## Mathematics, Department of

## Document Type

Article

## Citation

Eager, E.A. Modeling and mathematical analysis of plant models in ecology. PhD thesis, University of Nebraska, Lincoln, 2012.

## Abstract

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for *Cirsium canescens*. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to changes in kernel elements and show that these sensitivities can also vary considerably between the two models.

In Chapter 3 we study the global asymptotic stability of a general model for a plant population with an age-structured seed bank. We show how different assumptions for density-dependent seed production (contest vs. scramble competition) can change whether or not the equilibrium population is globally asymptotically stable. Finally, we consider a more difficult model that does not give rise to a positive system, complicating the global stability proof.

Finally, in Chapter 4 we develop a stochastic integral projection model for a disturbance specialist plant and its seed bank. In years without a disturbance, the population relies solely on its seed bank to persist. Disturbances and a seed's depth in the soil affect the survival and germination probability of seeds in the seed bank, which in turn also affect population dynamics. We show that increasing the frequency of disturbances increases the long-term viability of the population but the relationship between the mean depth of disturbance and the long-term viability of the population is not necessarily monotone for all parameter combinations. Specifically, an increase in the probability of disturbance increases the long-term mean of the total seed-bank population and decreases the probability of quasi-extinction. However, if the probability of disturbance is too low, a larger mean depth of disturbance can actually yield a smaller mean total seed-bank population and a larger quasi-extinction probability, a relationship that switches as the probability of disturbance increases.

Advisers: Richard Rebarber and Brigitte Tenhumberg

#### Included in

Analysis Commons, Biology Commons, Dynamic Systems Commons, Non-linear Dynamics Commons, Numerical Analysis and Computation Commons, Other Applied Mathematics Commons, Population Biology Commons, Probability Commons, Science and Mathematics Education Commons

## Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professors Richard Rebarber and Brigitte Tenhumberg. Lincoln, Nebraska: August, 2012

Copyright (c) 2012 Eric Alan Eager