A General Model of Optimal Reproduction in a Randomly Varying Environment

Because of natural selection, the reproductive behaviour of natural populations is expected to be such as to maximize their long term expectation of survival. Predictions can be made, therefore, about the natural reproductive behaviour of a population, by constructing for it a representative model, and finding the behaviour which maximizes the long-term survival expectation of the population in the model. The results of such predictions can then be compared with known reproductive behaviour in different organisms, and some general conclusions can be reached about the validity of the model and of the hypotheses on which it was based. An organism may at any time interval either initiate reproduction, or remain in a nonreproductive phase, which may be either a dormant phase or an active vegetative phase. In general, the viability of a reproducing organism is reduced relative to that of a comparable non-reproducing organism. In extreme cases, a reproducing organism may die after completing the reproductive cycle. In this paper, we consider the problem of the optimal fraction of a population which should attempt to reproduce at any given time interval, when viability, reproduction and differential viability of reproducing organisms are random variables with constant distributions, i.e. in a 'stationary' environment. Our main aim is to show what are the conditions which determine when it is optimal for the whole population to reproduce, and when it is optimal for only a fraction of the population to attempt to reproduce. The numerical values of the optimal reproducing fractions in populations are computed for different representative distributions of viability, reproduction and differential viability of reproducing organisms.