Moran model
The Moran model describes evolution in a finite population through repeated birth–death replacement events.
At each event, one individual is chosen to reproduce and one is chosen to die, so population size remains constant.
Neutral transition probabilities
If \(i\) of \(N\) individuals carry allele \(A\), then
\[P(i\to i+1)=\frac{i}{N}\frac{N-i}{N},\]\[P(i\to i-1)=\frac{N-i}{N}\frac{i}{N}.\]The state may also remain unchanged if reproduction and death involve the same type.
Comparison with Wright–Fisher
Wright–Fisher replaces an entire generation at once. Moran dynamics change the population one replacement event at a time.
Key idea. The Moran model provides a simple event-by-event representation of genetic drift in a population of fixed size.