ANALYSIS OF MONOTONE AND NON-MONOTONE TRAVELING WAVES IN A SYSTEM FOR SOCIAL OUTBURSTS

Rioting events in the last several years in the United States, such as the Ferguson riots of 2014 and the Baltimore riots of 2015, captured the attention of the entire nation and have increased scrutiny of racial and social tension. The strength and duration of these riots leads to a question: is there a mathematical model that can reproduce the spread and intensity of rioting behavior observed over time and space in events like these? The goal of this work is to prove the existence and stability of traveling wave solutions to a model for the spread of rioting and social outbursts given by a reaction-diffusion system which captures the relationship between two variables: intensity of rioting behavior and social tension. This model was first introduced by Berestycki, Nadal, and Rodríguez in 2015. To prove the existence and stability of the traveling wave solutions, we use existence and stability theory for monotone systems. In the case of parameter values that yield a non-monotone system, we establish the stability of traveling wave solutions by proving that the spectrum of the linear operator is not located in the closed, deleted, right half plane. We analyze the spectrum of the linear operator by finding the essential spectrum, placing a bound on the location of the point spectrum, and numerically searching for point spectra within this bounded region using the Evans function. In addition to proofs for the existence and stability of these traveling wave solutions, we provide a thorough exploration of different parameter regimes for the system, including an analysis of the stability of stationary points for the spatially homogeneous system, numerical approximations of traveling wave solutions and their wave speeds, and an analysis of the asymptotic behavior of traveling wave solutions for particular parameter sets.Doctor of Philosoph