Curriculum Vitae

Publications

Graduate Research

The motivation for my graduate research is the problem of invasive species. The term usage is somewhat disputed, but is understood here to refer to recently introduced non-indigenous species. Problems posed by this issue can be devastation of crops, drastic changes in landscape, and the extinction of native species due to habitat loss.

The introduction of grasses from Africa to Southern Arizona for the purpose of grazing cattle is the test case here. Native grasses tend to grow in smaller patches leaving plenty of open area, but this can not sustain ranching in the desired manner leading to the introduction of foreign species such as Eragrostis Lehmanniana and Pennisetum ciliare. One result is that the ground cover is much more dense leaving behind more fuel for fires in the dry season[Anable92]. Habitats thus created are preferred less by native species of insects, birds, and mammals[Flanders06, Anable92, Bock85]. These species have spread far beyond what was previously thought to be their limits and is predicted to expand its available habitat even further according to climate models[Schussman06,Geiger03]. My dissertation is centered around the creation of models for the purpose of understanding the spatial spread of an introduced plant species and the development of a general theory for a larger class of models.

The traditional approach to this problem is to use a multi-type contact process and simulate it to equilibrium and then collect data on the equilibrium state. The first problem with this approach is that it is not easy to say exactly when equilibrium is reached. Coupling from the past using the Propp and Wilson algorithm solves this problem[Propp96]. Although if the process is not monotone, another problem arises. This can take a massive number of simulations forcing each possible initial configuration to be tested. A major goal of my dissertation was to create a modified version of the multi-type contact process which remains monotone in biologically meaningful situations. Knowing exactly when equilibrium is reached allows the study of the transient behavior, the path to invasion, and has resulted in a general theory of monotonicity for a larger class of models.

Contact Process

The contact process is a continuous-time Markov process. The state space for the d-dimensional contact process is

This is also known as configuration space, and each state is a configuration of 1's and 0's on the given integer lattice. It is a pure jump process which means that each site, x, changes its value in discrete jumps from 0 to 1 or vice versa after a random waiting time with exponential distribution. These rates are given by a function known as the flip rate:

For my purposes I am interested in the 2D contact process. Below are some movies from simulations that I have done. Click here for instructions on the movie controls.

• Spread from the center: Center Spreading (λ = 0.8) (3.8 MB)
• Spread from the edge: Edge Spreading (λ = 0.6) (5.5 MB)
• Starting form fully occupied and dying out: Complete Die Out (λ = 0.3) (470 KB)

Monotonicity

This is a monotone process, which means that there exists a coupling which preserves the natural partial ordering of states for all time. Due to the property of monotonicity, the Propp-Wilson algorithm can be used to sample exactly from the stationary distribution. This algorithm uses coupling from the past and is computationally intense.

Although for the purposes of studying plant invasions, it is desirable to introduce a second particle type, 2. Now the state space will be

The problem with this is that it is no longer monotone, and so there is no simple way to sample from the stationary distribution. Thus I have developed a new model which includes 2 species but remains monotone.

New 2-Particle Monotone Model

The new model changes the state space {0, 1, 2}^Λ to {1−, 1+, 2−, 2+}^Λ where the minus states are considered virtual and the plus states are considered real. The minus states communicate with each other and the corresponding plus state. The plus states do not communicate with each other.

Here is a movie from the Propp-Wilson Algorithm for the new model: Path to Stationarity. (560 KB)

The goal of this research is to examine the path to stationarity and use this to learn how different aspects of biological situations affect an invasion. Fecundity, mortality, and spatial/temporal heterogeneity are key aspects being studied among others.