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Hamilton_J_Theor_Biol_1964

Hamilton_J_Theor_Biol_1964 - 36 J T heoret Biol(1964 7...

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Unformatted text preview: 36 J. T heoret. Biol. (1964) 7, l~lG .’ The Genetical Evolution of Social Behaviour. I W. D. HAMILTON \ The Galton Laboratory; (University College, London, W.C.:2 (Received 13 May 1963; and in revisedform 24 February 1964) A genetical mathematical model is described which allows for inter- actions between relatives on one another’s fitness. Making use of Wright’s Coefficient of Relationship as the measure of the proportion of replica genes in a relative, a quantity is found which incorporates the maximizing property of Darwinian fitness. This quantity is named “inclusive fitness”. Species following the model should tend to evolve behaviour such that each organism appears to be attempting to maximize its inclusive fitness. This implies a limited restraint on selfish competitive behaviour and possibility of limited self-sacrifices. ‘ Special cases of the model are used to show (a) that selection in the social situations newly covered tends to be slower than classical selection, (13) how in populations of rather non~dispersive organisms the model may apply to genes afl‘ecting dispersion, and (c) how it may apply approximately to competition between relatives, for example, within sihships. Some artificialities of the model are discussed. ‘ 1. introduction With very few exceptions; the only parts of the theory of natural selection which have been supported by mathematical models admit no possibility of the evolution of any characters which are on average to the disadvantage of the individuals possessing them. if natural selection followed the classical models exclusively, species would not show any behaviour more positively social than the coming together of the sexes and parental care. Sacrifices involved in parental care are a possibility implicit in any model in which the definition of fitness is based, as it should be, on the number of adult ofi‘spring. in certain circumstances an individual may leave more adult ,Yofi'spring by expending care and materials on its offspring already born than by reserving them for its own survivaland further fecundity. A gene causing its possessor to give parental care will then leave more replica genes in the next generation than an allele having the opposite tendency. The selective advantage may be seen to lie through benefits conferred in~ differently on a set of relatives each of which has a half chance of carrying the gene in question. 4 7.13. ‘ I. . | 1 .4 t.) W. D. HAMILTON From this point of view it is also seen, however, that there is nothing special about the parent-offspring relationship except its close degree and a /,vcertain fundamental asymmetry. The full-sib relationship is just as close. If an individual carries a certain‘ gene the expectation that a random sib will carry a replica of it is again one-half. Similarly, the half-sib relationship is equivalent to that of grandparent and grandchild with the expectation of replica genes, or genes “identical by descent” as they are usually called, standing at one quarter; and so on. , Although it does not seem to have received very detailed attention the possibility of the elentiQn of characters benefitting descendants more remote than immediate offspring has often been noticed. Opportunities for benefitting relatives, remote or not, in the same or an adjacent generation (i.e. relatives like cousins and nephews) must be much more common than opportunities for benefitting grandchildren and further descendants; As a first step towards a general theory that would take into account all kinds of relatives this paper will describe a model which is particularly adapted to deal with interactions between relat' e eneration. The model includes the classical model for “nanmymlappinggenerations” as a special case. An excellent summary of the general properties of this classical model has been given by Kingman (1961b). It is quite beyond the author’s power to give an equally extensive survey of the properties of the present model but certain approximatedeterministic implications of biological interest will be pointed out. As is already evident the essential idea which the model is going to use is quite simple. Thus although the following account is necessarily somewhat mathematical it is not surprising that eventually,‘ allowing certain lapses from mathematical rigour, we are able to arrive at approximate principles which can also be expressed quite simply and in non-mathematical form. The most important principle, as it arises directly from the model, is outlined in the last section of this paper, but a fuller discussion together with some attempt to evaluate the theory as a whole in the light of biological evidence will be given in the sequel. ' 1 2. The Model X; The model is restricted to the case of an organism which reproduces once and for all at the end of a fixed period. Survivorship and reproduction can both vary but it is only the consequent variations in their product, net repro- duction, that are of concern here. All genotypic effects are conceived as increments and decrements to a basic unit of reproduction which, if possessed by all the individuals alike, would render the population both stationary and non—evolutionary. Thiiis\the fitness (1° of an individual is treated as the sum \ THE GENETXCAL EVOLUTION OF SOCIAL BEHAVIOUR. I 3 of his basic unit, the elTect 50 of his personal genotype and the total (3" of effects on him due to his neighbours which will depend on their genotypes: a°=1+6a+e°. ‘ (1) The index symbol ° in contrast to ° will be used consistently to denote the inclusion of the personal effect 5a in the aggregate in question. Thus equation (1) could be rewritten , a°=1+e'.' In equation (I), however, thesymbol ° also serves to distinguish this neigh- bour modulated kind of fitness from the part of it a =1+5a which is equivalent to fitness in the classical sense of individual fitness. The symbol 6 preceding a letter will be used to indicate an effect or total of effects due to an individual treated as an addition to the basic unit, as typified in “ ' ‘ l , e = 1+5a. The neighbours of an individual are considered to be affected differently according to their relationship with him. Genetically two related persons dilfer from two unrelated members of the population in their tendency to carry replica genes which they have both inherited from the one or more ancestors they have in common. If we consider an autosomal locus, not subject to selection, in relative B with respect to the same locus in the other relative A, it is apparent that there are just three possible conditions of this locus in B, namely that both, one only, or neither of his genes are identical by descent with genes in A. We denote the respective probabilities of these conditions by c2, c1 and co. They are independent. of the locus considered; and since ‘ cz+c1+c0 = l, . the relationship is completely specified by giving any two of them. Li & Sacks (1954) have described methods of calculating these probabilities adequate for any relationship that does not involve inbreeding. The mean number of genes per locus i.b.d. (as from now on we abbreviate the phrase “identical by descent”) with genes at the same locus in A for a hypothetical population of relatives like B is clearly 2c2+c,. One half of this number, 02—bit}, maytherefore be called the expected fraction of genes i.b.d. in a relative. It can be shown that it is equal. to Sewall Wright’s Coefficient of Relationship r (in a non-inbred population). The standard methods of calculating r without obtaining the complete distribution can be found in Kempthorne (1957). Tables of , f: {it = ’;‘(C2+%C1) and F = c2 Ii W. I). HAMILTON for a large class of relationships can be found in Haldane & Jayakar (1962). Strictly, a more complicated metric of relationship taking into account the parameteis of selection is “necessary for a locus undergoing selection, but the following account based on use of the above coefficients must give a good approximation to the truth when selection 15 slow and may be hoped to give some guidance even when it is not Consider now how the effects which an arbitrary individual distributes to the population can be summarized. For convenience and generality we will include at this stage certain effects (such as effects on parents’ fitness) which must be zero under the restrictions of this particular model, and also others (such as effects on offspring) which although not necessarily zero we will not attempt to treat accurately in the subsequent analysis. The effect of A on specified B can be a variate. In the present deterministic treatment. however, we are concerned only with the means of such variates. Thus the effect which we may write (6a,,,,,,),, is really the expectation of the effect of A upon his father but for brevity we will refer to it as the effect on the father. The full array of effects like (50111111519111 (éaspecmcdsim)A, etc., we will denote {tsarcl.}A' From this array we can construct the simpler array {5ar.cz}A by adding together all effects to relatives who have the same values for the pair of coefficients (r, c2). For example, the combined effect 6aM might contain efi'ects actually occurring to grandparents, grandchildren, uncles, nephews and half— brothers. From what has been said above it is clear that as regards changes in autosomal gene—frequency by natural selection all the consequences of the full array are implied by this reduced array—at least, provided we ignore (a) the effect of previous generations of selection on the expected constitution of relatives, and (b) the one or more generations that, must really occur before effects to children, nephews, grandchildren, etc., are manifested. From this array we can construct a yet simpler array, or vector, {5a in, by adding together all effects with common r. Thus 5:1, would bring together effects to the above-mentioned set of relatives and effects to double- first cousins, for whom the pair of coefficients is (5;, fi). Corresponding to the effect which A causes to B there will be an effect of similar type on A. This will either come from B himself or from a person who stands to A in the same relationship as A stands to B. Thus corresponding to 1 THE GENETICAL EVOLUTION OF SOCIAL BEIiAVlOUR. I 5 an effect by A on his nephew there will be an effect on A by his uncle. The similarity between the effect which A dispenses and that which he receives is clearly an aspect of the problem of the correlation between relatives. Thus the term 6" in equation (1) is not a constant for any given genotype of A since it will depend on the genotypes of neighbours and therefore on the gene- frequencies and the mating system. Consider a single locus. Let the series of allelomorphs be G1, G2, G3,... ,G ,, and their gene- frequencies pl, 122', p3, .. ., 1)". With the genotype G G} associate the array {541121. },j; within the limits of the above—mentioned approximations natural selection in the model is then defined. If we were to follow the usual approach to the formulation of the progress due to natural selection in a generation, we should attempt to give formulae for the neighbour modulated fitnesses a3. In order to formulate the expecta- tion of that element of eff which was due to the return effect of a relative B we would need to know the distribution of possible genotypes of B, and to obtain this we must use the double measure of B’s relationship and the gene- frequencies just as in the problem of the correlation between relatives. Thus the formula for eff will involve all the arrays {1511”,}, and will be rather unwieldy (see Section 4) An..alternative approach, however, shows that the arrays {6a,}U are sufficient to define the selective effects. Every effect on reproduction which is due to A can be thought of as made up of two parts: an effect on the repro- duction of genes i.b.d. with genes in A, and an effect on the reproduction of unrelated genes. Since the coefficient r measures theexpected fraction of genes i. b d. in a relative, for any particular degree of relationship this break- down may be written quantitatively: (6:11.111 - r(50111)1+(1—r~)(5a1 1.1 112119, The total of effects on reproducriion which are (due to A may be treated similarly: Z (M1111 = Z reamai + 211— r1 (M11111, rel. rel. ref. 2 (Sam = z 2‘(5a1)11 + g 11- r116a.1.1, which we rewrite briefly as or where 612' is accordingly the total effect on genes i.b. d in relatives of A, and (SSA is the total effect on their other genes. The reason for the omission of an index symbol from the last term is that here there is, in effect, no question of whether or not the self-effect is to be in the summation, for if it is included it has to be multiplied by zero. If index symbols were used 6 w. n. llAMil.TON we should have 5512553,, whatever the subscript; it therefore seems more cxphcrt to omit them throughout. if, therefore, all etl‘ects are accounted to the individuals that cause them, of the total effect «ST; due to an individual of genotype G‘G} a part 612;} will involve a specific contribution to the gene-pool by this genotype, while the remaining part 55],,- will involve an unspecific contribution consisting of genes in the ratio in which the gene-pool already possesses them. It is clear that it is the matrix of effects 612,: which determines the direction of selection progress in gene-frequencies; (SSU only influences its magnitude. In view of this importance of the (mg-it is convenient to give some name to the concept with which they are associated. in accordance with our convention let a o _ U = 1+6R‘j’ then R; will be called the inclusive fitness, 613,; the inclusive fitness effect and 65,,- the diluting eflect, of the genotype GiGj. Let TJ=1+5TJ- So far our discussion is valid for non-random mating but from now on for Simplicity we assume that it is random. Using a prime to distinguish the new gene-frequencies after one generation of selection we have ' gPtPJ-Rivj‘l'Pz-Jgpjpk‘mjk ZPjRafj'i—zlgpjpkasjk Pi: . =p J J: ijpk 134 i 21’ij 1;: J.k 1,1: The terms of this expression are clearly of the nature of averages over a part (genotypes containing Gi, homozygotes GiG, counted twice) and the whole of the existing set of genotypes in the population. Thus using a well lrnown subscript notation we may rewrite the equation term by term as , Rf_+5$n Pi = P; “To __ Pt 0 a pr-p. ——Api = F(R"+5S ~T) or H A __ pi RD 0 Pr-fi?‘+5so( 13"”) p (2) This form clearly differentiates the roles of the R}! and 65,} in selective progress and shows the appropriateness of calling the latter diluting effects. THE GENET'ICAL EVOLUTION OF SOCIAL BEHAVIOUR. I 7 For comparison with the account of the classical case given by Moran (1962), equation (2) may be put in the form Pi 15R: o) =~ —————R AP‘ 7:?(261); " where 6/817, denotes the usual partial derivative, written d/dp, byMoran. Whether the selective effect is reckoned by means of the a; or according to the method above, the'denominator expression must take in all effects occurring during the generation. Hence a: = T3. As might be expected from the greater generality of the present model the extension of the theorem of the increase of mean fitness (Scheuer & Mandel, 1959; Mulholland & Smith, 1959; a much shorter proof by Kingman, 1961a) presents certain difliculties. However, from the above equations it is clear that the quantity thatwill tend to maximize, if any, is Rf, the mean inclusive fitness. The followingxbrief discussion uses Kingman’s approach. Themean inclusive fitness in the succeeding generation is given by l! I .I' r l ' ,5; R... = izptPjRiej = 211’s"; if;PinRi’JiR:+53..)(R.°j+55..)' , * ,1 , .. . R}. —R.‘. = AR: = fest}: pirRthtanzas“ >1 prijth’.g+ .. 1.: w +R,°,6si —-R,',Tf,2}- Substituting R... +58:- for T: in the numerator expression, expanding and rearranging: '~ ' AR:= 51%“; PinRij,’_Rf’j_Re3) + I .. . + 26S” (2P1P1R3Rf.~gf_2)}_ ‘71 .' 1 We have ( ) 2 0 in both cases. The first is the proven inequality of the classical model. The second follows from ZPstRi’jRi. = g as}? 2 (; Mir = R??- h} ‘ . Thus a sufficient condition for AR: 2 O is (SSH 2 0. That AR". 2 0 for positive dilution is almost obvious if we compare the actual selective changes with those which would occur if {R5} were the fitness matrix in the classical model. . ' t It follows that R: certainly maximizes (in the sense of reaching a local maximum oflRf) if it never occurs in the course of selective changes that 68” < 0. Thus Rf certainly maximizes if all (SSH 2 O and therefore also if all (dared-j 2 0. It still does so even if some or all 5a,] are negative, for, as we have seen 58,-, is independant of 5a”. 8 W. D. HAMILTON Here then we have discovered a quantity, inclusive fitness, which'under the conditions of the model tends to nuiximize in much the same way that fitness tends to maximize in the simpler classical model. For an important class of genetic effects where the individual is supposed to dispense benefits to his ‘ neighbours, we have formally proved that the average inclusive fitness in the population will always increase. 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,’l but may, for all the present analysis has told us, overshoot it in such a way as to produce a lower value. As to the nature of inclusive fitness it may perhaps help to clarify the notion if we now give a slightly different verbal presentation. Inclusive fitness may be imagined as the personal fitness which an individual actually expresses in its production of adult oflspring as it becomes after it has been first stripped and then augmented in a certain way. It is stripped of all com- ponents which can be considered as due to the individual’s social environ- ment, leaving the fitness which he would express if not exposed to any of the harms or benefits of that environment This quantity is then augmented by certain fractions of the quantities of harm and benefit which the individual himself causes to the fitnesses of his neighbours. The fractions in question are simply the coefficients of relationship appropriate to the neighbours whom he affects: unity for clonal individuals, one-half for sibs, one—quarter for half- sibs, one-eighth for cousins, . and finally zero for all neighbours whose relationship can be considered negligibly small. Actually, in the preceding mathematical account we were not concerned with the inclusive fitness of individuals as described here but rather with certain averages of them which we call the inclusive fitnesses of types. But the idea of the inclusive fitness of an individual is nevertheless a useful one. Just as in the sense of classical selection we may consider whether a given character expressed 1n an individual is adaptive in the sense of being 1n the interest of his personal fitness or not so in the present sense of selection we may consider whether the character or trait of behaviour 15 or is not adaptive 1n the sense of being 1n the interest of his inclusive fitness I , 1 3. Three Special Cases Equation (2) may be written 6R? —- 6R“ air-r- , <3> AP: = P1 1 That is it is aimed “uphill ” .that it need not be at all directly towards the local maximum is well shown in the classical example illustrated by Mulholland & Smith (1959). , “A? i;[,§rrifi“l‘<=flijW\ . z . Wrmfififlt . . THE GENETICAL EVOLUTION OF SOCIAL BEHAVIOUR. 1 9 Now 57‘; = Z (60,)” is the sum and 5R' = Z r(15a,),, is the first moment , about 1' = O of the array of effects {dam}, cause by the genotype G,G,; it appears that these two parameters are sufficient to fix the progress of the system under natural selection within our general approximation. _ Let i 5R3 ‘. rt]: {STE}, (6T;- gé O); * (4) and let 5R° . 139,-: 57—3, (5Toj 9'3 0)- ‘ _ (5) These quantities can be regarded as average relationships o...
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