Population size models based on cumulative size, with application to aphids

This paper investigates the use of a mechanistic model for describing the size of local aphid populations, specifically of the pecan aphid (Monellia caryella) and of the mustard aphid (Lipaphis erysimi). The mechanistic equation, like the logistic growth model, has parameters for a birth rate and a death rate, however the present mechanistic model generalizes the logistic growth model by incorporating the cumulative size of past generations into growth rate assumptions. A new non-linear regression model is derived, which solves the mechanistic model analytically and which may be fitted to data. The parameters of the regression model are the predicted peak size, Nmax⁡, the predicted time of peak, tmax⁡, and an approximate per capita birth rate, b, all of which are of interest in practical applications. The regression model is shown to fit diverse abundance curves adequately. The model also explains the population growth curves through the underlying rates. Simple approximations, based on these parameters, for the peak count and for the final cumulative aphid density are derived, and shown to be accurate. In general, this paper demonstrates the utility of analyzing local aphid population data using mechanistic models and their underlying rate parameters.