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

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.