Analytic approximation of invasion wave amplitude predicts severity of insect outbreaks

Abstract

Outbreaks of phytophagous forest insects are largely driven by host demographics and spatial effects of dispersal. We develop a structured integrodifference equation (IDE) outbreak model that tracks the demographics of sedentary hosts under insect infestation pressure. The model is appropriate for a spectrum of pests attacking the later age classes of long-lived hosts, including mountain pine beetle (MPB), spruce budworm, and spruce beetle, which, among them are responsible for more forest damage than fire. The model generates a train of periodic waves of infestation. We approximate the IDE with a partial differential equation and search for traveling wave solutions. The resulting ordinary differential equation predicts the shape of an outbreak wave profile and peak infestation as functions of wavefront speed, which can be calculated analytically. This culminates in the derivation of an explicit approximation of invasion wave amplitude based on net reproductive rate of the infesting insect and its host searching efficiency. Results are compared with observations taken during a recent MPB outbreak in the northern US Rocky Mountains.