Date of Award

2010

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

James Powell

Second Advisor

James Haefner

Abstract

In this paper, we begin by extending existing deterministic and individual-based ecological models for sexual dimorphism and adaptive speciation into density-based mathematical models describing the density or number of individuals with various trait values, or phenotypes. These density-based models describe the dynamics of a population of males and females using both clonal and sexual reproduction. Each generation, the populations are subject to mating, mutation, and ecological dynamics including infraspecific competition and carrying capacity of the environment. By avoiding individual-based models, we are able to avoid simulations and instead achieve repeatable results.

Implementing these models numerically, we are able to show how clonal reproduction readily results in sexual dimorphism or evolutionary branching. We compare the numerical results to a stability analysis for the case of a monomorphic population and showcase the need for a more complex stability analysis of a non-monomorphic population. We then evaluate the effect of the rate of mutation and, for a population mating sexually, show how the addition of shared material between genders eliminates sexual dimorphism and evolutionary branching whenever the ratio of shared material gets too large.

Adding the restriction that populations at the beginning of each generation follow a Gaussian distribution, we are able to utilize a method called Effective Particle Theory to analytically determine six difference equations relating the population, mean and variance of each gender, to the same parameters in the next generation. Comparing numerical results to the analytic Effective Particle Theory results, we show how the Effective Particle Theory can provide a good approximation of the numeric results in cases where the populations remain unimodal and how the method might be extended to describe bimodal or multimodal populations.

Included in

Mathematics Commons

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