Gompertz growth model in random environment with time-dependent diffusion.

Abstract

The Gompertz nonlinear growth (GNG) model with independently and identically distributed (i.i.d.) errors is often employed for describing growth data. However, the corresponding stochastic differential equation (SDE) variant is more realistic for modeling growth data, as it is capable of taking into account the effect of randomly fluctuating parameters, such as birth and death rates. However, one limitation of this prescription is that the diffusion term is assumed to be time independent. The purpose of this article is to generalize the Gompertz SDE model by taking the diffusion coefficient as time-varying. The resultant model is solved analytically and methodology for estimation of parameters, based on the method of maximum likelihood, is developed. Formulas for optimal predictors and prediction error variances and the linear Gompertz SDE (LGSDE) model and modified Gompertz SDE (MGSDE) model are also derived. Superiority of the proposed MGSDE model is shown over the LGSDE and GNG models for pig growth data.