Abstract
Studies of fixation dynamics in Markov processes predominantly focus on the mean time to absorption. This may be inadequate if the distribution is broad and skewed. We compute the distribution of fixation times in one-step birth-death processes with two absorbing states. These are expressed in terms of the spectrum of the process, and we provide different representations as forward-only processes in eigenspace. These allow efficient sampling of fixation time distributions. As an application we study evolutionary game dynamics, where invading mutants can reach fixation or go extinct. We also highlight the median fixation time as a possible analog of mixing times in systems with small mutation rates and no absorbing states, whereas the mean fixation time has no such interpretation.
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Notes
- 1.
Samuel Karlin (1924–2007) and James McGregor (1921–1988).
- 2.
All samples were generated on the same computer, a 2012 MacBook Air with 1.8 GHz i5 processor running OSX 10.10.4. Software used is Mathematica 9.0.1, and times were measured using the \(\texttt {AbsoluteTiming[]}\) function.
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Ashcroft, P. (2016). Fixation Time Distributions in Birth–Death Processes. In: The Statistical Physics of Fixation and Equilibration in Individual-Based Models. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41213-9_4
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