An alternative approach, however, shows that the arrays (Sa,>, 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 reproduction of genes i.b.d. with genes in A, and an effect on the reproduction of unrelated genes. Since the coefficient I measures the expected fraction of genes i.b.d. in a relative, for any particular degree of relationship this breakdown may be written quantitatively: (~aredA = r(hel.L + Cl- r)(JaredA. The total of effects on reproduction which are due to A may be treated similarly : IX, WreJA = IX, @adA + z, Cl- r) @areA or F (h), = F r@& + C Cl- r>(WA, r which we rewrite briefly as 6T; = 6Rf4 + 6S,, where 6Rz is accordingly the total effect on genes i.b.d. in relatives of A, and SS, 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 we should have SS~ =SSi, whatever the subscript; it therefore seems more explicit to omit them throughout.

If, therefore, all effects are accounted to the individuals that cause them, of the total effect 6Ti~ due to an individual of genotype GiGj a part SRG will involve a specific contribution to the gene-pool by this genotype, while the remaining part SSij 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 SRG which determines the direction of selection progress in gene-frequencies; SSij only influences its magnitude. In view of this importance of the SRG it is convenient to give some name to the concept with which they are associated.

In accordance with our convention let then RG will be called the inclusivefitness, SR,‘, the inclusive fitness efSect and SSij the diluting effect, of the genotype GiGj.

Let Tijl= 1+X$.

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 C Pi PjRTj + Pi C Pj Pk as,, pf= j & PjPlcx: 1 PjR$ +,& PjPkSSjk = pi j ,cxpjikT; ’

The terms of this expression are clearly of the nature of averages over a part (genotypes containing G,, homozygotes GiGi counted twice) and the whole of the existing set of genotypes in the population. Thus using a well known subscript notation we may rewrite the equation term by term as Rz +6S p; = pi I’+ T. or Ap, = R. Fas. WT. - Rf.1. .a . .

This form clearly differentiates the roles of the Rz and SSG in selective progress and shows the appropriateness of calling the latter diluting effects.

For comparison with the account of the classical case given by Moran (1962), equation (2) may be put in the form where alapi denotes the usual partial derivative, written dIdpi by Moran.

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.: = T.0.

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, 1961 a) presents certain difficulties. However, from the above equations it is clear that the quantity that will tend to maximize, if any, is R.:, the mean inclusive fitness. The following brief discussion uses Kingman’s approach. The mean inclusive fitness in the succeeding generation is given by RTI =CPIPR; = ~~~,pipjR(R:+sS..)(Rj+S..). i, j a. i,J . . . R.’-R.= AR” = ~~~ . . +R:@f -R’T;y . Substituting R.: + SS.. for T-y in the numerator expression, expanding and rearranging : Cpipj R~R:R:j-R03 + . . GJ 1

• 26s.. ( C PiPjRR.-R~2 . i,j We have ( ) > 0 in both cases. The first is the proven inequality of the classical model. The second follows from CpiPjRtR: =CpiRyT > (7 piR:)2 = RTf’* 4 j Thus a sufficient condition for AR.: > 0 is SS.. 3 0. That AR.: > 0 for positive dilution is almost obvious if we compare the actual selective changes with those which would occur if {RG} were the fitness matrix in the classical model. It follows that R.: certainly maximizes (in the sense of reaching a local maximum of R.:) if it never occurs in the course of selective changes that SS.. < 0. Thus R.y certainly maximizes if all SSij > 0 and therefore also if all (&,,r.)ij > 0. It still does SO even if some or all 6Uij are negative, for, as we have seen SS, is independant of liaij.

Here then we have discovered a quantity, inclusive fitness, which under the conditions of the model tends to maximize 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,? 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.