J. theor. Bid. (1979) 77, 379-381

LETTERS TO

THE

EDITOR

Kin Selection in Finite Sibships Charnov (1977) has recently given an elementary proof of Hamilton’s rule (cost/benefit < coefficient of relation) for the increase in frequency of an allele determining altruistic behavior. He assumes (i) altruism is controlled by a single locus with two alleles, (ii) families are very large (effectively infinite) in a population that is panmictic with discrete generations, and (iii) altruistic transactions occur exclusively between two relatives. To extend this model Charnov continues by relaxing the family size constraint with the objective of seeing if the rule is altered in sibships of finite size. For the specific case of a sexual haploid, he concludes that the ratio of benefit to cost necessary for the altruist allele to be favored increases as the family size decreases. This conclusion is incorrect. Consider a sibship of size K and calculate the conditional probability that an A (= altruist) genotype is helped given that the helper is derived from an altruist x non-altruist (A x a) mating. This probabihty is

which equals l/2 identically. Here the expression within the leftmost bracket is the frequency of A individuals found in sibships containing exactly j individuals of genotype A ; that in the second bracket is the probability that one of the other j- 1 altruist sibs benefits from the altruism of a specified A individual. This information can now be used to calculate the (unconditional) probability that an A genotype is the beneficiary of any act of altruism that occurs. This is Chamov’s equation (B3) which should read m = p + (1 - p)( l/2) where p is the frequency of the A allele in the population. The value of m so calculated may then be used in Chamov’s equation (9), c/b c (m - p)/( 1 - p), to yield c/b < l/2, which is Hamilton’s rule for full sibs. The conclusion is, therefore, that the finite sibship restriction is no restriction at all ; it does not alter the rule under these assumptions. 379 0022-5193/79/070379

+ 03 $02.00/O

,Q 1979 Academic

Press Inc. (London)

Ltd.

380

L. L. JOHNSON

In view of the foregoing, it might be conjectured that the same conclusion holds for the case having the more widespread biological interest, that of the sexual diploid. This is indeed true and can be shown by a straightforward extension of the reasoning used above. In brief, one calculates for each of the six possible mating classes the conditional probability that an altruist, AA or Au, is helped when the helper is AA and when the helper is Aa. All of these probabilities, except for those which are obviously 1 or 0, equal l/2 save for the probability that an AA is helped given that the helper is AA or Aa from an Aa x Aa mating. Both of these particular probabilities are l/4. By multiplying each of the conditional probabilities by the frequency of the parental mating class and summing, one obtains expressions analogous to (A3) and (A4) of Charnov (1977) for m and e in terms of p. When these are put into his equation (8) it can be simplified to c/b < 1/2just as in the haploid case. It is possible to construct a model according to the idea implied in Charnov’s (1977, Appendix B) finite family analysis. The assumption here is that altruistic acts occur, not as above, with equal probability among altruists regardless of sibship affiliation, but instead, with equal probability among sibships having at least one altruist. The sexual haploid case for this model leads to an expression which is, using the same symbols as before, (cf. equation (B3), Charnov, 1977)

m=

P

[ p+(1-p)(l-1/2K)

I[ +

p+(l

(1-P) -p)(l-

K ()I

2 1/2K) IL

2” 2K-l K-l

-1 .

Here the quantity within the first bracket is the probability that the altruist is from an A x A family; that within the second bracket, the probability that the altruist is from an A x a family. Each of these is multiplied by the probability that an altruist is helped given that the helper is from the specified family. Consideration of the limiting cases K = 2 and K = cc leads to m = (1 +p)/(3 -p) and m = 1/(2-p), respectively. When these values are used in the expression c/b < (m - p)/(l - p), as before, one obtains c/b < (1 - p)/(3 - p) and c/b < (1 - p)/(2 - p) for the respective cases. Thus in this model, the condition for the spread of the altruist allele is a function of its frequency in contrast to the model analysed above. If one assumes further, that the cost and benefit of altruism have fixed values independent of p, and then solves these inequalities to find limiting values of p, one obtains pc (3c-b)/(c-b) and pc (2c-b)/(c-b) when K = 2 and K =c.o, respectively. The object of these computations is simply to show that this

LETTERS

model leads to a polymorphism than 1.

TO THE

for altruism

The Jackson Laboratory. Bar Harhor, Maine 04609, U.S.A. (Received 24 MaI! 1978. atld in revised,form

REFERENCE CHARNOV.

E. L. (1977).

J. rhea.

Bid.

66, 541.

38

EDITOR

in families of all sizes greater

LAWRENCE

13 November 1978)

L.

JOHNSON