Bayesian Inference of Deterministic Population Growth Models

Abstract

Deterministic mathematical models play an important role in our understanding of population growth dynamics. In particular, the effect of temperature on the growth of disease-carrying insects is of great interest. In this chapter we propose a modified Verhulst—logistic growth—equation with temperature-dependent parameters. Namely, the growth rate r and the carrying capacity K are given by thermodynamic functions of temperature T, r(T) and K(T). Our main concern is with the problem of learning about unknown parameters of these deterministic functions from observations of population time series P(t, T). We propose a strategy to estimate the parameters of r(T) and K(T) by treating the model output P(t,T) as a realization of a Gaussian process (GP) with fixed variance and mean function given by the analytic solution to the modified Verhulst equation. We use Hamiltonian Monte Carlo (HMC), implemented using the recently developed rstan package of the R statistical computing environment, to approximate the posterior distribution of the parameters of interest. In order to evaluate the performance of our algorithm, we perform a Monte Carlo study on a simulated example, calculating bias and nominal coverage of credibility intervals. We then proceed to apply this approach to laboratory data on the temperature-dependent growth of a Chagas disease arthropod vector, Rhodnius prolixus. Analysis of this data shows that the growth rate for the insect population under study achieves its maximum around 26◦C and the carrying capacity is maximum around 25◦C, suggesting that R. prolixus populations may thrive even in non-tropical climates.