ARTICLE IN PRESS

Journal of Theoretical Biology 246 (2007) 551–554 www.elsevier.com/locate/yjtbi

Hamilton’s missing link Matthijs van Veelen CREED, Universiteit van Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands Received 16 October 2006; received in revised form 2 January 2007; accepted 2 January 2007 Available online 8 January 2007

Abstract Hamilton’s famous rule was presented in 1964 in a paper called ‘‘The genetical theory of social behaviour (I and II)’’, Journal of Theoretical Biology 7, 1–16, 17–32. The paper contains a mathematical genetical model from which the rule supposedly follows, but it does not provide a link between the paper’s central result, which states that selection dynamics take the population to a state where mean inclusive fitness is maximized, and the rule, which states that selection will lead to maximization of individual inclusive fitness. This note provides a condition under which Hamilton’s rule does follow from his central result. r 2007 Elsevier Ltd. All rights reserved. Keywords: Hamilton’s rule; Altruism; Inclusive fitness

1. Introduction Hamilton’s rule is by far the best-known prediction from kin selection theory. It captures the insight that the individual’s own fitness is not the only criterion for selection of traits, but that, weighted by measures of relatedness, the fitnesses of relatives matter too. Intuitively appealing as it may be, the question is justified if there is a model from which this prediction follows. The most natural place to look for the model behind the rule is the seminal paper by Hamilton (1964) in which the rule was presented. In part II and in the last section of part I of The Genetical Evolution of Social Behaviour, the rule can be found in a representation that is only slightly different from the way it is usually written now. The first sections of part I also contain a model and a central result that follows from it. There are however a few remarkable discrepancies between the rule and the central result. The most important one is that Hamilton’s central result states that selection dynamics take a population to a state where mean inclusive fitness is maximized, whereas Hamilton’s rule concerns the maximization of individual inclusive fitness. More generally, one can say that the central result is about the selection of genes and describes the outcome as a state at E-mail address: [email protected]. 0022-5193/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2007.01.001

the population level, while the rule on the other hand concerns phenotypes and describes selection of individual behaviour. When reduced to interactions between two individuals, Hamilton’s rule predicts that behaviour whereby an individual raises the fitness of another individual by b and lowers its own fitness by c will be selected for if rb  c40 and will be selected against if rb  co0, where r is a measure of relatedness. This rule has great intuitive appeal, and Hamilton’s rule is probably the only formula from biology that is known by many non-biologists too. It is hardly surprising that Hamilton’s rule gained much more popularity than his central result that is relatively inaccessible and rather rich in maths. Yet there is a gap to be bridged if we want to make Hamilton’s rule into a prediction that follows from his model, and that is what we will do in this note. At some places in the literature, authors have disregarded Hamilton’s model altogether, and derived his rule for special cases (see for example Charnov, 1977). Others have relaxed the assumptions of the model. While Hamilton (1964) assumes no frequency dependence and no inbreeding, alternative models do allow for frequency dependence (see for instance Cavalli-Sforza and Feldman, 1978), inbreeding (see for instance Michod, 1980; Uyenoyama, 1984; Lessard, 1992) or both (see Uyenoyama and

ARTICLE IN PRESS M. van Veelen / Journal of Theoretical Biology 246 (2007) 551–554

552

Feldman, 1980; Roze and Rousset, 2004). Again others have derived Hamilton’s rule with the use of the Price equation (see for instance Grafen, 1985, 2006; Taylor, 1989). In van Veelen (2005) I show that the latter approach regularly fails to produce a proper model. I also show how in some cases this problem can be overcome, but then the end result is a model that significantly differs from the model of Hamilton. In this note however we will look at Hamilton’s original model, and restrict ourselves to answering the question whether or not his rule indeed follows from his mathematical genetical model. It turns out that there are relatively mild conditions under which Hamilton’s rule can in fact be linked to his central result. Still there are cases where Hamilton’s result and his rule make different predictions and some simple examples are also given below. 2. Maximizing inclusive fitness Hamilton’s (1964) central result can be stated as follows. Consider a single locus and alleles G 1 ; . . . ; G n with frequencies p1 ; . . . ; pn . Behaviour might affect the fitness of the individual itself as well as the fitnesses of m  1 relatives. A genotype Gi G j therefore comes with a vector ðda1 ; . . . ; dam Þij of effects on itself and on relatives, which have relatednesses r1 ; . . . ; rm to the focal individual. If the focal individual itself gets indexed by the number 1, then obviously r1 ¼ 1. These effects are summed, with related nesses as weights, to produce inclusive Pm fitness Rij of  genotype G i G j as follows: Rij ¼ 1 þ k¼1 rk ðdak Þij . The dynamics are given as a set of difference equations with frequencies p1 ; . . . ; pn as  variables.  Frequencies can be written as a vector p ¼ p1 ; . . . ; pn on the unit simplex D. This simplex represents all possible compositions of the population, which implies that all gene-frequencies must be larger than or equal to zero and that they must sum up to one. The vector p represents a population state. For the central P result, P we quote Hamilton (1964). Here R:: is short for ni¼1 nj¼1 pi pj Rij . It follows that R:: certainly maximizes (in the sense of reaching a local maximum of R:: ) if it never occurs in the course of selective changes that dS:: o0. The term dS:: is the sum of ‘unspecific contribution consisting of genes in the ratio in which the gene pool already possesses them’ (see the paper itself for a precise definition). A sufficient condition for dS :: never to be smaller than 0 is that all genetic effects imply that actual benefits are dispensed to neighbours. So if we assume that ðdak Þij is positive for all i; j and all relatives ka1, then for Hamilton’s model the following theorem holds: Theorem 1. The take the population to a local P dynamics P maximum of ni¼1 nj¼1 pi pj Rij on D. Proof. See Hamilton (1964). One additional remark to be made is that in the paper, Hamilton shows that if dS:: X0 then DR:: X0. To show that the difference equation actually

reaches (or better: approaches) a local maximum, one however also needs to show that this inequality is strict at all points other than the local maxima, that is, DR:: 40 in all points p that are not a local maximum of R:: . He perhaps implicitly assumes that this is understood, but we do need the observation that Kingman (1961, p. 79, first two lines) makes to see that the first term on the right-hand side of the final equation in Hamilton’s (1964, p. 7) derivation is strictly positive in points p that are not local maxima of R:: . & Concerning the case where we do allow for negative effects on the fitness of neighbours, Hamilton states that For cases where individuals may dispense harm to their neighbours we merely know, roughly speaking, that the change in gene frequency in each generation is aimed somewhere in the direction of a local maximum of average inclusive fitness, but may, for all the present analysis has told us, overshoot it in such a way as to produce a lower value. The most noticeable feature of the dynamics is therefore that the quantity that will tend to maximize, if any, is mean inclusive fitness, that is, a weighted sum of inclusive fitnesses where the weighing factors are gene frequencies. The focus of this note is what that implies for individual inclusive fitness. In the examples below, matrix entries in row G i and column G j are values for inclusive fitness Rij . Matrices obviously are symmetric. The first one is G1 G2

G1

G2

1 1

1 2

P P The maximization can be written as maxp2D ni¼1 nj¼1 pi pj Rij and in this example the maximum is unique and it is attained at p ¼ ð0; 1Þ. Not surprisingly, the dynamics therefore take the population to a state with only G 2 in it. To link up with Hamilton’s rule as it was given in the introduction, it is useful to realize that rb  c is the difference in inclusive fitness caused by the behaviour. Therefore, an equivalent way of phrasing this rule is that behaviour with the highest inclusive fitness is selected. In this case, Hamilton’s rule makes the same prediction as Hamilton’s central result, for genotype G 2 G 2 indeed has the highest inclusive fitness. The second example is G1 G2

G1

G2

1 3

3 2

Here the maximum is also unique and equal to the population state p ¼ ð13; 23Þ. The dynamics therefore take the population to a mixture of both alleles. Here Hamilton’s rule makes a different prediction; in the example, 3 is apparently a value of inclusive fitness that

ARTICLE IN PRESS M. van Veelen / Journal of Theoretical Biology 246 (2007) 551–554

can be brought about by some behaviour, namely the behaviour of the heterozygote. Yet the outcome of selection is not that all individuals will display the according behaviour, for the mixture will also contain homozygotes with inclusive fitnesses 1 and 2, respectively. The third example is G1

G2

G1

1

0

G2

0

2

In this case, there are two local maxima; one at p ¼ ð1; 0Þ and one at p ¼ ð0; 1Þ. Selection therefore can get stuck in the local maximum with the behaviour that has inclusive fitness 1, even though an inclusive fitness of 2 is also possible. It is natural to call G1 dominant in the first example and speak of overdominance in the second and underdominance in the third. This is justified, if we realize that the numbers of the example are inclusive fitnesses and agree that dominance refers to these inclusive fitnesses. If we look at other ways to describe phenotypic expression of a trait, such as for instance amounts of food given, then examples can be made where effects of genes on giving behaviour are additive and nonetheless produce dominance in inclusive fitnesses. If we however assume effects to be small, this possible divergence vanishes. The question that remains to be answered is whether Hamilton’s rule follows from his central result if we exclude over- and underdominance. In the following theorem it is shown that if there is no pair of alleles for which we find over- or underdominance, then indeed the maximum from Hamilton’s central result is a state where all individuals have maximum inclusive fitness. Theorem 2. If for all combinations of i and j the following holds: minfRii ; Rjj gpRij ¼ Rji p maxfRii ; Rjj g Pn Pn  then at any local maximum of i¼1 j¼1 pi pj Rij on D, all genotypes have inclusive fitness maxi;j Rij . Proof. Without loss of generality we can assume that alleles are indexed such that inclusive fitnesses of homozygotes are non-decreasing Pn Pnin the index, that is: R11 pR22 p    pRnn . Let i¼1 j¼1 pi pj Rij have a local maximum at p . First assume that there is no k for which pk ¼ 1, which means that there is more than one allele with positive frequency. Then, for all pk ; pl that are strictly between 0 and 1, the following must hold: P P  P P  n n n n     q q i¼1 j¼1 pi pj Rij  i¼1 j¼1 pi pj Rij    ¼   qpk qpl     p¼p

p¼p

553

which implies that for all such k and l n X

pi Rki ¼

i¼1

n X

pi Rli .

i¼1

We want to disregard those terms with pi ¼ 0. One way to do this is to reduce the matrix of inclusive fitnesses by taking out the rows and columns for those alleles that are not in p and renumber the alleles. This is equivalent to assuming that pi 40 for i ¼ 1; . . . ; n and that nX2, that is, that we started with the reduced matrix already and it has at least two rows and columns. Now look at the condition for k ¼ 1 and l ¼ n n X i¼1

pi R1i ¼

n X

pi Rni .

i¼1

The restrictions on the inclusive fitnesses of the heterozygotes R1i pRii pRni for all i. This implies that Pn  imply Pthat n    First we assume R11 oRn1 . i¼1 pi R1i p i¼1 pi Rni . P Pn that n      Then either p1 ¼ 0 or i¼1 pi R1i o i¼1 pi Rni . The first contradicts that pi 40 for all i, the second contradicts the condition, that has to hold because pi is assumed to be a maximum. Therefore, R11 ¼ Rn1 .     assume Pnthat Rn1 ¼ R1n oRnn . Then either pn ¼ 0 or PThen n   i¼1 pi R1i o i¼1 pi Rni . The first contradicts that pi 40 for all i, the second contradicts that pi is a maximum. Therefore, Rn1 ¼ Rnn . But then also R11 ¼ Rnn and the conditions on the heterozygotes imply that all inclusive fitnesses are equal. In this case indeed all genotypes have the highest inclusive fitness. Now assume that there is a k for which pk ¼ 1 and that kan. If Rkk oRnn , then mean inclusive fitness can be  increased locally by taking p with p k ¼ 1   and pn ¼ .  This contradicts that pi is a maximum, and therefore Rkk ¼ Rnn . In this case genotype Gk G k indeed has the highest inclusive fitness. The last possibility is that pn ¼ 1. In this case the theorem obviously holds. & Hamilton’s central result in combination with this theorem states that, under restrictions on the inclusive fitnesses of heterozygotes, the behaviour with the highest inclusive fitness will spread through the population and outcompete the others, which is Hamilton’s rule. 3. Conclusion and discussion Theorem 2 provides a condition under which Hamilton’s rule, which suggests that selection favours phenotypes that maximize individual inclusive fitness, follows from a result concerning his mathematical genetical model, which states that selection dynamics take a population to a state where mean inclusive fitness is maximized. The condition is that there is no pair of alleles for which we have over- or underdominance. While this note is meant to serve the reader who wants to follow Hamilton’s original argument, it is worth being aware of how this relates to more recent models of kin selection. As

ARTICLE IN PRESS 554

M. van Veelen / Journal of Theoretical Biology 246 (2007) 551–554

a prime example, I would like to mention Roze and Rousset (2004) who develop a different model that is more general in the sense that it allows for inbreeding and less general in the sense that it concerns two rather than n alleles. They also use Taylor and Frank’s (1996) direct fitness formalism, deriving expressions that are first-order approximations of expected frequency changes. Hamilton on the other hand starts with (expected) frequency changes that need no approximating and directly derives properties of stable fixed points from them. Roze and Rousset’s fitness function W ij , therefore, is more general in the sense that it allows for frequency dependence. It is also less general in that they assume the population size to be constant, thereby excluding altruistic behaviour that improves efficiency. In their model, random mating is a special case, and if it is assumed that there is indeed no inbreeding, Roze and Rousset arrive at Eq. (13) (p. 216). For the two-allele case this equation predicts that if one homozygote has a higher inclusive fitness than the other, then the frequency of the first gene will increase if the dominance parameter h is between 0 and 1. This is a conclusion that is similar to what Theorems 1 and 2 predict for the n-allele case in Hamilton’s model. Acknowledgements I would like to thank the two referees for useful comments and the Netherlands’ Organisation for Scientific Research (NWO) for financial support.

References Cavalli-Sforza, L.L., Feldman, M.W., 1978. Darwinian selection and altruism. Theor. Popul. Biol. 14, 268–280. Charnov, E.L., 1977. An elementary treatment of the genetical theory of kin-selection. J. Theor. Biol. 66, 541–550. Grafen, A., 1985. A geometric view of relatedness. Oxford Surv. Evol. Biol. 2, 28–90. Grafen, A., 2006. Optimization of inclusive fitness. J. Theor. Biol. 238, 541–563. Hamilton, W.D., 1964. The genetical theory of social behaviour (I and II), J. Theor. Biol. 7, 1–16, 17–32. Kingman, J.F.C., 1961. On an inequality in partial averages. Q. J. Math. 12, 78–80. Lessard, S., 1992. Relatedness and inclusive fitness with inbreeding. Theor. Popul. Biol. 42, 284–307. Michod, R.E., 1980. Evolution of interactions in family-structured populations: mixed mating models. Genetics 96, 275–296. Roze, D., Rousset, F., 2004. The robustness of Hamilton’s rule with inbreeding and dominance: kin selection and fixation probabilities under partial sib mating. Am. Nat. 164, 214–231. Taylor, P.D., 1989. Evolutionary stability in one-parameter models under weak selection. Theor. Popul. Biol. 36, 125–143. Taylor, P.D., Frank, S.A., 1996. How to make a kin selection model. J. Theor. Biol. 180, 27–37. Uyenoyama, M.K., 1984. Inbreeding and the evolution of altruism under kin selection—effects on relatedness and group-structure. Evolution 38, 778–795. Uyenoyama, M., Feldman, M.W., 1980. Theories of kin and group selection—a population—genetics perspective. Theor. Popul. Biol. 17, 380–414. van Veelen, M., 2005. On the use of the Price equation. J. Theor. Biol. 237, 412–426.