seychelles warbler life history and model structure
The life history, study site and methodology for the Seychelles warbler study have been covered in detail elsewhere (Komdeur et al. 1995). In brief, this is a cooperative breeding species, endemic to the Seychelles archipelago. The warbler is purely insectivorous and maintains year-round territories. Territory quality was measured as estimated number of leaf insects present in a territory, and territories were divided into three categories of quality: low, medium or high (Komdeur 1992). The high-quality territories are in the island's centre, with medium, then low-quality territories forming approximately concentric surrounding bands (Fig. 1). Under natural conditions the size and number of these territories on the island remain approximately constant over the years (Komdeur 1996). Therefore, in this model we do not vary the numbers of these territories; a limitation that means the model can predict habitat occupancy close to demographic equilibrium, but cannot predict population dynamics in unsaturated environments. Although we do not incorporate changes explicitly in the sizes of territories, such changes are implicit in the model in that smaller groups have lower fecundities partly because they have smaller territories (Komdeur & Edelaar 2001b).
Territorial groups typically comprise a dominant pair (henceforth, simply: ‘breeders’), together with some retained offspring, 88% of whom are female (Komdeur 1999). Retained females (henceforth ‘helpers’) alloparent and achieve a minor share of the reproduction (Richardson et al. 2001). Retained males (henceforth ‘queuers’) rarely alloparent, with 78% instead attempting to acquire a breeding position on either their natal territory or on a territory adjacent to their natal territory following the death of the occupying male breeder (Komdeur & Edelaar 2001a). These queuing and helping strategies, open to offspring, lead to different dispersal choices. Male offspring must frequently choose between queuing and becoming non-resident ‘floaters’ in the low-quality habitat (Komdeur 1992), whereas females choose between the direct fitness benefits of their share of reproduction, combined with the inclusive fitness benefits of alloparenting, and the direct fitness benefits of seeking a territory vacancy. Together these decisions determine how many offspring are retained, or equivalently habitat occupancy.
The approach we use to predict habitat occupancy is an individual-based simulation. Dispersal costs are not included because distance to the vacancy has no effect on dispersal. Individual warblers of the various categories are known to sample different territories all over the island at a regular basis, by observations (Komdeur 1991) and by radiotelemetry studies (Komdeur J., Daan S., Madsen V. & Tinbergen JM, unpublished observations). For Seychelles warbler populations near saturation, there is usually only one nesting attempt in the spring (Komdeur 1996). Mortality occurs throughout the year, with some indication of a peak in the autumn during lean years (JK, unpublished data). Accordingly, a discrete time model framework was employed. Each year included the following sequence of events (to be read in conjunction with Fig. 2).
Figure 2. A schematic of the model structure used. Within the model death is a discrete event, thus we can derive population sizes only at their peak (immediately after breeding) and at their trough (immediately before breeding). The Seychelles warbler population is censused in December, with all adult birds being counted. Under the assumption that mortality rates do not vary seasonally, the average of Ni and is closely comparative with the census data, and it is this measure that we use throughout the paper.
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Death: when a randomly drawn number from the interval [0,1] was higher than the territory specific survivorships (Table A1), the individual died.
Dispersal (1): vacancies were filled by randomly selecting helpers, queuers or floaters from a pool of all those whose fitness would be increased by the move.
Eviction: helpers and queuers may leave their territories following the establishment of a new breeder (NB: the capacity of breeders to evict differs among model variants, see Table 1). Territories were considered in a random sequence, with helpers and queuers moving to either the best available breeding vacancy, or if there were none, to become floaters.
Table 1. Key to model codes and terminology. Model codes comprise two parts: the first (D?) indicates whether the breeder’s, or potential disperser's fitness is maximized by dispersals, and the second (ω?) the fitness measure used
|DIFD||All dispersal decisions maximize the disperser's fitness|
|Dij||Offspring related to breeders:|
|· Yearlings − all dispersal decisions maximize breeder's fitness|
| Helpers/queuers − all dispersal decisions maximize the disperser's fitness|
|Offspring not related to breeders:|
| Dispersal decisions may maximize either the breeder's or the disperser's fitness, with breeders able to control the dispersal of offspring of both sexes, or only the same sex, or neither. The following notation describes these scenarios; where ‘o’ denotes the fitness of the offspring, and ‘b’ that of the breeder, is maximized by the dispersal decision|
| 1st subscript (i) 2nd subscript (j)|
| Scenario abbreviation D0j D1j D2j Di0 Di1 Di2|
| Dispersal by queuers o o b o b b|
| Dispersal by helpers o b b o o b|
| The subscript's values follow from the annual probabilities of facing an incoming breeder, whose fitness interests would win out, are (1 – L0,1, or 2). Higher subscript values mean dispersal is more despotic|
|ωI||Lifetime reproductive success|
|ωII||ωI weighted by the depreciative effect, of a longer queue, on future offspring|
|ωIII||ωII weighted to account for male germlines typically staying in the same habitat for multiple generations|
|Breeder ||The dominant pair, who gain majority shares of the reproduction on each territory|
|Helper||Female residents, over 1 year old, who both help the breeders, and obtain minority shares of the reproduction|
|Queuer||Male non-breeding residents, over 1 year old, who have the prerogative to breeding vacancies on neighbouring territories|
|Floater||Non-breeders, over 1 year old, who float over the low-quality territories|
Breeding: for expected fecundities less than 1, if a randomly drawn number from a [0,1] interval was less than the expected fecundity, an offspring was born. Otherwise, births were drawn from a normal distribution using observed standard deviation data (Table A2). Births were then divided into males and females following Table A3, and added to the group as the next lowest helper or queuer. Following Richardson et al. (2001), a male breeder from a randomly chosen territory was given paternity 40% of the time, or else the dominant male took paternity. Following Richardson, Burke & Komdeur (2002), a helper randomly selected from the same territory was given maternity 26% of the time, or else the dominant female took maternity.
Dispersal (2): the year's offspring may disperse either to the best breeding vacancy or, failing this, to be floaters. They were considered in a sequence determined first by age (youngest first), and secondly by their territory (chosen randomly).
estimating fitness and model parameterization
This is an individual-based model in which individuals are born, have one or more opportunities to disperse, may become breeders, and finally die. With the exception of dispersal, all of these are determined directly by the empirical measures of survivorship and fecundity detailed in Tables A1–A3. Dispersal, by contrast, occurs when so doing increases an individual's fitness (which individual is an assumption we vary within the model; see Table 1), meaning that dispersal is determined only indirectly by survivorship and fecundity rates. Accordingly, we first derive estimates for the lifetime reproductive success of breeders and non-breeders in this system. Then, as lifetime reproductive success is founded on the simplification that all offspring are of equal value, we incorporate two weightings to generate a more accurate measure of fitness (also summarized in Table 1).
For a breeder with h helpers, in habitat k, with constant annual survival Lk and fecundity Mk,h lifetime reproductive success ωI is, by serial expansion:
- ((eqn 1))
where j= 0 before the breeding season and 1 after it. k is a habitat quality index. It varies from 1 to 4, not the 1–3 that might be expected given three habitat types (Fig. 1), because the prerogative to local vacancies that males enjoy means the quality of a territory depends not just on its intrinsic quality, but also on the quality of the adjoining territories. To this end, we assume all territories have six neighbouring territories with qualities as detailed in Table A4. Because some low-quality territories border medium-quality territories and some do not, we split this territory class into two: near-low (those adjoining medium quality territories) and far-low (those not doing so).
There is a simplification implicit in eqn 1 to the effect that current conditions are indicative of future conditions, in that an individual decides its dispersal preferences by choosing between the lifetime reproductive success with a group that is always of size x and a group that is always of size x − 1. Although this is not true for groups that are substantially too large, from the perspective of fitness maximization, it is approximately true for groups near to their optimal size. To predict habitat occupancy accurately, the key dispersal decisions are those where the costs and benefits are finely balanced, and this is the case only for those groups approaching their optimal size. Hence, although the constant-group-size assumption is in some cases flawed, these are not the cases that matter to predicting habitat occupancy.
Besides breeders, there are three other classes of individuals: helpers, queuers and floaters (described in Table 1). The choices they face are illustrated in Fig. 3. The fitness of helpers comprises their inclusive fitness benefits through alloparenting and their direct fitness benefits through cobreeding. Both these benefits can be derived directly from eqn 1, by incorporating relatedness discounts and share-of-reproduction discounts, respectively. Relatedness discounts reflect the average relatedness of a breeder to their offspring, i.e. 0·5 in models where there was no cuckoldry, appropriately less where there was. For both queuers and floaters, residual lifetime reproductive success is given by eqn 1 discounted by the probability of surviving long enough to obtain a vacancy. For queuers these probabilities were recalculated for each potential dispersal, depending on the number of local competitors (Fig. 3). While for floaters, the probabilities were estimated from the fraction of floaters that died while still floaters, averaged over the previous 10 years the model had run. Though highly robust (Houston et al. 1988), these dynamic-state-variable methods can be flawed. Specifically, these parameters might either converge on a stable value, but this value might be sensitive to initial conditions, or no stable value might be reached. However, we did not encounter either of these problems.
Figure 3. A representative sample of the dispersal decisions made within the model, illustrated with a subset of the territories modelled. In the left panel there is a high-quality territory, its two medium-quality neighbours, and the option of floating, and not being associated with a territory. Dispersal decision i relates to the newly fledged gamma male on the high-quality territory, who is deciding between staying and possibly ascending the queue, and leaving to float. Dispersal decision ii relates to the male beta on the high-quality territory, choosing between taking a vacancy on a neighbouring medium-quality territory, for which it must compete with all other male betas neighbouring the territory with the vacancy, and waiting for a vacancy on a high-quality territory. In the right panel there is a high-quality territory with three distant low-quality territories. Dispersal decision iii relates to the helper on the high-quality territory. The best female vacancy available is on a lower-quality territory, and thus she is choosing between this available vacancy and helping her parents. As the potential disperser in case i is newly fledged this is a Dispersal 2 decision (Fig. 2), while because the potential dispersers in cases ii and iii are adult helpers and queuers, respectively, these are Dispersal 1 decisions (Fig. 2).
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For individuals that chose to delay dispersal, there was a possibility that following the death of a parent, incoming breeders could evict them. When either a queuer or a helper wished to leave following the death of a parent, we assumed they could always do so. In cases where a helper or queuer wished to stay, but the incoming breeder's fitness was maximized by the helper or queuer leaving, we considered all possible model variants with respect to who's fitness was maximized by their leaving. These variants are summarized in Table 1.
Thus far we have described the simplest possible fitness estimates, because all offspring are treated as being of equal value. We now introduce two weightings which account for the variation in the prospects of offspring, and thus give more accurate measures of fitness. The first weighting depends on the number of queuers on a territory, because the number of queuing males increases and future male offspring will have to queue for longer before (possibly) obtaining a local breeding vacancy. This means the residual reproductive value of a breeder declines with the number of queuers, i.e. retained offspring depreciate future offspring. The calculation of offspring depreciation discounts (ODQ) to account for this effect are described in Ridley & Sutherland (2002).
The second weighting, Pk, also follows from the prerogative of queuers to local breeding vacancies. This localized dominance means that not only do breeders on the higher-quality habitat have relatively high lifetime reproductive success, but so too do their male descendants. Whenever there is variation in habitat quality, and offspring do not, in effect, join a pool from which they settle randomly across all habitat types, the fitness differentials among habitat types are underestimated by lifetime reproductive success (Rousset 1999). We use a dynamic state variable approach to calculate Pk. Pk were initially set to 1 in all four habitat types. A male born into each habitat was given an ‘allele’ detailing its natal habitat. These alleles were inherited by all the males’ offspring, and the model run until all individuals carried genes with the same value, i.e. the gene had gone to fixation. This cycle was iterated during a 100 000-year run of the model to yield the habitat-specific probabilities of new mutations going to fixation. These probabilities were then weighted by the habitat-specific proportion of offspring that were male and used as the starting point for the next iteration, and the whole process was iterated until fixation probabilities stabilized. Thus we have our most sophisticated measure of fitness:
- ωIII = ωII × Pk = ωI × ODQ × Pk((eqn 2))