A Stochastic Model for Insect Life History Data

Abstract

A stochastic model is developed for an insect’s life history: time may be measured on a chronological time scale or on some operational time scale such as degree-days. We assume a Brownian motion process, with drift, for development, which is similar to the development model underlying Stedinger et al. (1985), but its use in this paper is different and more theoretically appealing. Independent of development, the mortality process is defined by a two-state Markov process with nonhomogeneous mortality rate; biological considerations may suggest appropriate forms of the mortality rate. The life history process is obtained by superimposing the two stochastic processes and subsequently aggregating the state space; the aggregation is necessary because a development stage is observable, rather than the exact level of development. Stage occupancy is determined by the stage recruitment times, which are inverse Gaussian random variables. Defining stage occupancy in this manner allows for an interpretation at the microscopic level, and leads to a semi-Markov model for life history processes with inverse Gaussian stage transition rates if the mortality rate is constant. The proposed model is extended to incorporate recruitment to the first development stage. Stage-specific recruitment, sojourn times and mortality rates are expressed as functions of the model parameters. This model provides a bridge between the macroscopic models suggested by various authors (e.g. Manly 1974 and Stedinger et al. 1985) and the microscopic models developed by others (e.g. Read & Ashford 1968).

The model is fitted to longitudinal data on the prevalence of insects in the development stages assuming a product-Poisson likelihood for the counts. Maximum likelihood estimates of the parameters and their asymptotic standard errors are obtained via a Gauss-Newton algorithm for iteratively reweighted least squares. An example is provided and the resluts are compared to those obtained by Read & Ashford(1968), and Kempton (1979).