Homophily, payoff distributions, and truncation selection in replicator dynamics
This dissertation explores the field of replicator dynamics by examining extensions to and relaxations of the classical replicator equation and complimentary agent-based models. We extend the replicator equation by the incorporation of homophilic imitation, a form of tag-based selection. We show that though the equilibria are not affected by this modification, the population's diversity may increase or decrease depending on two invasion scenarios we detail, and there is significant impact on the rates of convergence to equilibria. Two important assumptions of the replicator equation that we relaxed are: mean payoffs, where all replicators earn the mean payoff of the underlying game; and proportional selection, where the probabilities for survival and reproduction are proportional to the difference between the fitness of a replicator and the mean fitness of the population. Our models thus comprise payoff distributions and two types of truncation selection: independent, where replicator above a threshold, φ, survive; and dependent, where the top τ of replicators survive. The reproduction rates are equal for all survivors. We show that the classical replicator equation is a special case of our independent truncation equation. Further, for any boundary fixed point, we may choose a φ such that that point is stable (or unstable). We observed complex and transient dynamics in both truncation methods. We applied this framework to evolutionary graphs that included diffusion, and show where cooperation is facilitated by these models in comparison to spatial and non-spatial proportional selection. Alfred Russel Wallace reasoned that the relatively unfit could coexist with the fit, and it has been argued that this would result in a genotypically diverse population resistant to extinction. This is because natural selection, rather than Spencer's
survival of the fittest,'' may be better encapsulated by the phrases: survival of the fit,'' or ``non-survival of the non-fit.'' We argue that truncation selection, here explored, can model this phenomenon, and thus is an important addition to the theoretical biology literature.