Moran model
The Moran model is a finite-population stochastic model in which evolution occurs through repeated birth–death replacement events.
State variable
Let
\[X_n=\text{number of individuals carrying type }A\text{ after event }n.\]For a population of fixed size \(N\),
\[X_n\in\{0,1,\ldots,N\}.\]If \(X_n=i\), then the frequency of \(A\) is
\[p_n=\frac{i}{N}.\]One neutral replacement event
In the simplest neutral model, every individual is equally likely to reproduce and every individual is equally likely to die.
For the number of \(A\) individuals to increase from \(i\) to \(i+1\), an \(A\) individual must reproduce and a non-\(A\) individual must die:
\[\boxed{P(i\to i+1)=\frac{i}{N}\frac{N-i}{N}}.\]For the number of \(A\) individuals to decrease, a non-\(A\) individual must reproduce and an \(A\) individual must die:
\[\boxed{P(i\to i-1)=\frac{N-i}{N}\frac{i}{N}}.\]No-change probability
The state remains unchanged if reproduction and death involve the same type. Therefore
\[P(i\to i)=\left(\frac{i}{N}\right)^2+\left(\frac{N-i}{N}\right)^2.\]Equivalently, because the three possible outcomes must sum to one,
\[\boxed{P(i\to i)=1-2\frac{i(N-i)}{N^2}}.\]Why the neutral process has no directional bias
The upward and downward transition probabilities are equal:
\[P(i\to i+1)=P(i\to i-1).\]Hence the expected one-event change is
\[E[\Delta X\mid X=i]=(+1)P(i\to i+1)+(-1)P(i\to i-1)=0.\]The process is neutral in expectation, even though individual trajectories fluctuate.
Absorbing states
If \(i=0\), there are no \(A\) individuals left, so the process remains at \(0\). If \(i=N\), all individuals are type \(A\), so the process remains at \(N\).
Thus
\[\boxed{0\text{ and }N\text{ are absorbing states}.}\]They correspond to loss and fixation.
Neutral fixation probability
For the neutral Moran process, the probability that type \(A\) eventually fixes starting from \(i\) copies is
\[\boxed{\rho_i=\frac{i}{N}}.\]Thus a single neutral mutant has fixation probability
\[\boxed{\rho_1=\frac1N}.\]Why Moran differs from Wright–Fisher
In Wright–Fisher, an entire generation is replaced at once. In Moran, only one replacement event occurs at a time.
| Wright–Fisher | Moran |
|---|---|
| whole generation is resampled | one birth–death replacement per event |
| allele count may jump by many copies | allele count changes by at most one |
| natural generation-scale time step | natural event-scale time step |
Adding selection
Suppose type \(A\) has relative fitness \(r>0\), while the other type has fitness \(1\). If \(i\) individuals are type \(A\), the probability that the reproducing individual is type \(A\) becomes
\[\boxed{\frac{ri}{ri+(N-i)}}.\]The probability that the reproducer is the other type is
\[\frac{N-i}{ri+(N-i)}.\]If death is chosen uniformly from the population, then
\[\boxed{P(i\to i+1)=\frac{ri}{ri+N-i}\frac{N-i}{N}},\]\[\boxed{P(i\to i-1)=\frac{N-i}{ri+N-i}\frac{i}{N}}.\]When \(r>1\), upward moves are favoured; when \(r<1\), downward moves are favoured.
Selected fixation probability
For the constant-fitness Moran process above, when \(r\ne1\), the fixation probability from \(i\) copies is
\[\boxed{\rho_i=\frac{1-r^{-i}}{1-r^{-N}}}.\]Taking the limit \(r\to1\) recovers the neutral result
\[\rho_i=\frac{i}{N}.\]Mutation
Mutation can be added by allowing offspring to change type during reproduction. Then the boundary states need not remain absorbing, because a lost type can be reintroduced by mutation.
Event time versus biological time
The discrete Moran model counts replacement events:
\[n=0,1,2,\ldots.\]This event index is not automatically the same as physical time. To compare with a continuous-time biological system, a separate event rate or time scaling must be specified.
Continuous-time Moran process
A continuous-time version can assign rates to upward and downward transitions. If \(b_i\) and \(d_i\) are the rates for \(i\to i+1\) and \(i\to i-1\), then the waiting time in state \(i\) is exponential with total rate
\[a_i=b_i+d_i.\]This connects the Moran framework to continuous-time Markov chains and birth–death processes.
Transition matrix
For the discrete event-by-event model, the transition matrix is tridiagonal because only
\[i-1,\qquad i,\qquad i+1\]can be reached from an interior state \(i\).
This local transition structure contrasts with Wright–Fisher, where one generation can move from \(i\) to any state \(j\in\{0,\ldots,N\}\).
Expected change under selection
For the selected model,
\[E[\Delta X\mid i]=P(i\to i+1)-P(i\to i-1).\]Substituting the transition probabilities gives
\[\boxed{E[\Delta X\mid i]=\frac{i(N-i)(r-1)}{N(ri+N-i)}}.\]This quantity is positive when \(r>1\), zero when \(r=1\), and negative when \(r<1\).
Why no-change events matter
In the Moran process, many replacement events do not alter allele count because reproduction and death can involve the same type.
Ignoring these events changes the time scale, even if the sequence of actual upward and downward jumps is unchanged.
Embedded jump chain
If one records only events that actually change \(i\), the resulting embedded chain moves between neighbouring states.
In the neutral case, conditional on a change occurring, upward and downward moves are equally likely:
\[P(i\to i+1\mid\text{change})=P(i\to i-1\mid\text{change})=\frac12.\]However, the waiting number of replacement events between actual changes depends on \(i\).
Simulation
A discrete Moran simulation repeatedly performs three steps: choose a reproducer according to the reproductive rule, choose an individual to die, and update the allele count.
Repeating many simulations gives fixation probabilities and distributions of fixation times.
When the Moran model is useful
The model is useful when gradual replacement is a more natural description than complete generational turnover, or when one wants a simple finite-state birth–death structure with explicit fixation behaviour.