Stochastic Genetics

Bachelor thesis (2021)

Authors

D.F.K. Brouwers Electrical Engineering, Mathematics and Computer Science

Contributors

R.C. Kraaij Applied Probability - EEMCS (mentor)
C. Kraaikamp Applied Probability - EEMCS (graduation committee member)
R. van der Toorn Mathematical Physics - EEMCS (graduation committee member)
Faculty
Electrical Engineering, Mathematics and Computer Science, Electrical Engineering, Mathematics and Computer Science
Copyright: © 2021 David Brouwers
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Published Date

26-07-2021

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

In this bachelor thesis we use a stochastic model to aspire to explain biodiversity patterns in different ecosystems with selection advantage. The stochastic model we use is an extension of the mean­-field voter model where we include a selection factor. In the model individuals with two different types of alleles in two different ecosystems are considered. The model is a stochastic Markov process that describes interactions of individuals with each­other over time. This means that the ratio of individuals with certain alleles stochastically drifts over time. The main goal of this bachelor thesis is investigate whether is it possible that individuals with two different types of alleles can coexist (a stable equi­librium) in two populations. We do this by taking the limit of this Markov process such that we can show convergence to ordinary differential equations. By studying these differential equations we ob­tain results: vector fields with equilibrium points. We conclude that, under certain conditions and with a selection advantage, coexistence of individuals with two different alleles in two different ecosystems is possible (see Section 4.4) in the form of a stable equilibrium. Furthermore we claim that the typical time to absorption, reaching an absorbing state where all the individuals have the same allele, of the two­-dimensional mean­-field voter model with selection scales exponentially with the system size.