THEORETICAL
POPULATION
BIOLOGY
Darwinian
14, 268-280 (1978)
Selection
and “Altruism”*
L. L. CAVALLI-SFORZA Department
of Genetics, Stanford University, School of Medicine, Stanford, Cahalifornia 94305 AND
M. Department
of Biological
W.
FELDMAN
Sciences, Stanford Received
University,
December
Stanford,
California
94305
29, 1977
Models are proposed for evolution at a single locus affecting altruistic behavior in which genotypic fitnesses are Darwinian and frequency (but not density) dependent. The fitnesses are composed, either in a multiplicative or an additive way, of factors which depend on the receipt and donation of altruistic behavior. The factors are determined from the matrices of conditional probabilities which describe the genotypes of relatives. Since selection occurs, these probabilities are in terms of genotype frequencies. The relationship between the risk to helper and benefit to recipient which allows altruism to evolve is shown to depend on the kinship coefficient between helper and helped, the particular fitness function proposed and the degree of dominance of the altruism. The commonly accepted criteria of W. D. Hamilton [I. Theor. Biol. 7 (1964), 1-16, 17-521 apply only in the additive case. A second class of models of social cooperation independent of relationship and its evolutionary dynamics are discussed.
The theory of the evolution of genetically determined altruistic behavior faces, at the outset, the central problem that altruists are, by definition, at greater risk to their lives than nonaltruists and are therefore in greater danger of elimination by natural selection. Haldane (1932, 1955) clearly perceived two possible resolutions of the dilemma. The first, Kin selection, postulates that the altruism is directed towards close relatives; an altruist may die but, by virtue of the altruistic behavior, his (or her) genes are more likely to be represented in the relatives, who, as the recipients of the altruism, have an increased chance of survival. Hamilton (1964, 1972) and Maynard Smith (1964), in quantifying this argument, proposed that the maximum ratio of the risk to the altruist to the advantage to * This research supported in part by NSF Grant DEB 7705742, NIH and AEC Grant AT(O4-3)-326.
268 0040-5809/78/0142-0268$02.00/0 Copyrighf 0 1978 by Academic Press, Inc. AlI rights of reproduction in any form reserved.
Grant 10452-15,
DARWINIAN
SELECTION AND ‘LALTRUISM"
269
the recipient, above which altruism cannot be maintained in the population, is determined by the probability that the parties share genes identical by descent. Hamilton’s notion of selection was couched in terms of an individual’s “inclusive fitness.” This has recently been defined as the expected number of genes in a future generation which are identical by descent to a gene in the individual in question (Oster et al., 1977; Wofsy, 1978). Most studies subsequent to those of Hamilton and Maynard Smith (e.g., Oster et al., 1977; Wofsy, 1978; Trivers, 1971) have involved game theoretic or other optimization arguments rather than dynamic models of genetic evolution. However, Oster et aZ. reconcile these approaches (at least for the problem of initial increase) in studying the problem of sex ratios in insect societies. Haldane’s other possibility for the evolution of altruism is group selection. Here the presence of altruists in a group increases the “fitness” of the group. In this context, altruism is a trait which is advantageous for the group as a whole but disadvantageous for its carrier in competition with nonaltruists. Following Wright (1945) Haldane suggested that the success of this formulation requires partitioning of the population into small groups (see also Maynard Smith, 1976.) That such models of local genetic drift can, under certain conditions on the level of exchange between the groups, allow the evolution of altruism has been shown by Eshel (1972). In the studies by Cohen and Eshel (1976) Matessi and Jayakar (1976) D. S. Wilson (1974), and Eshel (1977) individual fitnesses are functions of the group size and the number of altruists in the group. The models of group selection proposed by D. S. Wilson (1974) and Matessi and Jayakar (1976) do not invoke local genetic drift. In assessing the former Maynard Smith (1976) suggests that it should be regarded as a model of kin selection rather than group selection. Levitt (1975) developed dynamic genetic models of sib-altruism in which the fitness of an individual depends on the frequency of altruistic litter mates. Before Hamilton, Williams and Williams (1957) had proposed a somewhat similar within-brood selection scheme but did not develop the resulting dynamics (see also Maynard Smith, 1965). These, and those of Wilson (1974) and Matessi and Jayakar (1976) are perhaps the studies closest in spirit to that presented here. 1Ve develop models for the evolutionary dynamics of kin selection using Darwinian fitness rather than inclusive fitness. Other models of Darwinian selection, based on interactions between individuals, which can assure the maintenance of altruism are also described.
FITNESS UNDER ALTRUISM
TOWARD KIN
In these models altruistic behavior is determined at a single locus with alleles -4, and -4,. The altruistic phenotype is denoted by A while N represents the nonaltruistic or selfish phenotype. If A, is dominant then ;4 can be either AiA,
270
CAVALLI-SFORZA
AND
FELDMAN
or A,&; if -4, is recessive then only the genotype A,A, is altruistic. The Darwinian fitnesses of the genotypes A,A, , ,4,A,, and A,A, are constructed for the case of parent-to-offspring altruism, in which good and poor parental care characterize the altruistic and selfish phenotypes, respectively. A completely, analogous process can be applied to any degree of relationship between the donor and recipient of altruism. The fitness of a child of genotype AiAj arising from parental care is a function of the gain constant, p, as follows:
+ij = 1 + Pfij;
/3> 0.
(1)
The fij values are the probabilities that an offspring of genotype Ai.4j has an altruistic parent. If the parent is an altruist its offspring’s fitness is increased by the relevant bfij . To evaluate fil , fiz , and fz2for the parent-child relationship we construct the matrix of conditional probabilities that a parent is one of the three genotypes given the offspring’s genotype. Because selection is operating, the parents atmating are not in Hardy-Weinberg equilibrium. The frequencies of A,A, , A,A,, and A,A, in the parental generation are denoted by u, v, and w, respectively; with p = u + v/2 and q = 1 - p. Before selection the offspring are, of course, in Hardy-Weinberg proportions with gene frequencies p and q. The conditional probability matrix is then
\ Offspring
Parent 44
4%
44
WI
4P
Vl2P
0
4%
Ul2P
@Pq
w/&l
44
0
43l
wl!l
\
(2)
Now A,A, is an altruist. Let h be the degree of dominance of the altruistic trait so that, with probability h, A,A, will also be an altruist. If the child is A,A, then the probability that its parent is altruistic isf,, = u/p + hv/2p. In the same way the other fij values can be computed for case of parent-to-offspring altruism:
fu = 4~ + W%
(34
fiz = ~12~ + h+bq,
(3b)
fiz = hvl2q.
(3c)
(Note: h = 1 implies complete dominance of A, to A,; h = 0 implies that A, is completely recessive to A, .)
DARWINIAN
SELECTION
AND
“ALTRUISM”
271
We now postulate that a fitness loss y is incurred by the genotype Aid, in the performance of altruism. In the case where A, is completely dominant to A,, so that A,il, and A,A, are both of phenotype A the fitness loss to these genotypes would be the same, y. However, if the degree of dominance is h (0 < h < 1) then the loss will be y for A,,;Z, but only hy for A,,g, . In principle it is not essential to view fi as influencing only mortality up to maturity, and y only as a measure of potential loss of offspring. The above details should be regarded as one way to assign fitnesses to the altruistic and selfish phenotypes. Indeed in situations other than parent-to-offspring altruism the interpretations must change somewhat. For instance in sib-to-sib altruism, where the altruistic sib is older than the recipient of altruism, it may be more appropriate to view y as increased prereproductive mortality than as a decrease in the potential number of offspring. In more symmetrical situations where the ages of the donor and recipient are not significantly different, it may be difficult to discriminate which of mortality or number of offspring are directly affected by the altruistic process. The final assumption of our model is that the contributions to the fitness of a genotype as a donor or recipient of altruism are independent. There are two senses in which this can be satisfied. First we discuss the multiplicative case: I. Multiplicative
Model
where the fir values are as in (3). The genotype frequencies after selection are then
u' = p2(l - r)[l + IqUIP+ W34/~7
(64
w’ = 2pq(l - hY)[l + Is(U/2P + W4P@l/$J,
(a)
w' = $11+ p42ql/~,
P-3
where 6 = 1 - r(P + 244 + B(u + hv) - BY[U(P + hq) + ~V(P + WI.
(64
Equations (6) describe the case of parent-to-offspring altruism. Another situation which has apparently led to most of the development of “sociobiology” is that of sib-to-sib altruism. To treat this casewhere only diploid individuals are considered, we can compute the probability that, for example, an individual of
272
CAVALLI-SFORZA
AND
FELDMAN
genotype &A, has a sib of genotype A,A, . It is 2er(u + w/4)/4~“. The matrix computed in this way analogous to (2) is recorded as (8) sib 2 \ 1 sib 44
44
AA
-W,
44
w2/16p2 2w@+ w/4)/4p2 (u + v/~)~/P~ 4~ + w/4)/4~q [k/4)” + (u + w/4)@ + 7d4)llpp a@ + d4)/4pq w2/16q2 2w(w + o/4)/4q2 (w + w%12
(8) Using (8) the functions fij are calculated as before. Hamilton’s (1964) proposal for the evolution of social behavior in insects relied on the haplo-diploid nature of the species to which he referred. Haplodiploidy has subsequently played a central role in evolutionary theories of behavior. In population genetic terms this system-is equivalent to sex linkage. The matrices for the probabilities of relationship in the sex-linked situation in the absence of selection are well known (see, e.g., Li, 1955, Chap. 3) and can be adapted to the situation including selection. The frequencies of A,A, , A,A, , and AzAz (females) are 11,w, and w as before, with p = 1 - q = u + w/2, while A, and A, males have frequencies x and y, respectively. Then the matrix describing relationships between sisters in (9), where z = xq + yp: Sister 2 \ Sister 1 \
44 4% 44
AC%
(u + 014)/P wx/4iz 0
44
wl4P [v + wx + W(X+ Y)/412 v/4q
44
0 VI42 (9) @ + w/4)/q
Of course this analysis must be made in terms of the five frequencies 1~,w, w, X, and y. Relationship between brothers, sister and brother, and brother and sister can be written similarly. Each of these sib-to-sib relations produces its own set of transformations with its characteristic form of frequency dependent selection. We have presented the case of parent-to-offspring altruism in detail but the other relationships may be treated similarly. Two evolutionary questions are asked of systems such as (6). First, when are the boundary equilibria (0, 0, 1) and (1, 0,O) locally stable ? In the case of (0, 0, I), its instability means that the altruistic phenotype increases when rare; it is “protected” from loss. Stability of (1, 0, 0) entails the fixation of the altruistic phenotype. From Hamilton (1964) (see also E. 0. Wilson, 1975, Chap. 5), it is expected that the stability conditions at these boundaries should be determined
DARWINIAN
SELECTION
AND
“ALTRUISM”
273
by the relationship of the ratio y//I to the “coefficient of relationship” between the donor and the recipient of the altruism. We indicate by F the coefficient of kinship between the two individuals, which is equal to the inbreeding coefficient of their progeny. Usually [but not always) the coefficient of relationship is twice the coefficient of kinship. The second question asked of (6) concerns the existence and stability of polymorphic equilibria. For models as complex as the present one, this is usually much more difficult to resolve. When dominance is intermediate (h # 0, l), the first of these questions is answered by conventional local linear analysis. For every degree of relationship between donor and recipient mentioned above we have (0, 0, 1) is stable if y > 2F/(/L-l + 2Fh),
(104
(1, 0,O) is stable if y < 2F/@-1 + 2Fh -+ 1).
(lob)
For every degree of relationship, when h = 0, local linear analysis at (1, 0,O) presents no problem and (lob) is valid with h = 0. Similarly when h = 1, (10a) gives the condition for stability of (0, 0, 1). But with h = 0, local linear analysis at (0, 0, 1) fails because the leading eigenvalue is unity. Similarly with h = 1, linear analysis near (1, 0,O) produces an eigenvalue of unity. Analogous difficulties with h = 0 and h = 1 were noted in their studies by Levitt (1975) and Matessi and Jayakar (1976). N evertheless a second-order analysis can be made in all of the usually considered cases: parent-to-offspring, sib-to-sib diploid, and sib-to-sib sex-linked (equivalently, haplodiploid). The details of the analysis will be presented elsewhere. The results are displayed in Table I with (1Oa) and (lob) included for comparison. This table presents the bounds for y, the fitness cost of altruism as a function of j3, the gain in fitness obtained by recipients, and the coefficient of kinship. For each relationship the first row shows the bounds on y below which altruism is protected from loss. If y is below the value in the second row, then altruism fixes. It is clear from Table I that as F increases, the bounds on y//3 become less stringent (for fixed 8). It is also clear that the relationship between y//? and 2F in this multiplicative model is not as the same as that suggested by Hamilton. It is true that as /3 --+ 0, the relationship between y//I and 2F approaches Hamilton’s rule. For larger values of /I it is more natural to use Table I rather than y//I to describe the stability conditions. The boundary stability results in Table I, except for the parent-to-offspring case, are exactly the same as those obtained using expressions analogous to (3) for the fii values but involving the gene frequency rather than the technically more correct genotype frequencies. Use of the well-known matrices of relationship in the absence of selection (see, e.g., Li, 1955, Chap. 3) allows fast enumeration of fij . With h = 0 and h = 1 the results in the parent offspring case for
274
CAVALLI-SFORZA
AND
FELDMAN
initial increase and fixation of altruism respectively are different from those in Table I when gene frequencies are used. TABLE Stability
Conditions
I
for the Multiplicative
Model”
Equilibrium
Dominant (h = 1)
(0, 0, 1)
i/W’
f
4)
;t/(B-’
+ h/2)
Q/W
+ g,
(1, 0, 0)
h/W’
+ 1)
&lW1
+ h/2 + 1)
4/C-’
+ 1)
Sister-to-sister (sex-linked) @= = 3)
a
0, 11, (0, 1)
~/U-l
+ $
$/Wl
+ 3h/4)
g/w
+ $1
(LO, 01, (1, 0)
f/@-l
+ 21
8/W’
+ 3h/4 + 1) $I@-’
Sister-to-brother (sex-linked) (F = $1
(0, 0, u al)
&W’
+ a,
m-1
+ h/4)
t/wl,
(1, ‘3%
iAS-’
+ 8,
$/(fl-’
+ h/4 + 1)
&(B-’
Brother-to-brotherb (sex-linked) (F = $1
640,
i4B-’
+ 4)
(l,O, 01, (LO)
t/w1
+ 1)
(090, I), (0, 1)
4/W’)
(I,&
iw-’
Relationship Parent-to-offspring (F = 4)
Intermediate (0 < h < 1)
Recessive (h = 0)
Sib-to-sib (diploid) (F = $1
Brother-to-sisterb (sex-linked) (F = 4) a For each relationship less than the given value. b Dominance irrelevant
(1, 0)
11, (0, 1)
Oh (1,O)
the upper equilibrium
is unstable
+ 1)
+ 1)
+ 9) and the lower
stable for y
here.
In this multiplicative model the polymorphic equilibria are expressible as the roots of higher order (generally fourth of higher) polynomials. Numerical analysis indicates that for y values between the pairs of values in Table I, a single stable equilibrium exists. In this range of y values the altruism increases when rare but does not proceed all the way to fixation. For y smaller than the values in the second row of the pairs in Table I, it appears that fixation of altruism is globally stable. In the case of brother-to-brother altruism the order of the polynomial is reduced and the above remarks have been validated analytically.
DARWINIAN
II. Additive
SELECTION
AND
“ALTRUISM”
275
Model
In the representation
of the additive
model the fitnesses are
where the fii values are defined through the matrices exemplified by (2), as in (3). With fitnesses (11) the genotype frequencies after selection are, in the case of parent-to-offspring altruism,
24'= [p2(1 - Y) + Pi@ + w41/$k
(124
2" = 2pq(l - hy) + P(uq+ W2)1/$57
(12b)
m' = [q2+ /3hfJqPl/&
(124
with
6 = 1 - y( p2 + 2&q)
+ B[P+ z@- $)I.
(124
Recursions similar to (12) can be written for each degree of relationship considered above. The equilibrium analysis can be summarized in a manner analogous to that of Table I; it is presented in Table II. The results here are seen to fit quite closely into the framework originally conjectured by Hamilton. Some anomalies arise, however. In the parent-to-offspring case, with h = 1, the point (0, 0, 1) is stable if y > /I/2. But the point (1, 0,O) is stable if y < [1+ /3 (1 - /F)l’*]/2. In the range p/2 < y < [1 + /I - (1 - B’)““]/2 both boundaries are stable and an unstable interior R = 0 in the parent-offspring case, y in the interval
/w > Y > P/(2 -t B)
equilibrium
(13) exists. When
(14)
allows the stability of both boundaries and the existence of an unstable interior point. With h = 0 or h = 1, the equilibria in the additive model are usually roots of quadratics but for 0 < h < 1, these become fourth degree polynomials. The usual state of affairs in the sib-to-sib situation (also sister-to-sister in the sex-linked case) is that with h = 1, for a small interval of values larger than 2F, two polymorphic equilibria exist. One of these is stable, the other unstable. With h = 0, there is a corresponding interval of y values smaller than 2F/? in which
276
CAVALLI-SFORZA AND FELDMAN TABLE Stability
Relationship
Conditions
tw
(0, 0, 1) ~(l,O,
Sib-to-sib (diploid) (F = $1
0)
for the Additive Dominant (h = 1)
Equilibrium
Parent-to-offspring (F = *>
II
[l +p-s-(l-~yl/2
Model” Intermediate (0 < h < 1)
Recessive (h = 0)
812
8/(2+~~
812
812
(0, 0, 1)
tw
iw
812
(l,fA
0)
I312
812
812
Sister-to-sister (sex-linked) (F = 8)
GAO, 11, (0, 1)
38/4
3814
3814
(1,O.O),U,O)
3/9/4
3814
3814
Sister-to-brotherb (sex-linked) (F = 9)
(O,O, I), (0, 1)
(/3/4)(1+8/4)
(8/4)/U+FW4)
814
(1,O,Oh(1,0)
(8/4)/1+-w/4)
w4M-t8(1+Jd4)1
63/4Ml+S)
Brother-to-brother (sex-linked) (F = t)
(0, 0, l), (0, 1)
812
(l,O, 01, (1,O)
812
Brother-to-sister” (sex-linked) (F = $1
(O,OP I), (0, 1)
B/2
(190, Oh (1,f-J
69/2)/(1+38/2)
a For each relationship the upper equilibrium is unstable and the lower stable for y less than the given value. * These cases are equivalent in the multiplicative and additive constructions.
one stable and one unstable polymorphic equilibrium exist. In these cases, therefore, a boundary and a polymorphic equilibrium are stable. Thus in the sister-to-sister
sex-linked
case, if altruism
is recessive,
it can increase
when
rare
but reach a polymorphic state rather than proceeding to fixation. There are some differences from this pattern, for example, in the brother-to-brother sex-linked case where no interior point exists. With 0 < h < 1 the equilibria seem, from numerical analysis, to obey these qualitative patterns which, it should be noted, are different from those described above for the multiplicative model. INTERACTION MODELS WITHOUT KIN SELECTION
The previous models translated the notions of kin selection into frequencydependent Darwinian selection. We turn now to another class of models in which
DARWINIAN
SELECTION
AND
277
L’ALTRUISI\l”
fitness can be attributed to the interactions of individuals with other, not necessarily related, members of the group. The simplest example of this sort considers only groups of size 2 in a population with phenotypes A and N. The groups can be (-4, A), (=I, N), and (N, N), and four types of interactions can occur as shown in Table III. where u is the frequency of ,4 and v = 1 - U, that of N. TABLE III Interaction Parameters Fitness increment to __--_ Helper Helped
Interaction between Frequency Helper
Helped
A
-
A
u2
-‘1
A
--f
N
uv
8,
B2
N
4
-4
IIt
N
+
N
73
Yl T 4
Yl 6,
‘72
In the paradigm with A = altruist and N = nonaltruist, Q~, /3, are positive; cyl , /$ are negative; yI , 6, can be zero or positive; yZ (6, can be zero or negative. The construction reflects “altruistic” behavior if the quantities a = (Ye+ (Ye and c = /?s + y1 are both positive with b = ,G1+ yZ and d = 6, + 6, both negative. Two simple fitness schemes can be proposed using Table III. In the first, each -4 individual has fitness increments according to the type of twoindividual group it is in. The relative fitnesses of A and N would then be WA = 1 + au + ba, W,v = 1 + cu + dv. In the second the fitness of -4 is incremented individuals, so that
by all encounters
(154 (15b) involving
--1
WA = 1 + UU2 + hua, I$‘,,, = 1 + CUZI+ dv2. Following
the interactions
we have
li = uw,/c
with
l+ = u WA + v WV .
(17)
We restrict ourselves here to the cases with a > 0, c > 0, b < 0, d < 0, and j b I < 1, 1d / < 1. Without the latter restrictions the recursions may conceivabl!
278
CAVALLI-SFORZA
AND
FELDMAN
leave the admissible frequency range. Under these assumptions (15) produces a monotone transformation such that if b < d and c < a both zi = 0 and zi = 1 are stable and are separated by an unstable equilibrium. If b > d and c > a then the interior equilibrium
ti = (6 - d)/(b - d + c - a)
(18)
is globally stable. The fitness specification and result (18) are originally due to Hamilton (1971). If A and N are regarded as the dominant and recessive phenotypes at a single locus (for example r2 = A-, B - au) then we can write WA- = 1 + a(1 - 4s) + bq2 fand W,, = 1 + c( 1 - q?) + dq? with Q the frequency cf the “a” gene. If further, a = b and c = - 1 with d = 0, the model reduces to a case of social coorperation as considered by Boorman and Levitt (1973), which in turn is a special case of the general model studied by Cockerham et cd. (1972). With the fitness specification (16) and a, c > 0, 6, d < 0, 1b 1 < 1, and 1d 1 < 1, ti = 0 is unstable and ti = 1 is stable. If, in addition, (b - c)% + 4ad > 0, two polymorphic equilibria exist. Again under the same restrictions as above, the resulting transformation is monotonic and the equilibrium
li = c - 6 - 2d - [(b - c)” + 4&]1’2 2(a - 6 + c - d)
(19)
is stable. Again it is clear how this second model may be placed in a dominantrecessive genetic context. CONCLUSIONS
The two types of model proposed here allow the evolution of altruism to be studied in a Darwinian selection framework via frequency-dependent fitnesses. This structure obviates the need to use the concept of “inclusive fitness,” the appropriate definition of which has been a stumbling block in the translation of Hamilton’s strategy arguments into evolutionary genetic models. In the first class of genetic models the unifying feature is the use of conditional matrices of probabilities that an individual of given genotype has relatives of given degree of a specified genotype. These are specified in terms of genotype frequencies in the same way that the well-known Hardy-Weinberg analogs are functions of gene frequencies. In these genetic models it is the case that the closer the relationship between donor and recipient the easier it is for altruism to increase when rare. But the relationship between the ratio r//3 of loss to gain and the kinship coefficient allowing the evolution of altruism depends on how the model is structured. In the multiplicative model the ratio r/p is insufficient to determine the outcome, and the magnitude of /) itself plays a role (Table I). This is significant, for example, in the comparison of sib-to-sib altruism with parent-to-
DARwINIAN
SELECTION AND “ALTRUIShI”
279
offspring altruism. The initial increase conditions are formally the same, but the gain, /3, to the recipient (child) of parental altruism would generally be larger than in the sib case. Unlike the multiplicative case the coefficient of kinship in the additive case (Table II) is sufficient to explain the boundary behavior of the altruism. But in the additive case an intringuing interior equilibrium phenomenon arises which interferes with a complete evolutionary explanation based on the coefficient of kinship. The models described here are the simplest among a whole class of fitness specifications involving interactions between y, /3, and fij . It should be recognized that when placed in the population genetic framework, models involving kin selection, or cooperation, become special cases of gene, genotype, or phenotype frequency-dependent selection. For this reason, attempts, such as that by Cockerham et al. (1972) to classify such models assume an importance not generally appreciated by “sociobiologists.” The assumption throughout the genetic models that the cost and benefit are independent is an extremely crucial one. In the case of parental care for example, the intensity of the experience (as an offspring) may later influence the behavior of the same individual as a parent. The second class of (nongenetic) frequency-dependent selection models introduced by Hamilton (1971) might be looked on as a case of “good guys” (A) and “bad guys” (N). Clearly such an interpretation is mediated by the choice of a, b, c, and d, in (11) and (12). Kinship is not required here for the evolution of the A phenotype which is at a disadvantage in its contacts with N individuals. The difference in the equilibrium behavior of the two examples indicates that more complex social interactions may result in still more complex equilibrium properties, and that equilibrium and dynamic behavior is extremely model sensitive. It may well turn out that the study of simple phenotypic examples may provide far more insight into the evolution of individually disadvantageous traits than the more complicated genetic models. Such models allowing nonbiological transmission are currently under study.
ACKNOWLEDGMENTS The authors are grateful to Dr. Marcy Uyenoyama for many stimulating discussions and to Professor S. Karlin and Dr. B. 0. Bengtsson for their critical comments on the manuscript. REFERENCES BOORMAN, S., AND LEVITT, P. R. 1973. A frequency dependent natural selection model for the evolution of social cooperation networks, PYOC.Nat. Acad. Sci. USA 70,187-l 89. COCKERHA~\I, C., BURROWS, P. M., YOUNG, S. S., AND PROUT, T. 1972. Frequencydependent selection in randomly mating populations, Amer. Nut. 106, 493-515.
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CAVALLI-SFORZA
AND
FELDMAN
COHEN, D., AND ESHEL, I. 1976. On the founder effect and the evolution of altruistic traits, Theor. Pop. Biol. 10, 276-302. ESHEL, I. 1972. On the neighbor effect and the evolution of altruistic traits, Theor. Pop. Biol. 3, 258-277. ESHEL, I. 1977. On the founder effect and the evolution of altruistic traits; an ecogenetical approach, Theor. Pop. Biol. II, 410-424. HALDANE, J. B. S. 1932. “The Causes of Evolution,” Longmans Green, London/N.Y. HALDANE, J. B. S. 1955. Population genetics, New Biology 18, 34-51. HAMILTON, W. D. 1964. The genetical evolution of social behavior, /. Theor. Biol. 7, I, l-16; II, 17-52. HAMILTON, W. D. 1971. Selection of selfish and altruistic behavior in some extreme models. In “Man and Beast: Comparative Social behavior” (J. F. Eisenberg and W. S. Dillon, Eds.), Smithsonian Press, Washington, D.C., pp. 57-91. HAMILTON, W. D. 1972. Altruism and related phenomena mainly in the social insects, Annu. Rev. Ecol. Syst. 3, 193-232. LEVITT, P. R. 1975. General kin selection models for the genetic evolution of sib altruism in diploid and haplodiploid species, Proc. Nat. Acad. Sci. USA 72, 4531-4535. LI, C. C. 1955. “Population Genetics,” Univ. of Chicago Press, Chicago. MATESSI, C., AND JAYAKAR, S. D. 1976. Conditions for the evolution of altruism under Darwinian selection, Theor. Pop. Biol. 9, 360-387. MAYNARD SMITH, J. 1964. Group selection and kin selection, Nature (London) 201, 1145-l 147. MAYNARD SMITH, J. 1965. The evolution of alarm calls, Amer. Nat. 94, 59-63. MAYNARD SMITH, J. 1976. Group selection, Quart. Rev. Biol. 51, 277-283. OSTER, G., ESHEL, I., AND COHEN, D. 1977. Worker queen conflict in the evolution of social insects, Theor. Pop. Biol. 12, 49-85. TRIVERS, R. L. 1971. The evolution of reciprocal altruism, Quart. Rev. Biol. 46, 35-57. WILLIAMS, G. C., AND WrLLrAhqs, D. C. 1957. Natural selection of individually harmful social adaptations among sibs with special reference to social insects, Evolution 1 I, 32-39. WILSON, D. S. 1974. A theory of group selection, Proc. Nat. Acad. Sci. USA 72, 143-146. WILSON, E. 0. 1975. “Sociobiology,” Belknap Press, Cambridge, Mass. WOFSY, C. 1978. Reproductive success for social hymenoptera, Theor. Pop. Biol., to WRIGHT, S. 1945. Tempo and Mode in evolution: A critical review, Ecology 26, 415-419.
POPULATION
BIOLOGY
Darwinian
14, 268-280 (1978)
Selection
and “Altruism”*
L. L. CAVALLI-SFORZA Department
of Genetics, Stanford University, School of Medicine, Stanford, Cahalifornia 94305 AND
M. Department
of Biological
W.
FELDMAN
Sciences, Stanford Received
University,
December
Stanford,
California
94305
29, 1977
Models are proposed for evolution at a single locus affecting altruistic behavior in which genotypic fitnesses are Darwinian and frequency (but not density) dependent. The fitnesses are composed, either in a multiplicative or an additive way, of factors which depend on the receipt and donation of altruistic behavior. The factors are determined from the matrices of conditional probabilities which describe the genotypes of relatives. Since selection occurs, these probabilities are in terms of genotype frequencies. The relationship between the risk to helper and benefit to recipient which allows altruism to evolve is shown to depend on the kinship coefficient between helper and helped, the particular fitness function proposed and the degree of dominance of the altruism. The commonly accepted criteria of W. D. Hamilton [I. Theor. Biol. 7 (1964), 1-16, 17-521 apply only in the additive case. A second class of models of social cooperation independent of relationship and its evolutionary dynamics are discussed.
The theory of the evolution of genetically determined altruistic behavior faces, at the outset, the central problem that altruists are, by definition, at greater risk to their lives than nonaltruists and are therefore in greater danger of elimination by natural selection. Haldane (1932, 1955) clearly perceived two possible resolutions of the dilemma. The first, Kin selection, postulates that the altruism is directed towards close relatives; an altruist may die but, by virtue of the altruistic behavior, his (or her) genes are more likely to be represented in the relatives, who, as the recipients of the altruism, have an increased chance of survival. Hamilton (1964, 1972) and Maynard Smith (1964), in quantifying this argument, proposed that the maximum ratio of the risk to the altruist to the advantage to * This research supported in part by NSF Grant DEB 7705742, NIH and AEC Grant AT(O4-3)-326.
268 0040-5809/78/0142-0268$02.00/0 Copyrighf 0 1978 by Academic Press, Inc. AlI rights of reproduction in any form reserved.
Grant 10452-15,
DARWINIAN
SELECTION AND ‘LALTRUISM"
269
the recipient, above which altruism cannot be maintained in the population, is determined by the probability that the parties share genes identical by descent. Hamilton’s notion of selection was couched in terms of an individual’s “inclusive fitness.” This has recently been defined as the expected number of genes in a future generation which are identical by descent to a gene in the individual in question (Oster et al., 1977; Wofsy, 1978). Most studies subsequent to those of Hamilton and Maynard Smith (e.g., Oster et al., 1977; Wofsy, 1978; Trivers, 1971) have involved game theoretic or other optimization arguments rather than dynamic models of genetic evolution. However, Oster et aZ. reconcile these approaches (at least for the problem of initial increase) in studying the problem of sex ratios in insect societies. Haldane’s other possibility for the evolution of altruism is group selection. Here the presence of altruists in a group increases the “fitness” of the group. In this context, altruism is a trait which is advantageous for the group as a whole but disadvantageous for its carrier in competition with nonaltruists. Following Wright (1945) Haldane suggested that the success of this formulation requires partitioning of the population into small groups (see also Maynard Smith, 1976.) That such models of local genetic drift can, under certain conditions on the level of exchange between the groups, allow the evolution of altruism has been shown by Eshel (1972). In the studies by Cohen and Eshel (1976) Matessi and Jayakar (1976) D. S. Wilson (1974), and Eshel (1977) individual fitnesses are functions of the group size and the number of altruists in the group. The models of group selection proposed by D. S. Wilson (1974) and Matessi and Jayakar (1976) do not invoke local genetic drift. In assessing the former Maynard Smith (1976) suggests that it should be regarded as a model of kin selection rather than group selection. Levitt (1975) developed dynamic genetic models of sib-altruism in which the fitness of an individual depends on the frequency of altruistic litter mates. Before Hamilton, Williams and Williams (1957) had proposed a somewhat similar within-brood selection scheme but did not develop the resulting dynamics (see also Maynard Smith, 1965). These, and those of Wilson (1974) and Matessi and Jayakar (1976) are perhaps the studies closest in spirit to that presented here. 1Ve develop models for the evolutionary dynamics of kin selection using Darwinian fitness rather than inclusive fitness. Other models of Darwinian selection, based on interactions between individuals, which can assure the maintenance of altruism are also described.
FITNESS UNDER ALTRUISM
TOWARD KIN
In these models altruistic behavior is determined at a single locus with alleles -4, and -4,. The altruistic phenotype is denoted by A while N represents the nonaltruistic or selfish phenotype. If A, is dominant then ;4 can be either AiA,
270
CAVALLI-SFORZA
AND
FELDMAN
or A,&; if -4, is recessive then only the genotype A,A, is altruistic. The Darwinian fitnesses of the genotypes A,A, , ,4,A,, and A,A, are constructed for the case of parent-to-offspring altruism, in which good and poor parental care characterize the altruistic and selfish phenotypes, respectively. A completely, analogous process can be applied to any degree of relationship between the donor and recipient of altruism. The fitness of a child of genotype AiAj arising from parental care is a function of the gain constant, p, as follows:
+ij = 1 + Pfij;
/3> 0.
(1)
The fij values are the probabilities that an offspring of genotype Ai.4j has an altruistic parent. If the parent is an altruist its offspring’s fitness is increased by the relevant bfij . To evaluate fil , fiz , and fz2for the parent-child relationship we construct the matrix of conditional probabilities that a parent is one of the three genotypes given the offspring’s genotype. Because selection is operating, the parents atmating are not in Hardy-Weinberg equilibrium. The frequencies of A,A, , A,A,, and A,A, in the parental generation are denoted by u, v, and w, respectively; with p = u + v/2 and q = 1 - p. Before selection the offspring are, of course, in Hardy-Weinberg proportions with gene frequencies p and q. The conditional probability matrix is then
\ Offspring
Parent 44
4%
44
WI
4P
Vl2P
0
4%
Ul2P
@Pq
w/&l
44
0
43l
wl!l
\
(2)
Now A,A, is an altruist. Let h be the degree of dominance of the altruistic trait so that, with probability h, A,A, will also be an altruist. If the child is A,A, then the probability that its parent is altruistic isf,, = u/p + hv/2p. In the same way the other fij values can be computed for case of parent-to-offspring altruism:
fu = 4~ + W%
(34
fiz = ~12~ + h+bq,
(3b)
fiz = hvl2q.
(3c)
(Note: h = 1 implies complete dominance of A, to A,; h = 0 implies that A, is completely recessive to A, .)
DARWINIAN
SELECTION
AND
“ALTRUISM”
271
We now postulate that a fitness loss y is incurred by the genotype Aid, in the performance of altruism. In the case where A, is completely dominant to A,, so that A,il, and A,A, are both of phenotype A the fitness loss to these genotypes would be the same, y. However, if the degree of dominance is h (0 < h < 1) then the loss will be y for A,,;Z, but only hy for A,,g, . In principle it is not essential to view fi as influencing only mortality up to maturity, and y only as a measure of potential loss of offspring. The above details should be regarded as one way to assign fitnesses to the altruistic and selfish phenotypes. Indeed in situations other than parent-to-offspring altruism the interpretations must change somewhat. For instance in sib-to-sib altruism, where the altruistic sib is older than the recipient of altruism, it may be more appropriate to view y as increased prereproductive mortality than as a decrease in the potential number of offspring. In more symmetrical situations where the ages of the donor and recipient are not significantly different, it may be difficult to discriminate which of mortality or number of offspring are directly affected by the altruistic process. The final assumption of our model is that the contributions to the fitness of a genotype as a donor or recipient of altruism are independent. There are two senses in which this can be satisfied. First we discuss the multiplicative case: I. Multiplicative
Model
where the fir values are as in (3). The genotype frequencies after selection are then
u' = p2(l - r)[l + IqUIP+ W34/~7
(64
w’ = 2pq(l - hY)[l + Is(U/2P + W4P@l/$J,
(a)
w' = $11+ p42ql/~,
P-3
where 6 = 1 - r(P + 244 + B(u + hv) - BY[U(P + hq) + ~V(P + WI.
(64
Equations (6) describe the case of parent-to-offspring altruism. Another situation which has apparently led to most of the development of “sociobiology” is that of sib-to-sib altruism. To treat this casewhere only diploid individuals are considered, we can compute the probability that, for example, an individual of
272
CAVALLI-SFORZA
AND
FELDMAN
genotype &A, has a sib of genotype A,A, . It is 2er(u + w/4)/4~“. The matrix computed in this way analogous to (2) is recorded as (8) sib 2 \ 1 sib 44
44
AA
-W,
44
w2/16p2 2w@+ w/4)/4p2 (u + v/~)~/P~ 4~ + w/4)/4~q [k/4)” + (u + w/4)@ + 7d4)llpp a@ + d4)/4pq w2/16q2 2w(w + o/4)/4q2 (w + w%12
(8) Using (8) the functions fij are calculated as before. Hamilton’s (1964) proposal for the evolution of social behavior in insects relied on the haplo-diploid nature of the species to which he referred. Haplodiploidy has subsequently played a central role in evolutionary theories of behavior. In population genetic terms this system-is equivalent to sex linkage. The matrices for the probabilities of relationship in the sex-linked situation in the absence of selection are well known (see, e.g., Li, 1955, Chap. 3) and can be adapted to the situation including selection. The frequencies of A,A, , A,A, , and AzAz (females) are 11,w, and w as before, with p = 1 - q = u + w/2, while A, and A, males have frequencies x and y, respectively. Then the matrix describing relationships between sisters in (9), where z = xq + yp: Sister 2 \ Sister 1 \
44 4% 44
AC%
(u + 014)/P wx/4iz 0
44
wl4P [v + wx + W(X+ Y)/412 v/4q
44
0 VI42 (9) @ + w/4)/q
Of course this analysis must be made in terms of the five frequencies 1~,w, w, X, and y. Relationship between brothers, sister and brother, and brother and sister can be written similarly. Each of these sib-to-sib relations produces its own set of transformations with its characteristic form of frequency dependent selection. We have presented the case of parent-to-offspring altruism in detail but the other relationships may be treated similarly. Two evolutionary questions are asked of systems such as (6). First, when are the boundary equilibria (0, 0, 1) and (1, 0,O) locally stable ? In the case of (0, 0, I), its instability means that the altruistic phenotype increases when rare; it is “protected” from loss. Stability of (1, 0, 0) entails the fixation of the altruistic phenotype. From Hamilton (1964) (see also E. 0. Wilson, 1975, Chap. 5), it is expected that the stability conditions at these boundaries should be determined
DARWINIAN
SELECTION
AND
“ALTRUISM”
273
by the relationship of the ratio y//I to the “coefficient of relationship” between the donor and the recipient of the altruism. We indicate by F the coefficient of kinship between the two individuals, which is equal to the inbreeding coefficient of their progeny. Usually [but not always) the coefficient of relationship is twice the coefficient of kinship. The second question asked of (6) concerns the existence and stability of polymorphic equilibria. For models as complex as the present one, this is usually much more difficult to resolve. When dominance is intermediate (h # 0, l), the first of these questions is answered by conventional local linear analysis. For every degree of relationship between donor and recipient mentioned above we have (0, 0, 1) is stable if y > 2F/(/L-l + 2Fh),
(104
(1, 0,O) is stable if y < 2F/@-1 + 2Fh -+ 1).
(lob)
For every degree of relationship, when h = 0, local linear analysis at (1, 0,O) presents no problem and (lob) is valid with h = 0. Similarly when h = 1, (10a) gives the condition for stability of (0, 0, 1). But with h = 0, local linear analysis at (0, 0, 1) fails because the leading eigenvalue is unity. Similarly with h = 1, linear analysis near (1, 0,O) produces an eigenvalue of unity. Analogous difficulties with h = 0 and h = 1 were noted in their studies by Levitt (1975) and Matessi and Jayakar (1976). N evertheless a second-order analysis can be made in all of the usually considered cases: parent-to-offspring, sib-to-sib diploid, and sib-to-sib sex-linked (equivalently, haplodiploid). The details of the analysis will be presented elsewhere. The results are displayed in Table I with (1Oa) and (lob) included for comparison. This table presents the bounds for y, the fitness cost of altruism as a function of j3, the gain in fitness obtained by recipients, and the coefficient of kinship. For each relationship the first row shows the bounds on y below which altruism is protected from loss. If y is below the value in the second row, then altruism fixes. It is clear from Table I that as F increases, the bounds on y//3 become less stringent (for fixed 8). It is also clear that the relationship between y//? and 2F in this multiplicative model is not as the same as that suggested by Hamilton. It is true that as /3 --+ 0, the relationship between y//I and 2F approaches Hamilton’s rule. For larger values of /I it is more natural to use Table I rather than y//I to describe the stability conditions. The boundary stability results in Table I, except for the parent-to-offspring case, are exactly the same as those obtained using expressions analogous to (3) for the fii values but involving the gene frequency rather than the technically more correct genotype frequencies. Use of the well-known matrices of relationship in the absence of selection (see, e.g., Li, 1955, Chap. 3) allows fast enumeration of fij . With h = 0 and h = 1 the results in the parent offspring case for
274
CAVALLI-SFORZA
AND
FELDMAN
initial increase and fixation of altruism respectively are different from those in Table I when gene frequencies are used. TABLE Stability
Conditions
I
for the Multiplicative
Model”
Equilibrium
Dominant (h = 1)
(0, 0, 1)
i/W’
f
4)
;t/(B-’
+ h/2)
Q/W
+ g,
(1, 0, 0)
h/W’
+ 1)
&lW1
+ h/2 + 1)
4/C-’
+ 1)
Sister-to-sister (sex-linked) @= = 3)
a
0, 11, (0, 1)
~/U-l
+ $
$/Wl
+ 3h/4)
g/w
+ $1
(LO, 01, (1, 0)
f/@-l
+ 21
8/W’
+ 3h/4 + 1) $I@-’
Sister-to-brother (sex-linked) (F = $1
(0, 0, u al)
&W’
+ a,
m-1
+ h/4)
t/wl,
(1, ‘3%
iAS-’
+ 8,
$/(fl-’
+ h/4 + 1)
&(B-’
Brother-to-brotherb (sex-linked) (F = $1
640,
i4B-’
+ 4)
(l,O, 01, (LO)
t/w1
+ 1)
(090, I), (0, 1)
4/W’)
(I,&
iw-’
Relationship Parent-to-offspring (F = 4)
Intermediate (0 < h < 1)
Recessive (h = 0)
Sib-to-sib (diploid) (F = $1
Brother-to-sisterb (sex-linked) (F = 4) a For each relationship less than the given value. b Dominance irrelevant
(1, 0)
11, (0, 1)
Oh (1,O)
the upper equilibrium
is unstable
+ 1)
+ 1)
+ 9) and the lower
stable for y
here.
In this multiplicative model the polymorphic equilibria are expressible as the roots of higher order (generally fourth of higher) polynomials. Numerical analysis indicates that for y values between the pairs of values in Table I, a single stable equilibrium exists. In this range of y values the altruism increases when rare but does not proceed all the way to fixation. For y smaller than the values in the second row of the pairs in Table I, it appears that fixation of altruism is globally stable. In the case of brother-to-brother altruism the order of the polynomial is reduced and the above remarks have been validated analytically.
DARWINIAN
II. Additive
SELECTION
AND
“ALTRUISM”
275
Model
In the representation
of the additive
model the fitnesses are
where the fii values are defined through the matrices exemplified by (2), as in (3). With fitnesses (11) the genotype frequencies after selection are, in the case of parent-to-offspring altruism,
24'= [p2(1 - Y) + Pi@ + w41/$k
(124
2" = 2pq(l - hy) + P(uq+ W2)1/$57
(12b)
m' = [q2+ /3hfJqPl/&
(124
with
6 = 1 - y( p2 + 2&q)
+ B[P+ z@- $)I.
(124
Recursions similar to (12) can be written for each degree of relationship considered above. The equilibrium analysis can be summarized in a manner analogous to that of Table I; it is presented in Table II. The results here are seen to fit quite closely into the framework originally conjectured by Hamilton. Some anomalies arise, however. In the parent-to-offspring case, with h = 1, the point (0, 0, 1) is stable if y > /I/2. But the point (1, 0,O) is stable if y < [1+ /3 (1 - /F)l’*]/2. In the range p/2 < y < [1 + /I - (1 - B’)““]/2 both boundaries are stable and an unstable interior R = 0 in the parent-offspring case, y in the interval
/w > Y > P/(2 -t B)
equilibrium
(13) exists. When
(14)
allows the stability of both boundaries and the existence of an unstable interior point. With h = 0 or h = 1, the equilibria in the additive model are usually roots of quadratics but for 0 < h < 1, these become fourth degree polynomials. The usual state of affairs in the sib-to-sib situation (also sister-to-sister in the sex-linked case) is that with h = 1, for a small interval of values larger than 2F, two polymorphic equilibria exist. One of these is stable, the other unstable. With h = 0, there is a corresponding interval of y values smaller than 2F/? in which
276
CAVALLI-SFORZA AND FELDMAN TABLE Stability
Relationship
Conditions
tw
(0, 0, 1) ~(l,O,
Sib-to-sib (diploid) (F = $1
0)
for the Additive Dominant (h = 1)
Equilibrium
Parent-to-offspring (F = *>
II
[l +p-s-(l-~yl/2
Model” Intermediate (0 < h < 1)
Recessive (h = 0)
812
8/(2+~~
812
812
(0, 0, 1)
tw
iw
812
(l,fA
0)
I312
812
812
Sister-to-sister (sex-linked) (F = 8)
GAO, 11, (0, 1)
38/4
3814
3814
(1,O.O),U,O)
3/9/4
3814
3814
Sister-to-brotherb (sex-linked) (F = 9)
(O,O, I), (0, 1)
(/3/4)(1+8/4)
(8/4)/U+FW4)
814
(1,O,Oh(1,0)
(8/4)/1+-w/4)
w4M-t8(1+Jd4)1
63/4Ml+S)
Brother-to-brother (sex-linked) (F = t)
(0, 0, l), (0, 1)
812
(l,O, 01, (1,O)
812
Brother-to-sister” (sex-linked) (F = $1
(O,OP I), (0, 1)
B/2
(190, Oh (1,f-J
69/2)/(1+38/2)
a For each relationship the upper equilibrium is unstable and the lower stable for y less than the given value. * These cases are equivalent in the multiplicative and additive constructions.
one stable and one unstable polymorphic equilibrium exist. In these cases, therefore, a boundary and a polymorphic equilibrium are stable. Thus in the sister-to-sister
sex-linked
case, if altruism
is recessive,
it can increase
when
rare
but reach a polymorphic state rather than proceeding to fixation. There are some differences from this pattern, for example, in the brother-to-brother sex-linked case where no interior point exists. With 0 < h < 1 the equilibria seem, from numerical analysis, to obey these qualitative patterns which, it should be noted, are different from those described above for the multiplicative model. INTERACTION MODELS WITHOUT KIN SELECTION
The previous models translated the notions of kin selection into frequencydependent Darwinian selection. We turn now to another class of models in which
DARWINIAN
SELECTION
AND
277
L’ALTRUISI\l”
fitness can be attributed to the interactions of individuals with other, not necessarily related, members of the group. The simplest example of this sort considers only groups of size 2 in a population with phenotypes A and N. The groups can be (-4, A), (=I, N), and (N, N), and four types of interactions can occur as shown in Table III. where u is the frequency of ,4 and v = 1 - U, that of N. TABLE III Interaction Parameters Fitness increment to __--_ Helper Helped
Interaction between Frequency Helper
Helped
A
-
A
u2
-‘1
A
--f
N
uv
8,
B2
N
4
-4
IIt
N
+
N
73
Yl T 4
Yl 6,
‘72
In the paradigm with A = altruist and N = nonaltruist, Q~, /3, are positive; cyl , /$ are negative; yI , 6, can be zero or positive; yZ (6, can be zero or negative. The construction reflects “altruistic” behavior if the quantities a = (Ye+ (Ye and c = /?s + y1 are both positive with b = ,G1+ yZ and d = 6, + 6, both negative. Two simple fitness schemes can be proposed using Table III. In the first, each -4 individual has fitness increments according to the type of twoindividual group it is in. The relative fitnesses of A and N would then be WA = 1 + au + ba, W,v = 1 + cu + dv. In the second the fitness of -4 is incremented individuals, so that
by all encounters
(154 (15b) involving
--1
WA = 1 + UU2 + hua, I$‘,,, = 1 + CUZI+ dv2. Following
the interactions
we have
li = uw,/c
with
l+ = u WA + v WV .
(17)
We restrict ourselves here to the cases with a > 0, c > 0, b < 0, d < 0, and j b I < 1, 1d / < 1. Without the latter restrictions the recursions may conceivabl!
278
CAVALLI-SFORZA
AND
FELDMAN
leave the admissible frequency range. Under these assumptions (15) produces a monotone transformation such that if b < d and c < a both zi = 0 and zi = 1 are stable and are separated by an unstable equilibrium. If b > d and c > a then the interior equilibrium
ti = (6 - d)/(b - d + c - a)
(18)
is globally stable. The fitness specification and result (18) are originally due to Hamilton (1971). If A and N are regarded as the dominant and recessive phenotypes at a single locus (for example r2 = A-, B - au) then we can write WA- = 1 + a(1 - 4s) + bq2 fand W,, = 1 + c( 1 - q?) + dq? with Q the frequency cf the “a” gene. If further, a = b and c = - 1 with d = 0, the model reduces to a case of social coorperation as considered by Boorman and Levitt (1973), which in turn is a special case of the general model studied by Cockerham et cd. (1972). With the fitness specification (16) and a, c > 0, 6, d < 0, 1b 1 < 1, and 1d 1 < 1, ti = 0 is unstable and ti = 1 is stable. If, in addition, (b - c)% + 4ad > 0, two polymorphic equilibria exist. Again under the same restrictions as above, the resulting transformation is monotonic and the equilibrium
li = c - 6 - 2d - [(b - c)” + 4&]1’2 2(a - 6 + c - d)
(19)
is stable. Again it is clear how this second model may be placed in a dominantrecessive genetic context. CONCLUSIONS
The two types of model proposed here allow the evolution of altruism to be studied in a Darwinian selection framework via frequency-dependent fitnesses. This structure obviates the need to use the concept of “inclusive fitness,” the appropriate definition of which has been a stumbling block in the translation of Hamilton’s strategy arguments into evolutionary genetic models. In the first class of genetic models the unifying feature is the use of conditional matrices of probabilities that an individual of given genotype has relatives of given degree of a specified genotype. These are specified in terms of genotype frequencies in the same way that the well-known Hardy-Weinberg analogs are functions of gene frequencies. In these genetic models it is the case that the closer the relationship between donor and recipient the easier it is for altruism to increase when rare. But the relationship between the ratio r//3 of loss to gain and the kinship coefficient allowing the evolution of altruism depends on how the model is structured. In the multiplicative model the ratio r/p is insufficient to determine the outcome, and the magnitude of /) itself plays a role (Table I). This is significant, for example, in the comparison of sib-to-sib altruism with parent-to-
DARwINIAN
SELECTION AND “ALTRUIShI”
279
offspring altruism. The initial increase conditions are formally the same, but the gain, /3, to the recipient (child) of parental altruism would generally be larger than in the sib case. Unlike the multiplicative case the coefficient of kinship in the additive case (Table II) is sufficient to explain the boundary behavior of the altruism. But in the additive case an intringuing interior equilibrium phenomenon arises which interferes with a complete evolutionary explanation based on the coefficient of kinship. The models described here are the simplest among a whole class of fitness specifications involving interactions between y, /3, and fij . It should be recognized that when placed in the population genetic framework, models involving kin selection, or cooperation, become special cases of gene, genotype, or phenotype frequency-dependent selection. For this reason, attempts, such as that by Cockerham et al. (1972) to classify such models assume an importance not generally appreciated by “sociobiologists.” The assumption throughout the genetic models that the cost and benefit are independent is an extremely crucial one. In the case of parental care for example, the intensity of the experience (as an offspring) may later influence the behavior of the same individual as a parent. The second class of (nongenetic) frequency-dependent selection models introduced by Hamilton (1971) might be looked on as a case of “good guys” (A) and “bad guys” (N). Clearly such an interpretation is mediated by the choice of a, b, c, and d, in (11) and (12). Kinship is not required here for the evolution of the A phenotype which is at a disadvantage in its contacts with N individuals. The difference in the equilibrium behavior of the two examples indicates that more complex social interactions may result in still more complex equilibrium properties, and that equilibrium and dynamic behavior is extremely model sensitive. It may well turn out that the study of simple phenotypic examples may provide far more insight into the evolution of individually disadvantageous traits than the more complicated genetic models. Such models allowing nonbiological transmission are currently under study.
ACKNOWLEDGMENTS The authors are grateful to Dr. Marcy Uyenoyama for many stimulating discussions and to Professor S. Karlin and Dr. B. 0. Bengtsson for their critical comments on the manuscript. REFERENCES BOORMAN, S., AND LEVITT, P. R. 1973. A frequency dependent natural selection model for the evolution of social cooperation networks, PYOC.Nat. Acad. Sci. USA 70,187-l 89. COCKERHA~\I, C., BURROWS, P. M., YOUNG, S. S., AND PROUT, T. 1972. Frequencydependent selection in randomly mating populations, Amer. Nut. 106, 493-515.
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