Estimation and evaluation methods for population growth models with spatial diffusion: Dynamics of mountain pine beetle

Abstract

This study approaches a problem of estimation and evaluation of population models from the viewpoint that a forest ecosystem manager needs to know how the pest (mountain pine beetle) and related tree (pine) mortality, will spread to surrounding stands over time. A model is proposed, based on a diffusion equation (second order, partial differential equation) to describe in spatial terms the functional relationships between pest and host. Pest density, food availability for the pest, and the diffusivity of the pest in space are the variables used in the model. To estimate parameters, the differential equation model was approximated as a difference equation model, and a first-order autoregressive error structure with contemporaneous correlation between cross sections was assumed. A generalized least squares ( gls) estimation was applied and the estimates were used to predict mountain pine beetle ( mpb) outbreaks. Three types of maps ( mpb density surfaces) were constructed for a certain area: one showing the actual pest condition for 1984 ( am), one showing the output of a predictive model that uses food availability, lagged density, a covariate and diffusivity as the predictors (diffusion-advection, pm1), and one using only the diffusivity and the lagged density as the predictor variables (simple diffusion, px2). The results indicate that 3-year history of mpb population dynamics is adequate to provide an 1-year projection of mpb densities. The evaluation of the models was performed by translating and rotating the predictive maps ( pm1, pm2) over the observed map ( am) and calculating geometric distortions. A variogram analysis and several aggregation indices, as well as contagious distributions, were used to test the agreement between the spatial patterns of the maps. These comparisons suggest that the prediction generated by the simple two-dimensional diffusion model ( pm2) is the best. Including terms for resource availability (food in form of phloem of living lodgepole pine) does not improve the precision of the projection. This does not imply that food is unimportant to mpb ecology, but rather that it is less important than prior population density for short-term projections.