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Evolutionary graph theory: Difference between revisions
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→Moran process
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This process is repeated until either one of the two absorbing states is reached: a homogeneous state of either all residents (mutant went extinct) or all mutants (resident displaced). No other stable equilibrium state is possible. Whenever an absorbing state is reached, mutants (residents) are said to have reached fixation. The corresponding fixation probability and fixation time are important markers to characterize the evolutionary process. The Moran process represents a specific balance between selection and drift: advantageous mutations have a certain chance - but no guarantee - of fixation, whereas disadvantageous mutants are likely - but again, no guarantee - to become extinct. | This process is repeated until either one of the two absorbing states is reached: a homogeneous state of either all residents (mutant went extinct) or all mutants (resident displaced). No other stable equilibrium state is possible. Whenever an absorbing state is reached, mutants (residents) are said to have reached fixation. The corresponding fixation probability and fixation time are important markers to characterize the evolutionary process. The Moran process represents a specific balance between selection and drift: advantageous mutations have a certain chance - but no guarantee - of fixation, whereas disadvantageous mutants are likely - but again, no guarantee - to become extinct. | ||
===Fixation probability=== | |||
In unstructured populations the state of the population is fully determined by the number of mutants \(i\), which changes at most by \(\pm 1\) in every time step of the Moran process. With probability \(T^+\) the number of mutants increases from \(i\) to \(i+1\), with probability \(T^-\) it decreases to \(i-1\) and with probability \(1-T^+-T^-\) the number of mutants remains unchanged. | |||
\begin{align} | |||
T^+ &= \frac{i\cdot r}{i\cdot r+(N-i)}\cdot\frac{N-i}N\\ | |||
T^- &= \frac{N-i}{i\cdot r+(N-i)}\cdot\frac iN | |||
\end{align} | |||
The first factor of \(T^+\) (\(T^-\)) indicates the probability that a mutant (resident) is chosen for reproduction and the second factor denotes the probability that the offspring replaces a resident (mutant). Note that the ratio of the transition probabilities \(T^+/T^- = r\) is independent of the number of mutants in the population. This leads to a simple recursive formula for the fixation probability \(\rho(i)\) of the mutant in a population with \(i\) mutants: | |||
\begin{align} | |||
\label{eq:recursive} | |||
\rho(i) = \frac r{1+r}\ \rho(i-1)+\frac 1{1+r}\ \rho(i+1). | |||
\end{align} | |||
Thus, the dynamics corresponds to a biased random walk with absorbing boundaries. Eq. 1 admits two solutions \(\rho = 1\) and \(\rho = 1/{r^i}\). The absorbing boundaries additionally require \(\rho(0)=0\) and \(\rho(N)=1\). For \(r\neq 1\), the fixation probability of a single mutant \(\rho_1\) then becomes | |||
\begin{align} | |||
\rho_1 = \frac{\displaystyle 1-\frac 1r}{\displaystyle 1-\frac 1{r^N}}. | |||
\end{align} | |||
Assuming that mutations are rare events \(\rho_1\) is of particular interest. It is easy to see that a neutral mutant (\(r=1\)) has a fixation probability of \(\rho_1=1/N\): eventually the entire population will have a single common ancestor but in terms of fitness mutants and residents are indistinguishable and so every member of the population has equal chances to be the chosen one. Evolution is said to favor a mutant if the fixation probability of the mutant exceeds the fixation probability of a neutral mutant, \(\rho_1 >1/N\). | |||
===Spatial Moran process=== | ===Spatial Moran process=== | ||
