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Sibling competition and the advantage of mixed families
J. theor. Biol. (1986) 120, 381-387
Sibling Competition and the Advantage of Mixed Families N. H. BARTON
Dept of Genetics and Biometry, University College London, London NW1 2HE, U.K. AND
R. J. POST
Dept of Medical Entomology, Liverpool School of Tropical Medicine, Liverpool L3 5QA, U.K. (Received 2 July 1982, and in revised forrn 9 January 1986) It is noted that the sibling competition model for the evolution of sex and recombination, as it has been developed so far, involves truncation selection. After briefly reviewing aspects of the development and behaviour of such models an analytical treatment is presented which involves additive selection. Additive selection, as compared with truncation selection, decreases the advantage of sex to such an extent that it is unlikely that sibling competition could overcome its intrinsic two-fold cost, although it could still be important in promoting family variability produced by other mechanisms, such as polyandry.
Introduction Sexual reproduction allows the production o f genetic variation within families; the mother brings in this variation by diluting her genes with those o f the father, a process which gives sex an inherent two-fold disadvantage. It has been proposed that one factor which may have allowed sex to evolve despite this obstacle is sib competition; if siblings compete with each other as well as with unrelated individuals, variation between them will tend to raise the fitness o f the eventual survivor, and therefore give it a better chance against individuals emerging from less variable parthenogenetic families. Several specific models have been set up to quantify this idea, and to find out how severe sibling competition must be to allow sex to evolve (Williams & Mitton, 1973; Maynard-Smith, 1978; Taylor, 1979; Bulmer, 1980). In this paper we will extend these models in two ways, whilst remaining within their general framework. Firstly, we consider the effect of different selection schemes on the proposed advantage of variable families under sibling competition. All the above models rely on truncation selection, in which the individual best adapted to its particular environmental patch is the individual which survives. Although models o f this sort have been proposed in order to account for the maintenance o f extensive variation by balancing selection without excessive genetic loads (e.g. Milkman, 1967), we feel that it may be more realistic to consider genetic variation which is maintained by selection acting independently (i.e. additively) at many loci. In such an additive model the chance o f an individual surviving is the sum o f the effects of 381
0022-5193/86/120381+07 $03.00/0
© 1986 Academic Press Inc. (London) Ltd
382
N. H. BARTON
AND
R. J. P O S T
the many independent loci on fitness. The advantage of variable families seems likely to depend upon the mode of selection, and we shall try to compare this advantage under both truncation and additive selection. Secondly, many aspects of population structure will affect family variability. We will investigate one such feature, using both the truncation models of Bulmer et al. and our additive models to find the advantage of polyandry over monogamy; a female may produce more varied offspring by mating with several males. (The same advantage accrues to a male which mates with many females, but since polygyny directly raises males' fitness, and since males' progeny will usually be left in different patches, the effect of sib competition on polygyny will be negligible.) Furthermore, comparing these mating strategies within a sexual population will also allow us to consider more general mechanisms. Under many mechanisms an asexual population will maintain less variation than a sexual population, and so the two cannot easily be compared. For example, if heterozygotes are fittest, an asexual population may become purely heterozygous, and will then always displace a sexual population whatever the level of sib competition (see Hebert et al., 1972, for an example from nature). However, the type of mating system within a sexual population is less likely to alter the overall amount of variation and so the effect of heterozygous advantage, relative to disruptive selection between patches, can be compared unambiguously under sibling competition. We will consider a restricted class o f models, in which a large random-mating population leaves offspring scattered over many patches. In each patch there are R families, each containing N individuals. These compete to produce a single survivor, which enters the wider population to begin the next generation. We will therefore not consider models where the number of survivors depends on the family variation. For example, there might be competition for resources within patches such that mixed families produce more offspring even when isolated from all other families (e.g. Prrez-Tom6 & Toro, 1982). The outcome in such cases will depend on the details of the selection, and so generalisations would be difficult. Truncation Models
Before developing a model based on additive genetic variation we will describe the ways in which the current models based on truncation selection have been generalised by successive authors. Williams & Mitton (1973), and Williams (1975) described several models which gave a short-term advantage to sex; Maynard-Smith (1978) pointed out that the essential element in these models was not the unpredictability of the environment, which served only to maintain genetic variation, but rather, intense competition between siblings. He set up a specific model in which the environment varied in five ways; a patch could be A or a, B or b . . . . , E or e. This variability maintained polymorphisms at five corresponding loci, so that genotypes A A and A a were adapted to environment A, a a to environment a, and so on. The survivor in each patch was the individual which was adapted to the largest number of environmental features. Using Monte-Carlo simulation, it was found that sib competition gave sex
SIB COMPETITION
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VARIABILITY
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an advantage whenever R and N were greater than 1, and that RN, the number of organisms competing in each patch, had to be greater than 30-40 for this advantage to outweigh the two-fold "cost" of sex. This conclusion clearly depends on the number of loci involved, the degree of dominance, and the way the loci interact to determine fitness. Taylor (1979) generalised the model analytically; he used the same assumptions as Maynard-Smith, but considered variation at a number of loci large enough that the number of genes which matched the environment in any individual could be assumed to be normally distributed. If the probability of an allele at any one locus matching the corresponding environmental factor was 0.5 (as in Maynard-Smith's model), Taylor showed that the number of loci did not affect the advantage of sex. If the matching probability was greater than 0-5, as would usually be the case, the advantage of sex would be less (since the genetic variation in fitness would be reduced) and would increase slightly with the number of loci involved. Bulmer (1980) further generalised the model, using the methods of quantitative genetics, to deal with variation in some underlying "fitness potential", generated by additive variation at many loci; the survivor in each patch was chosen as the individual with the highest "potential", as in the previous models. The exact mechanism by which variation was maintained was not specified, and would not affect the outcome. However, one might imagine it to be maintained by random variation in the optimum value of the "potential" between patches. Having summarised these truncation models we should make some points about their general behaviour. Firstly, they all assume that recombination does not reduce the average value of the underlying "fitness potential", as would happen if, for example, several loci determined the ability to match each environmental factor (see Maynard-Smith, 1978). This effect could give a strong disadvantage to sex (see Mukai, 1977, for an experimental illustration). Secondly, the advantage of variability arises solely from competition between individuals within patches: it is due to the fact that the expectation of a concave function of some variable (in this case, of the "fitness potential") increases with its variance. The survival and reproduction of individuals after they leave the patches is assumed to be independent of the outcome of this competitive process. Finally, the effect of the number of loci underlying the character was only explored by Taylor, and then only for a number so large that the normal approximation to the binomial could be used. If only a few loci determine the competitive components of fitness, each family might stand a strong chance of producing the best possible adapted genotype, and the advantage of variability would disappear. This effect of the number of loci can be illustrated by a simple one-locus computer model. Maynard-Smith's frequency-dependent disruptive selection model was slightly modified to cover only a single locus and analysed deterministically. Following Maynard-Smith, two alleles (one dominant to the other) at a single locus yield two phenotypes, each adapted to one of two possible types of patch. The initial phenotype frequencies were equal and matched the relative frequencies of the two patches. Sexual populations began in Hardy-Weinberg equilibrium. All possible combinations of a given number of sexual and parthenogenetic families in each
384
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BARTON
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R. J. P O S T
is completely inherited; the heritability of fitness depends on the degree to which U varies. In Bulmer's model, on the other hand, fitness is determined entirely by the ordering of the variates y; the heritability depends here on the heritability of y. Thus, in the first model, environmental variation is introduced through the random element in choosing which genotype survives, whilst in the second, it is introduced through variability in the ordering of "fitness potentials". To compare the two models, then, we must decide on what levels of environmental variation are equivalent; this is, in general, difficult. We will proceed by restricting the comparison to small genetic variances; in the truncation model, it seems unlikely that the ordering of the individuals would be anywhere near perfect. Taking the limit of Bulmer's model where a proportion ( 1 - q ) of the genetic variance in y, Vs, is between families, and a proportion q within families, we have
Vb=(1-q)V~
Vw=I-(1-q)Vs(Vg<
(5)
By inserting these values into Bulmer's formula (11), and taking the limit as Vg tends to zero, we find that the advantage of the new strategy is
RVs(q'- q) (tz~rN(o'2_l + 1) + p. R-I (o'2-1)) exp (--/z2-1/(1 + 0"2-1)) o.~vv/2~.(1 + ¢2_,)3
(6)
(Here,/~M and tr 2M are the mean and variance of the largest of M standard normal variates.) So, with truncation selection, the advantage is proportional to the change in the heritable component o f y within families ( Vg(q'- q)): just as with additive selection, the advantage is proportional to the change in the heritable component of U within families. Table 2 shows the advantage per unit increase in within-family variance, for various R and N. TABLE 2
The advantage of sex under truncation selection, per unit increase in within-family variance (see text for explanation) N u m b e r of families ( R ) N u m b e r o f siblings (N)
2
3
6
11
2 5 10
0"19 0-49 0"75
0"20 0"51 0"76
0.12 0.28 0"42
0"04 0-09 0-13
The first point to make is that if survival after emergence from the patches is independent of survival within patches, additive selection gives no advantage at all to variability: in general, the advantage to variability depends on the concavity of the relation between fitness potential and fitness itself. However, if survival after
SIB COMPETITION
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survival (we use U to denote this characteristic to emphasise that it is not equal to the actual fitness, W). If there are ajg individuals of genotype g in family j (j = 1 . . . R; ~s aj, = N ) , then the probability of survival of some individual from family 1 is X atgV~/EX ajgUg. (1) g
J g
With this linear relation between U and the probability of emerging from the patch, there is no immediate advantage in variability: the survival probability is simply proportional to the expected value of U. However, if we suppose that characteristics which favour survival within patches also favour survival or reproduction later in life, then variable families wilt be at an advantage. This is because the average value of U for a family is increased by selection within patches, in direct proportion to the variance in that family. Suppose that one family has a new strategy, which alters the distribution of genotypes, aig; we will denote this new strategy by '. If we assume that the change o f strategy does not directly increase the average, U, before competition, then Fisher's (1930) "Fundamental Theorem" shows that the average after competition is increased by (var' ( U ) - v a t ( U))/U.
(2)
This simple formula, which will be accurate as long as the strategies under consideration do not affect the mean fitness, can easily be applied to specific models. For example, if variation is maintained by heterozygote advantage, with fitnesses ( 1 s : 1 : 1 - t), at K loci, then the advantage of polyandry over monogamy is:
K(N-1)s2t 2 4N(s + t) 2 "
(3)
Similarly, if the variation is maintained by environmental variation, with fitnesses (1 + s : 1 : 1 - s) in half o f the patches, and (1 - s : 1 : 1 + s) in the rest of the patches, then the advantage is K ( N 1)s 2 (4) 16N -
In general, it seems that the final advantage will be small unless most of the variance in fitness is due to competition within patches, and unless success within patches is strongly correlated with success in later life.
Comparison with Truncation Models How does this model, which assumes additive selection, compare with those based on truncation selection? We will concentrate on Bulmer's (1980) infinite locus model, because it is the most general, and because the strategies are described in terms of the variances within and between families, as above. However, there is still a difficulty in comparing t h e two modes of selection. One of the most important parameters determining the effect o f sib competition is the proportion of the fitness variation which is inherited. In the additive model, fitness is determined by a variate U, which
386
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habitat patch were written down, and the overall frequency with which sexuals won was calculated. The proportionate increase of sexual families over one generation is given as a measure of their advantage (as in Taylor, 1979 and Bulmer, 1980; see Table 1). TABLE 1
The advantage of sexual reproduction under frequency dependent disruptioe selection at a single locus N u m b e r o f families per patch
Proportional change in frequency o f sexuals
Sexual
Parthenogenetic
with 2 × disadvantage for sex
with 1 × disadvantage for sex
1 1 1 1 2 3 2
1 2 3 8 1 1 2
1-0467 0-9825 0"8847 0"6133 0"9696 0-9381 0"8966
1-1920 1"2284 1"2102 1-0855 1"0885 1-0476 1-1127
Including the two-fold disadvantage o f sex, which is normally obtained under a dioecious breeding system, the results indicate a small net advantage for sex, but only if there are no more than two families per patch. If there is more than one parthenogenetic family, the parthenogens have, together, enough variability to increase the number of times they produce the winner and hence, on average, beat the sexuals. If there are more sexual families per patch, we have a dilution effect in that an increased proportion of sexuals does not proportionally increase the chances of a sexual family producing the winner. Thus, there is a trade-off between producing variability, and larger numbers of offspring. Even with only very few families, the chances of producing the exact best adapted in both sexual and asexual families is very good, and hence the advantage o f producing variability is exhausted: it becomes better to produce larger families. With more loci it would take many more families per patch to reach this stage, because the best adapted genotype will be so much rarer. This is why, in Maynard-Smith's five-locus model, we can see an apparent increase in the advantage o f sex with more families. Maynard-Smith did not test a situation with large numbers of families, when we would expect sex to become disadvantageous. Bulmer used an infinite number of loci and found a greater advantage for sex when using equivalent numbers of families per patch than Maynard-Smith, as we would expect if the number o f loci were important. Additive Models We have a quite general set o f models dealing with truncation selection within patches; here, we will investigate a rather different method for selecting the survivor. We suppose that each genotype, g, has a characteristic U~ for its probability of
SIB COMPETITION
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emergence is correlated with survival within patches, there is an advantage to variability under both additive and truncation selection. If we assume that the relation between fitness potential and fitness is the same before and after emergence (i.e. additive before and after, or truncation before and after), then in both cases, this advantage is proportional to the within-family variance. In the extreme case where survival before and after are completely correlated, the advantage of variability per unit increase in variance is 1 (c.f. Table 2). However, such a strong correlation seems highly unlikely; in nature, the advantage of sib competition under additive selection is likely to be small. Secondly, under truncation selection models the advantage of variability increases with N, and decreases with R (for R > 2). This contrasts with the cases tabulated by Bulmer, where environmental variation was tow; there, the advantage increased with R, at least, for R < 10. Bulmer's Table 2d, where environmental variation is high showed a slight decline in the advantage for large R, which supports this conclusion. This difference in behaviour is important; when the environmental variance is low, the advantage of variability can be very high when many families compete in a patch. However, when the environmental variance is high, variability only becomes advantageous when a few large families compete. Thirdly, under additive selection, the genetic basis of the variance in fitness (in particular, the number o f loci) is irrelevant, whereas Taylor (1979) has shown that with truncation selection, unless the probability of an allele matching the corresponding environmental factor is half, the advantage of sex decreases with the number of loci. Furthermore, when only a few loci are involved, the advantage decreases still further since each family may then produce the best adapted genotype. Thus, the advantages given by Bulmer under truncation selection are upper limits with respect to the number of genes. So, both additive selection and environmental variation greatly reduce the power of sib competition to promote family variability. Although the mechanisms may still be important in promoting polyandry, which has no clear countervailing disadvantage, it seems unlikely that it could be strong enough to overcome strong obstacles, such as the two-fold cost o f sex. We would like to thank M. Bulmer for his helpful comments. R. J. Post was supported during this work by an MRC Postdoctoral Fellowship, and N. Barton by an SRC Postdoctoral Fellowship. REFERENCES BULMER, M. G. (1980). J. theor. Biol. 82, 335. FISHER, R. A. (1930). The Genetical Theory of Natural Selection. Oxford: Oxford University Press. HEBERT, P., WARD, R. • GIBSON, J. (1972). Genet. Res. 19, 173. MAYNARD-SMITH, J. (1978). The Evolution of Se~c Cambridge: Cambridge University Press. MILKMAN, R. D. (1967). Genetics 55, 493. MUKAI, T. (1977). In: Lecture Notes in Biomathematics 19, "Measuring selection in natural populations", (Christiansen, F. B. & Fenchel, T. M. eds). Berlin: Springer Verlag. P[REZ-TOMI~, J. M. & TORO, M. (1982). Nature 299, 153. TAYLOR, P. D. (1979). Z theor. Biol. 81, 407. WILLIAMS, G. C. (1975). Sex and Evolution. Princeton: Princeton University Press. WILLIAMS, G. C. & MrvI'ON, J. I. (1973). Jr. theor. BioL 38, 545.
Sibling Competition and the Advantage of Mixed Families N. H. BARTON
Dept of Genetics and Biometry, University College London, London NW1 2HE, U.K. AND
R. J. POST
Dept of Medical Entomology, Liverpool School of Tropical Medicine, Liverpool L3 5QA, U.K. (Received 2 July 1982, and in revised forrn 9 January 1986) It is noted that the sibling competition model for the evolution of sex and recombination, as it has been developed so far, involves truncation selection. After briefly reviewing aspects of the development and behaviour of such models an analytical treatment is presented which involves additive selection. Additive selection, as compared with truncation selection, decreases the advantage of sex to such an extent that it is unlikely that sibling competition could overcome its intrinsic two-fold cost, although it could still be important in promoting family variability produced by other mechanisms, such as polyandry.
Introduction Sexual reproduction allows the production o f genetic variation within families; the mother brings in this variation by diluting her genes with those o f the father, a process which gives sex an inherent two-fold disadvantage. It has been proposed that one factor which may have allowed sex to evolve despite this obstacle is sib competition; if siblings compete with each other as well as with unrelated individuals, variation between them will tend to raise the fitness o f the eventual survivor, and therefore give it a better chance against individuals emerging from less variable parthenogenetic families. Several specific models have been set up to quantify this idea, and to find out how severe sibling competition must be to allow sex to evolve (Williams & Mitton, 1973; Maynard-Smith, 1978; Taylor, 1979; Bulmer, 1980). In this paper we will extend these models in two ways, whilst remaining within their general framework. Firstly, we consider the effect of different selection schemes on the proposed advantage of variable families under sibling competition. All the above models rely on truncation selection, in which the individual best adapted to its particular environmental patch is the individual which survives. Although models o f this sort have been proposed in order to account for the maintenance o f extensive variation by balancing selection without excessive genetic loads (e.g. Milkman, 1967), we feel that it may be more realistic to consider genetic variation which is maintained by selection acting independently (i.e. additively) at many loci. In such an additive model the chance o f an individual surviving is the sum o f the effects of 381
0022-5193/86/120381+07 $03.00/0
© 1986 Academic Press Inc. (London) Ltd
382
N. H. BARTON
AND
R. J. P O S T
the many independent loci on fitness. The advantage of variable families seems likely to depend upon the mode of selection, and we shall try to compare this advantage under both truncation and additive selection. Secondly, many aspects of population structure will affect family variability. We will investigate one such feature, using both the truncation models of Bulmer et al. and our additive models to find the advantage of polyandry over monogamy; a female may produce more varied offspring by mating with several males. (The same advantage accrues to a male which mates with many females, but since polygyny directly raises males' fitness, and since males' progeny will usually be left in different patches, the effect of sib competition on polygyny will be negligible.) Furthermore, comparing these mating strategies within a sexual population will also allow us to consider more general mechanisms. Under many mechanisms an asexual population will maintain less variation than a sexual population, and so the two cannot easily be compared. For example, if heterozygotes are fittest, an asexual population may become purely heterozygous, and will then always displace a sexual population whatever the level of sib competition (see Hebert et al., 1972, for an example from nature). However, the type of mating system within a sexual population is less likely to alter the overall amount of variation and so the effect of heterozygous advantage, relative to disruptive selection between patches, can be compared unambiguously under sibling competition. We will consider a restricted class o f models, in which a large random-mating population leaves offspring scattered over many patches. In each patch there are R families, each containing N individuals. These compete to produce a single survivor, which enters the wider population to begin the next generation. We will therefore not consider models where the number of survivors depends on the family variation. For example, there might be competition for resources within patches such that mixed families produce more offspring even when isolated from all other families (e.g. Prrez-Tom6 & Toro, 1982). The outcome in such cases will depend on the details of the selection, and so generalisations would be difficult. Truncation Models
Before developing a model based on additive genetic variation we will describe the ways in which the current models based on truncation selection have been generalised by successive authors. Williams & Mitton (1973), and Williams (1975) described several models which gave a short-term advantage to sex; Maynard-Smith (1978) pointed out that the essential element in these models was not the unpredictability of the environment, which served only to maintain genetic variation, but rather, intense competition between siblings. He set up a specific model in which the environment varied in five ways; a patch could be A or a, B or b . . . . , E or e. This variability maintained polymorphisms at five corresponding loci, so that genotypes A A and A a were adapted to environment A, a a to environment a, and so on. The survivor in each patch was the individual which was adapted to the largest number of environmental features. Using Monte-Carlo simulation, it was found that sib competition gave sex
SIB COMPETITION
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VARIABILITY
383
an advantage whenever R and N were greater than 1, and that RN, the number of organisms competing in each patch, had to be greater than 30-40 for this advantage to outweigh the two-fold "cost" of sex. This conclusion clearly depends on the number of loci involved, the degree of dominance, and the way the loci interact to determine fitness. Taylor (1979) generalised the model analytically; he used the same assumptions as Maynard-Smith, but considered variation at a number of loci large enough that the number of genes which matched the environment in any individual could be assumed to be normally distributed. If the probability of an allele at any one locus matching the corresponding environmental factor was 0.5 (as in Maynard-Smith's model), Taylor showed that the number of loci did not affect the advantage of sex. If the matching probability was greater than 0-5, as would usually be the case, the advantage of sex would be less (since the genetic variation in fitness would be reduced) and would increase slightly with the number of loci involved. Bulmer (1980) further generalised the model, using the methods of quantitative genetics, to deal with variation in some underlying "fitness potential", generated by additive variation at many loci; the survivor in each patch was chosen as the individual with the highest "potential", as in the previous models. The exact mechanism by which variation was maintained was not specified, and would not affect the outcome. However, one might imagine it to be maintained by random variation in the optimum value of the "potential" between patches. Having summarised these truncation models we should make some points about their general behaviour. Firstly, they all assume that recombination does not reduce the average value of the underlying "fitness potential", as would happen if, for example, several loci determined the ability to match each environmental factor (see Maynard-Smith, 1978). This effect could give a strong disadvantage to sex (see Mukai, 1977, for an experimental illustration). Secondly, the advantage of variability arises solely from competition between individuals within patches: it is due to the fact that the expectation of a concave function of some variable (in this case, of the "fitness potential") increases with its variance. The survival and reproduction of individuals after they leave the patches is assumed to be independent of the outcome of this competitive process. Finally, the effect of the number of loci underlying the character was only explored by Taylor, and then only for a number so large that the normal approximation to the binomial could be used. If only a few loci determine the competitive components of fitness, each family might stand a strong chance of producing the best possible adapted genotype, and the advantage of variability would disappear. This effect of the number of loci can be illustrated by a simple one-locus computer model. Maynard-Smith's frequency-dependent disruptive selection model was slightly modified to cover only a single locus and analysed deterministically. Following Maynard-Smith, two alleles (one dominant to the other) at a single locus yield two phenotypes, each adapted to one of two possible types of patch. The initial phenotype frequencies were equal and matched the relative frequencies of the two patches. Sexual populations began in Hardy-Weinberg equilibrium. All possible combinations of a given number of sexual and parthenogenetic families in each
384
N.H.
BARTON
AND
R. J. P O S T
is completely inherited; the heritability of fitness depends on the degree to which U varies. In Bulmer's model, on the other hand, fitness is determined entirely by the ordering of the variates y; the heritability depends here on the heritability of y. Thus, in the first model, environmental variation is introduced through the random element in choosing which genotype survives, whilst in the second, it is introduced through variability in the ordering of "fitness potentials". To compare the two models, then, we must decide on what levels of environmental variation are equivalent; this is, in general, difficult. We will proceed by restricting the comparison to small genetic variances; in the truncation model, it seems unlikely that the ordering of the individuals would be anywhere near perfect. Taking the limit of Bulmer's model where a proportion ( 1 - q ) of the genetic variance in y, Vs, is between families, and a proportion q within families, we have
Vb=(1-q)V~
Vw=I-(1-q)Vs(Vg<
(5)
By inserting these values into Bulmer's formula (11), and taking the limit as Vg tends to zero, we find that the advantage of the new strategy is
RVs(q'- q) (tz~rN(o'2_l + 1) + p. R-I (o'2-1)) exp (--/z2-1/(1 + 0"2-1)) o.~vv/2~.(1 + ¢2_,)3
(6)
(Here,/~M and tr 2M are the mean and variance of the largest of M standard normal variates.) So, with truncation selection, the advantage is proportional to the change in the heritable component o f y within families ( Vg(q'- q)): just as with additive selection, the advantage is proportional to the change in the heritable component of U within families. Table 2 shows the advantage per unit increase in within-family variance, for various R and N. TABLE 2
The advantage of sex under truncation selection, per unit increase in within-family variance (see text for explanation) N u m b e r of families ( R ) N u m b e r o f siblings (N)
2
3
6
11
2 5 10
0"19 0-49 0"75
0"20 0"51 0"76
0.12 0.28 0"42
0"04 0-09 0-13
The first point to make is that if survival after emergence from the patches is independent of survival within patches, additive selection gives no advantage at all to variability: in general, the advantage to variability depends on the concavity of the relation between fitness potential and fitness itself. However, if survival after
SIB COMPETITION
AND
VARIABILITY
385
survival (we use U to denote this characteristic to emphasise that it is not equal to the actual fitness, W). If there are ajg individuals of genotype g in family j (j = 1 . . . R; ~s aj, = N ) , then the probability of survival of some individual from family 1 is X atgV~/EX ajgUg. (1) g
J g
With this linear relation between U and the probability of emerging from the patch, there is no immediate advantage in variability: the survival probability is simply proportional to the expected value of U. However, if we suppose that characteristics which favour survival within patches also favour survival or reproduction later in life, then variable families wilt be at an advantage. This is because the average value of U for a family is increased by selection within patches, in direct proportion to the variance in that family. Suppose that one family has a new strategy, which alters the distribution of genotypes, aig; we will denote this new strategy by '. If we assume that the change o f strategy does not directly increase the average, U, before competition, then Fisher's (1930) "Fundamental Theorem" shows that the average after competition is increased by (var' ( U ) - v a t ( U))/U.
(2)
This simple formula, which will be accurate as long as the strategies under consideration do not affect the mean fitness, can easily be applied to specific models. For example, if variation is maintained by heterozygote advantage, with fitnesses ( 1 s : 1 : 1 - t), at K loci, then the advantage of polyandry over monogamy is:
K(N-1)s2t 2 4N(s + t) 2 "
(3)
Similarly, if the variation is maintained by environmental variation, with fitnesses (1 + s : 1 : 1 - s) in half o f the patches, and (1 - s : 1 : 1 + s) in the rest of the patches, then the advantage is K ( N 1)s 2 (4) 16N -
In general, it seems that the final advantage will be small unless most of the variance in fitness is due to competition within patches, and unless success within patches is strongly correlated with success in later life.
Comparison with Truncation Models How does this model, which assumes additive selection, compare with those based on truncation selection? We will concentrate on Bulmer's (1980) infinite locus model, because it is the most general, and because the strategies are described in terms of the variances within and between families, as above. However, there is still a difficulty in comparing t h e two modes of selection. One of the most important parameters determining the effect o f sib competition is the proportion of the fitness variation which is inherited. In the additive model, fitness is determined by a variate U, which
386
N . H. B A R T O N
AND
R. J. P O S T
habitat patch were written down, and the overall frequency with which sexuals won was calculated. The proportionate increase of sexual families over one generation is given as a measure of their advantage (as in Taylor, 1979 and Bulmer, 1980; see Table 1). TABLE 1
The advantage of sexual reproduction under frequency dependent disruptioe selection at a single locus N u m b e r o f families per patch
Proportional change in frequency o f sexuals
Sexual
Parthenogenetic
with 2 × disadvantage for sex
with 1 × disadvantage for sex
1 1 1 1 2 3 2
1 2 3 8 1 1 2
1-0467 0-9825 0"8847 0"6133 0"9696 0-9381 0"8966
1-1920 1"2284 1"2102 1-0855 1"0885 1-0476 1-1127
Including the two-fold disadvantage o f sex, which is normally obtained under a dioecious breeding system, the results indicate a small net advantage for sex, but only if there are no more than two families per patch. If there is more than one parthenogenetic family, the parthenogens have, together, enough variability to increase the number of times they produce the winner and hence, on average, beat the sexuals. If there are more sexual families per patch, we have a dilution effect in that an increased proportion of sexuals does not proportionally increase the chances of a sexual family producing the winner. Thus, there is a trade-off between producing variability, and larger numbers of offspring. Even with only very few families, the chances of producing the exact best adapted in both sexual and asexual families is very good, and hence the advantage o f producing variability is exhausted: it becomes better to produce larger families. With more loci it would take many more families per patch to reach this stage, because the best adapted genotype will be so much rarer. This is why, in Maynard-Smith's five-locus model, we can see an apparent increase in the advantage o f sex with more families. Maynard-Smith did not test a situation with large numbers of families, when we would expect sex to become disadvantageous. Bulmer used an infinite number of loci and found a greater advantage for sex when using equivalent numbers of families per patch than Maynard-Smith, as we would expect if the number o f loci were important. Additive Models We have a quite general set o f models dealing with truncation selection within patches; here, we will investigate a rather different method for selecting the survivor. We suppose that each genotype, g, has a characteristic U~ for its probability of
SIB COMPETITION
AND VARIABILITY
387
emergence is correlated with survival within patches, there is an advantage to variability under both additive and truncation selection. If we assume that the relation between fitness potential and fitness is the same before and after emergence (i.e. additive before and after, or truncation before and after), then in both cases, this advantage is proportional to the within-family variance. In the extreme case where survival before and after are completely correlated, the advantage of variability per unit increase in variance is 1 (c.f. Table 2). However, such a strong correlation seems highly unlikely; in nature, the advantage of sib competition under additive selection is likely to be small. Secondly, under truncation selection models the advantage of variability increases with N, and decreases with R (for R > 2). This contrasts with the cases tabulated by Bulmer, where environmental variation was tow; there, the advantage increased with R, at least, for R < 10. Bulmer's Table 2d, where environmental variation is high showed a slight decline in the advantage for large R, which supports this conclusion. This difference in behaviour is important; when the environmental variance is low, the advantage of variability can be very high when many families compete in a patch. However, when the environmental variance is high, variability only becomes advantageous when a few large families compete. Thirdly, under additive selection, the genetic basis of the variance in fitness (in particular, the number o f loci) is irrelevant, whereas Taylor (1979) has shown that with truncation selection, unless the probability of an allele matching the corresponding environmental factor is half, the advantage of sex decreases with the number of loci. Furthermore, when only a few loci are involved, the advantage decreases still further since each family may then produce the best adapted genotype. Thus, the advantages given by Bulmer under truncation selection are upper limits with respect to the number of genes. So, both additive selection and environmental variation greatly reduce the power of sib competition to promote family variability. Although the mechanisms may still be important in promoting polyandry, which has no clear countervailing disadvantage, it seems unlikely that it could be strong enough to overcome strong obstacles, such as the two-fold cost o f sex. We would like to thank M. Bulmer for his helpful comments. R. J. Post was supported during this work by an MRC Postdoctoral Fellowship, and N. Barton by an SRC Postdoctoral Fellowship. REFERENCES BULMER, M. G. (1980). J. theor. Biol. 82, 335. FISHER, R. A. (1930). The Genetical Theory of Natural Selection. Oxford: Oxford University Press. HEBERT, P., WARD, R. • GIBSON, J. (1972). Genet. Res. 19, 173. MAYNARD-SMITH, J. (1978). The Evolution of Se~c Cambridge: Cambridge University Press. MILKMAN, R. D. (1967). Genetics 55, 493. MUKAI, T. (1977). In: Lecture Notes in Biomathematics 19, "Measuring selection in natural populations", (Christiansen, F. B. & Fenchel, T. M. eds). Berlin: Springer Verlag. P[REZ-TOMI~, J. M. & TORO, M. (1982). Nature 299, 153. TAYLOR, P. D. (1979). Z theor. Biol. 81, 407. WILLIAMS, G. C. (1975). Sex and Evolution. Princeton: Princeton University Press. WILLIAMS, G. C. & MrvI'ON, J. I. (1973). Jr. theor. BioL 38, 545.