A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth

The deterministic logistic model of population growth and its notion of an equilibrium ‘carrying capacity’ are widely used in the ecological sciences. Leading texts also present a stochastic formulation of the model and discuss the concept and calculation of an equilibrium population size distribution. This paper describes a new method of finding accurate approximating distributions. Recently, cumulant approximations for the equilibrium distribution of this model were derived [Biometrics 52 (1996) 980], and separately a simple saddlepoint (SP) method of approximating distributions using exact cumulants was presented [J. Math. Appl. Med. Biol. 15 (1998) 41]. This paper proposed using the SP method with the new approximate cumulants, which are readily obtained from the assumed birth and death rates. The method is shown to be quite accurate with three test cases, namely on a classic model proposed by Pielou [Mathematical Ecology, New York, Wiley, p. 304] and on two African bee models proposed previously by the authors [Biometrics 52 (1996) 980; Theor. Popul. Biol. 53 (1998) 16]. Because the new method is also relatively simple to apply, it is expected that its use will lead to a more widespread utilization of the stochastic model in ecological modeling.