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Evolutionary graph theory: Difference between revisions
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\end{align} | \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 | 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 | ||
\ | \[ | ||
\rho_1 = \frac{\displaystyle 1-\ | \rho_1 = \frac{\displaystyle 1-\frac1r}{\displaystyle 1-\frac1{r^N}}. | ||
\ | \] | ||
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, \(\rho_1 >1/N\). | 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, \(\rho_1 >1/N\). | ||
