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200931 https://hdl.handle.net/2286/R.2.N.200931 http://rightsstatements.org/vocab/InC/1.0/ All Rights Reserved 2025 123 pages Doctoral Dissertation Academic theses en Mercer, Max Andrew Lanchier, Nicolas Crook, Sharon Espanol, Malena Fishel, Susanna Gonzales Casanova, Adrian Motsch, Sebastien Arizona State University Partial requirement for: Ph.D., Arizona State University, 2025 Field of study: Applied Mathematics This dissertation examines three interacting particle systems, which are continuous-time Markov jump processes with spatial configurations as the state space, to express the importance of using spatially-explicit methods when modeling dynamical systems. The first model explored is the spatial model of allelopathy, introduced by Rick Durrett and Simon Levin. Qualitative results are proven about the long-term behavior of the process for different parameter regions, and monotonicity is used to show phase transitions along three parameters rigorously: the birth rates of the allelotoxin-susceptible and allelotoxin-resistant species, and a new parameter, reflecting the strength of the inhibitory effects. By incorporating spatial interactions, the outcomes previously influenced by the initial densityin a bistable region are instead determined by the parameter values. The second spatial model tracks the spread of disease with an intermediate asymptomatic state, with different rates of infectiousness for the asymptomatic and symptomatic populations. Analyzing the infinite-dimensional mean-field model reveals that an epidemic occurs when either rate is sufficiently large regardless of the rate of symptomatic expression; however, in the spatial model, when the rates of asymptomatic spread and symptomatic expression are sufficiently small, there will not be an epidemic, even when the symptomatic individuals are highly infectious. The final model is a spatial population model, introduced to study the benefit (or detriment) of community. The birth rate either increases or decreases with the number of neighbors, depending on the payoff. There is a phase transition in the directions of both the payoff and the natural birth rate. Contrary to the behavior of the mean-field heuristic, the population can survive by building communities, even when the initial density is zero. Mathematics Biology Epidemiology Contact Process Differential Equations Dynamical Systems Interacting Particle Systems Math Model Spatial Model Interacting Particle Systems in Biology