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Essays in Population Dynamics

Abstract

Populations comprising humans or agents following simple rules offer a varied landscape of dynamic behaviors and configurations. This dissertation presents three articles that follow an interdisciplinary approach where mathematical and statistical techniques are adapted from the pure mathematics and biology and applied to problems in the social sciences. A novel hypothesis, that American religious populations follow a lognormal distribution is presented and tested, and descriptive spatial statistics are employed to improve descriptions of this religious landscape. The next chapters offer complementary mathematical work in evolutionary dynamics and introduce mathematical models which may be employed on similar empirical problems. An example of a chaotic evolutionary dynamic is presented that is minimal in that it requires only two populations. This result highlights the link between theoretical social science models, evolutionary biology, and physics, and offers a new hypothesis consistent with observed chaotic phenomena in systems with dominance cycles. Chapter 4 considers the role local interaction structures have when a Stag Hunt coordination game is played by a population of best-responding agents. An infinite family of graphs is presented that are susceptible to invasions of the socially suboptimal strategy if the invasion begins on a vertex belonging to a proper subgraph. The projects are unified by a discussion of the philosophical and practical features of mathematical and interdisciplinary approaches to modeling in the social sciences.

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