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Abstract
This work comprises five research projects in the mathematics of evolution and biodiversity, divided into three parts.
The first part investigates how the structure of an evolutionary process influences the evolution of social traits. Modeling evolution as an abstract Markov process, we prove that the relevant properties of population structure can be summarized in two parameters: effective population size and effective degree of assortment between types. A formula is derived for the long-term evolution of game-theoretic traits in a structured population. We then investigate the cultural evolution of human cooperation over social networks. Previous work shows that spatial clustering on networks promotes the evolution of cooperation, but these studies did not incorporate the exploration and innovation present in human behavioral dynamics. Through pair analysis and simulation, we show that non-imitative strategy exploration dilutes the clustering effect of network structure and inhibits cooperation.
The second part investigates mutation rate evolution, focusing on conditions favoring high mutation rates. We explore this first using evolutionary game theory, in which the fitness of a phenotype depends on the frequencies of other phenotypes. When population dynamics are stable, mutation rates evolve to zero, but in cases of cyclical population dynamics generated by nontransitive competition, nonzero evolutionarily stable mutation rates may evolve. We turn next to mutation rate evolution in Escherichia Coli, natural and experimental populations of which often contain alleles with significantly elevated mutation rates. We investigate the production of mutator strains as an evolutionary strategy for exploring a fitness landscape. We show this strategy is successful at both finding beneficial mutations in novel or changing environments, and protecting against mutational load in stable environments.
The third part presents a new index for quantifying biodiversity, which combines species abundances and phylogenetic data in a formula inspired by physics and information theory. This new index, phylogenetic entropy, avoids a pitfall common to other indices combining such data, which may, counterintuitively, increase with elimination of a species. Applied to communities of phyllostomid bats in the Selva Lacedona, our measure gives diversity valuations that are better aligned with conservation priorities, compared to other established indices.
