Uniform bounding of probability generating functions and the evolution of reproduction rates in birds

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Abstract
Many species of birds have a characteristic clutch size which is either fixed atkor is of the formkork+ 1 for some appropriate integerk.In this paper we show, using a multitype Galton–Watson process to model a bird population, that such behaviour can correspond to maximization of the probability of survival of the species to timetfor each finitet.This is also a conclusion which might be drawn from the theory of natural selection and hence provides some mathematical evidence of the force of evolution. The results of the paper rest on a bounding of probability generating functions.