u

ontC 3

ry.c

Theoretical Population Biology 59, 7�9 (2001)

William D. Hamilton: A Trib

Sabin LessardDe� partement de Mathe� matiques et de Statistique, Universite� de MC.P. 6128, Succursale Centre-ville, Montre� al, Que� bec, Canada H3

Received August 23, 2000

In the presentation of his collected papers (Hamilton,1996), W. D. Hamilton writes that for most of his youngyears as a researcher he was extremely lonely. I did notknow the man, only some of his works. I met him oncein Santa Fe in 1987. One night I saw him, alone, walkingin the street looking at shop windows. I was with otherpeople and I invited him to join us. He declined,

I first became interested in Hamilton's works in 1980.I was then a postdoc working on sex ratio evolution withSam Karlin at Stanford University. Triggered by IlanEshel and Marc Feldman's study on evolutionary geneticstability (EGS) of the sex ratio, we were trying to providemore mathematical rigor to Fisher's verbal argument forthe evolution and maintenance of equal investment in thetwo sexes in Mendelian populations. The cases whereFisher's argument fails were of prime interest to a biologist.Hamilton's paper on extraordinary sex ratios (Hamilton,1967), published 13 years earlier was concerned withsituations where certain underlying assumptions ofFisher' s argument do not hold. These were sex-linkeddrive, which is a genetic factor, and local competition formates, which involves population structure. This paper isone of the more analytical ones W. D. Hamilton, a non-mathematician, ever wrote alone. It is also one of whichhe is more proud than almost any other: ``Perhaps this ismost of all because I believe it helped sex-ratio theory wellon its way to being the section of evolutionary theory thatbest proves the power and accuracy of the Neodarwinianparadigm as a whole.'' (Hamilton, 1996, p. 132). Hamiltonbelieved that sex ratio, an easily observable trait, had astatus practically unsurpassed in evolutionary theory asan exact science: it was more suitable than any other traitfor testing predictions about evolution.

One of the main contributions of Hamilton's paper onextraordinary sex ratios, besides the illustration of possibleconflicts within the genome and between different levelsof selection, is the introduction of game theoretic ideas to

doi:10.1006�tpbi.2000.1502, available online at http:��www.idealibra

7

te

re� al,J7

sex ratio evolution and the introduction of the notion ofan ``unbeatable strategy,'' a precursory to Maynard Smithand Price's notion of ESS, ``evolutionarily stable strategy'':``This required that, relative to a type I designated``unbeatable,'' no other type, no matter in what frequencyin a mixture, would be able to increase. Maynard Smith'sESS only required that it be shown that if almost allindividuals in the population are using the ESS, no otherstrategy starting from a low frequency can invade thatpopulation''(Hamilton, 1996, pp. 373�374). In haploid orhaploid-like models like the ones studied by Hamilton,no polymorphic equilibrium can exist and, when astrategy has a selective advantage over another at somefrequency, then it has a selective advantage over theother at any frequency. Consequently an ESS is de factounbeatable. But in general the conditions for an ESS ina defined microevolutionary context are less restrictive,since they bear only on the final strategies that can resistinvasions once they are fixed in the population. Thegenerality and simplicity of application of this concepthave made it very popular among evolutionary biologistsfor pointing out the presumed endpoints of evolution.

But Hamilton's formulation raises a more difficultquestion: will there be evolution toward the ESS bynatural selection when it exists? The EGS property, whenit holds, ensures that the population tends toward anESS, at least initially after enough generations havepassed, when a mutant allele invades the populationat any other equilibrium. When the next attainableequilibrium is closer to the ESS, as happens in some sexratio evolution models, we have the EGS property oversuccessive possible equilibria. But global convergenceaccording to such a scheme has been demonstrated onlyfor a few cases. The long-term evolutionary trends, asmutant genes are introduced one at a time into gametheoretic evolution models or sex ratio evolution models,remain exciting theoretical challenges.

om on

0040-5809�01 K35.00

Copyright ] 2001 by Academic PressAll rights of reproduction in any form reserved.

Like many others, I have been puzzled by kin selectiontheory, which is considered the major contribution ofW. D. Hamilton to evolutionary theory. This theory,which was introduced as an alternative to group selectionto explain the evolution of altruistic behaviors, has had aprofound influence on the way we think of, and treat,evolution today. It is remarkable that a Ph.D. thesiswritten by a solitary worker appeals to so many conceptsand tools that are at the heart of a discipline. Kin selec-tion theory is indeed in the main stream of populationgenetics. In his first paper on the subject, Hamilton refersonly to three authors, Fisher, Haldane, and Wright,nothing less than the fathers of population genetics. Theywere all concerned with a satisfactory explanation forcases ``where an animal behaves in such a way to promotethe advantages of other members of the species not itsdirect descendants at the expense of its own'' (Hamilton,1963). Haldane may have been the one who came closestto Hamilton's principle when he discussed whether agenetic trait causing someone to risk his life to save arelative in danger could evolve. But the principle wasalready foreshadowed in Fisher's remark of the evolutionof distastefulness in insects. Roughly speaking, Hamilton'sprinciple says that an altruistic behavior will evolve if theloss to the altruist is less than the total gain to its relativesweighted by coefficients of relatedness. The main point isthat the beneficiaries of the altruistic act must be relatedto the altruist. Hamilton (1964) proved that the inclusivefitness which transfers benefits of altruistic acts frombeneficiaries to actors, weighted by coefficients of related-ness, is maximized by natural selection in much the sameway that fitness is maximized in the simpler classicalmodel according to the fundamental theorem of naturalselection. When understood as an increase of the meanfitness, Hamilton's proof may lack some mathematicalrigor. But its corollary for the initial increase of analtruistic gene has generally proved to be sound when thecosts, benefits and coefficients of relatedness are definedin an appropriate manner and selection is weak enough.

As shown later on, the change in frequency of analtruistic gene is correlated to inclusive fitness via Price'scovariance formula (Hamilton, 1970), which leads to anapproximate adaptive topography in Wright's sense basedon inclusive fitness when selection is weak. Moreover, thecoefficient of relatedness originally defined for diploidoutbred populations as Wright's symmetric correlationcoefficient of relationship can be extended to haplodiploid

8

populations if reproductive values of the two sexes aretaken into account (Hamilton, 1972). It extends toinbred populations if defined as a regression coefficientor more generally as a covariance ratio which can be

expressed in terms of Male� cot's coefficient of kinship andGillois' coefficients of identity under weak selection(Michod and Hamilton, 1980). This covariance ratio canbe extended to the case of multiple alleles, but in generalit is not symmetric and depends on the allele frequenciesand the probabilities of performing an altruistic actaccording to the genotypes. Alternatively, it can bereplaced when selection is weak by the expected fractionof genes identical by descent to at least one of the genesat a single locus in the altruist, given that these genes areidentical by descent or not. Such a fraction correspondsto the original definition given by Hamilton for outbredpopulations.

Hamilton claimed several times that his theory appliesprovided selection is slow, as suggested by an inclusivefitness function which additively combines weightedchanges in fitness. This additive model is actually anapproximation when selection is weak, the linear part ofa Taylor expansion. Moreover, several authors have con-firmed that weak selection is a necessary ingredient forthe validity of Hamilton's theory. His theory hasmotivated many studies of selection models with kininteractions as multiplicative models in family-structuredpopulations and general frequency-dependent selectionmodels in group-structured populations. Thanks toHamilton, we know more today about all of thosemodels. Moreover, although some kin selection modelsmay be cast in the framework of fertility-viability selec-tion or group selection, the inclusive fitness approach isquite intuitive and has an undeniable appeal. It has sim-plified to a great extent the finding of optimal (actually,EGS) strategies or sex ratios and it has rapidly spreadamong population biologists. It has become an indis-pensable tool for advancing evolutionary theory.

But what 1 like most in kin selection theory is that itis a genetic theory. There is a pervasive idea among someevolutionary biologists that the underlying genetics is notdecisive in evolution, that the genetics does not reallycount to predict evolutionarily stable strategies. Kinselection theory takes the opposite view since it iscentered on relatedness, which is essentially based onidentity of genes. It is impossible to study kin selectionwithout knowing the population structure, the matingpattern, the genetic system, everything that can affectinbreeding and relationship. Population geneticists arestill in business.

That the evolution of sex was one of the more recent

Sabin Lessard

research interests of Hamilton (1990) is not surprising. Thisis one of the central unsolved questions in populationgenetics. That his explanation is based on host-parasitecoevolution is also not fortuitous. Hamilton spent a great

deal of his professional life studying interactions betweenindividuals. These latest works are typical of Hamilton'sview.

REFERENCES

Hamilton, W. D. 1996. ``Narrow Roads of Gene Land: The Collectedpapers of W. D. Hamilton, volume 1: Evolution of Social Behaviour''(W. H. Freeman, Ed.), Spektrum, Oxford, Macmillan Press.

Hamilton, W. D. 1967. Extraordinary sex ratios, Science 156, 477�488.

William Hamilton: A Tribute

Hamilton, W. D. 1963. The evolution of altruistic behavior, Am. Nat.97, 354�356.

Hamilton, W. D. 1964. The genetical evolution of social behaviour,I and II, J. Theor. Biol. 7, 1�52.

Hamilton, W. D. 1970. Selfish and spiteful behaviour in an evolutionarymodel, Nature 228, 1218�1220.

Hamilton, W. D. 1972. Altruism and related phenomena, mainly insocial insects, Ann. Rev. Ecol. Syst. 3, 193�232.

Michod, R. E., and Hamilton, W. D. 1980. Coefficients of relatedness insociobiology, Nature 288, 694�697.

Hamilton, W. D., Axelrod, R., and Tanese, R., 1990. Sexual reproduc-tion as an adaptation to resist parasites (A review), Proc. Natl. Acad.Sci. USA 87, 3566�3573.

9