The Model

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 reproduction, 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.

Thus the fitness d of an individual is treated as the sum of his basic unit, the effect 6a of his personal genotype and the total e” of effects on him due to his neighbours which will depend on their genotypes:

a*= 1+6a+e0. (1)

The index symbol ’ in contrast to ’ will be used consistently to denote the inclusion of the personal effect 6a in the aggregate in question. Thus equation

(1) could be rewritten

a’= l+e*.

In equation (I), however, the symbol ’ also serves to distinguish this neighbour modulated kind of fitness from the part of it

a=l+&

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

a = 1+6a.

The neighbours of an individual are considered to be affected differently according to their relationship with him.

Genetically 2 related persons differ 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

c,+c,+c, = 1,

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 2c, + cl. One half of this number, c,+$c,, may therefore 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=+r=+(cz++cl) and F=c, l-2

for a large class of relationships can be found in Haldane & Jayakar (1962).

Strictly, a more complicated metric of relationship taking into account the parameters 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 is 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 (&zfather)A 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 (&,ther)A, (8Uspecified sister)A, etc., we will denote bkel.1‘4. From this array we can construct the simpler array ih, czh by adding together all effects to relatives who have the same values for the pair of coefficients (r, cZ). For example, the combined effect da*,, might contain effects 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, {&IA, by adding together all effects with common Y. Thus da, would bring together effects to the above-mentioned set of relatives and effects to double-first cousins, for whom the pair of coefficients is (t, &). 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 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 e” 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 genefrequencies and the mating system.

Consider a single locus. Let the series of allelomorphs be G,, G,, GJ, . . . , G,, and their gene-frequenciesp,, p2, p3, . . . , p,,. With the genotype GiGj associate the array { ~arel.}ij; 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 a:. In order to formulate the expectation of that element of eFj 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 genefrequencies just as in the problem of the correlation between relatives.

Thus the formula for e$ will involve all the arrays {Bar,c2)ij and will be rather unwieldy (see Section 4).