The Handicap Principle in Sexual Selection
Graham Bell
Evolution, Vol. 32, No. 4. (Dec., 1978), pp. 872-885.
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http://www.jstor.org Tue Feb 26 12:52:33 2008 Evolution, 32(4), 1978, pp. 872-885
THE HANDICAP PRINCIPLE IN SEXUAL SELECTION
GRAHAMBELL Department of Biology, LM~GillUniversity, 1205 McGregor Avenue, LMontreal, Quebec, Canada H3A IBI
Received September 13, 19i7 Revised Januarq 10, 19i8 The theory of the selection of epigamic Fisher effect. Because Zahavi's idea color, structure and behavior was origi- would be of considerable interest if it nally due to Darwin (1874) and was later could be shown to be feasible, the purpose elaborated by Fisher (1930) and by of the present paper is a more extensive O'Donald (1962, 1963, 1967). It is well account of the behavior of models which known and is discussed below. An alter- incorporate the handicap principle. First, native account of this type of sexual selec- conditions that the handicap principle tion has been suggested recently by Za- should be effective during the first gener- havi (1975). The kernel of his idea is as ation of selection are obtained analytical- follows. Suppose that in a certain species ly. Secondly, the results of this analysis of animal the females are cryptically col- are compared with the results of computer ored, while the males may be either cryp- simulation tic or conspicuous; the argument will ap- ply as well to structure or behavior as to coloration. Conspicuous males are as- The analysis refers to autosomal loci in sumed to survive less well than cryptic an infinite haploid population with dis- males: if this were not so, natural selection crete generations. At the first ("A") locus, would suffice to explain the evolution of male bearers of the A gene are conspicu- conspicuousness. Conspicuous males are ous, whilst males bearing the a gene are thus the less fit, at least in terms of sur- cryptic; all females are cryptic, whatever vival. But those conspicuous males which their genotype. At the second ("B") locus, survive to reproduce despite this addition- individuals bearing the B gene are the al mortality must be, in other respects, more fit, in some unspecified manner not extraordinarily fit. This fitness "in other connected with conspicuousness, whilst respects" will be inherited without preju- those bearing the b gene are the less fit. dice by their daughters and may be suf- The frequency of the A gene is p, whilst ficient to counterbalance the lessened fit- the frequency of the B gene is q. ness of their male offspring. If this It is assumed that the requisite variation happens, then females will tend to evolve in female mating preference is generated a preference to mate with conspicuous at a third ("C") locus, where 'choosy' fe- males, since those females which produce males bearing the c gene, prefer to mate the most fit offspring will be favorably se- with conspicuous males, whilst 'undis- lected. The attraction of Zahavi's idea is criminating' females, bearing the c gene, its internal consistency and in particular display no preference between cryptic and the neatness with which the principal dif- conspicuous males. The frequency of the C ficulty of sexual selection is revealed to be gene isr. However, the nature of this varia- its motive force. tion is not made explicit in the analysis However. it is bv no means self-evident where it is simply assumed that some ge- that selectidn can act in this way, and the netic variance for mating preference exists. proposed mechanism has been severely The axiom of the analysis is that if females criticised by Maynard Smith (1976), who of given genotype produce fitter offspring was unable to find any combination of fit- by mating with males of given genotype, nesses and gene frequencies for which the then this type of mating will be favored by handicap principle would produce effec- selection; and that if the products of this tive sexual selection in the absence of the mating are fitter than the products of any SEXCAL SELECTION 873
other type of mating, the two genotypes clusively on selection acting at the A locus, involved will increase in frequency. as the result of which they will be elimi- Because the expression 'the A gene' is nated from the population because of their sex-limited, the table of fitnesses will be lower fitness. as follows: However, if gene effects are not multi- Males: Females: plicative, this conclusion may not hold. If A (con- a (all a, /3 and E are additive gene effects, the table of fitnesses is as follows: spicuous) -- (cryptic) cryptic) Males: Females: B (more fit) WAB W aB W aB A (con- a (all b (less fit) wAb Wah W(LD spicuous) --(cryptic) cryptic) The frequency of the B gene after selection B(morefit) a+~a+p+~~+P+E in cryptic and in conspicuous males re- b (less fit) a a+P a+P spectively is then: where again 0 s a,/3,~c 1, and in addi- frequency of B gene in a males: tion 0 s a + /3 + E c 1. Since the ef- qa 4 WaBi(qwaB + - q)~nh) fects on fitness are additive, WABW,,~f frequency of B gene in A males: wAhwrlBunless P = 0 or E = 0 or both. q~ = ~WAB/(P.WAB+ (1 - q)~~h) Conspicuous and cryptic males therefore differ with respect to the frequency of the Thus, qA = q, if W.4BWab = WAbWa~;that is, if the genes at the A and B loci have B gene after selection, provided that nei- multiplicative effects on fitness. For ex- ther allele at the B locus is fixed: ample, the genotype fitnesses might be as frequency of B gene in a males: follows: qr, = q(a + P + + P + q4 Males: Females: frequency of B gene in A males: I (1) A (con- a (all 44 = qia + + q4 i spicuous) (cryptic) cryptic) It is easily verified that q~ > q,, if P > 0 and E > 0 and vice versa. Maynard Smith B (more fit) cup a a (1976) uses a scheme in which the fitnesses b (less fit) ~PE CYE a~ of genotypes amongst the males are: Here, a is a 'baseline' fitness; P is the sex- WAB =(I - t); Wflh = (1 - s); WaB = 1; specific effect on fitness at the A locus; E WA~- (1 - s)(1 - ut). This scheme is is the locus-specific effect on fitness at the strictly multiplicative for u = 1, and cor- B locus. Since fitnesses are intended as responds to the additive scheme described rates of survival to reproductive age, we above for u = l/(l - s); between these have 0 a,p,t 1. The effects on fitness two points the fitnesses described are in- at the two loci are multiplicative, so that termediate in character between the mul- WAHW,~ = WA~W,B = a2Pe; hence q, = tiplicative and additive schemes. qa = aq/(aq + (1 - q)a~).Because the Using the additive model of fitnesses, frequency of the B gene after selection is we may now compute the expected mean the same in cryptic and in conspicuous' fitness of the offspring produced by the males, selection can make no distinction four possible matings with respect to the between genotypes at the A locus by acting B locus in females and the A locus in through genes at the B locus. Thus, the males. For example, the frequencies and fate of conspicuous males will depend ex- fitnesses of the zygote genotypes produced by the mating (B 9 x a 8)are as follows:
Zygote Frequency male fitness ---female fitness
AB (1 f qa)Pi4 a+€ a+P+E Ab (1 - qa)Pi4 a a+P aB (1 f qa)(2 - P)i4 a+p+~ ~+P+E ab (1 - qaI(2 - P)i4 a+P a+P 874 GRAHAM BELL
The mean fitness of the male offspring This pair of inequalities is easily shown to produced by this mating is therefore: reduce to the single sufficient condition:
This condition can be readily understood. The matings of A males produce the fittest offspring if the positive effect on fitness at the B locus (measured by E), weighted by the excess of B genes amongst A males (measured by q.4 - q,), should exceed the Similarly, the mean fitness of the female negative effect at the A locus (measured offspring is: by p). A factor of 2 appears because the expression of the A gene is limited to one of the two sexes. Because (qA - q,) will usually be a small number, the handicap Assuming an equal primary sex ratio, the principle will be effective only if E >> P; mean fitness of all offspring produced by that is, if selection is nearly truncate, the this mating will be: effect on fitness of the A gene being much smaller than that of the b gene. It is impracticable to carry the analyti- cal argument beyond this point, since un- der the non-random mating implied by the preferece of C females for A males the Performing the same calculation for the genes at the three loci will evolve into other three possible matings, we can write linkage disequilibrium, so that the as- down the mean fitness of the offspring sumption that a fraction q, of either choosy produced by a given mating: or of undiscriminating females bear the B gene will be invalidated. With linkage (B 9 X a 8): zi,,,, = (4a + (4 - p)P disequilibrium, the conditions for the + 2(1 + q,)e)/4 spread of the A gene are quite different. If only the coupling genotypes BC and bc (B 9 x A 6):Go,,= (4a + (3 - p)p are present, then the offspring of A males + 2(1 f q,)~)/4 will be the more fit if > assum- (b 9 x ad): = (4a + (4 - p)P ing that C females mate only with A + 2q,~)/4 males, and this is equivalent to 2~(1+ (b 9 x A 8): Go,,= (4a + (3 - p)P q.4 - qn) > P Since q, > q,, this is al- + 2q,~)/4 (2) ways satisfied if 26 > p, a very lax con- dition. However, if only the repulsion ge- A sufficient condition that B females notypes bC and Bc are then
fective during the first generation of selec- among females after selection, and pi is tion, given an initial population in linkage the frequency of the A gene among males equilibrium. The generation of linkage after selection. These rules guarantee that disequilibrium by the female mating pref- there is no automatic mating advantage; erence makes it necessary to use numerical an increase in the frequency of the C gene simulation on a com~uterin order to fol- does not of itself tend to cause an increase low the course of selection in subsequent in the frequency of the A gene. Fisherian generations. sexual selection cannot operate in a system Programme design.-There are two sets of this sort, since all males are eventually of input variables. The first comprises the mated to a single female, given that the initial frequencies of the A, B and C sex ratio is equal. No advantage can ac- genes, which I shall call the initial fre- crue to conspicuous males, other than quency set, IfS = (p,, q,, r,); unless through the increased mean fitness of their otherwise stated, this was IfS = (0.001, offspring, and therefore any increase in 0.001, 0.001). The second comprises the the frequency of conspicuous males can be three elements of the additive fitness unequivocally attributed to the handicap scheme described above, which I shall call principle. the fitness set, FS = {a,P, €1. The re- The polygamy mating rules permit only maining determinant of gene frequency is three of the four possible matings. Choosy the set of mating rules, which by describ- females mate only with conspicuous ing the preference of C females for A males, whilst undiscriminating females males governs the weight which is to be mate with cryptic and with conspicuous given to each of the four possible matings males in proportion to their frequencies in with respect to the C locus in females and the population. The four possible matings the A locus in males. Two different sets thus receive the weights: of rules were used in the simulations, the (C 9 x A 6) has weight r' first describing a monogamous and the (Coxad) 0 second a polygamous mating system. The (C P x A d) ~'(1- r') monogamy rules are as follows. Choosy (C 9 x ad) (1 - #')(I - r') females mate with conspicuous males, one to each male, so long as any remain. If The polygamy rules were used by May- there are more choosy females than there nard Smith (1976) in his critique of the are conspicuous males, the remaining handicap principle, and by O'Donald choosy females mate with cryptic males. (1962, 1963, 1967) in his extensive analyt- Undiscriminating females mate at random ical and numerical investigations of Fish- among those males remaining when the erian sexual selection. Conspicuous males choosy females have made their choice. have a mating advantage over cryptic The products of the four possible matings males because they are guaranteed all the are thus weighted as follows: matings of choosy females, while at the same time receiving a proportion of the matings of undiscriminating females; any ifr' > p' then (C 9 x Ad)has weightp' increase in the frequency of the C gene (C9 xa6) (rl-p') will thereby automatically cause an in- (c9xAd) 0 crease in the frequency of the A gene. An (C 9 x a d) (1 - r') increase in the frequency of conspicuous ifr' constructed in linkage equilibrium. The greatest observed value of D,,. not ex- values of r' and p' are found by operating ceeding lop4. If we put /3 < 0, then qA < on the zygote genotype frequencies with q, and DB,. < 0, indicating the expected survival rates (the 'fitnesses' in FS). For excess of repulsion genotypes. The failure each of the four possible matings, the gene to find any region of parameter space in frequencies after selection at the three loci which p, > p,, is at first sight disturbing, amongst the parents are computed, and since the analysis clearly indicates that are used to generate offspring genotypes such a region should exist. This result, in linkage equilibrium. The frequencies of however, follows from the design of the these offspring genotypes are then weight- program. ed according to the mating rules adopted The inequality (3) is not equivalent to for the simulation, using the values of r' a condition that the initial frequency of the and p' obtained previously, and summed A gene should be exceeded in the first or for all four possible matings. This proce- in any subsequent generation of selection, dure results in the calculation of genotype under the conditions of the simulation. We frequencies in the next zygote generation; begin the process in generation j = 0 with these are used as a source of output data, a population of zygotes in linkage equilib- and recycled to compute genotype fre- rium. Natural selection then acts on these quencies in the following generation. In zygotes through differential survival, as this manner, the genotype frequencies can the result of which the frequency of the A be followed for as many generations as is gene must decrease if conspicuous males desired. A listing of the program is avail- run some additional risk of mortality. able from the author on request; aside Since the A gene is under-represented in from the monogamy mating rules, it is the adults, it will also be under-represent- identical with that used by Maynard ed in the next zygote generation (j = I), Smith (1976). assuming as we have that the C gene con- Results.-I shall describe first the re- fers no automatic mating advantage. sults of simulations governed by the mo- Therefore the frequency of the A gene nogamy mating rules, by which means among zygotes will always decline during Fisherian sexual selection is excluded. If the first generation of selection, so that the simulation is initiated with IfS = {q,; p, < Po. However, if condition (3) is sat- po = ro) and FS = {a > 0, P > 0, E > isfied, the zygotes from the matings of A O), then after the first generation of selec- males will be on average more fit than tion we invariably find that: (i) p, < po; those from the matinns of a males, and (ii) q, > qo; (iii) rl = ro; (iv) Dnc. = the A gene will now ;end to increase in (PB,.Pb,. - Pn,.Pb()> 0. No selection has frequency. Since it is now over-represent- yet been applied at the C locus, where the ed in the adult population, it will also be alleles therefore do not change in frequen- over-represented in the next zygote gen- cy. The B gene is unconditionally favored eration (j = 2), so that p, > p,. This ar- and therefore increases in frequency. The gument suggests that, provided condition linkage disequilibrium parameter DBc, (3) is satisfied, a population which is ini- which measures the excess or deficiency of tially in linkage equilibrium will experi- coupling genotypes at the B and C loci, ence a decrease in the frequency of the A becomes positive for the following reason. gene between the 0th and 1st zygote gen- C females tend to mate with A males, erations under natural selection and an in- while the B allele is more frequent crease between the 1st and 2nd zygote amongst the A individuals than among the generations under sexual selection; if the male population as a whole, if q, > q,. simulation were to begin with a popula- In this manner, B and C tend to become tion of adults, then the A gene could in- associated, so that the population contains crease between the 0th and 1st zygote gen- an excess of coupling genotypes at these erations. However, this process will be two loci. The effect is a small one, the affected by the generation of linkage dis- SEXUAL SELECTION equilibrium and by the further constraints on fitness predicated by the mating rules, and must therefore be investigated through simulation. The analysis suggests that a necessary condition for sexual selection is E >> P, and in the following experiments FS = (0.01, 0.001, 0.1) was adopted. Condition (3) can be rearranged to read: From equations (1) it follows that: Substitution into (4) yields the quadratic inequality: "0 FIG. 1. Sexual selection under the handicap where A = -3~'; B = 42.5 - 2a - p); principle in a monogamous population. Ordinate = c=-a(a + p). This has two real posi- change in frequency X 10" Solid line = Lip; broken tive roots for the FS given: q = 0.016 and line = Ar, hp becomes positive over part of the range of q,,. An arrow indicates the value of q,, at 0.58. Sexual selection should be sufficient- which the maximum of hp is predicted by analysis. ly powerful to cause an increase in the fre- Input variables: IfS = {0.001, qo, 0.001); FS = quency of the A gene if q, lies between {0.01, 0.001, 0.I}. Change in frequency of A is mea- these limits. The greatest effect of sexual sured between 1st and 2nd zygote generations, so selection will occur at the value i which that Ap = pij = 2) - pij = 1). is defined by: zygote generation is formed. I emphasize that although the handicap principle is in- Performing the differentiation yields: deed capable of generating increases in the frequency of conspicuousness, these in- creases are both transient and very small, where X = ~(201+ P + E); Y = 2401 + and are unlikely to constitute an evolu- p); Z = -a(a + P). For the FS given. tionary process of general importance. 4 = 0.087. Thus, the analytical treatment Obviously, sexual selection will occur suggests that the A gene should be favor- more easily if the B gene takes longer to ably selected under the given FS in the pass through the population, since it will range 0.016 5 q, 5 0.58 and that this ef- then segregate in the appropriate range of fect should be strongest when go = 0.087, frequencies for a longer period of time. Figure 1 shows the results of simulation. The range of q, over which Ap > 0 in The point of maximum Ap is correctly Figure 1 might therefore be increased if predicted by the analysis, but the entire sexual selection is effective for E < 0.1. To range over which Ap > 0 is greatly over- investigate this possibility, experiments stated. Thus, the conditions shown by were performed with IfS = (0.001, 0.001, simulation are more stringent than those 0.001) and FS = (0.01, 0.001, E}; the re- suggested by analysis; this is because the sults are shown in Figure 2. Since the fit- FS implies such an enormous selective nesses are intended as rates of survival, E advantage for the B gene that if it begins cannot exceed 1 - (0.01 + 0.001) = at a frequency of more than 10% or so it 0.989. For values which approach this is almost fixed by the time that the second limit, Ap > 0 in the second zygote gen- 8'78 GRt-IHAIM BELL E 2 l/30, sexual selection becomes entirely ineffectual. This result indicates the central dilem- ma of sexual selection under the handicap principle. Sexual selection occurs rather easily when E is very large relative to P, but this is a very short-lived process be- cause it implies a very rapid transit of the B gene through the population. If E is de- creased in order to slow down the passage of the B gene, then we quickly enter a region where E is insufficiently large to procure any increase in the frequency of the A gene. The fixation of the B gene can also be delayed by making cr large relative to E, whilst keeping the ratio EIPlarge, as for the example in FS = {0.5, 0.001, 0.1} . However, this strategem has the ad- ditional effect of reducing the rate of ac- cumulation of B genes by A males and for this reason fails to provide a solution to the dilemma. 1 2 3 4 5 6 7 8 Although it is not difficult to confirm that transient increases in the frequency i of the A gene may occur, in no instance FIG. 2. Sexual selection under the handicap was it found that the A gene could exceed principle in a monogamous population. Afi varies its initial frequency; that is, although pi with differing strengths of the effect at the h locus. > Input variables: IfS = {0.001,0.001,0.001);FS = pi-_,(for some j) for some IfS and FS, pi < {0.01,0.001.e}.j = generation. p,, (for all j > 0) for all IfS and FS in- vestigated. It is not inconceivable that there may exist regions of parameter space in which p, > p,,, but these must be very eration, which is consistent with the an- small, if they exist at all. I conclude that alytical theory. For progressively smaller a sustained increase in the frequency of values of E, the following pattern of events conspicuous males will occur only if dif- emerges. There is an excess of coupling ferent 'fitness genes' (e.g. those at the B genotypes (DB(.> 0) in the first zygote locus) are segregating within a narrow generation, as expected, and this excess band of frequencies for long periods of increases as long as (q, - q,) increases. time. This implies not only very powerful Eventually, q becomes so large that (q, directional selection in each generation, - q,:)diminishes in time, and as it does so but also a continuous supply of mutants the excess of coupling genotypes also di- with very large fitness differentials on minishes, finally leading to DB( = 0 when which such selection can act. q = 1. As (q, - q,) and DB( reach their These results are very sensitive to the maximum values, Ap becomes positive, two given of the model other than IfS and and these positive values persist for be- FS: the mating rules and the initial link- tween one and three generations. As E de- age equilibrium. Consider first the effect creases, Aq decreases and the maxima of of substituting the polygamy for the mo- (qA - q,) and DB(.are shifted backwards nogamy mating rules. If we have IfS = in time, so that the region of increase in (0.001, 0.001, 0.1) and FS = (0.01, the frequency of the A gene occurs in suc- 0.001, 0. I), then sexual selection is very cessively later generations. Finally, when powerful, the A gene being fixed and the SEXUAL SELECTION 879 FIG. 3. The handicap principle in a polygamous population. Pure Fisherian sexual selection is mod- elled by e = 0. The handicap principle results in greater equilibrium frequency of A. Input variables: IfS = {0.001,0.001,0.001);FS = 10.01,0.001,€1. C gene reaching an equilibrium frequency of about 0.39. However, the high initial frequency of the C gene relative to that of the A gene gives so great a mating advan- tage to the conspicuous males that a very FIG. 4. The handicap principle in a polygamous population. The handicap principle may cause an similar result would have followed from increase in the frequency of A, which decreases un- straightforward Fisherian sexual selection der a pure Fisherian process. Input variables: IfS = alone. This has been pointed out previ- {0.001,0.001, 5 x 10-9;FS = 10.01,0.001, €1. ously by Maynard Smith (1976). How- ever, in his simulation he set u = l (see description of his fitness model, above), gene increase; for E = 0.1, the A gene which is equivalent to the assumption that eventuallv comes close to fixation. This fitness effects are multiplicative; his state- result raises the possibility that there may ment that Fisherian selection works best be parameter values under the polygamy if there is no handicap is not necessarily mating rules for which pure Fisherian sex- correct if fitness effects are not multipli- ual selection is ineffective but for which cative. We may model a pure Fisherian a combination of the Fisher effect and the process by putting E = 0; the B gene is handicap principle may procure an in- now neutral, the genotypic fitnesses are crease in the frequency of conspicuous- obtained multiplicatively, and conspicu- ness. Specifically, as the initial frequency ous males can accumulate no excess of- of the C gene is reduced whilst the initial potentially advantageous genes at other frequency of the A gene is kept constant, loci. If we put IfS = (0.001, 0.001, there exists a point below which natural 0.001), the mating advantage of conspic- selection of given intensity will always be uous males is sufficiently great to procure effective in reducing the frequency of the an increase in the frequency of the A gene, A gene. For FS = (0.01, 0.001, 0) this which rises to an equilibrium frequency of point occurs in lop% rO> 9 X lo-;' between 0.09 and 0.10. By putting E > 0, (clearly, at rO= 10-"he 10% disadvan- we can study the effect of the handicap tage of the A gene under natural selection principle on this process. Figure 3 shows will be just exceeded by the slightly great- that as E increases both the rate of increase er than 10% advantage- under sexual se- and the equilibrium frequency of the A lection given by the polygamy mating FIG.6. Initial stages of sexual selection in a po- -2 Iygamous population. The strength of the A locus 3 necessary for the handicap principle to enable a Fish- erian process to become established. Input variables: 2 IfS = i0.001, 0.001, 5 x 10-'1; FS = (0.01, 0.001, €1. C and thereby increases the mating ad- vantage of A malcs under Fisherian sexual 1 5 10 15 selection. Both effects are small, but in i combination they are sufficiently powerful FIG.5. Initial stages of sexual selection in a po- to reverse thc inequality between natural lygamous population. The handicap principle may enable a Fisherian process to become established. and sexual selection. The initial stages in (Top) Change in frequency of A and C. (Middle) the evolution of the population are illus- Magnitude of 49.(Bottom) Increase in linkage dis- trated in Figure 5. While the B gene is equilibrium between A and C loci: D ,, = P,),P,,, - rare, Lip < 0;but by the third generation P,,P,, . Input variables: If5 = {0.001, 0.001. 5 x q has become large enough to make q, - lo-'); FS = (0.01, 0.001. €1. q,) sufficiently large to fuel the handicap principle, so that Ap > 0. As q - 1 the power of the handicap principle wanes rules). Thus, if we set r, = 5 X and and eventually disappears, but by this E = 0, the 4 gene declines from an initial time Y and D,4,. (the linkage disequilibri- frequency of 10-qo some lower equilib- um between the A and C loci, D,,,, = rium value (in this case, about 3 x P..i,P,,.- P.,,P,,.) have become sufficient- Pursuing the argument, we expect this ly large for the A gene to continue to be lower equilibrium value to increase with favorably selected by a pure Fisheriari E, and speculate that for some E it may process. excecd the initial frequency of the A gene. It is clear that for any given value of E Figure 4 shows that this is indeed the case: there will be a value of r, belolv which for E = 0.1 the A gene increases to an even the combination of Fisherian sexual eventual equilibrium at >I%. Two rather selection and the handicap principle will different processes are involved in this fail: for E = 0.1 this value is nearly lo-". event. First, the accumulation of B genes Thus, the effect of the handicap principle by '4 males lessens the disadvantage of in this experiment is to extend the range their A-bearing male progeny under nat- of frequencies of the C gene over which ural selection, and secondly, the passage sexual selection is effective downward by of the B gene through the population about one order of magnitude. Similarly, strengthens the coupling between A and we might ask what strength of effect at SEXUAL SELECTION 881 the B locus is required in order to over- low from less extreme initial linkage dis- come the overall disadvantage of A males. equilibria. Figure 6 illustrates the initial dynamics of selection for different values of E; if E > 0.01, then the eventual equilibrium fre- The analysis and the simulations re- quency of the A allele lies above its initial ported above demonstrate the formal pos- frequency. The eventual frequency reached sibility that the handicap principle can by the A gene when t = 0.1 exceeds 1%. generate effective sexual selection. The Maynard Smith (1976) did not find that conditions under which this possibility is the handicap principle would work when realized can be scrutinized in order to choosy females are rare relative to con- throw light on the applicability of the spicuous males, but in the only experiment handicap principle to real populations. that he reports he sets r, = pO/500.Since A non-multiplicative model of genotyp- the positive result reported above held ic fitnesses is essential to the operation of only between ro = pO/10and r0 = p,/100, the handicap principle, and the arguments it seems likely that his experiment merely presented above are all derived from a confirms that there is a region of r, << p, model of additive gene effects. The dis- in which even the combination of the tinction between the additive and multi- handicap principle and the Fisher effect plicative ways of combining effects at dif- cannot induce effective sexual selection. ferent loci is as follows. The additive The analytical results suggest that sex- model implies that effects on fitness are ual selection under the handicap principle mutually exclusive but not independent; will be easier if the population initially in- the multiplicative model implies that ef- cludes an excess of coupling genotypes, fects in fitness are independent but are not and this is readily confirmed by simula- mutually exclusive. Effects which are nei- tion. It appears that if the initial popula- ther independent nor mutually exclusive tion consists exclusively of coupling ge- are combined according to models inter- notypes (BC and bc), the A gene will mediate between the additive and multi- increase even under the monogamy mat- plicative. For example, Ricklefs (1977) ing rules if E > @. The initial rise in fre- uses an additive model of mortality when quency is not surprising. Since C is cou- 'reproductive' and 'non-reproductive' risks pled with B, C will increase in frequency are run concurrently, but he uses a mul- as B increases under natural selection. At tiplicative one when the exposure to these the same time, C will come into coupling two different sources of mortality is sep- with A because of the mating preference arated in time. If, in the present example, of C females for A males, and the resul- effects at the A and B loci were complete- tant coupling of A with B causes an in- ly separated in time, then the genes at the crease in the frequency of A, provided B locus would be neutral when selection that the average effect of the B gene (mea- was occurring at the A locus, and A males sured by t) exceeds that of the A gene could accumulate no excess of B genes. (measured by p). This is merely an ex- Thus, a necessary condition for the hand- ample of the 'hitch-hiker' effect. More sur- icap principle to operate at all is that the prisingly, although the A gene will sub- effects on fitness at the A and B loci sequently decline in most cases, there are should not be perfectly independent. This parameter values for which A continues is not unreasonable; one would expect it to increase until B is fixed. This is because to be satisfied, for instance, if the char- a population consisting only of ABC and acter controlled by genes at the B locus abc zygotes is equivalent to two non-in- were expressed during the breeding season terbreeding subpopulations, and A is cer- as well as outside it. J. Maynard Smith tain to be fixed if the fitness of AB indi- (pers. comm.) has pointed out to me that viduals exceeds that of ah individuals, a fitness scheme in which conspicuousness which requires E > @. Similar results fol- has a much more deleterious effect on h 882 GRAHAM BELL males than on B males may be more fa- same time a succession of genes equivalent vorable to the operation of the handicap in effect to the B gene sweep through the principle than the additive scheme used population; the process envisaged could above. However, there seems to be little either be the rapid fixation of genes at dif- biological justification for this arrange- ferent loci, or the alternation of selection ment. coefficients at a single locus. Such ex- Under the monogamy mating rules, a tremely powerful, prolonged directional transient increase of the A gene may occur selection does not seem very plausible. if FS = {a5 c >> .B].- The baseline fit- These stringent conditions on the op- ness cannot be large relative to the effect eration of the handicap principle are at the B locus, since d(qA - q,,)/dcr < 0, greatly relaxed if the B and C genes are so that the accumulation of B genes by A initially in coupling. However, although males is retarded by large a. The B locus it is perfectly possible that they should must therefore represent a major deter- happen to occur on the same chromosome, minant of fitness. At the same time, the the process is something of a deus ex effect at the B locus must be very large maclzina, and the handicap principle adds relative to the sex-limited effect at the A little to the 'hitch-hiker' effect that would locus; since the parameter space has four occur in any case. dimensions it is not possible to summarize I conclude that the handicap principle this condition with a single figure, but in- is of dubious value in explaining the evo- crease in the frequency of the A gene for lution of conspicuous secondary sexual two or more generations at a rate of 1% characters in strictly monogamous popu- or more per generation was observed only lations. There is, however, no other when E 100P. This implies that /3 must known mechanism by which sexual selec- be small relative to a, that is, that the tion can achieve this result (although the effect of conspicuousness on survival, in operation of sexual selection on other the absence of the B gene, must not be types of character, for example the date very great. When these conditions on the of breeding, in monogamous populations fitness set are satisfied, the A gene increas- is well known; see Darwin, 1874, and es in frequency under sexual selection in O'Donald, 1972 and 1973). If the sex ratio the range of q for which (9, - q,,) is suf- amongst adults is equal, then each male ficiently large. This range is not, however, will find a mating partner, and sexual se- very extensive (see Fig. I). Moreover, the lection can act only through the quality of condition FS = {a 5 E >> /3} implies a offspring, rather than through the number large unconditional advantage for the B of females inseminated. If conspicuous gene, which passes rapidly through the males are indeed at a disadvantage with population under natural selection, per- respect to natural selection (a conclusion sisting in the range throughout which sex- disputed by many authors, and notably by ual selection is effective for only one or a Thayer, 1909; and Cott, 1957), then their very few generations. The rate of passage occurrence in monogamous species cannot of the B gene can be slowed only by in- be due to natural selection but at the same creasing a or by diminishing E or both, time receives no adequate explanation and these operations make effective sexual through Darwinian or Fisherian sexual selection unlikely because of the condi- selection. Moreover, this is not a negligi- tions on the fitness set given above. ble phenomenon. About a quarter of the A sustained increase in the frequency of monogamous North American passerines the A gene was not observed, and is prob- listed by Verner and Willson (1969) are ably impossible under the conditions of sexually dimorphic, including such strik- the simulation, given the monogamy mat- ing and well-studied species as the cardi- ing rules and initial linkage equilibrium. nal, Richmondena cardinalis (male bright It could be achieved only if the conditions red, female predominantly yellow-brown), on the fitness set are satisfied, while at the and the rufous-sided towhee, Pipilo ery- SEXUAL SELECTION 883 thophthalmus (male with head and entire This mode of sexual selection is very upper body black, female dusky brown). powerful if the polygamy mating rules de- Polygamous mating systems are thought scribed above are given, since each A to generate powerful sexual selection, be- male recieves on average (r'ip') + (1 - r') cause there can be considerable variation females, whereas each a male receives on in the number of females inseminated by average only (1 - r') females. Counter- different males. This idea is originally due vailing natural selection will eliminate the to Darwin (1874), who regarded certain A males only if it is sufficiently intense to male characters as necessarily stimulating overcome the reproductive advantage the female to mate, without explaining generated by an additional (r'ip') matings. how this necessity arose or could be main- However, certain objections to the appli- tained. The modern version of Darwin's cability of Fisher's theory to natural pop- theory is due to Fisher (1930), who pointed ulations should be mentioned. The first is out that the female's abilitv to choose, as that if the current disadvantage of the well as the male's predisposition to be c'ho- male character with respect to natural se- sen, will be liable to selection. The male lection is granted, it is still required that character in question is supposed to be fa- on its first appearance, in a less exagger- vored by natural selection when it first ated form, the character should actually appears in the population. Because those have reduced the risk of mortality. Al- males which express the character are the though not inconceivable, it is not clear more fit, females which prefer to mate that this should be true for the majority with them will be favored by virtue of the of 'conspicuous' characters. Secondly, the increased mean fitness of their offspring. polygamy mating rules themselves may The males are now doubly favored, since not be ecologically plausible. the increment in fitness caused by the mat- Consider the initial stages of selection, ing preference is added to their prior ad- when the male character is still favored by vantage under natural selection. The male natural selection. If males bearing the character and the female preference character are to receive a mating advan- spread through the population; and as tage (in terms of the number of broods they do so, any alternative genes or mod- they sire) in a population where the sexes ifiers which exaggerate the expression of are equally frequent, then they must on the male character or increase the powers average mate with more than one female, of discrimination of the female are also while males not bearing the character at- favored. The exaggeration of the male tract only one female, or do not succeed character eventually renders it harmful to in mating at all. This implies that the in- its bearer, so that it begins to be exposed creased fitness of offspring more than to unfavorable natural selection. But by compensates a choosy female for any dis- this time the reproductive advantage that advantages incurred by sharing a single the character enjoys by virtue of the fe- male with other females. This is not im- male mating preference is sufficiently plausible if there is no lasting pair-bond, large to overpower its disadvantage under and if the female alone, or neither sex, is natural selection. Indeed, the overall ad- responsible for parental care of the young; vantage of the character tends to be di- but if the male alone or the male and fe- rectly frequency-dependent, since the re- male together care for the young, the her- productive advantage increases with the itable increase in the mean fitness of their frequency of choosy females, which in offspring must exceed the effect of any im- turn increases with the frequency of con- pairment of their offspring's survival and spicuous males. The result is a 'run-away' process by which the genes for conspicu- growth caused by diminished per capita ousness are rapidly fixed, or reach a stable parental care. Since the effect of polygyny equilibrium at which the effects of sexual on female fitness may be rather severe and natural selection cancel one another (e.g. Downhower and Armitage, 197l), 884 GRAHAM BELL this seems to require an unrealistically spicuousness in males. The male character high heritability of fitness. is obviously indifferent to selection and This objection, of course, applies equal- will remain at the same frequency ly to Fisherian sexual selection and to the throughout. Female choosiness, however, handicap principle: what is under attack is favored, and the genes responsible will is not the realitv of either of these Dro- increase in frequency. Suppose that genic cesses, but, rather, the reasonableness of variance for conspicuousness now arises, the polygamy mating rules in species for example by mutation a -+ A at the A where the male is involved in parental locus. If the population is strictly monog- care. It seems plausible to suppose that in amous, then conspicuousness is not fa- real populations with pair-bonds during vored except under the stringent condi- the breeding season a certain proportion tions described above. But if the of choosy females will prefer to mate with population is polygamous, then the rela- a cryptic male, rather than being the sec- tively high frequency of choosy females ond, third or n-th mate of a conspicuous will induce Fisherian sexual selection, and male. If this proportion is zero, the polyg- the A gene will increase in frequency. amy mating rules hold; if it is unity, the Thus, one major difficulty of Fisher's the- monogamy rules will be appropriate. As ory of sexual selection-the prior segre- the proportion passes from zero through gation of genes for female choosiness at intermediate values to unity, we can ex- appreciable frequency-can be removed pect two things to happen: first, the over- by supposing that genes for male conspic- all strength of sexual selection will decline; uousness mimic previously non-heritable and secondly, the importance of the hand- characters, without the need to suppose icap principle relative to the Fisher effect that the male character is advantageous will increase, until at the limit only the with respect to natural selection when ex- handicap principle is capable of producing pressed in a less exaggerated fashion. any increase in the frequency of genes for The interaction of the handicap princi- conspicuousness. This argument implies ple with Fisherian sexual selection may be the existence of a range of mating rules its most important, or only, contribution within which a pure Fisherian process will to the evolution of sexual dimorphism. be impotent to prevent the decline of The augmenting of Fisherian selection by genes for conspicuousness, while the com- the handicap principle might be critical if bination of Fisherian selection and the female choosiness were rare relative to handicap principle procures effective sex- male conspicuousness, if the mating rules ual selection. of the population tended to monogamy, or Maynard Smith (1976) has pointed out if there were originally no additive genetic that the handicap principle will be effec- variance for conspicuousness. For exam- tive in the absence of any Fisher effect, if ple, the handicap principle might play a there is no additive genetic variance for part in the evolution of male secondary male conspicuousness. The B genes ac- sexual characters in populations of birds cumulated by conspicuous males are then which although primarily monogamous passed on to the progeny of choosy fe- exhibited occasional polygyny. The criti- males, without any systematic disadvan- cal evaluation of these possibilities, and tage being conferred by inheritance of the their relevance to the field situation, handicap. Although this possibility is dis- awaits the exploration of models with dip- missed as being special pleading, it is not loid heredity and realistic mating rules. implausible, except that the characters of interest in real populations are probably nearly always inherited. However, con- Zahavi (1975) has suggested that fe- sider the course of evolution in a popula- males may prefer to mate with conspicu- tion where there is initially no additive ous males because these males, having genetic component to the variation of con- survived despite their handicap, must be SEXUAL SELECTION 885 extraordinarily fit in other respects. This previously non-heritable variation. It is paper examines the validity of the handi- concluded that any importance of the cap principle by studying the behavior of handicap principle in real populations is a simple haploid model. Three loci are in- likely to reside in its catalytic effect on volved: alternative genes at the A locus Fisherian sexual selection. determine male conspicuousness (A-con- spicuous; a-cryptic); those at the B locus determine some unrelated aspect of fitness This paper has grown out of a corre- (B-more fit; b-less fit); those at the C spondence with J. Maynard Smith, and locus determine the mating preference of could not have been written without his females (C-choosy, prefer to mate with advice. I am also indebted for comments A males; c-undiscriminating). If the ef- and criticisms to P. Handford, U. Motro, fects on fitness at the A locus (measured P. O'Donald, R. Shand, J. Turner and A. by j3) and at the B locus (measured by E) Zahavi. are multiplicative, the handicap principle cannot work. If these effects are additive, sexual selection in a monogamous popu- COTT. H. B. l95i. AdaptiL~eColouration in Ani- lation follows if 2~(q,- q,,) > p, where mals. Methuen, London. q~ and q, are the frequencies of the B DARWIN,C. R. 18i4. The Descent of Man and gene after selection in conspicuous and Selection in Relation to Sex. 2nd Edn. John Mur- cryptic males respectively. However, al- ray, London. DO\VNHO\VER,J. F., AND K. B. ARMITAGE.19il though this condition leads to small and The yellow-bellied marmot and the evolution of transient increases in the frequency of the polygamq-. Amer. Natur. 105:355-3i0. A gene, sustained increase is contingent FISHER,R. A. 1930. The Genetical Theory of Nat- not only on very powerful and prolonged ural Selection. Clarendon Press, Oxford. MAYNARDSMITH.J. i9i6. Sexual selection and the direction selection, but also on a contin- handicap principle. J. Theoret. Biol. 57:239-242. uous supply of mutants providing very O'DONALD,P. 1962. The theory of sexual selection. large fitness differentials on which this se- Heredity 17:541-552 lection can act. These conditions are so . 1963. Sexual selection for dominant and recessive genes. Heredity 18:45 1-45 i restrictive that the handicap principle act- -. 1967. A general model of natural and sex- ing alone is unlikely to be a significant ual selection. Heredity 22:499-5 18. evolutionary force. In polygamous popu- -. 19i2. Natural selection of reproductive lations, the handicap principle can be rates and breeding times and its effect on sexual thought of as modulating the dynamics of selection. Amer. Natur. 106:368-379. -. -. 1973. Frequency-dependent sexual selec- Fisherian sexual selection, leading in gen- tion as a result of \variations in fitness at breeding eral to higher equilibrium frequencies of time. Heredity 30:35 1-368. the A gene. In particular, the combination RICKLEFS,R. E. 197i. On the evolution of repro- of the handicap principle and Fisherian ductive strategies in birds: reproducti\re effort. Amer, Natur. 11 1:453-478. sexual selection can generate nontrivial in- THAYER.G. H. 1909. Concealing Coiouration in creases in the frequency of male conspic- the Animal Kingdom. MacMillan. New Vork. uousness in circumstances where a pure VERNER.J., AND M. F. WILLSON. 1969. Mating Fisherian process cannot: this may happen systems, sexual dimorphism, and the role of male if female choosiness is initially rare rela- North American passerine birds in the nesting cycle. Ornithological Monographs, No. 9. Amer- tive to male conspicuousness; if the mating ican Ornithologists' Union. system is intermediate between monog- ZAHAVI,A. 19i5. Mate selection-a selection for a amy and polygyny; or if the k gene mimics handicap. J. Theoret. Biol. 53:205-214. http://www.jstor.org LINKED CITATIONS - Page 1 of 1 - You have printed the following article: The Handicap Principle in Sexual Selection Graham Bell Evolution, Vol. 32, No. 4. (Dec., 1978), pp. 872-885. Stable URL: http://links.jstor.org/sici?sici=0014-3820%28197812%2932%3A4%3C872%3ATHPISS%3E2.0.CO%3B2-H This article references the following linked citations. If you are trying to access articles from an off-campus location, you may be required to first logon via your library web site to access JSTOR. Please visit your library's website or contact a librarian to learn about options for remote access to JSTOR. Literature Cited The Yellow-Bellied Marmot and the Evolution of Polygamy Jerry F. Downhower; Kenneth B. Armitage The American Naturalist, Vol. 105, No. 944. (Jul. - Aug., 1971), pp. 355-370. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28197107%2F08%29105%3A944%3C355%3ATYMATE%3E2.0.CO%3B2-Z Natural Selection of Reproductive Rates and Breeding Times and Its Effect on Sexual Selection Peter O'Donald The American Naturalist, Vol. 106, No. 949. (May - Jun., 1972), pp. 368-379. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28197205%2F06%29106%3A949%3C368%3ANSORRA%3E2.0.CO%3B2-D On the Evolution of Reproductive Strategies in Birds: Reproductive Effort Robert E. Ricklefs The American Naturalist, Vol. 111, No. 979. (May - Jun., 1977), pp. 453-478. Stable URL: http://links.jstor.org/sici?sici=0003-0147%28197705%2F06%29111%3A979%3C453%3AOTEORS%3E2.0.CO%3B2-X