The mathematical modeling and analysis of nonlocal ecological invasions and savanna population dynamics

Abstract

The main focus of this dissertation is the development and analysis of two new mathematical models that individually address major open problems in ecology. The first challenge is to characterize and model the processes that result in a savanna ecosystem as a stable state between grassland and forest, and the second involves modeling the non-local spread of a biological invader over heterogeneous terrain while incorporating the influence of a mass transportation network on the system. Both models utilize and compare work done in other, often more opaque, modeling paradigms to better develop transparent and application-ready solutions which can be easily adapted and inform ecological work done in the field. Savanna is defined by the coexistence of trees and grass in seasonally dry areas of the tropics and sub-tropics, but there is no consensus as to why savanna occurs as a stable state between tropical grassland and forest. To understand the dynamics behind the tree-grass relationship, we begin by reviewing and analyzing approaches in currently available savanna models. Next, we develop a mathematical model for savanna water resource dynamics based on FLAMES, an Australian process-based software model created to capture the effects of seasonal rainfall and fire disturbance on savanna tree stands. As a mathematically explicit dynamical system represented by coupled differential equations, the new model immediately has the advantage of being concise and transparent compared to previous models, yet still robust in its ability to account for different climate and soil characteristics. Through analytical analysis of the model, we show a clear connection between climate and stand structure, with particular emphasis on the length and severity of the dry season. As a result, we can numerically quantify the parameter space of year-by-year stochastic variability in stand structure based on rainfall and fire probabilities. This results in a characterization of savanna existence in the absence of extreme fire suppression based on the availability of water resources in the soil due to climate and ground water retention. One example of the model's success is its ability to predict a savanna environment for Darwin, Australia and a forest environment for Sydney, even though Sydney receives less annual rainfall than Darwin. The majority of this dissertation focuses on modeling the spread of a biological invader in heterogeneous domains, where invasion often takes place non-locally, through nearby human transportation networks. Since early detection and ecological forecasting of invasive species is urgently needed for rapid response, accurately modeling invasions remains a high priority for resource managers. To achieve this goal, we begin by revisiting a particular class of deterministic contact models obtained from a stochastic birth process for invasive organisms. We then derive a deterministic integro-differential equation of a more general contact model and show that the quantity of interest may be interpreted not as population size, but rather as the probability of species occurrence. We then proceed to show how landscape heterogeneity can be included in the model by utilizing the concept of statistical habitat suitability models which condense diverse ecological data into a single statistic. Next, we develop a model for vector-based epidemic transport on a network as represented by a strongly connected, directed graph, and analytically compute the exact optimal control for suppression of the infected graph vectors. Since this model does not require any special assumptions about the underlying spatiotemporal epidemic spread process, it should prove suitable in a variety of application contexts where network based disease vector dynamics need to be understood and properly controlled. We then discuss other methods of control for the special case of the integro-differential model developed previously and explore numerical results of applying this control. Finally, we validate model results for the Bromus tectorum invasion of Rocky Mountain National Park using data collected by ecologists over the past two decades, and illustrate the effect of various controls on this data. A final chapter concerns a problem of cognitive population dynamics, namely vowel pronunciation in natural languages. We begin by developing a structured population approach to modeling changes in vowel systems, taking into account learning patterns and effects such as social trends. Our model treats vowel pronunciation as a continuous variable in vowel space and allows for continuous dependence of vowel pronunciation on time and age of the speaker. The theory of mixtures with continuous diversity provides a framework for the model, which extends the McKendrick-von Foerster equation to populations with age and phonetic structures. Numerical integrations of the model reveal how shifts in vowel pronunciation may occur in jumps or continuously given perturbations such as the influx of an immigrant population.