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The evolution of brood reduction by siblicide in birds
J. theor. BioL (1990) 145, 163-175
The Evolution of Brood Reduction by Siblicide in Birds H. C. J. GODFRAYt AND A. B. HARPER:[:
t Department of Biology and Centre for Population Biology, Imperial College at Silwood Park, Ascot, Berkshire SL5 7 P Y and ~ Department of Zoology, South Parks Road, Oxford OX1 3 PS, U.K. (Received on 8 August 1989, Accepted in revised form on 9 March 1990) The young of a number of bird species frequently kill their siblings whilst still in the nest. We develop simple single-locus genetic models of the evolution of siblicide and non-siblicide in bird populations. Two cases are distinguished: (1) where there is an obvious runt which, when siblicide occurs, always perishes and (2) where a siblicidal individual chooses a victim at random from its nest mates. The genetic models are compared with an inclusive fitness analysis of siblicide due to O'Connor (1978, Anita. Behav. 26, 79-96). In the runt model, the genetic analysis and inclusive fitness analysis yield the same results though the genetic analysis reveals a large area of parameter space where both strategies are evolutionarily stable strategies (ESSs) but only one is a continuously stable strategy (CSS). In the model without a runt, the two analyses give different results and for a large area of parameter space both siblicidal and non-siblicidal behaviours are stable to invasion (both ESSs and CSSs) so that the observed state of a population will depend on its past history.
Introduction Siblicide occurs in birds when one member of a brood causes the death of a sibling. While not a c o m m o n feature o f bird biology, sibling aggression that culminates in the death of a nest mate has been recorded from a wide variety of taxa including pelicans, gannets, cranes, skuas and herons, though it is best known from large raptors (Lack, 1968; O ' C o n n o r , 1978). There are a number o f hypotheses about the selective advantage of siblicide. Lack (1966) suggested that siblicide may be advantageous to both parents and surviving offspring in years when environmental conditions make it difficult to successfully rear a complete brood. Once siblicide became established, parents might be selected to lay large clutches in the expectation that siblicide would lead to brood reduction in all but the very best years. A second reason why a mother may lay larger clutches than her y o u n g will tolerate is to compensate for any infertility in her eggs (Stinson, 1979). It has also been suggested that asynchronous incubation which leads to a hierarchy o f chick sizes may be an adaptation to facilitate early and efficient brood reduction. Hamilton (1964) and Trivers (1974) pointed out that there may be a conflict of interest between parents and offspring over such questions as when brood reduction should occur. The first analysis of this conflict in birds was that by O ' C o n n o r (1978) who used inclusive fitness arguments to determine the level o f benefit to the surviving offspring that would lead to selection for siblicide. At about the same time Stinson 163
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(1979) used a simple genetic model in his study of brood reduction in eagles where siblicide is particularly common. The value of the inclusive fitness approach is that the evolution of a trait can be studied without a detailed knowledge of its genetic basis (Hamilton, 1964). However, inclusive fitness theory assumes that the traits in question have relatively small, normally additive effects on fitness (e.g. Michod, 1982; Grafen, 1986), assumptions that are very likely to be untrue in the case of siblicide. Indeed, simple genetic models of the spread of siblicide and non-siblicide (of the type first used by Stinson) can give very different results to inclusive fitness arguments and explicitly genetic models have proved necessary in the analysis of other forms of sibling competition and parent-offspring conflict (e.g. Parker & MacNair, 1978, 1979; MacNair & Parker, 1978, 1979; Charnov, 1978; Stamps et al., 1978; Feldman & Eschel, 1982; Parker, 1985; Harper, 1986). In this paper we investigate the conditions that allow the evolution of siblicide and, in particular, ask how great a benefit must accrue to surviving brood members to lead to selection for siblicide. Our conclusions are based on simple but explicit genetic models which we compare with inclusive fitness arguments. We find that the predictions o f the model are very dependent on the presence or absence of a hierarchy in chick sizes and an easily eliminated runt. When a hierarchy is present, the normal case in birds, the inclusive fitness and genetic arguments give the same results. However, with no hierarchy, as occurs in some other animals, the situation is more complicated and the outcome may depend on the evolutionary history of a population. The rest of the paper is divided into four sections. In the first section we describe two different models of brood reduction through siblicide in birds: in one model a runt is always the victim whilst in the second model the victim is a random member o f the rest of the brood. In the next section, the benefits of siblicide to the actor are analysed using a simple inclusive fitness argument based on O'Connor (1978) (which ignores some of the assumptions of inclusive fitness theory listed above). The third section comprises genetic models of the spread of siblicidal and nonsiblicidal genes of varying degrees of penetrance in populations subject to either form of siblicide and the final section is a Discussion. Two Forms of Siblicide in Birds
Consider a bird that lays a clutch of c eggs. We assume that the fitness of any member of the clutch is 1. If an act of siblicide occurs, the resources that would have been consun~ed by the dead chick are distributed equally amongst the remaining siblings. We thtis assume the fitness of the remaining siblings increases by a factor b / ( c - 1 ) where b measures the benefit of siblicide to the whole of the remaining brood. We discuss two forms of siblicide. In the "Random victim" model, a siblicidal individual chooses a victim at random from the other members of the brood. If there is more than one potential siblicidal individual, then the actual aggressor is picked at random from this group with a probability proportional to its siblicidal
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tendency (i.e. the penetrance of the siblicidal gene). In the " R u n t " model, a particular member of the brood is singled out as a victim. If one or more members of the rest o f the brood are siblicidal, the runt dies. The identity o f the runt is determined purely by the environment and is uninfluenced by genotype. We assume that the runt is unable to harm other brood members. The first model might be appropriate to birds in which all eggs hatch synchronously while the second model is most appropriate to birds with asynchronous hatching and a hierarchy of chick sizes leading to an obviously identifiable runt. We assume that if a brood consists of a number of potentially siblicidal individuals then only one act of siblicide occurs, i.e. any siblicidal behaviour is repressed after brood reduction and the probability of two young being killed simultaneously is vanishingly small. The major difference between the two models is that non-siblicidal individuals may be protected from siblicidat clutch-mates in the runt model. For example, suppose a clutch o f two contains one siblicidal and one non-siblicidal individual. If the siblicidal individual is the runt than the second chick survives. In the random victim model, the non-siblicidal individual will always perish. There are a number of assumptions here that will not be valid for all biological examples of siblicide. We assume that the fitness of all members of the brood are equal including both the victim and the aggressor. In reality, the victim may be a less fit individual (particularly in the runt model). There may be costs associated with the act of siblicide such that it may pay an individual from refraining from siblicide in the hope that another sibling will carry out the deed [techniques suggested by Eschel & Motro (1988) and Motro & Eschel (1988) could be used to investigate this possibility]. We also assume that the probability of an individual being killed is independent of its own potential for being siblicidal. The models could easily be made more realistic by the inclusions of these features but we omit them in the belief that they will not qualitatively affect our conclusions and in order to keep our models as simple as possible.
Inclusive Fitness Arguments (After O'Connor 1978) The act o f siblicide will have both positive and negative influences on the fitness o f the brood members. In an inclusive fitness analysis, the total costs and benefits o f the act o f siblicide are compared, each cost and benefit weighted by the coefficient of relatedness of the siblicidal chick to the chick experiencing the change in fitness. If an individual kills a sibling, it gains an increment of fitness of b / ( c - 1), as do the ( c - 2 ) other members of the brood. The benefits to the other members of the brood are weighted by their coefficient of relatedness to the siblicidal individual which in the case of full sibs is ½. The only cost of siblicide is the loss of a sibling (fitness set at one); which again is weighted by the coefficient o f relatedness of ½. Thus siblicide should spread if the benefits outweigh the costs, b + 1 ( c _ 2 ) , ' b.,,,>l_ c-1 c-1 2
(1)
b > ( c - 1)/c.
(2)
or
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Siblicide spreads when the benefits to the remaining members of the brood are greater than (c - 1)/c. The condition for the spread o f a non-siblicidal trait through a population with siblicide is simply the reverse inequality b < ( c - 1 ) / c . Note that we have not specified whether a runt or a " r a n d o m individual"-is the victim of siblicide: the conditions for the spread of siblicide are identical in both cases. Brood reduction increases the inclusive fitness of a parent when b > 1, i.e. when the loss o f one individual is more than compensated by greater overall brood fitness. As clutch size increases, eqn (2) approaches b > 1 and there will thus be no conflict o f interest between parents and young. However, with small clutches siblicide can spread when (c - 1)/c < b < 1 indicating the possibility of parent-offspring conflict (Trivers, 1974). Genetic Models o f Siblicide
Suppose the population is composed o f individuals with genotype AA which are siblicidal with probability a. We ask under what conditions a rare dominant gene S, which leads to siblicidal behaviour with a different probability, s, will increase in frequency in the population. We are thus concerned with the invasion dynamics of the allele S which, when rare, will almost always be present as the heterozygous genotype A S and will nearly always mate with the common genotype AA. The rare allele will spread if more copies of the S gene are produced from an A S x A A mating than those o f an arbitrarily chosen A allele in an AA x AA mating. Unlike the inclusive fitness argument, it is now important to specify whether the victim is chosen randomly or is a runt. In the Appendix we describe general methods for studying the spread of the mutant S allele in the " R a n d o m victim" and " R u n t " models for arbitrary a and s. Analytic results are not possible for the general case and we concentrate here on the spread o f a rare siblicidal gene through a population consisting wholly o f non-siblicidal individuals (a = 0) and the spread of a rare non-siblicidal gene through a population consisting wholly of siblicidal individuals (a = 1)~ I-~hthe case of a rare siblicidal gene, we examine both a gene with complete penetrance (s = 1) and with arbitrarily small penetrance (s = e, e-~ 0). Similarly for the case o f a rare non-siblicidal gene, we examine a gene with complete penetrance (s = 0) and with arbitrarily small penetrance (s = 1 - e , e-~ 0). The results o f our analysis are given in Table 1. "'RANDOM
VICTIM"
MODEL
The criteria for the spread o f the rare gene in the " R a n d o m victim" model are shown in Fig. 1. The conditions for the spread o f a siblicidal gene with very low penetrance in a population composed o f non-siblicidal individuals is identical to the prediction from inclusive fitness theory. With arbitrarily small penetrance, the assumptions of weak selection and additivity necessary for the application of inclusive fitness are fulfilled. However, a non-siblicidal population is susceptible to invasion by a gene with larger penetrance when the benefits to siblicide are less than those predicted by the inclusive fitness arguments. For example, with a clutch size of two, a gene for obligate siblicide will spread if the benefits to the surviving
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TABLE 1
Conditions f o r a rare mutant exhibiting siblicidal behaviour with probability s to invade a population composed o f a genotype leading to siblicidal behaviour with probability a. The cases examined include siblicidal ( Sib ) and non-siblicidaI ( N S ) resident alleles while invading alleles are either siblicidal or non siblicidal with complete or arbitrarily small penetrance ( e -> O) Model
Resident allele
Invading allele
Random victim
NS (a = 0 )
Sib ( s = 1)
NS(a=0)
Sib ( s = e)
S i b ( a = 1)
NS(s=0)
Sib (a = 1)
NS(s= l-e)
NS(a=0)
S i b ( s = 1)
b > 1 - ( 0 . 5 ) c-l
NS(a=0)
S i b ( s = e)
c-1 b>-£
S i b ( a = 1)
NS(s=0)
b<0.5
Sib (a = 1)
NS(s=l-e)
b<0-5
Runt
Condition for the spread of the rare genotype b>
(c-1)[c-2+(0.5) c-'] n(n-2)+2-(0.5)
c-I
c-1 b>-c b~
( c - 1)[c(0-5)c-' - 1] 1 + c ( c - 2)(0-5) "-1 b<-(c-1)
offspring are b =½ while the low penetrance gene will only spread with b =½. Simulation studies suggest that if a siblicidal gene spreads when rare it will continue spreading until it reaches fixation. Once a population becomes fixed for siblicide in the " R a n d o m victim" model, it is very difficult for a non-siblicidal allele to invade. With a clutch size of two there must be no benefit to b r o o d reduction (b < 0) before the allele can spread and for larger clutch sizes there must actually be a benefit for the whole brood in not reducing the numbers o f birds in the nest. The reason for these stringent conditions is that, when rare, non-siblicidal genes very frequently find themselves in broods with siblicidal individuals where they suffer a high risk of elimination. In addition, those non-siblicidal genes that do survive, tend to do so in larger clutches than siblicidal genes and so suffer a relative fitness penalty. Genes with high penetrance can spread more easily than those with low penetrance because low penetrance genes are liable not only to be destroyed by the c o m m o n genotype but also by siblicidal individuals of the rare genotype. Simulation studies indicate that a nonsiblicidal gene that does spread when rare will achieve fixation. There is thus a hysteresis in the conditions for the spread of siblicide and non-siblicide. For a wide range of clutch sizes and benefits to b r o o d reduction, neither siblicidal nor non-siblicidal alleles can invade a population fixed at the alternative state. The presence o f siblicidal or non-siblicidal behaviour in any
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0 o
O~
d
m
-0.5
--[
i
2
3
4
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5
l l i l ] l l l t i l i l i l l l i l
6
7
8
9
Brood size
FXG 1. The spread of siblicidal and non-siblicidal alleles in the "'Random victim" model, Siblicide results in a benefit, b, shared between the remaining members of the brood. A value o f b = 1 is equivalent to one brood member so the parent favours brood reduction when b > 1. A siblicidal gene of arbitrarily small penetrance invades a population fixed at non-siblicidat behaviour in region a. A sibticidal gene with greater penetrance can spread over a wider area of parameter space: a gene with penetrance = 1 spreads in regions a and b. When siblicidal behaviour is fixed in the population, a non-siblicidal gene (with complete penetrance) can spread in region c. The negative value of b implies that brood reduction is actually harmful to the remaining chicks. Finally, in region d, neither a population fixed for siblicide nor non-siblicide can be invaded by alternative rare alleles and thus the state of the population will be determined by its past history.
particular species will depend on the evolutionary history of that species. Considering the restrictive conditions for the spread of non-siblicidal genes in species with large clutches, there would seem to be a real likelihood that bird species with "Random victim" siblicide might become caught in an evolutionary trap: brood reduction that increases the inclusive fitness of neither the parents nor the young. "'RUNT" MODEL
The criteria for the spread of the rare gene in the "'Runt" model are shown in Fig. 2. The conditions for the spread of a tow penetrance siblicide allele through a non-siblicidal population are identical both to the equivalent condition for the "Random victim" model and also to the prediction for the spread of siblicide from the inclusive fitness argument. In contrast to the last model, the conditions for the
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0,75
"o o
o
r,
,m
0.5 o ww-
o0
0.25
0 2
3
4
5
6
7
8
9
Brood size
FIG. 2. The spread of siblicidal and non-siblicidal alleles in the " R u n t " model. Axes as in Fig. 1 (though note different scale on the ordinate). A siblicidal gene with arbitrarily small penetrance invades a population fixed for non-siblicidal behaviour in regions a and b. Unlike the "'Random victim" model, an allele with high penetrance spreads in a narrower area of parameter space: an allele with penetrance = 1 spreads only in region a. A non-siblicidal allele of any penetrance invades a population fixed at siblicide in region c. In region d, both siblicide and non-siblicide are ESSs but, as explained in the text and Fig. 3, only non-siblicide is a CSS.
spread of obligate siblicide are more stringent than that of occasional siblicide (with the exception of clutches of two where the conditions are identical). However, we believe that once a low penetrance gene for sibticide has gone to fixation, a gene with a slightly higher penetrance will then be able to invade and so on until the population consists of obligate siblicides. The predictions of the spread of a siblicidal allele from the genetic and inclusive fitness arguments are thus the same. The criterion for the spread of a non-siblicidal allele of any degree of penetrance is simply that the benefits o f siblicide must be less than ½. In comparison with the " R a n d o m victim" model, it is considerably easier for a non-siblicidal allele to invade a population of obligate siblicides. The reason for this is that there is the possibility o f a non-siblicidal gene being protected from elimination by a siblicidal gene in a runt. As in the " R a n d o m victim" model, there appears to be an area of hysteresis in the switch between siblicidal and non-siblicidal behaviour [when 0 . 5 < b < ( 1 - ( 0 - 5 ) ~ - I ) ] . However, there is a difference in the form of the stability o f the
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siblicidal behaviour in the two models. This is best seen by examining the fate of a rare allele invading a population exhibiting different rates o f siblicide for parameter values in the "'area o f hysteresis" (Fig. 3). Consider a population which displays obligately non-siblicidal behaviour (i.e. in the lower left-hand c o m e r o f both figures). Any rare siblicidal mutant, whatever its degree o f penetranc¢, will fail to spread. If the population is perturbed slightly away from obligately non-siblicidal behaviour, the only rare mutants that will spread are those with even greater non-sublicidal tendencies and these will tend to return the population towards a complete absence of siblicide. In both cases, non-siblicidal behaviour is thus stable. Now consider a population displaying obligately siblicidal behaviour (i.e. the upper right-hand c o m e r o f both figures). In both cases, any less-siblicidal, mutant allele, whatever its degree of penetrance, will fail to spread. Thus, as we have concluded already, the obligately siblicidal behaviour is in some sense stable. Consider now a small perturbation away from obligate siblicide. In the case o f the " R a n d o m victim" model, the only mutants that can now spread are those that tend to move the population back towards obligate siblicide: in this model obligate siblicide is stable in a strong sense. However, in the " R u n t " model, a small perturbation away from obligate siblicide allows mutants with a smaller probability of being siblicidal to spread. These will run to fixation and themselves allow other alleles with even less silicidal tendencies to spread until, ultimately, the population becomes "Runt" model
"Random victim" model
ta
~5 EL
0
I
0
I
Probability that common genotype is siblicidal
FIG. 3. The fate o f a rare gene invading a population in regions d o f Figs 1 and 2. The population is a s s u m e d to be at fixation for an allele that is siblicidal with probability a (abscissa). A rare m u t a n t which is siblicidal with probability s (ordinate) invades the resident population in the shaded region o f parameter space. Note that obligate siblicide a n d non-siblicide are uninvasible to all other alternative strategies. S u p p o s e the populations are perturbed from obligate siblicidal or non-siblicidal behaviour. Alternative alleles are now able to spread but will tend to move the behaviour of the population back towards either obligate siblicide or non-siblicide. The exception is obligate siblicide in the " R u n t " model. A slight m o v e m e n t away from obligate siblicide in population behaviour allows alleles coding for even lower levels o f siblicide to invade and eventually the population will come to be c o m p o s e d o f obligately non-siblicidal individuals.
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completely non-siblicidal. Thus siblicidal behaviour in the " R u n t " model is only stable in a very weak sense for 0-5 < b < [ 1 - (0-5)c-1]: arbitrarily small perturbations in the resident population away from obligate siblicide lead to invasion of less siblicidal alleles and ultimately to an obligately non-siblicidal state. There is thus no hysteresis between siblicidal and non-siblicidal behaviour and the predicted boundary between the two activities is identical from genetic and inclusive fitness arguments. Eschel & Motro (1981) and Eschel (1983) describe a number of examples of evolutionary strategies that, at fixation, are stable to invasion by any rare mutant, yet if slightly perturbed away from the stable value are then susceptible to invasion. They describe such strategies as evolutionary stable strategies (i.e. as ESSs, Maynard Smith, 1982) but not continuously stable strategies (CSSs). The strategy of obligate siblicide, for parameter values in region d (Fig. 2) o f the " R u n t " model, is an example o f this type o f strategy. Discussion Predictions about the evolution o f siblicide can be obtained from either inclusive fitness or population genetic arguments. We have compared the two approaches and found that the assumption o f the form of siblicide (our runt and random victim models) is crucial to the predictions of the genetic model. Where the victim is a weak and non-aggressive runt, as seems to occur frequently in birds, siblicide is predicted by both arguments when the benefits to the surviving clutch members are (c-1)/c where c is clutch size and benefit is measured in units o f the fitness of one clutch member prior to siblicide. An alternative formulation of this criterion is that siblicide is favoured when the benefits to the actor are greater than 1/c. However, when the victim is chosen at random, the more exact genetic models predict very different results from those o f the inclusive fitness model. In particular, siblicidal behaviour evolves over a wider range o f parameter values, and having evolved, it is difficult for non-siblicidal behaviour to re-establish itself, even when the benefits o f siblicide are much reduced or negative. We have compared our genetic models with the inclusive fitness arguments of O ' C o n n o r (1978) as these have been widely discussed in the ornithological literature though we realize that it might be possible to construct a more sophisticated inclusive model o f siblicide. It should be noted that one of the critical differences between our two models o f siblicide is that the runt is unable to attack an elder, non-aggressive sibling. It is possible that runts in some species with obligate siblicide might be able to attack and kill elder non-aggressive siblings that had failed to kill them. Consider the spread o f a non-siblicidal allele through a population with obligate siblicide and with a clutch size of two. If the runt is unable to kill a tolerant elder sibling then non-siblicide spreads when b < 0 . 5 (our runt model, see Fig. 1). However, if the runt kills the tolerant elder sibling with probability q then it can be shown that non-siblicide spreads when b < 0-5q. Thus if the runt is always successful (q = 0), the condition for the spread o f siblicide is b < 0, the identical condition to that predicted by our random victim model (Fig. 2). Experimental studies on the dynamics
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of brood reduction are needed before more precise predictions about the evolution o f siblicide in different bird groups can be made. However, we believe it is at least possible that some bird species have evolved siblicidal behaviour o f a type that is very difficult for a non-siblicidal allele to invade. Possible candidate species are those eagles with obligate siblicide which appears to be unaffected by local food supplies (Meyburg, 1974; Gargett, 1978). Lack (1954) first suggested that hatching asynchrony and runt production may arise as an adaptation for efficient brood reduction in years when a parent is unable to rear its full clutch. Since then, no less than seven other hypotheses have been advanced to explain hatching asynchrony (reviewed in Lessells & Avery, 1989; Slagsvold & Lifjeld, 1989). One hypothesis suggests that asynchrony is selected because sibling rivalry is reduced in broods with a hierarchy of chick sizes in comparison with broods where all chicks are the same size: the hierarchy allows competitive disputes to be settled quickly without extended conflict (Hamilton, 1964; Hahn, 1981). Our models have shown that a hierarchy o f chick sizes can have a marked effect on the levels of parent-offspring conflict by making siblicide less likely. It would seem intuitively appealing that the parent might be selected to produce a hierarchy to minimize this form of sibling conflict. The problem with this argument is that the reduction in siblicide is not an immediate benefit that could act as a selection pressure leading to the evolution o f a chick hierarchy, but a benefit that occurs subsequent to its establishment. Siblicide is known in a number of other groups outside birds (Polis, 1981; Godfray, 1987a). In some cases such as coccinelid and chrysomelid beetles there is a hierarchy in the age of siblings and the elimination, quite often, of large numbers of younger individuals. Parker & Mock (1987) have examined inclusive fitness models appropriate to such animals. In contrast, Godfray (1987b) has argued that the random victim model is appropriate to cases o f siblicide in parasitoid wasps and that there is evidence for a hysteresis in the switch between siblicidal and non-siblicidal behaviour (le Masurier, 1987). We thus believe that the models presented here, suitably modified, may be applicable to other animal groups. The aim of this paper has been to resolve the differences between inclusive fitness and genetic arguments for the evolution o f siblicide by brood reduction and to highlight the most important assumptions affecting the predictions o f the model. We believe that future experimental work is essentially to determine which model (or blend o f models) is most appropriate for bird species with siblicide. Future theoretical work is needed to examine, among other things, the consequences of allowing a more complicated genetic basis to the trait; the effects o f the costs o f siblicide, especially in the random victim model; the evolution o f siblicide as an escalated form o f other types o f sibling interaction and the evolutionary response of parents to excess siblicide (see also Parker & Mock, 1987).
REFERENCES CHARNOV, E. L. (1978). Evolution of eusocial behaviour: offspring choice or parental parasitism? J. theor. Biol. 75, 451-456.
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ESCHEL, I. (1983). Evolutionary and continuous stability. Z theor. Biol. 103, 99-112. ESCHEL, I. & MOTRO, U. (1981). Kin selection and strong evolutionary stability of mutual help. Theor. pop. Biol. 19, 420-433. ESCHEL, I. & MOTRO, U. (1988). The three brothers' problem: kin selection with more than one potential helper. 1. The case of immediate help, Am. Nat. 132, 550-566. FELDMAN, M. W. & ESCHEL, 1. (1982). On the theory of parent-offspring conflict: a two-locus genetic model. Am. Nat. 119, 285-292. GARGETT, V. (1978). Sibling aggression in the Black Eagle in the Matapos, Rhodesia. Ostrich 49, 57-63. GO DFRAY, H. C. J. (1987 a). The evolution of clutch size in invertebrates. Oxf Suro. evol. Biol. 4, 117-154. GODFRAY, H. C. J. (1987b). The evolution of clutch size in parasitic wasps. Am. Nat. 129, 221-233. GRAEEN, A. (1986). A geometric view of relatedness. O x f Sure. evol. Biol. 2, 28-29. HAHN, D. C. (1981). Asynchronous hatching in the Laughing Gull: cutting losses and reducing rivalry. Anim. Behav. 29, 421-427. HAMILTON, W. D. (1964). The genetical theory of social behaviour. I. J. theor. Biol. 7, 12-45. HARPER, A. B. (1986). The evolution of begging: sibling competition and parent-offspring conflict. Am. Nat. 128, 99-114. LACK, D. (1954). The Natural Regulation of Animal Numbers. Oxford: Oxford University Press. LACK, D. (1966). Population Studies of Birds. Oxford: Clarendon Press. LACK, D. (1968). Ecological Adaptations for Breeding in Birds. London: Methuen. LESSELLS, C. M. & AVERY, M. I. (1989). Hatching asynchrony in European Bee-eaters Merops apiaster. J. Anita. Ecol. 58, 815-835. MACNAtR, M. R. & PARKER, G. A. (1978). Models of parent-offspring conflict. II. Promiscuity. Anita. Behav. 26, 111-122. MACNAIR, M. R. & PARKER, G. A. (1979). Models of parent-offspring conflict, llI. Intra-brood conflicL Anita. Behav. 27, 1202-1209. LE MASURIER, A. O. (1987). A comparative study of the relationship between host size and brood size in Apanteles spp. (Hymenoptera: Braconidae). Ecol. Ent. 12, 383-393. MAYNARD SMITH, J. (1982). Evolution and Theory of Games. Cambridge: Cambridge University Press. MEYaURG, B.-U. (1974). Sibling aggression and mortality among nestling eagles. Ibis 116, 224-228. MICHOD, R. E. (1982). The theory of kin selection. Ann. Rev. Ecol. Syst. 13, 23-56. MOTRO, U. & ESCHEL, I. (1988). The three brothers" problem: kin selection with more than one potential helper. 2. The case of delayed help. Am. Nat. 132, 567-575. O'CONNOR, R. J. (1978). Brood reduction in birds: selection for fratricide, infanticide and suicide. Anita. Behav. 26, 79-96. PARKER, G. A. (1985). Models of parent-offspring conflict. V. Effects of the behaviour of the two parents. Anita. Behav. 27, 1210-1235. PARKER, G. A. & MACNAIR, M. R. (1978). Models of parent-offspring conflict. I. Monogamy. Anita. Behav. 26, 97-110. PARKER, G. A. & MACNAIR, M. R. (1979). Models of parent-offspring conflict. IV. Suppression, evolutionary retaliation by the parent. Anim. Behav. 27, 1210-1235. PARKER, G. A. & MOCK, D. W. (1987). Parent-offspring conflict over clutch size. EvoL Ecol. 1, 161-174. POLLS, G. A. (1981). The evolution and dynamics of intraspecific predation. Ann. Rev. Ecol. Syst. 12, 225-251. SLAGSVOLD, T. & LIFJELD, J. T. (1989). Constaints on hatching asynchrony and egg size in Pied Flycatchers. J. Anita. Ecol. 58, 837-849. STAMP, J. A., METCALF, R. A. & KRISHNAN, V. V. (1978). A genetic analysis of parent-offspring conflict. Behav. EcoL Sociobiol. 3, 369-392. STINSON, C. H. (1979). On the selective advantage of fratricide in raptors. Evolution 33, 1219-1225. TRlVERS, R, L (1974). Parent-offspring conflict. Am. Zool. 14, 249-264. APPENDIX
"Runt" Model We need to calculate the number of S genes resulting from an AS x AA mating. The rare gene S will spread if more than c/2 copies are produced (c/2 are the number of copies of an arbitrarily marked A gene that are produced in an AA x AA mating).
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We divide the b r o o d into the runt and the c - 1 other members. Let i be the n u m b e r o f A S individual a m o n g these c - 1 chicks. The progeny from an A S x A A mating are AA or A S with equal probability and so i is binomially distributed with parameters c - 1 and 0.5. To determine whether the runt survives we need to know the p h e n o t y p e of the A S and AA individuals. The probability that an A S individual is potentially siblicidal is s and thus j, the n u m b e r of A S with the siblicidal phenotype, is binomially distributed with parameters i and s. By the same argument k, the n u m b e r o f AA individuals with the siblicidal phenotype, is binomially distributed with parameters ( c - 1 - i) and a. The joint distribution of siblicidal A S and AA individuals in a brood (excluding the runt), p(i,j, k, c - 1), is thus
I f the runt survives, (i + 0.5) copies of A S are produced from an A S x AA mating while if the runt dies, i[1 + b / ( c - 1 ) ] are produced. The runt survives if j = k = 0 . The S gene spreads if the total number of S genes arising from the mating is greater than c/2 or 2
p(i, 0,0, c - 1 ) { i + 0 . 5 } +
i=0
~. p ( i , j , k , c - 1 ) { i [ l + b / ( c - 1 ) ] } j=l
>
.
(A.2)
k=I
The entries in Table 1 are obtained by substituting into eqns (A.1) and (A.2) the appropriate values of a and s.
TABLE A.1 This table calculates the probability that an aggressor or a victim will have genotype A A or AS given that the brood is composed o f j potentially siblicidal AS individuals and k potentially siblicidal A A individuals. The broodproduces i AS individuals if the victim is A A and one less if the victim is AS. The sum o f the products o f the last two columns is the variable X which appears in eqn (A.3) Aggressor's genotype
Victim's genotype
AA
AA
AA
AS
AS
AA
AS
AS
Probability k n-l-i j + k n-1 k i j+kn-1 j
n--i
j+kn-1 j i-I j+kn-1
No. of AS produced i i- 1
i i- 1
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"Random Victim" Model
Let p(i,j, k, c) be the probability that a b r o o d of c individuals arising from an A S x A A mating contains i A S chicks, j of which are potentially siblicidal, and c - i A A chicks, k of which are potentially siblicidal. The probabilities are given by eqn (A.1) with c - 1 replaced by c. If there is no siblicide then i A S individuals survive. I f there is a siblicide, then either i[1 + b / ( c - 1)] or (i - 1)/[1 + b / ( c - 1)]AS individuals survive depending on the genotype of the victim. The probability of the victim being a particular genotype depends on the genotype of both the aggressor and the victim as listed in Table A.1. S will spread if the total n u m b e r of S genes produced by a clutch of size c from an A S x A A mating is greater than c/2 or
p(i,O, 0, c){i}+ Y. ~, p ( i , j , k , c ) { X [ l + b / ( c - l ) ] } i=O
j=l
>~,
(A.3)
k=l
where X is the sum o f the product of the last two columns in Table A.1. The entries in Table 1 are obtained by substituting into eqns (A.1) and (A.3) the appropriate values of a and s.
The Evolution of Brood Reduction by Siblicide in Birds H. C. J. GODFRAYt AND A. B. HARPER:[:
t Department of Biology and Centre for Population Biology, Imperial College at Silwood Park, Ascot, Berkshire SL5 7 P Y and ~ Department of Zoology, South Parks Road, Oxford OX1 3 PS, U.K. (Received on 8 August 1989, Accepted in revised form on 9 March 1990) The young of a number of bird species frequently kill their siblings whilst still in the nest. We develop simple single-locus genetic models of the evolution of siblicide and non-siblicide in bird populations. Two cases are distinguished: (1) where there is an obvious runt which, when siblicide occurs, always perishes and (2) where a siblicidal individual chooses a victim at random from its nest mates. The genetic models are compared with an inclusive fitness analysis of siblicide due to O'Connor (1978, Anita. Behav. 26, 79-96). In the runt model, the genetic analysis and inclusive fitness analysis yield the same results though the genetic analysis reveals a large area of parameter space where both strategies are evolutionarily stable strategies (ESSs) but only one is a continuously stable strategy (CSS). In the model without a runt, the two analyses give different results and for a large area of parameter space both siblicidal and non-siblicidal behaviours are stable to invasion (both ESSs and CSSs) so that the observed state of a population will depend on its past history.
Introduction Siblicide occurs in birds when one member of a brood causes the death of a sibling. While not a c o m m o n feature o f bird biology, sibling aggression that culminates in the death of a nest mate has been recorded from a wide variety of taxa including pelicans, gannets, cranes, skuas and herons, though it is best known from large raptors (Lack, 1968; O ' C o n n o r , 1978). There are a number o f hypotheses about the selective advantage of siblicide. Lack (1966) suggested that siblicide may be advantageous to both parents and surviving offspring in years when environmental conditions make it difficult to successfully rear a complete brood. Once siblicide became established, parents might be selected to lay large clutches in the expectation that siblicide would lead to brood reduction in all but the very best years. A second reason why a mother may lay larger clutches than her y o u n g will tolerate is to compensate for any infertility in her eggs (Stinson, 1979). It has also been suggested that asynchronous incubation which leads to a hierarchy o f chick sizes may be an adaptation to facilitate early and efficient brood reduction. Hamilton (1964) and Trivers (1974) pointed out that there may be a conflict of interest between parents and offspring over such questions as when brood reduction should occur. The first analysis of this conflict in birds was that by O ' C o n n o r (1978) who used inclusive fitness arguments to determine the level o f benefit to the surviving offspring that would lead to selection for siblicide. At about the same time Stinson 163
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(1979) used a simple genetic model in his study of brood reduction in eagles where siblicide is particularly common. The value of the inclusive fitness approach is that the evolution of a trait can be studied without a detailed knowledge of its genetic basis (Hamilton, 1964). However, inclusive fitness theory assumes that the traits in question have relatively small, normally additive effects on fitness (e.g. Michod, 1982; Grafen, 1986), assumptions that are very likely to be untrue in the case of siblicide. Indeed, simple genetic models of the spread of siblicide and non-siblicide (of the type first used by Stinson) can give very different results to inclusive fitness arguments and explicitly genetic models have proved necessary in the analysis of other forms of sibling competition and parent-offspring conflict (e.g. Parker & MacNair, 1978, 1979; MacNair & Parker, 1978, 1979; Charnov, 1978; Stamps et al., 1978; Feldman & Eschel, 1982; Parker, 1985; Harper, 1986). In this paper we investigate the conditions that allow the evolution of siblicide and, in particular, ask how great a benefit must accrue to surviving brood members to lead to selection for siblicide. Our conclusions are based on simple but explicit genetic models which we compare with inclusive fitness arguments. We find that the predictions o f the model are very dependent on the presence or absence of a hierarchy in chick sizes and an easily eliminated runt. When a hierarchy is present, the normal case in birds, the inclusive fitness and genetic arguments give the same results. However, with no hierarchy, as occurs in some other animals, the situation is more complicated and the outcome may depend on the evolutionary history of a population. The rest of the paper is divided into four sections. In the first section we describe two different models of brood reduction through siblicide in birds: in one model a runt is always the victim whilst in the second model the victim is a random member o f the rest of the brood. In the next section, the benefits of siblicide to the actor are analysed using a simple inclusive fitness argument based on O'Connor (1978) (which ignores some of the assumptions of inclusive fitness theory listed above). The third section comprises genetic models of the spread of siblicidal and nonsiblicidal genes of varying degrees of penetrance in populations subject to either form of siblicide and the final section is a Discussion. Two Forms of Siblicide in Birds
Consider a bird that lays a clutch of c eggs. We assume that the fitness of any member of the clutch is 1. If an act of siblicide occurs, the resources that would have been consun~ed by the dead chick are distributed equally amongst the remaining siblings. We thtis assume the fitness of the remaining siblings increases by a factor b / ( c - 1 ) where b measures the benefit of siblicide to the whole of the remaining brood. We discuss two forms of siblicide. In the "Random victim" model, a siblicidal individual chooses a victim at random from the other members of the brood. If there is more than one potential siblicidal individual, then the actual aggressor is picked at random from this group with a probability proportional to its siblicidal
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tendency (i.e. the penetrance of the siblicidal gene). In the " R u n t " model, a particular member of the brood is singled out as a victim. If one or more members of the rest o f the brood are siblicidal, the runt dies. The identity o f the runt is determined purely by the environment and is uninfluenced by genotype. We assume that the runt is unable to harm other brood members. The first model might be appropriate to birds in which all eggs hatch synchronously while the second model is most appropriate to birds with asynchronous hatching and a hierarchy of chick sizes leading to an obviously identifiable runt. We assume that if a brood consists of a number of potentially siblicidal individuals then only one act of siblicide occurs, i.e. any siblicidal behaviour is repressed after brood reduction and the probability of two young being killed simultaneously is vanishingly small. The major difference between the two models is that non-siblicidal individuals may be protected from siblicidat clutch-mates in the runt model. For example, suppose a clutch o f two contains one siblicidal and one non-siblicidal individual. If the siblicidal individual is the runt than the second chick survives. In the random victim model, the non-siblicidal individual will always perish. There are a number of assumptions here that will not be valid for all biological examples of siblicide. We assume that the fitness of all members of the brood are equal including both the victim and the aggressor. In reality, the victim may be a less fit individual (particularly in the runt model). There may be costs associated with the act of siblicide such that it may pay an individual from refraining from siblicide in the hope that another sibling will carry out the deed [techniques suggested by Eschel & Motro (1988) and Motro & Eschel (1988) could be used to investigate this possibility]. We also assume that the probability of an individual being killed is independent of its own potential for being siblicidal. The models could easily be made more realistic by the inclusions of these features but we omit them in the belief that they will not qualitatively affect our conclusions and in order to keep our models as simple as possible.
Inclusive Fitness Arguments (After O'Connor 1978) The act o f siblicide will have both positive and negative influences on the fitness o f the brood members. In an inclusive fitness analysis, the total costs and benefits o f the act o f siblicide are compared, each cost and benefit weighted by the coefficient of relatedness of the siblicidal chick to the chick experiencing the change in fitness. If an individual kills a sibling, it gains an increment of fitness of b / ( c - 1), as do the ( c - 2 ) other members of the brood. The benefits to the other members of the brood are weighted by their coefficient of relatedness to the siblicidal individual which in the case of full sibs is ½. The only cost of siblicide is the loss of a sibling (fitness set at one); which again is weighted by the coefficient o f relatedness of ½. Thus siblicide should spread if the benefits outweigh the costs, b + 1 ( c _ 2 ) , ' b.,,,>l_ c-1 c-1 2
(1)
b > ( c - 1)/c.
(2)
or
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Siblicide spreads when the benefits to the remaining members of the brood are greater than (c - 1)/c. The condition for the spread o f a non-siblicidal trait through a population with siblicide is simply the reverse inequality b < ( c - 1 ) / c . Note that we have not specified whether a runt or a " r a n d o m individual"-is the victim of siblicide: the conditions for the spread of siblicide are identical in both cases. Brood reduction increases the inclusive fitness of a parent when b > 1, i.e. when the loss o f one individual is more than compensated by greater overall brood fitness. As clutch size increases, eqn (2) approaches b > 1 and there will thus be no conflict o f interest between parents and young. However, with small clutches siblicide can spread when (c - 1)/c < b < 1 indicating the possibility of parent-offspring conflict (Trivers, 1974). Genetic Models o f Siblicide
Suppose the population is composed o f individuals with genotype AA which are siblicidal with probability a. We ask under what conditions a rare dominant gene S, which leads to siblicidal behaviour with a different probability, s, will increase in frequency in the population. We are thus concerned with the invasion dynamics of the allele S which, when rare, will almost always be present as the heterozygous genotype A S and will nearly always mate with the common genotype AA. The rare allele will spread if more copies of the S gene are produced from an A S x A A mating than those o f an arbitrarily chosen A allele in an AA x AA mating. Unlike the inclusive fitness argument, it is now important to specify whether the victim is chosen randomly or is a runt. In the Appendix we describe general methods for studying the spread of the mutant S allele in the " R a n d o m victim" and " R u n t " models for arbitrary a and s. Analytic results are not possible for the general case and we concentrate here on the spread o f a rare siblicidal gene through a population consisting wholly o f non-siblicidal individuals (a = 0) and the spread of a rare non-siblicidal gene through a population consisting wholly of siblicidal individuals (a = 1)~ I-~hthe case of a rare siblicidal gene, we examine both a gene with complete penetrance (s = 1) and with arbitrarily small penetrance (s = e, e-~ 0). Similarly for the case o f a rare non-siblicidal gene, we examine a gene with complete penetrance (s = 0) and with arbitrarily small penetrance (s = 1 - e , e-~ 0). The results o f our analysis are given in Table 1. "'RANDOM
VICTIM"
MODEL
The criteria for the spread o f the rare gene in the " R a n d o m victim" model are shown in Fig. 1. The conditions for the spread o f a siblicidal gene with very low penetrance in a population composed o f non-siblicidal individuals is identical to the prediction from inclusive fitness theory. With arbitrarily small penetrance, the assumptions of weak selection and additivity necessary for the application of inclusive fitness are fulfilled. However, a non-siblicidal population is susceptible to invasion by a gene with larger penetrance when the benefits to siblicide are less than those predicted by the inclusive fitness arguments. For example, with a clutch size of two, a gene for obligate siblicide will spread if the benefits to the surviving
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TABLE 1
Conditions f o r a rare mutant exhibiting siblicidal behaviour with probability s to invade a population composed o f a genotype leading to siblicidal behaviour with probability a. The cases examined include siblicidal ( Sib ) and non-siblicidaI ( N S ) resident alleles while invading alleles are either siblicidal or non siblicidal with complete or arbitrarily small penetrance ( e -> O) Model
Resident allele
Invading allele
Random victim
NS (a = 0 )
Sib ( s = 1)
NS(a=0)
Sib ( s = e)
S i b ( a = 1)
NS(s=0)
Sib (a = 1)
NS(s= l-e)
NS(a=0)
S i b ( s = 1)
b > 1 - ( 0 . 5 ) c-l
NS(a=0)
S i b ( s = e)
c-1 b>-£
S i b ( a = 1)
NS(s=0)
b<0.5
Sib (a = 1)
NS(s=l-e)
b<0-5
Runt
Condition for the spread of the rare genotype b>
(c-1)[c-2+(0.5) c-'] n(n-2)+2-(0.5)
c-I
c-1 b>-c b~
( c - 1)[c(0-5)c-' - 1] 1 + c ( c - 2)(0-5) "-1 b<-(c-1)
offspring are b =½ while the low penetrance gene will only spread with b =½. Simulation studies suggest that if a siblicidal gene spreads when rare it will continue spreading until it reaches fixation. Once a population becomes fixed for siblicide in the " R a n d o m victim" model, it is very difficult for a non-siblicidal allele to invade. With a clutch size of two there must be no benefit to b r o o d reduction (b < 0) before the allele can spread and for larger clutch sizes there must actually be a benefit for the whole brood in not reducing the numbers o f birds in the nest. The reason for these stringent conditions is that, when rare, non-siblicidal genes very frequently find themselves in broods with siblicidal individuals where they suffer a high risk of elimination. In addition, those non-siblicidal genes that do survive, tend to do so in larger clutches than siblicidal genes and so suffer a relative fitness penalty. Genes with high penetrance can spread more easily than those with low penetrance because low penetrance genes are liable not only to be destroyed by the c o m m o n genotype but also by siblicidal individuals of the rare genotype. Simulation studies indicate that a nonsiblicidal gene that does spread when rare will achieve fixation. There is thus a hysteresis in the conditions for the spread of siblicide and non-siblicide. For a wide range of clutch sizes and benefits to b r o o d reduction, neither siblicidal nor non-siblicidal alleles can invade a population fixed at the alternative state. The presence o f siblicidal or non-siblicidal behaviour in any
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04
0 o
O~
d
m
-0.5
--[
i
2
3
4
i
I
5
l l i l ] l l l t i l i l i l l l i l
6
7
8
9
Brood size
FXG 1. The spread of siblicidal and non-siblicidal alleles in the "'Random victim" model, Siblicide results in a benefit, b, shared between the remaining members of the brood. A value o f b = 1 is equivalent to one brood member so the parent favours brood reduction when b > 1. A siblicidal gene of arbitrarily small penetrance invades a population fixed at non-siblicidat behaviour in region a. A sibticidal gene with greater penetrance can spread over a wider area of parameter space: a gene with penetrance = 1 spreads in regions a and b. When siblicidal behaviour is fixed in the population, a non-siblicidal gene (with complete penetrance) can spread in region c. The negative value of b implies that brood reduction is actually harmful to the remaining chicks. Finally, in region d, neither a population fixed for siblicide nor non-siblicide can be invaded by alternative rare alleles and thus the state of the population will be determined by its past history.
particular species will depend on the evolutionary history of that species. Considering the restrictive conditions for the spread of non-siblicidal genes in species with large clutches, there would seem to be a real likelihood that bird species with "Random victim" siblicide might become caught in an evolutionary trap: brood reduction that increases the inclusive fitness of neither the parents nor the young. "'RUNT" MODEL
The criteria for the spread of the rare gene in the "'Runt" model are shown in Fig. 2. The conditions for the spread of a tow penetrance siblicide allele through a non-siblicidal population are identical both to the equivalent condition for the "Random victim" model and also to the prediction for the spread of siblicide from the inclusive fitness argument. In contrast to the last model, the conditions for the
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I
0,75
"o o
o
r,
,m
0.5 o ww-
o0
0.25
0 2
3
4
5
6
7
8
9
Brood size
FIG. 2. The spread of siblicidal and non-siblicidal alleles in the " R u n t " model. Axes as in Fig. 1 (though note different scale on the ordinate). A siblicidal gene with arbitrarily small penetrance invades a population fixed for non-siblicidal behaviour in regions a and b. Unlike the "'Random victim" model, an allele with high penetrance spreads in a narrower area of parameter space: an allele with penetrance = 1 spreads only in region a. A non-siblicidal allele of any penetrance invades a population fixed at siblicide in region c. In region d, both siblicide and non-siblicide are ESSs but, as explained in the text and Fig. 3, only non-siblicide is a CSS.
spread of obligate siblicide are more stringent than that of occasional siblicide (with the exception of clutches of two where the conditions are identical). However, we believe that once a low penetrance gene for sibticide has gone to fixation, a gene with a slightly higher penetrance will then be able to invade and so on until the population consists of obligate siblicides. The predictions of the spread of a siblicidal allele from the genetic and inclusive fitness arguments are thus the same. The criterion for the spread of a non-siblicidal allele of any degree of penetrance is simply that the benefits o f siblicide must be less than ½. In comparison with the " R a n d o m victim" model, it is considerably easier for a non-siblicidal allele to invade a population of obligate siblicides. The reason for this is that there is the possibility o f a non-siblicidal gene being protected from elimination by a siblicidal gene in a runt. As in the " R a n d o m victim" model, there appears to be an area of hysteresis in the switch between siblicidal and non-siblicidal behaviour [when 0 . 5 < b < ( 1 - ( 0 - 5 ) ~ - I ) ] . However, there is a difference in the form of the stability o f the
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siblicidal behaviour in the two models. This is best seen by examining the fate of a rare allele invading a population exhibiting different rates o f siblicide for parameter values in the "'area o f hysteresis" (Fig. 3). Consider a population which displays obligately non-siblicidal behaviour (i.e. in the lower left-hand c o m e r o f both figures). Any rare siblicidal mutant, whatever its degree o f penetranc¢, will fail to spread. If the population is perturbed slightly away from obligately non-siblicidal behaviour, the only rare mutants that will spread are those with even greater non-sublicidal tendencies and these will tend to return the population towards a complete absence of siblicide. In both cases, non-siblicidal behaviour is thus stable. Now consider a population displaying obligately siblicidal behaviour (i.e. the upper right-hand c o m e r o f both figures). In both cases, any less-siblicidal, mutant allele, whatever its degree of penetrance, will fail to spread. Thus, as we have concluded already, the obligately siblicidal behaviour is in some sense stable. Consider now a small perturbation away from obligate siblicide. In the case o f the " R a n d o m victim" model, the only mutants that can now spread are those that tend to move the population back towards obligate siblicide: in this model obligate siblicide is stable in a strong sense. However, in the " R u n t " model, a small perturbation away from obligate siblicide allows mutants with a smaller probability of being siblicidal to spread. These will run to fixation and themselves allow other alleles with even less silicidal tendencies to spread until, ultimately, the population becomes "Runt" model
"Random victim" model
ta
~5 EL
0
I
0
I
Probability that common genotype is siblicidal
FIG. 3. The fate o f a rare gene invading a population in regions d o f Figs 1 and 2. The population is a s s u m e d to be at fixation for an allele that is siblicidal with probability a (abscissa). A rare m u t a n t which is siblicidal with probability s (ordinate) invades the resident population in the shaded region o f parameter space. Note that obligate siblicide a n d non-siblicide are uninvasible to all other alternative strategies. S u p p o s e the populations are perturbed from obligate siblicidal or non-siblicidal behaviour. Alternative alleles are now able to spread but will tend to move the behaviour of the population back towards either obligate siblicide or non-siblicide. The exception is obligate siblicide in the " R u n t " model. A slight m o v e m e n t away from obligate siblicide in population behaviour allows alleles coding for even lower levels o f siblicide to invade and eventually the population will come to be c o m p o s e d o f obligately non-siblicidal individuals.
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completely non-siblicidal. Thus siblicidal behaviour in the " R u n t " model is only stable in a very weak sense for 0-5 < b < [ 1 - (0-5)c-1]: arbitrarily small perturbations in the resident population away from obligate siblicide lead to invasion of less siblicidal alleles and ultimately to an obligately non-siblicidal state. There is thus no hysteresis between siblicidal and non-siblicidal behaviour and the predicted boundary between the two activities is identical from genetic and inclusive fitness arguments. Eschel & Motro (1981) and Eschel (1983) describe a number of examples of evolutionary strategies that, at fixation, are stable to invasion by any rare mutant, yet if slightly perturbed away from the stable value are then susceptible to invasion. They describe such strategies as evolutionary stable strategies (i.e. as ESSs, Maynard Smith, 1982) but not continuously stable strategies (CSSs). The strategy of obligate siblicide, for parameter values in region d (Fig. 2) o f the " R u n t " model, is an example o f this type o f strategy. Discussion Predictions about the evolution o f siblicide can be obtained from either inclusive fitness or population genetic arguments. We have compared the two approaches and found that the assumption o f the form of siblicide (our runt and random victim models) is crucial to the predictions of the genetic model. Where the victim is a weak and non-aggressive runt, as seems to occur frequently in birds, siblicide is predicted by both arguments when the benefits to the surviving clutch members are (c-1)/c where c is clutch size and benefit is measured in units o f the fitness of one clutch member prior to siblicide. An alternative formulation of this criterion is that siblicide is favoured when the benefits to the actor are greater than 1/c. However, when the victim is chosen at random, the more exact genetic models predict very different results from those o f the inclusive fitness model. In particular, siblicidal behaviour evolves over a wider range o f parameter values, and having evolved, it is difficult for non-siblicidal behaviour to re-establish itself, even when the benefits o f siblicide are much reduced or negative. We have compared our genetic models with the inclusive fitness arguments of O ' C o n n o r (1978) as these have been widely discussed in the ornithological literature though we realize that it might be possible to construct a more sophisticated inclusive model o f siblicide. It should be noted that one of the critical differences between our two models o f siblicide is that the runt is unable to attack an elder, non-aggressive sibling. It is possible that runts in some species with obligate siblicide might be able to attack and kill elder non-aggressive siblings that had failed to kill them. Consider the spread o f a non-siblicidal allele through a population with obligate siblicide and with a clutch size of two. If the runt is unable to kill a tolerant elder sibling then non-siblicide spreads when b < 0 . 5 (our runt model, see Fig. 1). However, if the runt kills the tolerant elder sibling with probability q then it can be shown that non-siblicide spreads when b < 0-5q. Thus if the runt is always successful (q = 0), the condition for the spread o f siblicide is b < 0, the identical condition to that predicted by our random victim model (Fig. 2). Experimental studies on the dynamics
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of brood reduction are needed before more precise predictions about the evolution o f siblicide in different bird groups can be made. However, we believe it is at least possible that some bird species have evolved siblicidal behaviour o f a type that is very difficult for a non-siblicidal allele to invade. Possible candidate species are those eagles with obligate siblicide which appears to be unaffected by local food supplies (Meyburg, 1974; Gargett, 1978). Lack (1954) first suggested that hatching asynchrony and runt production may arise as an adaptation for efficient brood reduction in years when a parent is unable to rear its full clutch. Since then, no less than seven other hypotheses have been advanced to explain hatching asynchrony (reviewed in Lessells & Avery, 1989; Slagsvold & Lifjeld, 1989). One hypothesis suggests that asynchrony is selected because sibling rivalry is reduced in broods with a hierarchy of chick sizes in comparison with broods where all chicks are the same size: the hierarchy allows competitive disputes to be settled quickly without extended conflict (Hamilton, 1964; Hahn, 1981). Our models have shown that a hierarchy o f chick sizes can have a marked effect on the levels of parent-offspring conflict by making siblicide less likely. It would seem intuitively appealing that the parent might be selected to produce a hierarchy to minimize this form of sibling conflict. The problem with this argument is that the reduction in siblicide is not an immediate benefit that could act as a selection pressure leading to the evolution o f a chick hierarchy, but a benefit that occurs subsequent to its establishment. Siblicide is known in a number of other groups outside birds (Polis, 1981; Godfray, 1987a). In some cases such as coccinelid and chrysomelid beetles there is a hierarchy in the age of siblings and the elimination, quite often, of large numbers of younger individuals. Parker & Mock (1987) have examined inclusive fitness models appropriate to such animals. In contrast, Godfray (1987b) has argued that the random victim model is appropriate to cases o f siblicide in parasitoid wasps and that there is evidence for a hysteresis in the switch between siblicidal and non-siblicidal behaviour (le Masurier, 1987). We thus believe that the models presented here, suitably modified, may be applicable to other animal groups. The aim of this paper has been to resolve the differences between inclusive fitness and genetic arguments for the evolution o f siblicide by brood reduction and to highlight the most important assumptions affecting the predictions o f the model. We believe that future experimental work is essentially to determine which model (or blend o f models) is most appropriate for bird species with siblicide. Future theoretical work is needed to examine, among other things, the consequences of allowing a more complicated genetic basis to the trait; the effects o f the costs o f siblicide, especially in the random victim model; the evolution o f siblicide as an escalated form o f other types o f sibling interaction and the evolutionary response of parents to excess siblicide (see also Parker & Mock, 1987).
REFERENCES CHARNOV, E. L. (1978). Evolution of eusocial behaviour: offspring choice or parental parasitism? J. theor. Biol. 75, 451-456.
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ESCHEL, I. (1983). Evolutionary and continuous stability. Z theor. Biol. 103, 99-112. ESCHEL, I. & MOTRO, U. (1981). Kin selection and strong evolutionary stability of mutual help. Theor. pop. Biol. 19, 420-433. ESCHEL, I. & MOTRO, U. (1988). The three brothers' problem: kin selection with more than one potential helper. 1. The case of immediate help, Am. Nat. 132, 550-566. FELDMAN, M. W. & ESCHEL, 1. (1982). On the theory of parent-offspring conflict: a two-locus genetic model. Am. Nat. 119, 285-292. GARGETT, V. (1978). Sibling aggression in the Black Eagle in the Matapos, Rhodesia. Ostrich 49, 57-63. GO DFRAY, H. C. J. (1987 a). The evolution of clutch size in invertebrates. Oxf Suro. evol. Biol. 4, 117-154. GODFRAY, H. C. J. (1987b). The evolution of clutch size in parasitic wasps. Am. Nat. 129, 221-233. GRAEEN, A. (1986). A geometric view of relatedness. O x f Sure. evol. Biol. 2, 28-29. HAHN, D. C. (1981). Asynchronous hatching in the Laughing Gull: cutting losses and reducing rivalry. Anim. Behav. 29, 421-427. HAMILTON, W. D. (1964). The genetical theory of social behaviour. I. J. theor. Biol. 7, 12-45. HARPER, A. B. (1986). The evolution of begging: sibling competition and parent-offspring conflict. Am. Nat. 128, 99-114. LACK, D. (1954). The Natural Regulation of Animal Numbers. Oxford: Oxford University Press. LACK, D. (1966). Population Studies of Birds. Oxford: Clarendon Press. LACK, D. (1968). Ecological Adaptations for Breeding in Birds. London: Methuen. LESSELLS, C. M. & AVERY, M. I. (1989). Hatching asynchrony in European Bee-eaters Merops apiaster. J. Anita. Ecol. 58, 815-835. MACNAtR, M. R. & PARKER, G. A. (1978). Models of parent-offspring conflict. II. Promiscuity. Anita. Behav. 26, 111-122. MACNAIR, M. R. & PARKER, G. A. (1979). Models of parent-offspring conflict, llI. Intra-brood conflicL Anita. Behav. 27, 1202-1209. LE MASURIER, A. O. (1987). A comparative study of the relationship between host size and brood size in Apanteles spp. (Hymenoptera: Braconidae). Ecol. Ent. 12, 383-393. MAYNARD SMITH, J. (1982). Evolution and Theory of Games. Cambridge: Cambridge University Press. MEYaURG, B.-U. (1974). Sibling aggression and mortality among nestling eagles. Ibis 116, 224-228. MICHOD, R. E. (1982). The theory of kin selection. Ann. Rev. Ecol. Syst. 13, 23-56. MOTRO, U. & ESCHEL, I. (1988). The three brothers" problem: kin selection with more than one potential helper. 2. The case of delayed help. Am. Nat. 132, 567-575. O'CONNOR, R. J. (1978). Brood reduction in birds: selection for fratricide, infanticide and suicide. Anita. Behav. 26, 79-96. PARKER, G. A. (1985). Models of parent-offspring conflict. V. Effects of the behaviour of the two parents. Anita. Behav. 27, 1210-1235. PARKER, G. A. & MACNAIR, M. R. (1978). Models of parent-offspring conflict. I. Monogamy. Anita. Behav. 26, 97-110. PARKER, G. A. & MACNAIR, M. R. (1979). Models of parent-offspring conflict. IV. Suppression, evolutionary retaliation by the parent. Anim. Behav. 27, 1210-1235. PARKER, G. A. & MOCK, D. W. (1987). Parent-offspring conflict over clutch size. EvoL Ecol. 1, 161-174. POLLS, G. A. (1981). The evolution and dynamics of intraspecific predation. Ann. Rev. Ecol. Syst. 12, 225-251. SLAGSVOLD, T. & LIFJELD, J. T. (1989). Constaints on hatching asynchrony and egg size in Pied Flycatchers. J. Anita. Ecol. 58, 837-849. STAMP, J. A., METCALF, R. A. & KRISHNAN, V. V. (1978). A genetic analysis of parent-offspring conflict. Behav. EcoL Sociobiol. 3, 369-392. STINSON, C. H. (1979). On the selective advantage of fratricide in raptors. Evolution 33, 1219-1225. TRlVERS, R, L (1974). Parent-offspring conflict. Am. Zool. 14, 249-264. APPENDIX
"Runt" Model We need to calculate the number of S genes resulting from an AS x AA mating. The rare gene S will spread if more than c/2 copies are produced (c/2 are the number of copies of an arbitrarily marked A gene that are produced in an AA x AA mating).
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We divide the b r o o d into the runt and the c - 1 other members. Let i be the n u m b e r o f A S individual a m o n g these c - 1 chicks. The progeny from an A S x A A mating are AA or A S with equal probability and so i is binomially distributed with parameters c - 1 and 0.5. To determine whether the runt survives we need to know the p h e n o t y p e of the A S and AA individuals. The probability that an A S individual is potentially siblicidal is s and thus j, the n u m b e r of A S with the siblicidal phenotype, is binomially distributed with parameters i and s. By the same argument k, the n u m b e r o f AA individuals with the siblicidal phenotype, is binomially distributed with parameters ( c - 1 - i) and a. The joint distribution of siblicidal A S and AA individuals in a brood (excluding the runt), p(i,j, k, c - 1), is thus
I f the runt survives, (i + 0.5) copies of A S are produced from an A S x AA mating while if the runt dies, i[1 + b / ( c - 1 ) ] are produced. The runt survives if j = k = 0 . The S gene spreads if the total number of S genes arising from the mating is greater than c/2 or 2
p(i, 0,0, c - 1 ) { i + 0 . 5 } +
i=0
~. p ( i , j , k , c - 1 ) { i [ l + b / ( c - 1 ) ] } j=l
>
.
(A.2)
k=I
The entries in Table 1 are obtained by substituting into eqns (A.1) and (A.2) the appropriate values of a and s.
TABLE A.1 This table calculates the probability that an aggressor or a victim will have genotype A A or AS given that the brood is composed o f j potentially siblicidal AS individuals and k potentially siblicidal A A individuals. The broodproduces i AS individuals if the victim is A A and one less if the victim is AS. The sum o f the products o f the last two columns is the variable X which appears in eqn (A.3) Aggressor's genotype
Victim's genotype
AA
AA
AA
AS
AS
AA
AS
AS
Probability k n-l-i j + k n-1 k i j+kn-1 j
n--i
j+kn-1 j i-I j+kn-1
No. of AS produced i i- 1
i i- 1
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"Random Victim" Model
Let p(i,j, k, c) be the probability that a b r o o d of c individuals arising from an A S x A A mating contains i A S chicks, j of which are potentially siblicidal, and c - i A A chicks, k of which are potentially siblicidal. The probabilities are given by eqn (A.1) with c - 1 replaced by c. If there is no siblicide then i A S individuals survive. I f there is a siblicide, then either i[1 + b / ( c - 1)] or (i - 1)/[1 + b / ( c - 1)]AS individuals survive depending on the genotype of the victim. The probability of the victim being a particular genotype depends on the genotype of both the aggressor and the victim as listed in Table A.1. S will spread if the total n u m b e r of S genes produced by a clutch of size c from an A S x A A mating is greater than c/2 or
p(i,O, 0, c){i}+ Y. ~, p ( i , j , k , c ) { X [ l + b / ( c - l ) ] } i=O
j=l
>~,
(A.3)
k=l
where X is the sum o f the product of the last two columns in Table A.1. The entries in Table 1 are obtained by substituting into eqns (A.1) and (A.3) the appropriate values of a and s.