where C is a genotype’s mean clutch size, a is the primary sex ratio (male/female, assumed to be 1 in birds), l_x is the probability of surviving from birth (in birds, from the laying of the egg) to age x of those individuals from successful clutches or litters, a is the average age of first breeding, w is the age of last breeding, and    is the mean number of broods successfully reared during a breeding season. (Eq. 2 can be used with data from either sex; by convention, I will discuss it with respect to female reproduction.)

is the annual reproductive success (ARS(b)), measured as the number of broods reared,

where P_i is the probability of rearing brood i, c_i is the number of clutches laid per female in producing brood i, and s_i is the probability that a brood i clutch produces any young at all (Murray 1991a, 1991b).

The annual reproductive success (ARS(k)), measured in number of young reared, is

where k_i is the mean number of young produced by a successful brood i clutch.

is the mean number of breeding seasons of the females hatched in successful clutches. The mean lifetime reproductive success, then, in terms of the number of broods reared, is the product of and . Lifetime reproductive success, in terms of number of young reared, is the product of ,   and k_i .

These measures of lifetime reproductive success are quite different from that reported in the books edited by Clutton-Brock (1988) and by Newton (1989), as pointed out earlier (Murray 1992a).

According to Wootton et al. (1991), with whom I agree (Murray 1992b), Eq. 2 must be true, regardless of whatever hypothesis on the evolution of clutch size one supports. The evolutionary problem is to explain the cause-and-effect relationships implied by the equation.

From Law 2 and Eq. 2 we can infer that the evolution of a larger clutch size (C) must result in a reduced lifetime reproductive success  ( ). Conversely, the evolution of traits extending life expectancy (therefore increasing    ) or increasing the number of broods reared during a breeding season  ( ) should result in the evolution of a smaller clutch size.

My argument is that a female that lays a clutch of N eggs has a lower probability of producing any young to nest leaving than if she had laid N - 1 eggs in her clutch (Fig. 1). Because birds lay no more than one egg per day and because the probability of a nest’s contents’ surviving from one day to the next is <1, the probability of a brood i clutch producing any young to nest-leaving (s_i) decreases as the clutch size becomes larger (Murray 1994, 1996), as indicated by,

                                                                                                                                                                                                                                (5)

where f is the first day on which young leave the nest, d_s is the daily survival of the nests’ contents, and i = 1, 2, . . ., n for first, second, and nth brood, respectively.

Evidence indicates that females are more likely to lay a replacement clutch following a small clutch than a large clutch (e.g., Slagsvold 1984; Smith et al. 1987; Tinbergen 1987; Lindén 1988) and that the time interval between consecutive clutches is shorter for smaller than for larger clutches (McGillivray 1983; Slagsvold 1984; Smith et al. 1987). Additional advantages of a small clutch could be better protection of young by parents after their leaving the nest (Safriel 1975) and the production of heavier young, which may increase the survival of young through their first year (Perrins 1965, 1980; Lack 1966; von Haartman 1971; Loman 1977; Murphy 1978; Garnett 1981).

The apparent benefit of a larger brood — that more fledglings are produced per successful clutch, ignoring the losses of failed clutches — is at least questionable, inasmuch as I have shown that a greater fecundity does not necessarily result in a greater Malthusian parameter (Murray 1997). Biologists, nevertheless, tend to think that natural selection favours the successful rearing of more young. The proposed paradox of hatching asynchrony, for example, is based on this assumption (Stoleson & Beissinger 1995), but the notion that the evolution of asynchronous hatching results in reducing the number of young produced from a nest seems paradoxical only when we think that rearing more young in a brood is an accurate indicator of fitness.

I propose these as laws, conjectures, or guesses, as Popper (1968, 1979) and Feynman (1965) call them, as being universally applicable statements from which other, testable statements may be deduced. Whether these statements form part of a satisfactory causal explanation depends on what kinds of predictions we can deduce from them and how well these conform to empirical fact. I have in fact discussed several predictions of this theory and provided supporting evidence elsewhere (Murray 1979, 1985, 1991b).

Although Law 3 states that natural selection favours a small (i.e., replacement) clutch size, we can see that selection for a smaller clutch size is a consequence of selection for greater lifetime reproductive success (i.e., ). Traits that extend life or increase annual reproductive success evolve because their Malthusian parameters are greater than those of the prevailing traits and necessarily (because of Law 2) lead to the evolution of a smaller clutch size. Put in yet another way, I am suggesting that natural selection favours genotypes that maximise their possessors’ probability of successfully rearing just enough offspring to sustain a population with a long-term growth rate of zero because, in evolutionary time, a population can do no better. A genotype for a clutch size smaller than replacement, of course, cannot sustain a population and soon becomes extinct.

With this background, what can we say about the evolution of incubation behaviour and hatching patterns? Once we get over the idea that natural selection favours production of more young, and instead we begin to understand that what is favoured are genotypes with the greatest Malthusian parameter, we can begin to look at what traits could lead to greater reproductive success and greater Malthusian parameters.

Let us consider now how natural selection for hatching patterns could maximise . We have already noted that a smaller clutch increases s_i (Fig. 1) and, therefore, P_i , but the evolved clutch size cannot be smaller than replacement. Evolution toward greater s_i is constrained by the necessity of producing enough young to sustain the population in the face of predation, disease, and other sources of mortality.

Early incubation and asynchronous hatching increase s_i , the probability of rearing any young from a clutch (Fig. 2), and, thus, should be the norm among birds. Variations on this idea are the ‘predation’ or ‘nest failure’ hypothesis (Tyrväinen 1969; Hussell 1972, 1985; Clark & Wilson 1981; Stoleson & Beissinger 1995) or, more cumbersomely, the ‘reduction-of-total-nest-failure’ hypothesis (Murray 1994). The s_i for a clutch of a given size can be increased by beginning incubation prior to the laying of the last egg (Fig. 2), but this has a cost of reduced probability of survival of later laid eggs (Beissinger & Waltman 1991; Stoleson & Beissinger 1997), reducing  . The number of young surviving from a successful clutch (k_i) could be increased by starting incubation with the last egg, but this has the cost of reducing the probability of rearing any young at all (smaller s_i). Thus, both  and cannot both be simultaneously maximised. Incubation should start early enough to increase s_i but late enough for the parent(s) to rear, on average, the number of young that maximises the Malthusian parameter. The variation in the start of incubation—some species beginning incubation with the first egg, others with the last, and others in between—is, I hypothesise, a balance between and in maximising .

In addition, because the survival of young from a successful clutch affects all subsequent values of l_x, we should expect to find adaptations that increase the number of young that leave a successful nest, increasing   . One such adaptation would be the reduction of brood size when the parents are confronted with a limited food supply (i.e., the ‘brood reduction hypothesis’ [Lack 1947, 1954]). According to my theory, asynchronous hatching and brood reduction are independent adaptations, the first increasing and the second increasing .

Unfortunately, the kinds of data necessary for testing this theory are lacking, but we can consider the extreme cases quasi-quantitatively. First, some species have precocial young, which leave the nest shortly after hatching. Selection should favour synchronous or nearly synchronous hatching, especially if chicks and parents move some distance from the nest shortly after hatching (Clark & Wilson 1981; Magrath 1990; Slagsvold 1990; Murray 1994). Synchronous hatching may seem contradictory to the theory, but synchronous hatching is favoured in precocial, nidicolous species because    would be greatly reduced if chicks hatched more than slightly asynchronously.

Incubation, however, does not always begin simultaneously for all eggs in nidifugous species. Variation in the start of incubation may be an incidental consequence of such things as warm temperatures stimulating development (Ward 1965; Jayakar & Spurway 1965) or parents being present on the nest while protecting its contents from predators (Bruning 1973) or other nest intruders (Beissinger et al. 1998). Nevertheless, in species with precocial young, synchrony of hatching seems enhanced by clicking sounds associated with breathing or vibrations emanating from the eggs in the few days before hatching (Vince 1969; Freeman & Vince 1974). These sounds seem to speed up or slow down development and synchronise hatching.

At the other extreme are species, such as the cavity-nesting Green-rumped Parrotlet Forpus passerinus, which begins incubating its large clutch from the day the first egg is laid (Beissinger & Stoleson 1991; Beissinger & Waltman 1991). This species has a long nest period, an average of 20 days incubation and 31 days brooding, before young leave the nest (Beissinger & Waltman 1991). In natural cavities, the mean s_i is about 0.44 (Beissinger & Bucher 1992). This corresponds to a mean d_s of about 0.984, assuming that one egg is laid each day and incubation and brooding periods in larger clutches are not longer than in smaller clutches. These assumptions are not exactly so (Stoleson & Beissinger 1997; Beissinger et al. 1998), but if we assume they are for the purpose of illustration, s_i is 0.4393 when incubation begins on day 1 and 0.3739 when it begins on day 10 (Fig. 3), a drop of 15%. Surely, it is better for parrotlets to begin incubation before the last day.

The specific day on which incubation should start is more difficult to predict, simply because we do not have enough information. The general prediction, however, is that incubation should start at the time that results in maximising .

In manipulation experiments, Stoleson and Beissinger (1997) showed that parrotlet parents were capable of rearing as many as eight chicks, even with simulated synchronous hatching. Although these results are interesting, I doubt that they falsify the theory because we do not know c_i or s_i for either natural or experimental clutches of different size.

In between the synchronously hatching precocial, nidifugous species and the completely asynchronously hatching parrotlet are the many species in which incubation begins after the first egg and before the last egg is laid. For example, in many passerines, incubation begins with the penultimate egg. Thus, most chicks are ready to leave the nest earlier than if incubation had started with the final egg, increasing the probability that one or more young would survive, for example, a predator- or weather-induced early departure from the nest (increasing s_i and   ), while decreasing the probability that all chicks will survive to nest-leaving (decreasing k_i and ). If no predators arrive and the weather is benign, nestlings could remain in the nest until the younger chicks are ready to leave, increasing k_i and . According to this hypothesis, is maximised.

From the theory we might predict that, among nidicolous species with the same nest period, asynchronous hatching should be greater in species with a lower probability of surviving another day (d_s) and, therefore, a lower s_i. This expectation may not occur, however, because we have to consider the effect of early incubation on k_i, that is on the success of late hatched chicks.

For example, the Prairie Warbler Dendroica discolor has an s_i of between 0.20 and 0.22, and the nest period is about 26 days for a four-egg clutch (Nolan 1978). From Eq. 5, mean d_s is about 0.942 (i.e., s_i on day 26 is 0.212). Both d_s and s_i are lower than in the parrotlet, yet hatching spans from less than 12 hours in some nests to more than 24 hours (Nolan 1978), so asynchrony of hatching is slight. Nevertheless, a head start of even a few hours may enable an earlier hatched nestling to survive a predator attack that occurs just before the later hatched chicks are ready to leave the nest. In the Prairie Warbler the advantages of an even earlier start to incubation may well be offset by increased probability of losses of later hatched chicks (i.e., a reduced k_i).

The nesting cycle of the Least Flycatcher Empidonax minimus is similar to that of the Prairie Warbler: from the laying of the first egg to nest leaving of the first chick is only slightly longer, about 28 days for a four-egg clutch (Briskie & Sealy 1989). Nest survival (s_i), however, is higher (0.38), as is mean d_s (about 0.965). Nevertheless, incubation begins earlier and hatching spread is longer (mean, 36.4 hours for four-egg clutch).

This between-species comparison of hatching success and degree of hatching asynchrony would seem not to agree with an apparent expectation from the theory—that a higher probability of nest failure should be associated with greater asynchrony. Just the reverse occurs in these species. Evolution, however, is a within-species process. We should be comparing the patterns between genotypes. Within each species we need to know which hatching pattern maximises , and we do not.

Within a species, in which mean d_s would apply to all members of a population, we should expect an earlier start to incubation and more asynchronous hatching in larger clutches, for which there is some evidence (Hussell 1972, 1985; Slagsvold 1986; Skagen 1987; Briskie & Sealy 1989), again increasing s_i.

Finally, with regard to theoretical expectations, this theory could explain the well-known within-species decrease in clutch size toward the end of the breeding season (Lack 1954, 1966; Klomp 1970) and the possible earlier start to incubation and greater asynchrony in late season broods compared with early season broods (Gibb 1950; Slagsvold 1986; Skagen 1987). A smaller clutch and earlier incubation would each increase s_i, which would increase the probability of reproductive success of the individual bird and increase for the genotype. Thus, the reduction in clutch size and the earlier incubation could be adaptations that increase the likelihood of successful breeding as the breeding season draws to a close. It is better to rear a few young rather than to rear no young at all in trying to rear a bigger brood.

In order to test these predictions we need high quality natural history data, such as good empirical values for l_x (the probability of surviving from birth to age x of individuals from successful clutches), a (mean age of first breeding), c_i (the number of clutches laid per female in producing a successful brood i clutch), s_i (the probability that a brood i clutch will be successful), P_i (the mean number of brood i clutches reared per female), and k_i (the mean number of young reared in successful brood i), especially for clutches of different size. We need these data because they are the parameters that are an explicit part of the theory, and we could make much better guesses about the effect of differences in these parameters on the Malthusian parameter, but these data are essentially unreported in the literature. The theory cannot be tested with idiosyncratic measures of ‘reproductive success’ or ‘fitness.’ Many investigators with long-term life-history data could calculate values for these parameters.

Second, we need accurate data on the lengths of the incubation and brooding periods. I think it is possible that young birds may remain in the nest after they are physically capable of living outside the nest. I am sure that many ornithologists have had the experience, as I have, of approaching a nest, only to have the young explode out of it with the slightest touch. Were these young ready to leave the nest the day before but remained in the nest until the youngest chick was ready, or did I ‘fledge’ them prematurely (i.e., before they were ready)? I think they were ready but waiting. In this regard, Briskie and Sealy (1989) report that the oldest Least Flycatcher nestlings usually do not leave the nest until a day or two after they are capable of leaving successfully. Therefore, I wonder whether, when we are careful to measure incubation and brooding periods only in ‘undisturbed’ nests, we may in fact be overestimating the nest period. Fully ready chicks waiting in their nests for siblings to complete development may be an adaptation that increases k_i (and ) without much additional risk to themselves.

Without more accurate data, testing any theory could only be difficult. I point out, however, that a hypothesis, applicable to species A but not to species B, and another hypothesis, applicable to species B but not to species A, are not falsifiable because each is ad hoc (Popper 1968, 1979). All we could do is compile lists of species for which hypothesis A or hypothesis B is applicable. With regard to incubation behaviour and asynchrony of hatching we would have 20 such lists (Stoleson & Beissinger 1995; Slagsvold et al. 1995). My philosophy of science makes me favour my universal theory.

By approaching the problem of understanding incubation behaviour and asynchronous hatching from the point of view of Popper (1968, 1979)—by constructing falsifiable hypotheses in the form of the deductive-nomological model (Hempel & Oppenheim 1948) and utilizing laws assumed to be universally applicable—I propose that early incubation and asynchronous hatching are adaptations that increase the probability of rearing any young from a clutch (i.e., increases s_i and ), which has been referred to as the ‘predation’ or ‘nest failure’ hypothesis (Clark & Wilson 1981; Hussell 1972, 1985; Stoleson & Beissinger 1995) and the ‘reduction-of-total-nest-failure’ hypothesis (Murray 1994). Brood reduction is an adaptation that maximises the number of young that can be reared from a successful brood (i.e., increasing k_i and ). The incubation and hatching patterns are fine-tuned, and thus vary among species, by selection favouring the combination of behaviours that has the greatest . This variation results from specific differences in length of the total nesting period, intensity of predation, inclement weather, and other sources of nest loss, the length of the breeding season, and many other factors.

I thank S. R. Beissinger for his helpful comments on an earlier manuscript.

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