Exact Solutions to Certain Stochastic Differential Equation Models of Population Growth

Abstract

The large number of mathematical models of population growth, if nothing else, is a testimony to the complexity and subtlety of the processes involved. The problems associated with models containing a probabilistic component have proved to be especially difficult. For example, one might expect that the case of density independent exponential growth with a stochastic term would be well analyzed by virtue of its usefulness as a null hypothesis in testing for the presence of regulation in populations. However, recent reviews by Lewontin and Cohen (1969) and Brockelman and Fagen (1972) have pointed out some misconceptions and difficulties which arise in both the biological concepts and the mathematical analysis of this case. In this paper we treat exactly a mixed deterministic-stochastic model of density-independent population growth and two models of density-dependent growth. The analyses lead to closed-form expressions for the probability density function of population size as a function of time, the intrinsic rate of natural increase of the population, and the autocovariance function of a stochastic environmental term.