,I. theor. Bid (1978) 70, 245-250
LETTERSTOTHE EDITOR
On the Handicap Principle-A
Critical Defence
In a response to a criticism of Zahavi’s Handicap principle, a quantitative model which enables the evolution of preference of a handicap is suggested. In his article “Mate Selection-a selection for a handicap” (1975), Zahavi suggests that many attributes of sexual attraction are rather conspicuous handicaps, designed to guarantee the high quality of their carriers in other respects. He proposes that only a male who is fit in all other respects (namely, in this context, a male of high quality) can survive to breed despite the presence of his handicap. And, so long as quality is inheritable in the population, a female can improve the quality of her own offspring by choosing a handicapped male as a mate [for the possible advantage of such a choice in increasing thefertility of the choosing female, see Zahavi (1977), Motro (1977)]. Unfortunately, however, if the handicap itself is inherited (which is obviously th(e interesting case), a female who chooses a handicapped male as a mate (from here on, a choosing ,&en&e) will decrease the fitness of those of her olfspring who inherit the handicap. This raises the crucial quantitative question of whether it is possible for average fitness to be higher among the offspring of handicapped males so as to make their choice as mates selectively advantageous. Employing simple quantitative models, Maynard Smith (1976) and Davis &. O’Donald (1976) attempted to show that whenever the handicap is inherited by both sexes, its sexual preference cannot possibly be selected for in natural populations-and it is most unlikely to be selected for, even if th.e handicap is passed exclusively to male offspring. However, following models quite similar to theirs, but concentrating on a specific range of parameters which seem to be most relevant to the problem, I arrived at different results. Therefore, a short argument about the biological characterization of the parameters is in order. It appears to be an irrefutable truth of natural selection (a truth that has been validated again by the computer tests of Maynard Smith) that the average fitness of offspring born tcl a random defective individual is generally below the average fitness of the population. The exception-namely, the quality marker Itandicap-if it exists at all (and it is the objective of this note to show it can) should, inevitably, be of a very special sort (which. theoretically. may not mean its rarity in a natural population once it bccomcs sexually advantageous).
246
I.
ESHEL
On one hand, direct selection against the candidate for a quality marker handicap must not be too high. On the other hand, its correlation with other favourable characteristics (namely, the quality) should be sufficiently high, at least at mating time, to over-compensate for the offspring’s deficiency in fitness due to inheriting the handicap. But then it is claimed that a substantial correlation between the handicap and the quality can be maintained onI> with high selection pressure on the handicap (as well as on the quality loci). And it seems to follow, from the results of Davis & O’Donald and. indirectly, from those of Maynard Smith, that the required selection pressure is always too high to be compensated for by the relatively higher quality of the handicapped. More specifically, Davis & O’Donald have shown that unless for unrealistically high selection pressure (and at least in their model) the average fitness of offspring born to a handicapped male is always lower than the average of the population. Unfortunately, Davis & O’Donald considered only that part of the handicap-quality correlation that may develop ~~ith;~ one generation. Tacitly (but quite crucially for their results), they assumed a permanent linkuge equilibrium at birth-an assumption which, even in their model, becomes false after one generation. Moreover, presumably for the sake of mathematical convenience, they restricted their analysis to a very special situation in which the effect of the handicap is limited to a linear exaggeration of the carrier’s deficiency in quality. Therefore. what they have shown is that without the cumulative effect of linkage disequilibrium. a linear effect of the handicap on the fitness of its carrier is not sufficient for the evolution of that handicap as a quality-marker. It appears that a more discriminative effect of the handicap on the fitness of individuals of different quality is needed to enable the development of :I substantial quality-handicap correlation, without affecting too drastically the average fitness of the handicapped offspring. Such a discriminative effect may be typical of many known attributes of sexual attraction or social dominance: bright colouration may hardly affect the survival probability of a bird which is fast enough to evade its potential predators, but may be fatal to a slow one. And exaggerated antlers may have only minimal effect on a strong buck, so long as it can actively protect itself against predators, e.g. Mech (1970) and see Wilson (1975) for many other examples. Since the model of Maynard Smith (unlike that of David & O’Donald) allows for the most general effect of the handicap on the various types, we adopted it initially without modification. We concentrated on the situation in which the effect of a handicap on the fitness of high-quality individuals is minor, but its effect on the fitness of low-quality individuals is fairly drastic. Starting with the case which was less favourable for the handicap
LETTERS TO THE EDITOR
24’
principle-the one in which the handicap affects both sexes-we assumed that high quality was determined by a single dominant allele .4 at one locus. and that the handicap was determined by a single dominant allele B at another, non-linked locus (later, we introduce linkage into the model). Setting A for the type of AA and Aa. B for BB or Bh, we denote the fitness of the four possible combinations AB, Abh, aaB and aahb by I -I, I. (I -hs)(I -t) and I --s respectively. with 0 < f, .P< I and I < /I < I 1. The requirement h > 1 stands for super-multiplicity of the deleteriou\ effects of aa and i? [note that in Maynard Smith’s original work, the fitnesh of aaB is denoted by (1 -s)(l -ut)]. Finally, we assumeda source of variance at the locus A (a more realistic model for this may be that of many loci with a mutation pressure at each, e.g. Eshel (1971). However. Maynard Smith’\ model is preferable for its mathematical simplicity). Like Davis & O’Donald, we started by investigating the possibleadvantage\ of choosing handicapped males at the first appearance of the handicap, i.e. \i,e concentrated on the choice of a non-handicapped female when the handicapped male was assumed to be helerozygote. We also assumed linkage equilibrium; then we studied the effxt of linkage disequilibrium. especialI\ with tight linkage. Let /) bc the frequency of the recessive allele ~1in the population at the time of birth. With linkage equilibrium, this is also the frequency of thi\ :illele among handicapped newborn offspring. Since ;L \urplus of (I --.s) 01 the an non-handicapped individuals arc dying before maturity, the proportiorl of the allele u among mature non-handicapped individual< i\: p, =
I -- ps ~-f p.
I -- p s
(Ii
In the sameway, the proportion of this allele among mature handicapped individual< is: I --psh pr = ~~~--j-- ,‘. 1--p-s/? If the allele B is rare, it may be assumedthat virtually no offspring of a non-choosing female will be handicapped, e.g. see Davis & O’Donald, but pi of her offspring will be of the low quality type an. Therefore, the average fitness among offspring of a non-choosing female is: wl(p, s) = 1-pf.s.
(3)
On the other hand, only p,p2 of the offspring of a choosing female will be of the low quality type aa. But, with the handicapped male being heterozygote, half of the offspring (either high or low quality) will be handicapped.
248
I.
ESHEI.
The average fitness of the progeny of a choosing female will therefore
be:
Now, when t is close to 0, and when sh is close to I, i.e. when the effect of the handicap is negligible for high-quality individuals. hut almost fatal to low-quality individuals, this average fitness tends to : wz(p, s, 0, l/s) = I - !-”
p1p2.
(5)
Inserting the values of p, and /)2 as expressed in equations ( I ) and (2). we seethat the condition: w,(p,
s, 0, l/s) > w,(p, s).
(6)
is satisfied if and only if:
provided that the right-hand side < 1, i.e. p > \I 2 - I. In this case, choosing females will always leave more surviving offspring, and natural selection will favour a preference for handicapped maleseven before linkage disequilibrium develops. We thus get: Corollary
I:
If the frequency p of the recessive disadvantageous allele is more than &-- 1, if the damage inflicted by this allele on its homozygous form is more than l/&+2), i.e. if the genetic load is at least (t/i- 1)/(~6+ I), and if the damage caused by a rare handicap is sufficiently low for high-quality individuals and sufficiently high for low-quality individuals, natural selection will favour, from the first appearance of the handicap, the tendency of females to choose handicapped males even if the handicap is expressed in both sexes. It is expected that if the handicap is expressed in both sexesand if the development of a linkage disequilibrium is taken into consideration (possibly with a flux of mutation to keep variance at the A locus), the range of parameters for which a preference for handicapped malescan evolve will increase; and it will increase more with tight linkage. It can be shown that among these factors, tight linkage has the most substantial effect on the evolution of a sexual preference for handicapped males.
LETTERS
TO
THE
249
EDITOR
In order to study the effect of linkage disequilibrium with tight linkage, we start, again, by assuming no recombination. Let ~1 be a steady rate of mutation from A to a. The frequency of the allele a among non-handicapped individuals will be: 1; With complete linkage, the frequency of this allele among handicapped individuals (provided that the handicap is expressed in both sexes) will be:
‘The average fitness of offspring
born to non-choosing
females will be:
1 -sp2. ‘The average fitness of offspring
born to choosing females will be:
For I = 0, this becomes:
which for any /I > I is larger than the average fitness I -s$ By arguments of continuity, we readily obtain : Corollary
in the population.
2:
For any frequency p of the low-quality allele in the population and fat any damage inflicted by the handicap B on the low-quality individuals, if the linkage between the handicap and the quality-determining locus is tight enough and if the damage inflicted by the handicap on the high-quality individuals is small enough, there will be a selective advantage to sexual preference for handicapped males. Results similar to Corollary 1 can also be obtained from the model of David & O’Donald, provided that slight modifications are made in order to allow for a discriminative enough effect of the handicap on individuals of different qualities. In all cases, it is shown that although a sexual preference for handicapped males can be favoured by natural selection, such a phenomenon is restricted to a very special sort of handicap. Maynard Smith’s computer tests demonhandicap, even if it was limited to strated that for almost any “random” one sex. natural selection favoured a sexual preference for non-handicapped
2.50
I.
1:SHEL
males (except for the Fisher effect). But if the exception, even though rare during evolution, became sexually favourable (and we see that it can be so) it may not be that rare in current populations. Note that in the original model suggested by Maynard Smith, the frequent) of the low-quality allele a tends to zero and selection for non-choosing females must always take over in the long run. I believe that without an> substantial source of variance in the heritable fitness in the population, it is inevitable that the situation in nature and a handicap can not then evolve. However, we followed the assumption of Maynard Smith only through one generation, postulating, further, that a fixed frequency of low quality gems always exists in the population. This may be feasible, for example. in a multilocus model with a minor flux of mutation at each quality IOCLIL Unfortunately, such a model, even though easy to conceive. is rather hard to analyse and the 3-10~~s model should be understood a> a convenient simplification. What remains to bc considered is the question of which assumption is qualitatively more relevant to natural situation and, to my judgement, the controversy is not yet settled by the suggested models. Departmellt of’ Statistics, Tel-Aviv University, Israel
I LAN ESH~L
(Received 30 Jmre 1977) REFERENCES DAVIS, J. W. F & O’DONALD, P. (1976). J. them. Biol. 57, 355. ESHEL, I. (1970). Theor. pop. Biol. 2, 2, 209. FISHER, R. A. (1930). The Genetical Theory of Natural Selection. Oxford: Clarendon MAYNARD SMITH, J. (1976). J. theor. Biol. 57, 239. MITCH, L. D. (1970). The Wolf The Ecology and Behavior of‘an Endan,pered Species.
City, New York: Natural History Press. MOTRO, U. (1977). The Courtship Handicap-Phenotypic Ed&r. WILSON, E. 0. (1975). Sociobiology. Cambridge, Mass.: Harvard Univ. Press. ZAHAVI, A. (1975). J. theor. Biol. 53, 205. ZAHAVI, A. (1977). In Evolutionary Eco/o.gy, (Stonehouse & Perrins, eds). In press.
Press. Garden
LETTERSTOTHE EDITOR
On the Handicap Principle-A
Critical Defence
In a response to a criticism of Zahavi’s Handicap principle, a quantitative model which enables the evolution of preference of a handicap is suggested. In his article “Mate Selection-a selection for a handicap” (1975), Zahavi suggests that many attributes of sexual attraction are rather conspicuous handicaps, designed to guarantee the high quality of their carriers in other respects. He proposes that only a male who is fit in all other respects (namely, in this context, a male of high quality) can survive to breed despite the presence of his handicap. And, so long as quality is inheritable in the population, a female can improve the quality of her own offspring by choosing a handicapped male as a mate [for the possible advantage of such a choice in increasing thefertility of the choosing female, see Zahavi (1977), Motro (1977)]. Unfortunately, however, if the handicap itself is inherited (which is obviously th(e interesting case), a female who chooses a handicapped male as a mate (from here on, a choosing ,&en&e) will decrease the fitness of those of her olfspring who inherit the handicap. This raises the crucial quantitative question of whether it is possible for average fitness to be higher among the offspring of handicapped males so as to make their choice as mates selectively advantageous. Employing simple quantitative models, Maynard Smith (1976) and Davis &. O’Donald (1976) attempted to show that whenever the handicap is inherited by both sexes, its sexual preference cannot possibly be selected for in natural populations-and it is most unlikely to be selected for, even if th.e handicap is passed exclusively to male offspring. However, following models quite similar to theirs, but concentrating on a specific range of parameters which seem to be most relevant to the problem, I arrived at different results. Therefore, a short argument about the biological characterization of the parameters is in order. It appears to be an irrefutable truth of natural selection (a truth that has been validated again by the computer tests of Maynard Smith) that the average fitness of offspring born tcl a random defective individual is generally below the average fitness of the population. The exception-namely, the quality marker Itandicap-if it exists at all (and it is the objective of this note to show it can) should, inevitably, be of a very special sort (which. theoretically. may not mean its rarity in a natural population once it bccomcs sexually advantageous).
246
I.
ESHEL
On one hand, direct selection against the candidate for a quality marker handicap must not be too high. On the other hand, its correlation with other favourable characteristics (namely, the quality) should be sufficiently high, at least at mating time, to over-compensate for the offspring’s deficiency in fitness due to inheriting the handicap. But then it is claimed that a substantial correlation between the handicap and the quality can be maintained onI> with high selection pressure on the handicap (as well as on the quality loci). And it seems to follow, from the results of Davis & O’Donald and. indirectly, from those of Maynard Smith, that the required selection pressure is always too high to be compensated for by the relatively higher quality of the handicapped. More specifically, Davis & O’Donald have shown that unless for unrealistically high selection pressure (and at least in their model) the average fitness of offspring born to a handicapped male is always lower than the average of the population. Unfortunately, Davis & O’Donald considered only that part of the handicap-quality correlation that may develop ~~ith;~ one generation. Tacitly (but quite crucially for their results), they assumed a permanent linkuge equilibrium at birth-an assumption which, even in their model, becomes false after one generation. Moreover, presumably for the sake of mathematical convenience, they restricted their analysis to a very special situation in which the effect of the handicap is limited to a linear exaggeration of the carrier’s deficiency in quality. Therefore. what they have shown is that without the cumulative effect of linkage disequilibrium. a linear effect of the handicap on the fitness of its carrier is not sufficient for the evolution of that handicap as a quality-marker. It appears that a more discriminative effect of the handicap on the fitness of individuals of different quality is needed to enable the development of :I substantial quality-handicap correlation, without affecting too drastically the average fitness of the handicapped offspring. Such a discriminative effect may be typical of many known attributes of sexual attraction or social dominance: bright colouration may hardly affect the survival probability of a bird which is fast enough to evade its potential predators, but may be fatal to a slow one. And exaggerated antlers may have only minimal effect on a strong buck, so long as it can actively protect itself against predators, e.g. Mech (1970) and see Wilson (1975) for many other examples. Since the model of Maynard Smith (unlike that of David & O’Donald) allows for the most general effect of the handicap on the various types, we adopted it initially without modification. We concentrated on the situation in which the effect of a handicap on the fitness of high-quality individuals is minor, but its effect on the fitness of low-quality individuals is fairly drastic. Starting with the case which was less favourable for the handicap
LETTERS TO THE EDITOR
24’
principle-the one in which the handicap affects both sexes-we assumed that high quality was determined by a single dominant allele .4 at one locus. and that the handicap was determined by a single dominant allele B at another, non-linked locus (later, we introduce linkage into the model). Setting A for the type of AA and Aa. B for BB or Bh, we denote the fitness of the four possible combinations AB, Abh, aaB and aahb by I -I, I. (I -hs)(I -t) and I --s respectively. with 0 < f, .P< I and I < /I < I 1. The requirement h > 1 stands for super-multiplicity of the deleteriou\ effects of aa and i? [note that in Maynard Smith’s original work, the fitnesh of aaB is denoted by (1 -s)(l -ut)]. Finally, we assumeda source of variance at the locus A (a more realistic model for this may be that of many loci with a mutation pressure at each, e.g. Eshel (1971). However. Maynard Smith’\ model is preferable for its mathematical simplicity). Like Davis & O’Donald, we started by investigating the possibleadvantage\ of choosing handicapped males at the first appearance of the handicap, i.e. \i,e concentrated on the choice of a non-handicapped female when the handicapped male was assumed to be helerozygote. We also assumed linkage equilibrium; then we studied the effxt of linkage disequilibrium. especialI\ with tight linkage. Let /) bc the frequency of the recessive allele ~1in the population at the time of birth. With linkage equilibrium, this is also the frequency of thi\ :illele among handicapped newborn offspring. Since ;L \urplus of (I --.s) 01 the an non-handicapped individuals arc dying before maturity, the proportiorl of the allele u among mature non-handicapped individual< i\: p, =
I -- ps ~-f p.
I -- p s
(Ii
In the sameway, the proportion of this allele among mature handicapped individual< is: I --psh pr = ~~~--j-- ,‘. 1--p-s/? If the allele B is rare, it may be assumedthat virtually no offspring of a non-choosing female will be handicapped, e.g. see Davis & O’Donald, but pi of her offspring will be of the low quality type an. Therefore, the average fitness among offspring of a non-choosing female is: wl(p, s) = 1-pf.s.
(3)
On the other hand, only p,p2 of the offspring of a choosing female will be of the low quality type aa. But, with the handicapped male being heterozygote, half of the offspring (either high or low quality) will be handicapped.
248
I.
ESHEI.
The average fitness of the progeny of a choosing female will therefore
be:
Now, when t is close to 0, and when sh is close to I, i.e. when the effect of the handicap is negligible for high-quality individuals. hut almost fatal to low-quality individuals, this average fitness tends to : wz(p, s, 0, l/s) = I - !-”
p1p2.
(5)
Inserting the values of p, and /)2 as expressed in equations ( I ) and (2). we seethat the condition: w,(p,
s, 0, l/s) > w,(p, s).
(6)
is satisfied if and only if:
provided that the right-hand side < 1, i.e. p > \I 2 - I. In this case, choosing females will always leave more surviving offspring, and natural selection will favour a preference for handicapped maleseven before linkage disequilibrium develops. We thus get: Corollary
I:
If the frequency p of the recessive disadvantageous allele is more than &-- 1, if the damage inflicted by this allele on its homozygous form is more than l/&+2), i.e. if the genetic load is at least (t/i- 1)/(~6+ I), and if the damage caused by a rare handicap is sufficiently low for high-quality individuals and sufficiently high for low-quality individuals, natural selection will favour, from the first appearance of the handicap, the tendency of females to choose handicapped males even if the handicap is expressed in both sexes. It is expected that if the handicap is expressed in both sexesand if the development of a linkage disequilibrium is taken into consideration (possibly with a flux of mutation to keep variance at the A locus), the range of parameters for which a preference for handicapped malescan evolve will increase; and it will increase more with tight linkage. It can be shown that among these factors, tight linkage has the most substantial effect on the evolution of a sexual preference for handicapped males.
LETTERS
TO
THE
249
EDITOR
In order to study the effect of linkage disequilibrium with tight linkage, we start, again, by assuming no recombination. Let ~1 be a steady rate of mutation from A to a. The frequency of the allele a among non-handicapped individuals will be: 1; With complete linkage, the frequency of this allele among handicapped individuals (provided that the handicap is expressed in both sexes) will be:
‘The average fitness of offspring
born to non-choosing
females will be:
1 -sp2. ‘The average fitness of offspring
born to choosing females will be:
For I = 0, this becomes:
which for any /I > I is larger than the average fitness I -s$ By arguments of continuity, we readily obtain : Corollary
in the population.
2:
For any frequency p of the low-quality allele in the population and fat any damage inflicted by the handicap B on the low-quality individuals, if the linkage between the handicap and the quality-determining locus is tight enough and if the damage inflicted by the handicap on the high-quality individuals is small enough, there will be a selective advantage to sexual preference for handicapped males. Results similar to Corollary 1 can also be obtained from the model of David & O’Donald, provided that slight modifications are made in order to allow for a discriminative enough effect of the handicap on individuals of different qualities. In all cases, it is shown that although a sexual preference for handicapped males can be favoured by natural selection, such a phenomenon is restricted to a very special sort of handicap. Maynard Smith’s computer tests demonhandicap, even if it was limited to strated that for almost any “random” one sex. natural selection favoured a sexual preference for non-handicapped
2.50
I.
1:SHEL
males (except for the Fisher effect). But if the exception, even though rare during evolution, became sexually favourable (and we see that it can be so) it may not be that rare in current populations. Note that in the original model suggested by Maynard Smith, the frequent) of the low-quality allele a tends to zero and selection for non-choosing females must always take over in the long run. I believe that without an> substantial source of variance in the heritable fitness in the population, it is inevitable that the situation in nature and a handicap can not then evolve. However, we followed the assumption of Maynard Smith only through one generation, postulating, further, that a fixed frequency of low quality gems always exists in the population. This may be feasible, for example. in a multilocus model with a minor flux of mutation at each quality IOCLIL Unfortunately, such a model, even though easy to conceive. is rather hard to analyse and the 3-10~~s model should be understood a> a convenient simplification. What remains to bc considered is the question of which assumption is qualitatively more relevant to natural situation and, to my judgement, the controversy is not yet settled by the suggested models. Departmellt of’ Statistics, Tel-Aviv University, Israel
I LAN ESH~L
(Received 30 Jmre 1977) REFERENCES DAVIS, J. W. F & O’DONALD, P. (1976). J. them. Biol. 57, 355. ESHEL, I. (1970). Theor. pop. Biol. 2, 2, 209. FISHER, R. A. (1930). The Genetical Theory of Natural Selection. Oxford: Clarendon MAYNARD SMITH, J. (1976). J. theor. Biol. 57, 239. MITCH, L. D. (1970). The Wolf The Ecology and Behavior of‘an Endan,pered Species.
City, New York: Natural History Press. MOTRO, U. (1977). The Courtship Handicap-Phenotypic Ed&r. WILSON, E. 0. (1975). Sociobiology. Cambridge, Mass.: Harvard Univ. Press. ZAHAVI, A. (1975). J. theor. Biol. 53, 205. ZAHAVI, A. (1977). In Evolutionary Eco/o.gy, (Stonehouse & Perrins, eds). In press.
Press. Garden
