Printed in Great Britain Some effects of fluctuating offspring distributions on the survival of a gene

In this paper we study the fate of a gene that has just been introduced into a large population that is stationary in size. I t is assumed that lines descended from two or more genes of this type develop independently of each other. But in contrast to the usual treat-ment of this subject, offspring distributions are not assumed to stay the same from genera-tion to generation. Particular attention is paid to two cases: (a) cyclical fluctuation of the offspring distribution, (6) the case in which a new gene has an advantage which is gradually lost as time passes. 1. INTBODXTOTION Several authors, beginning with Fisher (1922), have studied mathematically the fate of a mutant gene which is originally present in one individual in a very large population. In work of this sort two basic assumptions are made. The first is that lines descended from two or more such genes develop independently of each other. This is reasonable because, if the mating system is at all close to random mating, it is unlikely that two genes of the rare mutant type enter the same individual, and by the time the gene becomes common enough to make this assumption not valid, its fate has already been decided. The second assumption is that the distribution of offspring of an individual bearing the gene does not change from generation to generation. These assumptions lead to the following mathematical formulation. Let pn r = P(r descendants in generation n of a single mutant gene in generation 0) and Pn)(t)=Zpnrtr. (1-1) r-0 Because of the assumption of independence, (1-2) r-