not a deterministic event. The Darwinian fitness associated with stotting is clearly dependent on the avoidance of predation. The probability of avoiding predation in an encounter with a predator, V(x), will be equal to the sum of the probability of avoiding predation by outrunning a predator (i.e. the probability of evading the predator if pursued, P(x), times the probability of pursuit, 1 - D ( x ) ) plus the probability of avoiding predation by deterring pursuit (i.e. D(x)). In other words, V(x) = P(x)(1 - D(x)) +D(x). Assuming some genetic basis to stotting, the stotting intensity which maximizes the avoidance of predation, ~, will be favored by the action of natural selection. Consider three types of gazelles, those in good, average and poor condition. Assuming equal intensity of stotting, a poor-condition gazelle will be somewhat less able to evade capture than a good-condition gazelle (Fig. 4). As illustrated in Fig. 4, each type (condition) of gazelle adheres to its own P(x) curve: Po(x), P~(x) and P2(x). To simplify the model we assume that predators only select prey on the basis of their stotting behavior. Hence, one D(x) curve applies to all three gazelle-types. For each gazelle-type, a different stotting intensity will maximize survival: x0, Xl and x2 for poor-, average- and good-condition gazelles. This outcome is similar to the multiplicative model presented above (section 2). The height and rate at which gazelles stot serve as a reliable indicator of the condition (and thus the evasion ability) of the prey. In determining ~i for each /-type gazelle, we assume that there exists a maximal stotting intensity (x maX) for each type gazelle and, in Fig. 4, suggest that this intensity corresponds to the point where P(x) drops to zero. A gazelle which jumps at x r"ax is one which has invested all its effort into stotting and has no reserves left for escape. The conclusions that were derived from the multiplicative model were robust with respect to the shape of F(x) and S(x) (i.e. most linear, concave, convex or sigmoid curves would suffice). This is not the case with the additive model presented here: the evolutionary stability of stotting behavior as we have described it depends critically on the characteristics of the D(x) and P(x) functions. The additive model will be discussed in greater detail elsewhere. For the time being, it is sufficient to note that the set of values for P(x) and D(x) which allow for the evolution of stotting as previously discussed is limited. In the first place, limitations are placed on the maximum values that D(x) can attain. Were D(x) to vary from 0.25 to 0.75 (as in Fig. 4) or from 0.1 to 0.9 (Fig. 5), the model would be tenable, but not if D(x) were to vary from 0 to 1.0.

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If D(x) did reach 1 (i.e. deterrence could be achieved with complete cetainty) then maximal stotting would be favored for some individuals in the population and hence P(x) would be zero; stotters would now be preferred by predators, stotting would be evolutionarily unstable (see below). The second limitation is that the P(x) curve can be neither linear nor concave. Even where convex, there are additional constraints placed on this function. Most importantly, whether the P(x) function is admissible depends on whether -(dP(x)/dx) becomes sufficiently large (in absolute terms) when 0 < x < x max. In other words, only where the change in probability of escape if pursued (with a unit increment in x) is small when x is small (thus favoring stotting over non-stotting) but becomes large when x becomes large (thus favoring less-than-maximal stotting; hence P ( ~ ) > 0), will stotting evolve as we have discussed above. That is, the P(x) curve must be one which plateaus and then drops precipitously, rather than gradually. An example of such a P(x) curve is presented in Fig. 5 (Pa in figure), together with a counter-example, Pb. The Pb curve would lead to an evolutionarily unstable situation. Whereas Pa produces a maximum (I~'o) at a value of ~ < x max such that P~(~) > 0, Pb which only gradually declines as x increases, produces a maximum (r~b) at x m'x. Because Pb(~)=0, predators would be selected to favor stotters over non-stotters, thus the D(x) curve might even be reversed. As a result, stotting would not be maintained in the population. It is apparent, therefore, that to establish the applicability of the additive model of the handicap principle to stotting behavior (or any other behavior) requires precise quantitative knowledge

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of the P(x) and D(x) functions. In the absence of such information, we can only conclude that, at best, the model proposed is plausible; its generality remains to be established. We have asserted that ~o < ~t <.~2, provided that P(x) and D(x) are of the form depicted in Fig. 4. If this is so, will it also be the case that Po(x) < Pl(X)< P2(x)? This inequality is evident in Fig. 4, but we claim that the inequality is of general validity based on the following argument, which assumes predator preference is dynamical and adaptive. If the inequality did not hold, then P0(~)>/P~(~) and/or P~(~)-> P2(x). It would then not pay a predator to select gazelles of poorer-condition (those with low x) in preference to better-condition gazelles; it may even pay predators to prefer "better" gazelles. With a shift in the predator's prey-preference, the deterrence function D(x) is, by definition, altered and hence £ will shift as well. Assuming ~o is set at a particular value (which in the extreme case, when poor-condition individuals do not stot, will be equal to zero), ~t will accordingly decrease (relative to its value before the change in predatorpreference) until the point that P~(£) is greater than Po(x) while at the same time ~ > ~o. (Such a region of x must exist since G(x) lies everywhere above Po(x).) This will reestablish the deterrence function so that it is again monotonically increasing. While stotting has been variously interpreted (see Pitcher, 1979, for references), only rarely has it been seen as a means of deterring pursuit (Zahavi in Dawkins, 1976; Woodland, Jaafar & Knight, 1980). Estes & Goddard (1967) state a commonly held view that stotting is "undoubtedly a warning signal". We feel this is unlikely in view of the observations that stotting spreads like a wave throughout the herd, young gazelles stot most often, and gazelles stot in response to wild dogs and hyenas but not cheetahs and lions (Estes & Goddard, 1967; Walther, 1969; Kruuk, 1972). Were stotting a warning signal to conspecifics, we would expect instead that: having been warned, herd-mates would immediately run with utmost speed, older gazelles would also stot, and gazelles would stot in response to all predators. Reported observations support the interpretation presented above. Once one or more gazelles begin to stot, others in the herd are expected to follow, because predators are expected to select any non-stotting gazelles in the presence of the stotters (see above). Storring will only be expressed where there is some possibility of thereby deterring the predator. Cheetahs never waver after beginning a chase but hyenas do: "Hyenas occasionally 'try out' gazelles by chasing one victim after another" (Kruuk, 1972, p. 189). Wild dogs will also choose their prey after some testing of the animal: "As far as we could tell the prey animal was never singled out until after the pack or at any rate the leader(s) had broken into

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a run" (Estes & Goddard, 1967, p. 58). Hyenas also concentrate on young gazelles (fawns and subadults, Kruuk, 1972, p. 98); thus older gazelles may not stot because they are not at risk. Beyond the deterrence stage, i.e. when the pursuing predator is close to his prey, stotting is never observed (Walther, 1969), as we would expect. It is worthwhile noting that in our view stotting is a behavior directed at a predator, but which involves competition among gazelles. (As such the model parallels the epigamic-character model in which males compete with each other so as to be selected by females.) A lone gazelle would not benefit by stotting if no alternative prey existed which a predator might select instead. Thus we would predict that stotting should depend on the social context of the gazelle: lone gazelles should stot less frequently and/or at a lower intensity than those in a herd. The warning-signal hypothesis would make the same prediction, but not Pitcher's (1979) hypothesis that gazelles stot in order to detect potential ambushes. According to the latter hypothesis, gazelles should stot whenever a hunting hyena or wild dog has been spotted. 5. Further Applications of the Additive Model: Roaring in Red Deer Stags

The additive model developed for stotting behavior can be applied to the evolution of other traits, for example, the roaring of red deer stags, Cervus elaphus (review in Clutton-Brock & Albon, 1979). In brief, the confrontation between two males consists of an approach followed by roaring after which one male may withdraw. If neither male withdraws, a physical struggle (shoving, pushing, etc.) may follow. The situation for the participants parallels that described for stotting: since a physical struggle is potentially dangerous for a stag, a male should avoid fighting males to whom it would likely lose. Roaring ability, for example as measured by duration, amplitude or the inverse of pitch, will likely be positively correlated with the size and development of the chest-cavity and the thoracic muscles and therefore be positively correlated with fighting ability (for evidence, see Clutton-Brock & Albon, 1979). If we measure roaring ability by the number of roars per minute (following Clutton-Brock & Albon, 1979), then we posit a positive correlation between roaring rate (x) and the likelihood of winning a male-male conflict on the basis of roaring alone, that is deterring challengers, D(x). On the other hand, roaring itself is exhausting (CluttonBrock, in Dawkins & Krebs, 1978) and therefore the rate of roaring would be inversely correlated with the probability of winning a physical struggle, should the fight escalate beyond roaring, P(x). In this example, roaring "handicaps" fighting ability, which is related to mating success, rather than survival, of the male. The probability of winning an encounter, V(x), will

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equal the sum of the probability of winning by roaring alone (no struggle ensues), D(x), plus the probability of winning by fighting (i.e. roaring followed by struggling), ( 1 - D ( x ) P ( x ) ; that is, V(x)=D(x) +P(x)(l-D(x)). Consider three types o f stags: those in good-, average- and poor-condition. The outcome of the physical struggles between two stags depends on the condition of the stags (Clutton-Brock et al., 1979) and thus for a given roaring rate, we expect P(x) to be lowest for stags in poor condition and highest for those in good condition. On the other hand, deterrence is based on roaring performance but is not related to fighting ability per se and thus is not likely related to the condition of the stag. For each type (condition) stag, a separate V(x) curve will be generated: Vo(x), Vl(x) and V2(x). We assume that there exists a maximum roaring rate for each type stag and that x~aX< xTaX< xT ax. The results parallel exactly those obtained for stotting in gazelles, if we make the same assumptions regarding the form of D(x) and P(x) as we did in section 4. The optimal roaring rate (~) will be greatest for good-condition males, and lowest for poor-condition males, with average-condition males in between. As a result, plasticity in the expression of roaring will be favored: stags in the best condition will roar the most, while it will not pay poor-condition stags to "cheat" by roaring more than their own optimum. Hence, roaring serves as a reliable indicator of condition (and thus fighting ability). As noted in section 4, deterrence is probabilistic not deterministic. That is, even with a high rate of roaring, males cannot be assured of deterring their opponent: there always exists a finite possibility ( 1 - D(x)) that the opponent will challenge and a fight will ensue. This is the case even when the opponents differ in condition and thus differ in their expressed roaring rates. Where a stag confronts a stag in better condition, the better stag can expect, on average, to win a fight but this is not guaranteed: the " p o o r e r " male may win due to an accident or a moment of carelessness. It might seem that the system is open to "bluffing"; were there no challenges from other males ( D = 1.0), "bluffers" would indeed be expected to benefit and the system would collapse. Instead, as is required by our model, fights among males are not uncommon (Clutton-Brock & Albon, 1979); in such circumstances a "bluffer" would risk grave injury (Clutton-Brock, 1982). An additional factor insuring the reliability of the signal is that stags do not merely rely on roaring but also assess their rivals in other ways, for example, during the parallel march the condition of the body and legs can be observed. Males whose signals do not match are suspect. For example, Rohwer & Rohwer (1978) observed that, among Harris's sparrows

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Zonotrichia guerela, subordinate males whose features were dyed (so as to signal dominant status), were strongly attacked by other males. Our interpretation o f roaring in red deer has much in c o m m o n with Clutton-Brock & Albon (1979), who also interpret roaring as a means of assessing one's rivals. One difference between their approach and ours lies in the answer to the question: What assures the reliability of the criterion of assessment? Clutton-Brock & Albon (1979) in essence assume that roaring is a signal that cannot be faked, i.e., less-strong individuals are not physically capable of roaring intensely. Instead, we argue (as does Zahavi, 1977b) roaring is reliable because it is a signal which incurs a cost. On the one hand, individuals will only incur a cost to the extent that they can "afford" to: natural selection will act against stags which roar to the point where lesser rivals can defeat them. On the other hand, it is in the interest of stags to minimize the cost they incur. This may explain why roaring duels begin at low intensities and escalate only slowly. In this way, a stag may win (or safely back out of) a contest at little cost and thus husband his stamina in case challenged by a third stag. For Clutton-Brock & Albon, the cost associated with roaring is incidental whereas in our formulation it plays a crucial role. The differences between the two approaches can be clarified with reference to Fig. 4. In essence Clutton-Brock & Albon maintain that x max is reached before P(x) begins to drop precipitously (call this presumed x re"x, C). In other words, the region of P(x) beyond C (see Fig. 4) does not exist. They also claim that x mx for e a c h / - t y p e (condition) male correlates strongly with his immediate fighting ability. If this were the case it is apparent that the cost o f roaring is of minor importance in affecting the expressed roaring rate of stags, and thus we would expect stags to always roar at their maximal rates. In contrast our model predicts that ~i will generally be less than x~ ~x and furthermore that ~i will shift in response to any change in the D(x) and P(x) functions. If stags consistently roared at the maximal rate of which they are capable, this would support Clutton-Brock & Albon's interpretation but such does not appear to be the case (Clutton-Brock & A l b o n , 1979). 6. Discussion INDIVIDUAL DIFFERENCES AND PHENOTYPIC PLASTICITY The models presented depend, at minimum, on significant differences among males with respect to "condition", intrinsic viability, and other measures of male "quality", and the potential for plasticity in the expression of male characters. We have assumed significant heterogeneity in the male

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population but evidence for or against this is scanty. Large differences in reproductive success among males have often been reported (e.g., Le Boeuf, 1974), but differences with respect to survival are much harder to document. Boag & Grant (1981) observed large survival differences among phenotypic classes of male and female Geospiza fortis: during a drought larger individuals were more likely to survive. Moreover, difference in size in this species are largely heritable (see references in Boag & Grant, 1981). Rohwer & Rohwer (1978) observed significant intrinsic differences in fighting ability among male Zonotrichia guerela: even when experimentally altered to appear like subordinates, dominant individuals proved themselves superior fighters. Where males contribute in some way to parental care, differences in male "quality" need not be genetic for the models to be applicable. It may be objected that in such cases a female will not prefer a male whose very handicap may diminish his ability to contribute toward care of the young. However, as was argued above (section 2), and as demonstrated in Fig. 3, the survival of "good" males will be greater than "poorer" males, even though the better males possess the greater handicap. Application of this principle in the field may enable us to test for the evolution of phenotypic plasticity: where phenotypic plasticity of a trait is expressed, individuals with the greatest handicap will survive the best. In the absence of adaptive plasticity, we would expect those with the smallest handicap to survive the best. If we consider "clutch size" a trait which handicaps female survival (Nur, 1984), the theoretical result we have obtained may explain why female magpies (Pica pica) with the largest clutches were observed to survive the best (Hrgstedt, 1981). Conversely, it should be noted that phenotypic plasticity may act to obscure the deleterious effects of a handicap in such a way that a true handicap may appear to have beneficial effects. The only reliable alternative is to experimentally manipulate the handicap and/or the condition of individuals. Many "handicapping" characters exhibit some form of plasticity in expression which would lend support to our models. Deer antlers, for example, increase in size each year as the stag enters his prime years, and significantly, the antlers diminish in size with each succeeding year as the stag enters senescence (Grzimek, 1972). On the island of Rhum, increased population size was correlated with a decrease in antler size (Clutton-Brock et al., 1982). The absence of antlers in some males (who are termed "hummels") appears to reflect adverse environmental conditions endured during the period of growth rather than genotypic differences (Lincoln & Fletcher, 1977; cited in Clutton-Brock, 1982). Clutton-Brock (1982) summarizes evidence that male dominance status and antler weight are significantly

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correlated. Though no definitive conclusion is yet possible, the evidence supports the view that dominance status determines the investment in antlers rather than antler size determining the dominance status of the individual (Clutton-Brock, 1982). Eberhard (1982) presents indirect evidence that dimorphism in horn size in dynastine beetles (Dynastidae) represents facultative response to body size (itself related to diet) and not the expression of genotypic differences. In the earwig Fornicula auricularia, the manipulation of nymphal diet is sufficient to determine forcep size in the adult (Diakonov, 1925; Kuhl, 1928). Male dimorphism has been observed in many species (see Gadgil, 1972, for reference), but in general little effort has been made to establish the genetic basis of these differences; in the past the differences were often assumed to be genetically based but closer study is called for. Plumage in birds also evidences plasticity in expression. In many species, first-year adults adopt juvenile (generally cryptic) coloration, though they may be reproductively mature. Only older, perhaps more skilled adult males adopt brilliant and conspicuous plumage. Further study may reveal that individual variation in plumage is widespread in birds. For example, in the blue tit Parus caeruleus, first-year adults vary widely in the extent to which they have molted into mature adult plumage (N. J. Phillips & N. Nur, unpublished). Wing length in birds (which in turn reflects feather length) is similarly plastic in expression. For example, van Balen (1967, p. 56) notes that in the great tit Parus major, "The mean wing length in a population varied strongly from year to year, probably influenced by feeding conditions and possibly by the weather during the moulting season". Breeding coloration in fish provides an excellent example of phenotypic plasticity consistent with our model. In pupfish (Cyprinodon spp.), with the establishment of a territory (which the experimenter could manipulate by introducing suitable substrate), males rapidly adopted intense nuptial coloration (Kodric-Brown, 1978). Male nuptial coloration also depends on body size and the number of aggressive encounters with other males (Kodric-Brown, 1978, and references therein). This is what we would expect given that nuptial coloration increases a fish's predation risk (Haskins et aL, 196 I) while at the same time increasing his mating success (Kodric-Brown, 1983). MODELS

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Two models were presented, the multiplicative model, in which the benefit is expressed in terms of fecundity and the cost in terms of survival, and the additive model, in which the costs and benefits were expressed in terms of

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either fecundity (e.g. roaring in stags) or survival (e.g. stotting in gazelles) but not both. In general, the multiplicative model will be appropriate for morphological characters while the additive model will be appropriate for behaviors, but this need hot be so. For example, fighting among stags is a behavior best modelled multiplicatively, since it involves a fecundity benefit (access to females) and a survival cost (death due to injury). Andersson (1982a) presents a model which is remarkably similar to our multiplicative model (though the two were developed independently). His conclusion, that an individual's optimal handicap size will be positively correlated with its phenotypic "quality", closely parallels ours. Our analysis extends his treatment in two important ways. First, Andersson considered only one possible survival function and two possible mating success functions, thus the generality of his conclusion was not established. We have shown that the qualitative conclusions reached are fairly independent of the exact shapes of the F and S functions (Fig. 3). Second, the crux of Andersson's model is that a handicap will be more costly (in relative and absolute terms) to a poor-condition than to a good-condition individual. This may indeed be the case (as Andersson argues) but we have shown that this assumption is unnecessary. With the linear and parabolic functions used in Fig. 3, either additivity in the effect of the handicap on survival (i.e. parallel S-curves, as shown in that figure), or a condition-by-handicap interaction of the sort postulated by Andersson, would produce the same qualitative results. This is a boon for field workers, as it means application of the multiplicative model does not require demonstrating that poorcondition individuals are m o r e adversely affected by a handicap than others, only that they are not less adversely affected. ADAPTIVE VARIABILITY NOT RANDOM

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Eshel (1978), who investigated a model of a genetically-based handicap, made the point that the handicap requires a "special" relationship between the handicap and "quality", in particular, tight linkage between the handicap and "quality". A random defect won't suffice, as Maynard Smith (1976) has demonstrated. In our models, the linkage is not genetic but is provided through the action of phenotypic plasticity. Phenotypic plasticity itself has a genetic basis, but the expression of the handicap will depend most directly on the environment, not the alleles at one or several loci. It is worth pointing out that by "phenotypic plasticity" we are not referring to random variability in the expression of a genotype, but rather predictable, adaptive variability. The example of a handicap suggested by Halliday (1978), of an individual with a broken leg, would not qualify as a phenotypically plastic handicap.

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7. Conclusion

We have discussed a variety of situations in which the "handicap principle" might be applicable. Beginning with paradigms previously suggested by Zahavi (1975, 1977a, b), we have formulated the applications in terms of explicit (albeit simple) models. Examination of the models reveals that the multiplicative model is relatively robust, but the additive model is less so, since it hinges on a particular quantitative relationship between the handicap and its cost (the decrease in the ability of an individual to evade a predator or defeat a rival). This function, P(x) in our model, has not yet been measured. In other words, the handicap principle, as we have treated it here, appears to be of great general value provided that the trait in question handicaps survival while promoting the ability to acquire mates (i.e. sexual selection and natural selection conflict). Where selection acts only on viability or mating success, caution is necessary in applying the handicap principle; its validity will be limited to specific conditions. Our models indicate why and in what manner the evolution of phenotypic plasticity in the expression of "handicapping" characters is favored. The adaptive significance of phenotypic plasticity in animals has only recently begun to be appreciated (see Eberhard, 1982; Clutton-Brock, 1982; Andersson, 1982a). Until now the discussion of the evolution of"handicapping" characters has been dominated by the view that these characters are genetically fixed: the notion of plasticity has either been dismissed (Halliday, 1978; Maynard Smith, 1976, 1978a) or ignored (e.g. Lande, 1981; Kirkpatrick, 1982; Harvey & Arnold, 1982; but see Hamilton & Zuk, 1982). For some the notion of a phenotypically plastic handicap is not "interesting" (Eshel, 1978). Our models indicate why in nature, where individual differences exist with respect to "condition" (related to social status, territory, nutritional history), those males most successful at defeating their rivals, or most successful at attracting females, will at the same time be the males demonstrating the highest survivorship. We conclude that the handicap principle, as we have elaborated it, deserves to be seriously reconsidered by evolutionary biologists. We hope that this contribution will encourage further study, especially the determination of the dynamics of the evolution of phenotypically plastic handicapping traits. The senior author acknowledges support of a George S. Wise postdoctoral fellowship while writing this paper. While developing these models, the junior author was a student of Amotz Zahavi. The better ideas expressed here we wish to credit to him; we apologize if we have unwittingly failed to properly cite any of his original ideas. We thank T. Caraco, P. Klopfer, R. Michod, S. Rohwer, U. Motto, A. Zahavi

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and an anonymous referee for their comments; however, for any errors in the paper, we take full responsibility.

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