## Introduction

Protandry refers to earlier arrival of males than females (Morbey & Ydenberg 2001). Here ‘arrival’ may broadly mean completion of a migratory journey (Myers 1981), production of offspring of different sexes (Kranz *et al*. 1999), emergence from a developmental stage (Fagerström & Wiklund 1982) or other similar processes. Protandry is found widely in migratory birds, while its opposite − protogyny, earlier arrival of females − has been found in only a few sex-role-reversed bird species (Oring & Lank 1982; Reynolds, Colwell & Cooke 1986).

In their review of seven different hypotheses to explain protandry in various taxa, Morbey & Ydenberg (2001) state that two have become accepted for migratory birds. The hypothesis with the strongest support so far for migratory, territorial birds is the rank advantage hypothesis (*sensu *Morbey & Ydenberg 2001). This hypothesis is based on the benefits of gaining priority access to territories. If early males obtain superior territories selection will advance arrival dates, in some cases up to quite risky periods in the year (Kokko 1999). Using the terminology of Morbey & Ydenberg (2001), selection for protandry is indirect under this hypothesis. The arrival times of the two sexes evolve independently and protandry is a consequence of stronger selection on early arrival in males than in females.

The mate opportunity hypothesis (*sensu *Morbey & Ydenberg 2001) also has some support in the ornithological literature. This hypothesis has its roots in studies of insect emergence times, and has attracted substantial attention by theoreticians. Female fitness depends typically on the number of matings less strongly than male fitness (Andersson 1994). If males arrived simultaneously with females (but with some variation around the mean), individual males would lose mating opportunities as they cannot mate with females who are receptive before the male has arrived or emerged (Fagerström & Wiklund 1982; Bulmer 1983; Iwasa *et al*. 1983; Morbey 2002). Thus males arrive earlier. Morbey & Ydenberg (2001) classify the mate opportunity hypothesis as a direct selective advantage, the relative timing of male and female arrival being directly under selection.

Morbey & Ydenberg (2001) point out that the hypotheses for protandry in birds are not mutually exclusive, that future modelling work would benefit from considering multiple selective pressures simultaneously and that theoretical work has concentrated mainly on the mate opportunity hypothesis. The problem is particularly severe for studies of migratory birds, where the rank advantage hypothesis is widely believed to be appropriate. Morbey & Ydenberg (2001) cite Kokko (1999) for providing the theoretical backbone for the rank advantage hypothesis, yet the model by Kokko (1999) does not consider the difference between male and female arrival times. Instead, it includes only competition within a single sex, usually interpreted as males. However, female fitness will suffer similarly if late arrival forces females to occupy poor quality breeding sites (Bensch & Hasselquist 1991), thus both sexes should arrive early to occupy the best territories (Smith & Moore 2005). No theoretical work to date explicitly predicts arrival time differences based on the rank advantage hypothesis.

The goal of the current paper is to fill in this gap and to model multiple selective pressures. We employ two different methods with slightly different assumptions and research foci. Modelling results become considerably more robust if similar conclusions can be drawn from two completely different approaches. The models allow us to assess the prospects for protandry when the rank advantage hypothesis is operating alone, or together with processes that relate to the mate opportunity hypothesis, such as biased adult sex ratios or extra-pair paternity that leads to sperm competition.

### modelling arrival times of migratory birds

We present first a simplified numerical analysis, then proceed to an individual-based simulation. Any modelling method has advantages and drawbacks (Levins 1966). In our particular case, an individual-based simulation allows more flexibility and thus heightened biological realism, but it also has the unavoidable drawback that precise conclusions − such as determining the exact direction of protandry vs. protogyny when differences in arrival times are small − are hard to achieve (e.g. Łomnicki 1999). It is therefore good practice to begin with a simpler modelling technique to expose the logic of an argument. In both models a bird's strategy is its ‘target’ arrival date, denoted *t*, and small values of *t* indicate that the bird arrives early, on average. Also, in both models no bird can decide on an exact arrival date because chance events such as adverse weather conditions or other environmental factors can cause delays.

### model 1. numerical analysis of quality-dependent arrival times

In our first model, we derive predictions for a population consisting of four different types of individuals: in addition to being male or female, individuals of either sex are also of high or low quality. We also divide territories into two quality categories: good and poor. The model tracks changes in numbers of birds on each day from *d* = 0 to *d* = *D*, including deaths, and calculates the expected numbers of individuals in each category that result in one of 16 (for males) or 12 (for females) different states (Appendix I). The reason for a different number of states for the two sexes is a sexual difference in territory acquisition behaviour: only males can occupy a territory without yet having found a mate. Therefore, for example, the state ‘low quality male on a poor territory without a female’ is possible, while females cannot have a territory without a mate.

Prior to the onset of spring migration, there are *n*_{M1}, *n*_{M0}, *n*_{F1} and *n*_{F0} individuals (high- and low-quality males, high- and low-quality females, respectively). The habitat has *T*_{1} good and *T*_{0} poor territories. If *n*_{M1} +*n*_{M0} > *T*_{1} + *T*_{0}, and if many males survive migration, it is possible that some males cannot breed at all; analogously for females (*n*_{F1} + *n*_{F0} > *T*_{1} + *T*_{0}).

Individuals know their own quality, and can make migration decisions accordingly. The ‘decision’ is modelled as the target arrival date, *t*, which is given independent values for each of the four classes of individuals, using individual category as a subscript (e.g. *t*_{M1}). On each day of a migration season that lasts from *d* = 0 to *d* = *D*, non-arrived birds move to one of the ‘arrived’ states (Appendix I) according to the geometric probability distribution *P*(*d*, *p*) (Fig. 1). For example, the number of high-quality males that arrive on day *d* is *P*(*d*, *p*_{M1}) *n*_{M1}. The interpretation of the geometric distribution is that significant numbers of birds arrive as soon as the migration season begins, but the exact fraction arriving on day 0, as well as the length of the ‘tail’ of later-arriving birds, depends on the target arrival date. The above definition makes the average arrival date equal the target date, *t*.

After having arrived, the bird may undergo several changes in state. The state changes are detailed in Appendix II; here we give a verbal description. On each day, deaths occur among arrived birds. Corresponding to the biologically realistic assumption that environmental conditions improve as spring progresses, we assume that the daily mortality risk follows a declining function α exp(–β*t*), where 0 < α < 1 and β > 0. Parameter α indicates mortality on day 0, and parameter β the speed of the decrease of daily mortality from its value at day 0. Following Kokko (1999), these two parameters can take different values for birds of different quality such that high-quality birds suffer less from adverse conditions early in the season (thus they are expected to have lower α and higher β than low-quality birds). Surviving males who do not yet have a territory may then obtain one, but it may take some time before territories are found and ownership is settled; only after this period is it possible for a male to acquire a female. The rate of settling, *g*, differs between high- (*g*_{1}) and low-quality males (*g*_{0}).

Similar to settling in a territory, pair formation is assumed to take time: for a given number of settled males and arrived females, the number of pairings is proportional to the parameter 0 < γ < 1 (as a borderline case we may set γ = 1, which assumes minimal delay: all available individuals mate on their first day if opposite-sex individuals are available). Within each day pairings occur in an ordered fashion, which reflects mate choice: males residing on good territories are mated first; within a territory, quality class males of high individual quality are mated first, and within each territory–male combination, females of high quality are allowed to pair first. The exact number of pairs that follows from these assumptions is given in step 4 of Appendix II. This order of events reflects our assumption that females base their choice primarily on good quality habitat, but that they also pay attention to mate quality, and that high-quality females have priority access to territories and mates. The process is repeated for each day of the arrival season, after which the fate of birds is known and fitness can be assigned: the model tracks how many birds are alive and paired, and the quality of the territory and mate. Having a high-quality mate and/or a good territory by the end of the migratory season both independently improve fitness (for a full list of fitness values see Appendix I). Expected fitness is the weighted mean of fitness of individuals in each state. The weights are the state-specific numbers of individuals at the end of the arrival season, which simultaneously indicates the probability that a single individual ends up in the focal state. Individuals who died obviously have zero fitness in the calculation.

Settling, pairing and deaths are modelled as fractions of individuals changing state, which means that the model incorporates biologically realistic stochasticity: despite her priority access, a high-quality female sometimes mates later than a low-quality female. We are able to calculate exact fitness consequences as the arrival date distribution and the expected fates of individuals can be computed precisely, and we always found a single stable equilibrium for target arrival dates *t*_{M1}, *t*_{M0}, *t*_{F1} and *t*_{F0}.

A typical model outcome shows the importance of the adult sex ratio (Fig. 2). In Fig. 2a, 100 males compete for 100 territories, while the number of females varies from 50 (relative number of females *x* = 0·5, Fig. 2a) to 200 (*x* = 2). Male-biased sex ratios in Fig. 2a yield a positive difference between female and male arrival dates, i.e. protandry. High-quality males arrive first, followed by either high-quality females or low-quality males, depending on the exact value of *x*. With more than 0·9 females per male, the patterns are reversed: within each quality class, females arrive before males. Conclusions remain similar when male rather than female numbers are varied (Fig. 2b): male-biased sex ratios lead to protandry, although once again the switchpoint is not exactly at a 1 : 1 sex ratio.

Thus, the more numerous sex has to compete for breeding opportunities more intensely, and arrives earlier. In this respect, Fig. 2 could be seen to simply describe the conditions for sex-role reversal in arrival times (Reynolds *et al*. 1986). However, we built into the model an assumption of sexual asymmetry: males gain territories first, and females settle only in territories that are already defended by a male. Therefore, the model is not a simple symmetrical description of sex roles. Instead, it examines how an assumed asymmetry in sex roles concerning breeding site acquisition is reflected in sex-specific arrival times. If the rank advantage argument explains protandry, our model should predict that overall males arrive earlier, unless other factors such as a strongly female-biased sex ratio interfere.

Figure 2 provides little evidence that the rank advantage hypothesis *per se* promotes early male arrival beyond that of females. First, even though we assumed that males need time to settle in territories, the zones of protandry (positive value of ‘overall difference’, Fig. 2) are not larger than those of protogyny. Secondly, the rank advantage hypothesis should operate at its purest at sex ratios of unity: as soon as there is a sex ratio bias, the mate opportunity hypothesis interferes as one of the sexes experiences limited access to mates. However, unbiased sex ratios predict protogyny (Fig. 2), females arriving on average 1 day before males.

Figure 2 uses particular parameter values, and one could hypothesize that if settling on territories and/or mate acquisition take longer than assumed in Fig. 2, the rank advantage hypothesis could become stronger. To avoid presenting a narrow focus of particular parameter combinations only, we ran the model at unity sex ratio (100 males, 100 females competing for 100 territories of variable quality), using a set of 1000 randomly and independently chosen parameter combinations in the biologically feasible range. All the following were chosen as independently and uniformly distributed random numbers (range in brackets): the proportion of males that are high quality (0 … 1), the proportion of females that are high quality (0 … 1), the proportion of good territories (0 … 1, the total number of territories being 100), length of the arrival season *D* (20 … 60), pairing speed γ (0 … 1) and the fitness benefits *w*_{T}, *w*_{F} and *w*_{M} (1 … 5, 0 … 0·2 and 0 … 0·2, respectively). Mortality parameters α and β were drawn independently twice (0 … 1), with the larger value of α and the smaller of β chosen to represent poor quality individuals; similarly for the rate of settling on territories *g* (range 0 … 1, lower of the two values chosen to represent low-quality individuals).

Randomized trials with unbiased sex ratios failed to produce protandry more often than protogyny (Fig. 3a), and yielded no support for the idea that longer periods of settling or pair formation favour protandry. If anything, faster settling of low-quality males shows a statistical relationship with protandry, but this effect is too weak for any degree of biological significance (Fig. 3b). Repeating this procedure with varying sex ratios, however, shows that adult sex ratio performs well when predicting protandry or protogyny (Fig. 3c). The exceptions (i.e. the anomalous solutions in the top-left or bottom-right of Fig. 3c) do not have territory settling or pairing rates that differ from the distribution from which they are drawn (*P* > 0·05 in *t*-tests testing against an exactly known mean). Instead, exceptions occur when high- and low-quality individuals are very different in terms of territory settling rate *g* or mortality patterns α and β, if the sex ratio differs little from 0·5, if the arrival season is short, or the fitness effects of being paired to a high-quality bird are small (*P <* 0·05 in each case; however, multiple variables were tested in Fig. 3, and the last two variables are not significant if a Bonferroni correction is applied). These exceptions shift protogyny to protandry or vice versa approximately equally often, thus the rank advantage hypothesis produces no consistent bias towards protrandry. Trials with distributions other than a geometric distribution for the arrival date did not change this conclusion (not shown).

### model 2. individual-based simulation with extra-pair paternity

Models that assume that males can mate with multiple females often produce protandry (Bulmer 1983; Iwasa *et al*. 1983). In socially monogamous birds, multiple mating means sperm competition and extra-pair paternity. We built an individual-based simulation that allows sperm competition to take place, which also allows us to examine possible limitations of the numerical approach above.

We assume a population that consists of a variable number of males and females. The probability of arriving on day *d*, conditional on not having arrived yet, equals *P*(*d*, *t*) = {1 + exp[−2(*d–t*)]}^{−1}. The constant 2 in the expression specifies the shape of the arrival curve, and is chosen such that birds are given the ability to control their arrival date much more precisely than in the previous numerical analysis (Fig. 1). Each individual possesses two loci controlling their target arrival date *t*. One, *a*, is expressed in males only and specifies the male's target arrival date (see Fig. 1 for the relationship to the actual date), the other, *b*, is the target date expressed in females. Both loci can take any allelic value between 0, the start and *D*, the end of the migration season. The conditional probabilities imply that the mean arrival date in the second model is close to, but not exactly equal to, the target date; we report mean arrival dates as these correspond to what can be observed in nature.

The details of the arrival process are given in Appendix III; the main assumptions are summarized here. We assume a constant number (*T*) of breeding sites, that vary in quality from 1 to *Q*_{max} (where *Q*_{max} is an integer ≥ 1) as an integer-valued uniform random distribution, i.e. the probability of each value is 1/*Q*_{max}. As in model 1, we assume that daily mortality on breeding grounds declines exponentially with time *t*, the rate of decline given by a parameter β (step 4 in Appendix III). For simplicity, we ignore quality differences between individuals and do not include variation in mortality on day 0 (i.e. model 2 includes no α). Biases in the adult sex ratio are introduced by setting upper ceilings for the numbers of females and males that can exist in the population (step 10 in Appendix 3).

Arrived females choose mates among males who have a territory but not yet a social mate. The proportion of extra-pair young is *p*_{E}, such that each offspring produced by the female is an extra-pair young (EPY) with probability *p*_{E}. The extra-pair sire is chosen randomly among the males who have arrived, and are alive, at the time the female arrives on the breeding grounds. The offspring have mutations in the loci that determine migratory behaviour with a low probability µ.

For visual clarity, the simulation results are depicted as the average arrival time difference between females and males, given as the average genotypic value of *a* minus the average genotypic value of *b* after a minimum of 100 years of simulation in 50 independent replicates. Note that although the target arrival date does not strictly equal the mean of the arrival date distribution, the differences between two target dates equal the average differences between two observed dates. The arrival time difference is positive if males arrive first (protandry), and negative values indicate protogyny. To ensure that distributions had stabilized, simulations were run until the correlation coefficient of the arrival time difference against time (generations) was in the range 0 ± 0·005; however, simulations were always run for at least 100 generations.

When there is no extra-pair paternity, the results (Fig. 4) are in line with the numerical approach (Fig. 2). Unbiased sex ratios yield no significant protandry or protogyny (Fig. 4a: protandry = −0·13 ± 0·10, Student's *t* = −1·31, *P* = 0·20; Fig. 4b: protandry = 0·06 ± 0·07, *t* = 0·87, *P* = 0·39). Male-biased sex ratios lead to protandry, and female-biased ratios lead to protogyny (Fig. 4, filled dots). Pronounced protogyny is only found in Fig. 4b, where males are in short supply in such a way that the total number of males falls below that of territories. In the rightmost points in Fig. 4a males are similarly in short supply, but now the number of males equals that of territories. Both sexes are thus limited by the availability of good habitat, and are selected to arrive almost equally early.

Increasing the proportion of EPY always shifts the solutions towards protandry (Fig. 4). However, very high proportions of EPYs are required to overturn the effects of adult sex ratio. At 25% EPY there is a discernible effect on protandry, but one that that makes little qualitative difference. Fifty per cent EPY yields protandry regardless of adult sex ratios.

Trials with different choices for parameters β, µ, *Q*_{max} and *T* yielded similar results. To reach an overview of the effects of these parameters, we investigated their joint effect in a randomized trial similar to that used in model 1. We ran single simulation runs with a set of 500 choices for the rate of mortality decline over time, β (uniform distribution with range 0 … 1), mutation probability µ (0 … 0·2), number of males *N*_{M} (0 … 0·5), number of females *N*_{F} (50 … 200), proportion of EPY *p*_{E} (100 … 200), number of territories *T* (50 … 200) and the variation in territory quality *Q*_{max} (1 … 10, integer values only). Consistent with Fig. 4, the degree of protandry increased significantly with male numbers *N*_{M} (*r* = 0·25, *P* < 0·001), proportion of EPY *p*_{E} (*r* = 0·11, *P* = 0·018), and decreased with female numbers *N*_{F} (*r* = −0·55, *P* < 0·001) and with available variation in territory quality *Q*_{max} (*r* = −0·19, *P* < 0·001). The effect of other parameters remained non-significant. Protandry remained linked strongly to adult sex ratio but was overall more common than protogyny, as the randomized trials included sperm competition with an average proportion 25% of EPY (Fig. 5).