Diffusion processes in population genetics

Diffusion Processes in Genetics

The theory of evolution provides examples of stochastic processes which have not yet been treated systematically. We find here many open problems whose solution promises new insights into the general theory. There exists a huge literature on the mathematical theory of evolution and statistical genetics, but existing methods and results are due almost entirely to R. A. Fisher and Sewall Wright.'...

Ancestral processes in population genetics-the coalescent.

A special stochastic process, called the coalescent, is of fundamental interest in population genetics. For a large class of population models this process is the appropriate tool to analyse the ancestral structure of a sample of n individuals or genes, if the total number of individuals in the population is sufficiently large. A corresponding convergence theorem was first proved by Kingman in ...

Convergence of one-dimensional diffusion processes to a jump process related to population genetics.

A conjecture on the convergence of diffusion models in population genetics to a simple Markov chain model is proved. The notion of bi-generalized diffusion processes and their limit theorems are used systematically to prove the conjecture. Three limits; strong selection-weak mutation limit, moderate selection-weak mutation limit, weak selection-weak mutation limit are considered for typical dif...

Diffusion Processes and a Class of Markov Chains Related to Population Genetics

Time Reversal of Some Stationary Jump Diffusion Processes from Population Genetics

Wedescribe the processes obtained by time reversal of a class of stationary jump diffusion processes that model the dynamics of genetic variation in populations subject to repeated bottlenecks. Assuming that only one lineage survives eachbottleneck, the forwardprocess is a diffusion on [0, 1] that jumps to the boundary before diffusing back into the interior. We show that the behavior of the ti...