 
Summary: For Math 635, Spring 2012.
David F. Anderson
January 30, 2012
The WrightFischer Model
We consider a model from population genetics, which was first developed by Fischer, and
later extended by Wright. In this model we assume the existence of N diploid (two copies
of each gene) individuals. Thus, there are a total of 2N genes in the gene pool. We make
the following assumptions:
1. The number of individuals remains constant at N from generation to generation.
2. The genes for any individual in the (n + 1)st generation are randomly selected (with
replacement) from the pool of genes in the nth generation.
Note that the last assumption allows us to disregard the individuals, and only consider the
gene pool itself.
We suppose we have two alleles of the gene in question, which we denote by A and
a. We let Xn {0, 1, . . . , 2N} denote the number of alleles of type A in the entire gene
pool at time n. Oftentimes A is assumed to be a mutant that has entered the population.
We are interested in the probabilities associated with fixation, meaning when the system is
homogeneous in A, which occurs when Xn = 2N and A has overtaken the population, or in
a, which occurs when Xn = 0. Using the terminology developed so far in the class, we can
set
