We consider integrodifference equations (IDEs), which are of the form N_{t+1}(x) = \int K(x-y)F(N_t(y))dy, where K is a probability distribution and F is a growth function. It is already known that for monotone growth functions, solutions of the IDE will have spreading speeds and are sometimes in the form of travelling waves. We are interested in the case where F has a stable 2-point cycle, namely for the Ricker function and the logistic function [May, 1975]. It was claimed in [Kot, 1992] that the solution of this IDE alternates between two profiles, all the while moving with a certain speed. However, simulations revealed that not only do the profiles alternate, but the solution is a succession of two travelling objects with different speeds. Using the theory from [Weinberger, 1982], we can prove the existence of two speeds and establish their theoretical formulas. To explain the succession of travelling objects, we relate to the concept of dynamical stabilization [Malchow, 2002].

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: the Susceptible-Infested-Juvenile (SIJ) model. Stability of fixed points is analyzed as a function of 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. To assess the vulnerability of natural resources to climate change, we develop a thermally-driven mechanistic model to predict MPB population growth rates using a distributional model of beetle phenology in conjunction with criteria for successful tree colonization. The model uses projected daily minimum and maximum temperatures for the years 2025 to 2085 generated by three separate global climate models. Growth rates are calculated each year for an area defined by latitude range 42° N to 49° N and longitude range 108° W to 117° W on a Cartesian grid of approximately 4km resolution. Using these growth rates, we analyze how the optimal thermal window for beetle development is changing with respect to elevation as a result of climate change induced warming. We also use our combined model to evaluate if thermal regimes exist that would promote life cycle bivoltinism and discuss how yearly growth rates would change as a result. Outbreaks of MPB are largely driven by host tree stand demographics and spatial effects of beetle dispersal. We augment the SIJ model to account for the spatial effects of MPB dispersal throughout a forest landscape by coupling it with a Gaussian redistribution kernel. The new model generates a train of sustained solitary waves of infestation that move through a forest with constant speed. We convert the resulting integrodifference equation into a partial differential equation and search for travelling wave solutions. The resulting differential equation provides predictions of the shape of an outbreak wave profile and of peak infestation as functions of wave speed, which can be calculated analytically. These results culminate in the derivation of an explicit formula for predicting the severity of an outbreak based on the net reproductive rate of MPB and host searching efficiency.