Diversity in Evolving Systems : Scaling and Dynamics of Genealogical Trees

Diversity is a fundamental property of all evolving systems. This thesis examines spatial and temporal patterns of diversity. The systems I will study consist of a population of individuals, each with a potentially unique state, together with a dynamics consisting of copying or reproduction of individual states with small modifications to them (innovations). I show that properties of diversity can be understood by modelling the evolving genealogical tree of the population. This formulation is general enough that it captures interesting features of a range of natural and artificial systems, though I will pay particular attention to genetic diversity in biological populations, and discuss the implications of the results to conservation. I show that diversity is unevenly distributed in populations, and a disproportionate fraction is found in small sub-populations. The evolution of diversity is a dynamic process, and I show that large fluctuations in diversity can result purely from the internal dynamics of the population, and not necessarily from external causes. I also show how diversity is affected by the structure of the population (spatial or well-mixed), and determine the scaling of diversity with habitat area in spatial systems. Predictions from the model agree with existing experimental genetic data on global populations of bacteria. I then apply the method of modelling the genealogical tree of a population to further questions in evolution. Using a generic model of a pathogen evolving to coexist with a population of hosts, I show that the evolutionary dynamics of the system can be better understood by considering the dynamics of strains (groups of individuals descended from a common ancestor) rather than individuals. A fundamental question in the study of evolution is how selection can operate above the level of the individual, and these results suggests a more general mechanism for such selection.