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

Persistent Link:
http://hdl.handle.net/10150/194861
Title:
A Stochastic Spatial Model for Invasive Plants and A General Theory of Monotonicity for Interaction Map Particle Systems
Author:
Stover, Joseph Patrick
Issue Date:
2008
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
Keywords:
Contact Process; Monotone; Attractive; Coupling
Degree Name:
PhD
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Watkins, Joseph C.
Committee Chair:
Watkins, Joseph C.

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleA Stochastic Spatial Model for Invasive Plants and A General Theory of Monotonicity for Interaction Map Particle Systemsen_US
dc.creatorStover, Joseph Patricken_US
dc.contributor.authorStover, Joseph Patricken_US
dc.date.issued2008en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © 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.en_US
dc.description.abstractAwareness 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.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectContact Processen_US
dc.subjectMonotoneen_US
dc.subjectAttractiveen_US
dc.subjectCouplingen_US
thesis.degree.namePhDen_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorWatkins, Joseph C.en_US
dc.contributor.chairWatkins, Joseph C.en_US
dc.contributor.committeememberKennedy, Thomas G.en_US
dc.contributor.committeememberCushing, Jim M.en_US
dc.identifier.proquest2828en_US
dc.identifier.oclc659749890en_US
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