Mathematical elements of attack risk analysis for mountain pine beetles

Three different mathematical approaches are combined to develop a spatial framework in which risk of mountain pine beetle (MPB) attack on individual hosts may be assessed. A density-based partial differential equation model describes the dispersal and focusing behavior of MPB. A local projection onto a system of ordinary differential equations predicts the consequences of the density equations at individual hosts. The bifurcation diagram of these equations provides a natural division into categories of risk for each host. A stem-competition model links host vigor to stand age and demographics. Coupled together, these models illuminate spatial risk structures which may also shed light on the role of climatic variables in population outbreaks. Preliminary results suggest that stand microclimate has much greater influence on risk of attack than host vigor and stand age.