INVESTIGATION OF STOCHASTIC REACTION-DIFFUSION PARTIAL DIFFERENTIAL EQUATIONS AND OF CONSISTENT STOCHASTIC DIFFERENTIAL EQUATION MODELS FOR ONE-LOCUS AND TWO-LOCI POPULATION GENETICS

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Title: INVESTIGATION OF STOCHASTIC REACTION-DIFFUSION PARTIAL DIFFERENTIAL EQUATIONS AND OF CONSISTENT STOCHASTIC DIFFERENTIAL EQUATION MODELS FOR ONE-LOCUS AND TWO-LOCI POPULATION GENETICS
Author: Dogan, Elife
Abstract: There are two main parts in this work separated into chapters 2 and 3 and chapters 4 and 5 , respectively . In the first part , stochastic partial differential equations are derived for the reaction -diffusion process in one , two and three dimensions . Specifically , stochastic partial differential equations are derived for the random dynamics of particles that are reacting and diffusing in a medium . In the derivation , a discrete stochastic reaction -diffusion equation is first constructed from basic principles , i .e . , from the changes that occur in a small time interval . As the time interval goes to zero , the discrete stochastic model leads to a system of Ito stochastic differential equations . As the spatial intervals approach zero , a stochastic partial differential equation is derived for the reaction -diffusion process . The stochastic reaction -diffusion equation can be solved computationally using numerical methods for systems of Ito stochastic differential equations . In the second part , stochastic ordinary and partial differential equations are derived for randomly varying populations of haploid and diploid individuals under genetic changes with one , two and a large number of alleles . Specifically , stochastic differential equations are derived for the genotype population distributions . In the derivation , a discrete stochastic population genetics equation is first constructed from basic principles . A similar procedure is applied to find stochastic ordinary differential equations for population genetics . For a large number of alleles , a stochastic partial differential equation is derived for the population genetics . Stochastic differential equation models for interacting populations undergoing genetic changes provide straightforward unifying models for understanding the dynamics of the population genetics problems . Comparisons between numerical solutions of the stochastic differential equations and independently formulated Monte Carlo calculations for the reaction diffusion and population genetics problems support the accuracy of the derivations .
URI: http : / /hdl .handle .net /2346 /ETD -TTU -2011 -08 -1628
Date: 2011-08

Citation

INVESTIGATION OF STOCHASTIC REACTION-DIFFUSION PARTIAL DIFFERENTIAL EQUATIONS AND OF CONSISTENT STOCHASTIC DIFFERENTIAL EQUATION MODELS FOR ONE-LOCUS AND TWO-LOCI POPULATION GENETICS. Texas Tech University. Available electronically from http : / /hdl .handle .net /2346 /ETD -TTU -2011 -08 -1628 .

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