No abstract

Abstract.-If a population is growing in a randomly varying environment, such that the finite rate of increase per generation is a random variable with no serial autocorrelation, the logarithm of population size at any time t is normally distributed. Even though the expectation of population size may grow infinitely large with time, the probability of extinction may approach unity, owing to the difference between the geometric and arithmetic mean growth rates.A problem of recurrent interest to population ecology is the question of "density-dependent" control of population numbers. The literature on this subject is so vast and so well known to ecologists that it cannot and need not be referenced here. Briefly, the question is to what extent the actual growth rates of populations are affected by the population density, and so to what extent density and resource shortage must be taken into account in explaining the observed history of population numbers.The solution to this difficult and not always well-defined problem involves, among other things, an adequate description of how a population would change its numbers if its growth rate were not related to number, but only to variations in the independent environment (including other species, of course). Thus, we ask whether the observed variation in numbers of a species could be satisfactorily explained by supposing that at all times numbers are growing by the simple exponential growth law, but that the exponential rate of increase r is varying according to some extrinsic law unrelated to N, the population number. As stated, this proposition about variation in r is so general that no theoretical or experimental distinction could possibly be made between this hypothesis and that of density dependence. More specification is required. We assume a stationary distribution for r. Specifically, then, we want to ask how population numbers will change if the population is undergoing simple geometric increase or decrease with a rate that varies at random independently of both N and t.Although many sophisticated treatments of stochastic population growth exist for a variety of models,1' 2 it is our purpose in this note to point out a peculiarity of multiplicative population growth which is apparently not widely appreciated and which gives rise to some confusion.Consider a population of size Ne at time t, which in a single reproductive period has a multiplicative increase It so that Nt+1 = Ndl(1) and in general Nt = No Hi.

No abstract

No abstract

No abstract