Kot, MarkGilbertson, Nora2023-01-212023-01-212023-01-212022Gilbertson_washington_0250E_25014.pdfhttp://hdl.handle.net/1773/49594Thesis (Ph.D.)--University of Washington, 2022As humans rapidly alter the environment around them, impacts such as habitat disturbances and fragmentation, range shifts, and invasions are increasingly common. In the face of these events, knowledge of population dynamics and how these dynamics may respond to changing environmental factors is of particular relevance. Mathematical models can help analyze such population behavior, and their predictions can aid decision-making regarding species conservation, habitat design, biological control, and other matters related to ecosystem management. In this dissertation, I build and analyze several discrete-time population models, including developing a method for analyzing the spatial models known as integrodifference equations (IDEs). First, I describe the ecological underpinnings of the models and the mathematical components of both nonspatial and spatial discrete-time models. Next, I examine a reduced successional community. Succession may become more common with increased habitat disturbances and with climate change opening new ecological niches for species. I present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations, where the climax population is subject to an Allee effect. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. I fully characterize the long-term dynamics of the model, uncovering diverse sets of potential behaviors including some behaviors not previously seen in pioneer–climax models. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur at both fixed population-densities and fluctuating densities. The abrupt loss of a coexistence state, shifting the system to an exclusion state, is also possible. Then I consider IDEs, popular models for exploring a variety of problems related to population persistence and spread. I present a novel method for approximating the equilibrium population-distributions of IDEs with strictly-increasing growth, for populations on a finite habitat-patch. This method approximates the growth function of the IDE with a piecewise-constant function, and I call the resulting model a block-pulse IDE. I write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. I characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including stability analyses and an exploration of bifurcation structure. I demonstrate the use of block-pulse IDEs by using three-, five-, and ten-step block- pulse IDEs to approximate models with compensatory Beverton-Holt growth and depensatory, or Allee-effect, growth. For the IDE with Allee-effect growth, I also calculate the critical patch-size needed for persistence for several dispersal kernels, showing that this patch size depends on the choice of dispersal kernel when an Allee effect is involved. Finally, I conclude with some of the implications of these results. I discuss open problems and extensions of the current models and methods, and motivate paths for future exploration.application/pdfen-USnoneDifference equationsDynamical systemsIntegrodifference equationsMathematical ecologyPopulation dynamicsApplied mathematicsApplied mathematicsThesis