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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 selfsacriﬁces. ‘ 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 parentoffspring relationship except its close degree and a /,vcertain fundamental asymmetry. The fullsib 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 onehalf. Similarly, the halfsib 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 nonmathematical 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 noninbred 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 abovementioned 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 selfeffect 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 genepool by this genotype, while
the remaining part 55],, will involve an unspeciﬁc contribution consisting of
genes in the ratio in which the genepool already possesses them. It is clear
that it is the matrix of effects 612,: which determines the direction of selection
progress in genefrequencies; (SSU only inﬂuences its magnitude. In view of this importance of the (mgit 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 nonrandom mating but from now on for
Simplicity we assume that it is random. Using a prime to distinguish the new
genefrequencies after one generation of selection we have ' gPtPJRivj‘l'PzJgpjpk‘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
prp. ——Api = F(R"+5S ~T)
or H
A __ pi RD 0
Prﬁ?‘+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 (daredj 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, onehalf for sibs, one—quarter for half sibs, oneeighth 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“
airr , <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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