Stochastic Genetics
D.F.K. Brouwers (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Richard C. KRAAIJ – Mentor (TU Delft - Applied Probability)
Cor Kraaikamp – Graduation committee member (TU Delft - Applied Probability)
R van der Toorn – Graduation committee member (TU Delft - Mathematical Physics)
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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.