Abstract
The relationship between breeding time and fitness in the Arctic skua is described in terms of a fitness function estimated from the empirical data. There is no significant variation in the fitness function between pairs who There is no significant evidence for selection between different phases of females. Models of the mechanism of sexual selection by female mating preferences are fitted to the distributions of breeding time. About 50 per cent of females must exercise mating preferences to maintain the differences observed. The predictions of the models are discussed in relation to the data at present available.

There is no significant evidence for selection between different phases of females. Models of the mechanism of sexual selection by female mating preferences are fitted to the distributions of breeding time. About 50 per cent of females must exercise mating preferences to maintain the differences observed. The predictions of the models are discussed in relation to the data at present available.
INTRODUCTION
THE Arctic skua is a sea bird that breeds in colonies from the North of Scotland up into the Arctic Circle. It is polymorphic with pale, intermediate and dark phases in its populations. The frequencies of the phases form a dine with pale phases generally more abundant in the northern parts of the range and dark phases more abundant in the southern parts.
INTRODUCTION
It breeds monogamously, though about 15 per cent of birds mating together for the first time change their mates in the next breeding season. After a pair have bred together for 2 or 3 years, however, they then normally stay together.
INTRODUCTION
The colony of Arctic skuas on Fair Isle in the Shetlands has been intensively studied for many years. The colour phases of the birds, their breeding time and breeding success and their changes of mate were recorded in detail. The data, which have already been partially analysed (O'Donald, 1962 (O'Donald, , 1972a Berry and Davis, 1970) , suggest that sexual selection may favour the darker males. Because the Arctic skua is monogamous, sexual selection cannot take place simply as a result of the favoured males mating with more females than the others. To gain an advantage in a monogamous species, a sexually favoured male must mate with a female who will bear him more offspring. Darwin (1871) put forward a subtle theory of how this could happen. He suggested that the females who are the first to breed at the start of the breeding season do so because they are in 33/1-A 1 a better nourished state than the others who breed later. For the same reason they would also be able to rear more and fitter offspring. The variation in the breeding condition of females would depend on how well they had survived the previous winter. The males who gained an advantage would then be those whom the females preferred to mate with. Or by direct competition with other males for territories, they might have the first choice of the females who were ready to breed. Thus they would gain a selective advantage by mating with the earlier females who would then bear them more offspring. O'Donald (1962) showed that, as Darwin's theory requires, clutch size and fledging success are greater among the earlier pairs of Arctic skuas. He later analysed the data (O'Donald, 1972a) in terms of a "fitness function" of the relationship between breeding time, measured by the date of hatching of the first chick, and fitness, measured by the number of chicks a pair managed to fledge successfully. A computer model of Darwin's theory, using the fitness function calculated for the breeding times of the Arctic skua, showed that the sexual selection is necessarily frequency-dependent. It is positively frequency-dependent if most of the females have a mating preference for a particular male phenotype: the selective advantage of the favoured males then increases as they increase in frequency. But if only a few of the females have mating preferences, the selection is negatively frequency-dependent, the selective advantage declining as the favoured males increase and spread through the population. Berry and Davis (1970) in their analysis of the data showed that among pairs breeding together for the first time darker males tend to breed earlier: dark males breed on average before the intermediates who breed before the pales. There is no significant difference, however, between the mean breeding times of the different phases of female. O'Donald (I 972c) used Berry and Davis's values of the means and variances of the different phases of males to calculate the selective coefficients of the sexual selection, assuming the fitness function was the same for all pairs. If the dark-phased males are given a relative fitness of one and a selective coefficient of zero, the selective coefficients of intermediates and pales, measured relative to the darks, are 0.13 and 0•34 respectively. Since new pairs are about 36 per cent of all pairs the overall selective coefficients of intermediates and pales are 0047 and 0l2 by this calculation.
INTRODUCTION
In this paper, the whole of the data, collected on Fair Isle, is fully analysed to determine the variation in both breeding times and fitnesses among the males and females of the different phases. The results are used to test O'Donald's models of sexual selection. Models of specific mating preferences by which the males are sexually selected are also tested by fitting them to the distributions of breeding times of newly mated pairs.
THE ORIGIN OF THE FAIR ISLE DATA
The skuas on Fair Isle were first studied by K. Williamson when he became Warden of the Fair Isle Bird Observatory. Williamson (1965) gives a general account of Fair Isle and its birds. P. E. Davis, who became Warden in 1957, continued the study. From 1958 to 1960, he was assisted by P. O'Donald who was then a research student supported by a Nature Conservancy Studentship. O'Donald (1962) in his Ph.D. thesis analysed the data he had helped to collect and it was from these data that the fitness function of breeding time, used in the computer models, was calculated (O'Donald, 1 972a). R. J. Berry very generously made available to us his tabulation of all the original data from Fair Isle. The data were then coded for storage on a disk-file in the Titan Computer of the University of Cambridge.
THE FITNESS FUNCTION OF BREEDING TIME
In order to calculate the mean fitnesses of different phenotypes, which are determined by their different distributions of breeding times, the overall relationship between breeding time and fitness must be known. Table 1 shows the empirical relationship between the breeding time given by the 
THE FITNESS FUNCTION OF BREEDING TIME
where w is the mean fitness of individuals breeding at time x. At x = 0, w is at its maximum value of 1 -cc. The mean and variance in fitness can be found in terms of parameters cc, K and 0 and the moments of the distribution of x. For relative fitnesses with a maximum at w = 1, we may put cc = 0 giving the relative fitness function w = where qb = 1/K. The mean relative fitness is therefore given by It is clear from earlier work (O'Donald, 1962) that new pairs with no experience together breed much later on average than pairs who have several years of previous experience breeding together. The new pairs produce fewer offspring. This may be a result solely of their later breeding; or they may produce fewer offspring even when they breed at the same time as experienced pairs. If their reduced fitness is solely caused by their late breeding, then the overall fitness function can be used to calculate the coefficients of sexual selection operating on new pairs. If not, a separate fitness function valid for new pairs would have to be used.
THE FITNESS FUNCTION OF BREEDING TIME
In order to compare the fitnesses of pairs with different breeding experience, it is necessary to group the data into weekly intervals. Even so, there are hardly any individuals with several years' experience in the last interval. Table 2 shows the values of the fitness in those intervals in which there were enough pairs to provide reasonable estimates of fitness. When there were too few pairs of greater experience they were lumped with those of less experience as the table shows. The mean fitness in each interval is simply the mean number of chicks reared by the total number of pairs in each interval. Heterogeneity of fitness with experience is tested by x2. Given the number of pairs of different experience in an interval, the number of chicks they are expected to produce can be calculated from the mean fitness of all pairs in the interval (assuming the null hypothesis that in a given interval there is no variation of fitness with experience). 2 cannot be There is a similar sequence in the variances, the pale males being much the most variable in breeding time and the dark males the least variable. But these differences between the phases almost completely disappear in pairs breeding for 2 or more years together. And no such differences are found between phases in females. These facts strongly suggest that sexual selection favours the darker males. Sexual selection can only take place when pairs mate for the first time. If the females prefer the darker males or if the darker males compete more successfully for the females, then the distribution of breeding times of the darker males should have an earlier mean and a smaller variance than the distribution of the paler males. But these differences in the distributions of breeding times should disappear after the first year of breeding. Table 3 shows that there are no significant differences in the means and variances of the breeding times of the phases after the first year. Given the means and variances of the breeding times, the fitnesses of the different phases can be calculated from the fitness function. 
Sp = O34
From the data given in this paper, the sexual selective coefficients can be calculated directly from the empirical fitnesses given in table 1. The breeding times of new pairs of males are shown in fig. 1 These are the coefficients of sexual selection. The overall differences in fitness between the phases can be calculated empirically. Tables 3 and 4 give a complete analysis of fitness by phase, sex and breeding experience. The mean fitnesses averaged over pairs with different breeding experience are shown in table 5. Most of the variation in fitness in the males comes from the new pairs and a considerable part of their variation in fitness is caused by their variation in breeding time and hence by sexual selection.
Sp = O34
An analysis of variance of the differences in fitness can be carried out treating the pairs with different breeding experience as " blocks" in a 
MATING PREFERENCES FOR THE PHASES
The selective differences we have demonstrated to exist between the phases of males are determined by the number of offspring the different pairs succeed in fledging. A considerable component of the selection depends on the breeding times of new pairs. This may be defined as sexual selection because it is caused by mating behaviour and not by fertility or the chances of survival. The mechanism which causes the variation in the chances of mating during the breeding season is not of course known. The females may have preferences for mating with dark or intermediate males caused by differences in the males' mating behaviour; or the males may compete with each other through their threat displays. Models of sexual selection are more easily described in terms of mating preferences while being valid for selection by direct competition between males. Mating preferences provide the more likely mechanism, however, and we shall describe four specific models, in these terms. data from different years have been adjusted to a common median, but this adjustment has only a slight effect. As table 3 proves, in new pairs, pales have a much later mean breeding time with a higher variance than intermediates and darks. But the difference almost disappears with more breeding experience. In fitting the models to the distributions, the breeding times were grouped into weekly intervals as shown in table 6. Model 1. This model has already been described in detail by O'Donald (1973a) . It was the model which showed that sexual selection in monogamous birds is necessarily frequency-dependent (O'Donald, 1 972b, 1 973a).
MATING PREFERENCES FOR THE PHASES
It assumes that there are two phenotypes of male. Some females, who represent a proportion of all females, have a mating preference for one of the male phenotypes. They always mate with the preferred males if they n n can, but they will mate with the other males if none of their preferred males are available. The remaining females, a proportion 1 -of the total, then mate at random with the two male phenotypes. In a given interval of time, i, in the breeding season, there is assumed to be a proportion p of females darks and intermediates were added together to form a class of not-pale males phenotypically preferred as mates to the pales. 08846. Since the marginal totals were fixed, 7 degrees of freedom were lost, and an additional degree of freedom was also lost by lumping the numbers in the first and second week. Thus 4 degrees of freedom were left. However, because the numbers of degrees of freedom are greater for the other models, we decided to use as an approximation to the log likelihood and obtain the 2-unit support limits of the log likelihood. The log likelihood is Model 2. Like model 1, this model uses a single parameter a for the proportion of females who have a mating preference. However the females with the preference mate first with dark males; if there are no darks left, they then mate with intermediates, and finally if no intermediates are left either, they mate with pales. The remaining 1 -a of the females mate at random among remaining unmated males. The minimum x2 of 1216 occurs at a = 0-12. The observed numbers were grouped into 13 classes from which 7 degrees of freedom have been lost. The minimum x2with 6 degrees of freedom is not quite significant representing a probability of P = 006. However the values of x2 are all well outside the approximate 2-unit support limits given by the other models. Model 2 is therefore much the least likely of the four models. 
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There is indeed clear evidence that the pales are pure breeding and therefore homozygous, but intermediates and darks may be misclassified as heterozygotes and homozygotes (O'Donald and Davis, 1959) . During the course of selection in model 4, the favoured allele starts with a selective advantage that is more or less additive, the heterozygote being at less of an advantage than the favoured homozygote. But as equilibrium is approached, the overall advantage passes to the heterozygote and maintains the equilibrium. In all the other models, the favoured allele spreads through the population to complete fixation.
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In using model 4 to simulate the distributions of breeding times it was assumed that y = 0. Then X2mln = 4.375 at the point a 0l8 = 029 which gives an equilibrium gene frequency OfPe = O68. The minimum x2 is slightly smaller for this model than for model 3: model 4 is therefore the model with the greatest likelihood. In both models the 2-unit support limits of the total mating preference U = The actual gene frequency of the dark allele is lower than this. It varies between males and females and is slightly higher in males as we should expect if the theory is true. And it can be estimated in two ways. We can assume that darks and pales are both homozygous and find the total proportion of dark alleles, or we can assume that the genotypes are approximately in the Hardy-Weinberg proportions and calculate the frequency of the pale allele from the proportion of pale phenotypes. The frequencies in females give the following estimates:
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= 0475 (assuming darks and pales are homozygous) = 05l8 (by Hardy-Weinberg Law).
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In males the corresponding estimates are:
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= 0558 (darks and pales homozygous) = 0586 (Hardy-Weinberg Law).
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These estimates were obtained from the numbers of different individuals in the skua colony. There are no significant differences in gene frequency in individuals with different breeding experience.
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The gene frequencies do not lie in the range of equilibrium values predicted by model 4. However in the Fair Isle population, equilibrium might not have been reached: the darks may still have been increasing in frequency as a result of selection. Any selective changes are certain to be slow from one year to the next for adult skuas live a long time: a generation may be as long as 10 years. This would explain why the frequencies of the phases remained more or less constant in the 14 years that the colony was (p(l_p) + (p(l_p) n / \
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The heterogeneity in mortality in phases of different sex is given by x = 292.
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studied. At the same time it is possible that natural selection resulting from differences in mortality between the phases maintains the equilibrium at its present frequency by opposing the sexual selection. As O'Donald (1973a) has shown, such equilibria are stable provided the mating preferences are not shared by more than 40-45 per cent of the females. An equilibrium maintained by the opposing forces of natural and sexual selection is stable only if the sexual selection is negatively frequency dependent. This is true of mating preferences involving no more than about 45 per cent of females. In both models 3 and 4, mating preferences involving only 19 per cent of the females are allowable within the 2-unit support limits. There is evidence given in table 7 of differences in mortality between the phases but the numbers of individuals are too few for the differences to be significant. If they were significant, the differences in mortality would certainly give rise to natural selection that could balance the sexual selection at a point of stable equilibrium.
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The four models compared. As we have shown, model 2 is the least likely, while model 4 is slightly more likely than model 3. Model 2 can certainly be rejected in comparison with the others because all its values of x2 lie well outside the 2-unit support limits of models 3 and 4. Fig. 3 Values of a and fi lying inside the closed areas are within the 2-unit support limits.
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The symbol" + marks the point of minimum x2 in model 4. The symbols x mark the points of minimum x2 in model 3 when a> 0 and a = 0.

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