Mathematical models for the dynamics of competitive systems, with applications to religious shift and ferromagnetism

The dynamics of two-group competition may be derived by applying a simple mass-action law to the population fraction belonging to each group. This results in a differential equation describing how group size changes with time, where system behavior is determined by the probability of switching between groups. Using a deterministic model for this process instead of agent-based simulations allows for analysis using the tools of dynamical systems theory. Additionally, non-trivial interactions between individuals may be introduced using a general coupling function, the form of which can be specified to approximate global, local, or any intermediate-range coupling. This allows for direct comparison between model predictions, stochastic simulations, and real-world data.;This approach to modeling social systems may be applied to the study of religious shift, through comparison to a new international data set tracking religious non-affiliation. By choosing an appropriate form for the probability of switching and finding the best-fit parameters, it is possible to determine key characteristics of this data set and make predictions for future trends. When non-global social interactions are included, asymptotic and numerical techniques show that the system remains robust to polarizing perturbations. Therefore, using the original equation with only global interactions is a well-justified simplification, even though social group affiliation might otherwise be thought of as a complex, non-global process. This helps explain why real-world data matches so well with the simple model.;When ferromagnetism is considered as an example of two-group competition, it may be studied using similar techniques. The Ising model has served for almost a century as a basis for understanding ferromagnetism, and has become a paradigmatic model for phase transitions in statistical mechanics. A continuum model is derived here by combining the Ising energy arguments with techniques previously applied to sociophysics. This formulation results in an integro-differential equation that has several advantages over the traditional version, most significantly that it allows for asymptotic analysis of phase transitions, material properties, and the dynamics of the formation of magnetic domains.