## Dissertations, Theses, and Student Research Papers in Mathematics

#### Date of this Version

Summer 2009

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Ful filment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics. Under the Supervision of Professors Richard Rebarber and Brigitte Tenhumberg.

#### Abstract

Asymptotic and transient dynamics are both important when considering the future population trajectory of a species. Asymptotic dynamics are often used to determine whether the long-term trend results in a stable, declining or increasing population and even provide possible directions for management actions. Transient dynamics are important for estimating invasion speed of non-indigenous species, population establishment after releasing biocontrol agents, or population management after a disturbance like fire. We briefly describe here the results in this thesis.

(1) We consider asymptotic dynamics using discrete time linear population models of the form n(t + 1) = An(t) where A is a population projection matrix or integral projection operator, and n(t) represents a structured population at time t. Within the model are the underlying parameters. Some of these parameters are of more interest to us: either ones which have a large confidence interval or are the easiest to manage from a conservation managers point of view. Using these parameters of interest, we next divide the parameter space into two parts: on one side the population grows and on the other side the population declines. We call this dividing hypersurface the growth-decline boundary and use this as a tool to show how senescence affects the prediction of a matrix model for the Serengeti cheetah (Acinonyx jubatus). We next analyze an integral projection model for thistles (Onopordum illyricum) again using this growth-decline boundary.

(2) We consider how temperature effects the transient (short term) dynamics for an ectothermal species such at the pea aphid (Acyrthosiphon pisum). We define "transient amplification" and develop two different models to explore the effect of temperature on this concept. We estimate model parameters (survivorship and fecundity) at two different temperatures, and then scrutinize both model predictions by comparing observed and predicted transient population growth rates and the projection of population size over 20 days. Both models predict that temperature affects the short term transient amplification. The degree day model predicts no effect of temperature on the asymptotic transient amplification while the results of the ordinal day model were inconclusive.

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