There are many variants.

See Population genetics

Assumptions:

• Non-overlapping generations
• ...

Haploid Wright-Fisher model with selection

The definition can be found here:

Definition (Haploid Wright-Fisher model with selection): In a panmictic, haploid population of constant size $N$, where individuals are of type $a$ and $A$: if generation at time $t$ consists of $k$ individuals of type $a$, and $N-k$ of type $A$, then, according to the Wright-Fisher model with selection, the generation at time $t+1$ is formed by $N$ individuals, each of which has a probability to be of type $a$ given by:

$P(\text{type a}) = \frac{k(1+s)}{k(1+s)+N-k}$

and is of type $A$ otherwise. The process is called sampling with replacement, because we are, in effect, replacing each individual of the previous population by a new one, which follows a given distribution of alleles (type). $s$ is called the selection coefficient, and $1+s$ is the fitness of type $a$. If, we give a fitness $1+s'$ to type $A$, then we use

$P(\text{type a}) = \frac{k(1+s)}{k(1+s)+(N-k)(1+s')}$

And one can see how this would be generalized for more possible types in the model.

The way this probability comes about is:

• The denominator is just to normalize the probability
• In the numerator, $k$ is the number of individuals of type $a$. The factors $(1+s)$ and $(1+s')$ determine the relative average number of offspring per individual. By this I mean that $\frac{\text{average number of offspring of an type a individual}}{\text{average number of offspring of an type a individual}} = \frac{1+s}{1+s'}$. The average number of offspring of type $a$, for instance is $P(\text{type a}) N$, as it is for a Bernoulli distribution (in this case, for the number of $a$-type individuals), or a multinomial distribution, if more than two types are being considered.

If $s=0$ for all types, selection doesn't play a role, and the model describes genetic drift only.

Haploid Wright-Fisher model with selection and mutation

Also described here.

Starting from the same setup as above (for the Haploid Wright-Fisher model with selection), the definition for the model with mutation is:

Definition (Haploid Wright-Fisher model with 'selection and mutation): If there are $k$ individuals of type $a$ among parents (and $N-k$ individuals of type $A$), and we have mutation rates $\mu_1$ for $a \rightarrow A$, and $\mu_2$ for $A \rightarrow a$, then, the probability of type $a$ (also called the proportion of potential offspring, in frequentist language, used often in biology) is:

$\psi_k = \frac{k(1+s)(1-\mu_1)}{k(1+s)+N-k} + \frac{(N-k)\mu_2}{k(1+s)+N-k}$

As above, as each of the individuals in the next generation (offspring) have a type independently following this distribution. The number of type $a$ offspring follows a binomial distribution $Bin(N, \psi_k)$

Fixation

Diffusion approximation

See page 326 in here for instance

See this question

https://link.springer.com/chapter/10.1007/978-3-319-52045-2_2