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    A Stochastic Spatial Model for Invasive Plants and A General Theory of Monotonicity for Interaction Map Particle Systems

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    Author
    Stover, Joseph Patrick
    Issue Date
    2008
    Keywords
    Contact Process
    Monotone
    Attractive
    Coupling
    Advisor
    Watkins, Joseph C.
    Committee Chair
    Watkins, Joseph C.
    
    Metadata
    Show full item record
    Publisher
    The University of Arizona.
    Rights
    Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
    Abstract
    Awareness of biological invasions is becoming widespread and several mathematical tools have been used to study this problem. Interacting particle systems, specifically the contact process, have been used to study systems with invasion/infection type dynamics. The Propp-Wilson algorithm is a method for exact sampling from the stationary distribution of an ergodic monotone Markov chain using a method called coupling from the past. The contact process is monotone so we can sample exactly from the stationary distribution of a modified finite grid version using the Propp-Wilson algorithm. In order to study an invasion, we would like to include at least 2 species; however, monotonicity is not well defined for contact processes with more than 2 particle types. Here we develop a general theory of monotonicity for interaction map particle systems, which are interacting particle systems with contact process type dynamics. This allows us to create monotone models with any number of particles and to use the Propp-Wilson algorithm for not only sampling from the stationary distribution, but analyzing the path of invasion leading to equilibrium. Virtual particle invasion models that fall into this new theoretical framework, which we develop here, present a wide range of biological dynamics. Computer simulation of the stochastic system and mean field analysis are two powerful tools that we use for analyzing these types of models. Statistics gathered along the path to invasion help us understand the spatial dynamics of this ecological process and what the stationary behavior looks like. This allows us to understand when the invasion is successful or if coexistence occurs and how these depend on the transition rates and interactions within the process.
    Type
    text
    Electronic Dissertation
    Degree Name
    PhD
    Degree Level
    doctoral
    Degree Program
    Applied Mathematics
    Graduate College
    Degree Grantor
    University of Arizona
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