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Spatial Moran process: Difference between revisions
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==Fixation probability== | ==Fixation probability== | ||
In contrast to unstructured (well-mixed) populations, the state of the population is not only determined by just the number of mutants (or residents) but rather needs to take the spatial distribution of mutants and residents into account. Rather surprisingly, however, it turns out that for a large class of graphs, called circulations, the spatial distribution of mutants and residents does not affect their fixation probabilities. This forms the basis for the conjecture by [[#References|Maruyama (1970)]] and [[#References|Slatkin (1981)]] speculating that population structure leaves fixation probabilities unchanged. Indeed this is true for structures as diverse as complete graphs (every vertex is connected to every other vertex), lattices or random regular graphs (see e.g. [[#References|Bollobás, 1995]]). Further examples and interactive demonstrations are covered in [[ | In contrast to unstructured (well-mixed) populations, the state of the population is not only determined by just the number of mutants (or residents) but rather needs to take the spatial distribution of mutants and residents into account. Rather surprisingly, however, it turns out that for a large class of graphs, called circulations, the spatial distribution of mutants and residents does not affect their fixation probabilities. This forms the basis for the conjecture by [[#References|Maruyama (1970)]] and [[#References|Slatkin (1981)]] speculating that population structure leaves fixation probabilities unchanged. Indeed this is true for structures as diverse as complete graphs (every vertex is connected to every other vertex), lattices or random regular graphs (see e.g. [[#References|Bollobás, 1995]]). Further examples and interactive demonstrations are covered in [[Moran graphs]]. | ||
===Circulation graphs=== | ===Circulation graphs=== | ||
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Note that even though the fixation probabilities remain unchanged on circulation graphs, the corresponding fixation times are very sensitive to the details of the population structure and pose a much harder problem. The circulation theorem only ensures that the ''ratio'' of the transition probabilities \(T^+/T^− = r\) remains unchanged but even on circulation graphs \(T^+\) and \(T^−\) depend not only on the number but also on the distribution of mutants. Generally, population structures tend to substantially increase the fixation times because the structure limits the possibilities for mutants to conquer new nodes. | Note that even though the fixation probabilities remain unchanged on circulation graphs, the corresponding fixation times are very sensitive to the details of the population structure and pose a much harder problem. The circulation theorem only ensures that the ''ratio'' of the transition probabilities \(T^+/T^− = r\) remains unchanged but even on circulation graphs \(T^+\) and \(T^−\) depend not only on the number but also on the distribution of mutants. Generally, population structures tend to substantially increase the fixation times because the structure limits the possibilities for mutants to conquer new nodes. | ||
Finally, the circulation theorem indirectly suggests that some population structures do affect the fixation probabilities. Thus it becomes an intriguing question whether population structures may act as [[ | Finally, the circulation theorem indirectly suggests that some population structures do affect the fixation probabilities. Thus it becomes an intriguing question whether population structures may act as [[evolutionary suppressors]] that suppress selection and enhance random drift (\(\rho<\rho_1\) for \(r > 1\)) or, conversely, as [[evolutionary amplifiers]] that enhance selection and suppresses random drift (\(\rho>\rho_1\) for \(r > 1\))? | ||
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