A Model for Mountain Pine Beetle Outbreaks in an Age-Structured Forest: Predicting Severity and Outbreak-Recovery Cycle Period

Abstract

The mountain pine beetle (MPB, Dendroctonus ponderosae), a tree-killing bark beetle, has historically been part of the normal disturbance regime in lodgepole pine (Pinus contorta) forests. In recent years, warm winters and summers have allowed MPB populations to achieve synchronous emergence and successful attacks, resulting in widespread population outbreaks and resultant tree mortality across western North America. We develop an age-structured forest demographic model that incorporates temperature-dependent MPB infestations. Stability of fixed points is analyzed as a function of (thermally controlled) MPB population growth rates and indicates the existence of periodic outbreaks that intensify as growth rates increase. We devise analytical methods to predict outbreak severity and duration as well as outbreak return time. After incorporating a spatial aspect and controlling initial stand demographic variation, the model predicts cycle periods that fall within observed outbreak return time ranges. To assess future MPB impact on forests, we use climate model projected temperatures with our model-based approximation methods to predict potential severity of future outbreaks that reflect the effects of changing climate.

Analytic Approximation of Invasion Wave Amplitude Predicts Severity of Insect Outbreaks

“…We analyze the age-structured host tree (lodgepole pine, Pinus contorta) MPB outbreak model developed by Duncan et al [15], j 1,n+1 (x) = (1 − s)J n (x) + I n−2 (x), (1) j k+1,n+1 (x) = sj k,n (x), k = 1, . .…”

“…A detailed development of this model, including fixed point stability analysis and outbreak severity approximation, can be found in Duncan et al [15]. The salient results are that the infestation-free state loses stability for R ≥ 1; an endemic fixed point comes into existence and is stable for values of R larger than 1 but less than some critical threshold, above which the system settles into large oscillations corresponding to periodic outbreaks.…”

“…The salient results are that the infestation-free state loses stability for R ≥ 1; an endemic fixed point comes into existence and is stable for values of R larger than 1 but less than some critical threshold, above which the system settles into large oscillations corresponding to periodic outbreaks. We note that parameter values must satisfy T > 2 ln R β (derived from the outbreak severity prediction in Duncan et al [15]) to preclude the model from predicting infested tree densities that exceed the carrying capacity of the forest. For realistic parameters range estimates this condition is always true.…”

“…Estimates for K and d were determined using reference values consistent with field observation. See Duncan et al [15] for details.…”

“…Duncan et al [15] provide a detailed temporal analysis of local outbreaks generated by a similar host demographic model sans explicit insect dispersal. In fact, were there no dispersal component in our model, i.e., σ 2 = 0, then the PDE in (27) [15] as well as with infestation data from the SNRA outbreak. The traveling wave approximation is the numerical solution of (28).…”