Mathematical models for the frequency-dependent transmission of
cultural traits

Abstract

Cultural evolutionary theory is concerned with the social transmission of behaviours, beliefs or ideas that constitute culture. In humans, transmission of culture may be from one generation to the next or between individuals of the same generation. This thesis contains three models for the transmission of cultural traits, subject to frequency-dependent social learning. All models are formulated as a system of differential equations that cannot be solved analytically. By finding the equilibria of the systems and analysing their stability, the long-term behaviour of the systems may be determined.

A mathematical model for the spread of drinking behaviour is presented, with a focus on total recovery. The equilibria of the system are found and a local stability analysis is performed. The system is found to have a parameter-dependent threshold at which the two equilibria switch stability. This indicates a change in the long-term system behaviour. Consequently, whether drinking behaviour dies out or becomes endemic may be predicted from the values of the model parameters. The rate at which individuals take up drinking behaviour is found to have the greatest effect on whether it becomes endemic.

A model for both the linear and nonlinear frequency-dependent transmission of a cultural trait, with potential applications to binge drinking behaviour, is then investigated. The system equilibria cannot be found explicitly in terms of the model parameters. However, by considering different cases corresponding to regions of parameter space, qualitative differences in the long-term behaviour of the system are determined. By comparing the linear and nonlinear frequency-dependent models, the effect of conformity is determined for different regions of parameter space.

Finally, a reaction-diffusion model for two competing languages, u and v, with a focus on language coexistence is presented. Language u is assumed to confer a status advantage to its speakers, thus switching languages is one-directional from v to u. Four constant system equilibria are found and global instability and stability thresholds are found for each solution. The coexistence of languages u and v is found to be globally stable, subject to certain parameter constraints and a sufficiently small initial population of speakers.