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  Fixation times in graph-structured populations

Hindersin, L. (2013). Fixation times in graph-structured populations. Master Thesis, Universität zu Lübeck, Lübeck.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0014-C095-8 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0018-EDFA-A
Genre: Thesis

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Hindersin, Laura1, Author              
Traulsen, Arne1, Referee              
Affiliations:
1Research Group Evolutionary Theory, Max Planck Institute for Evolutionary Biology, Max Planck Society, ou_1445641              

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 Abstract: The Moran process is widely used for modeling stochastic dynamics of finitely large populations. It describes the invasion process of a novel mutant into a resident population. Generally, the population is assumed to be well-mixed, which is a rather strong assumption. Studying the Moran process on graphs instead of unstructured populations is a recent approach to overcome this assumption. Some graph structures increase the fixation probability of a mutant that has a fitness advantage compared to the resident population. Graphs with this property are called amplifiers of selection. However, simulations show that the time until fixation increases considerably on those graphs. The objective of this thesis is to analyze different graphs of small size with respect to the fixation time. Simulations support the results for larger population size, where analytical approaches are unfeasible. We show that depending on the initial graph structure, the removal of one link can either lead to an increase or decrease in fixation time. This result is surprising and counterintuitive. Another interesting finding is that the shortest average fixation time does not only depend on the mutant’s starting node. But instead, different starting nodes are preferable, depending on the mutant’s fitness.

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Language(s): eng - English
 Dates: 20132013-11-06
 Publication Status: Published in print
 Pages: VIII, 56 S.
 Publishing info: Lübeck : Universität zu Lübeck
 Table of Contents: 1 Introduction 1
2 Background and Methods 5
2.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Canonical Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The Moran Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Fixation Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Isothermal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 Amplification and Suppression of Selection . . . . . . . . . . . . . . . . . . 13
2.3.4 Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.5 Effective Rate of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Small Population Size 17
3.1 Graph Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Fixation Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1 Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Analytical and Simulated Fixation Probability . . . . . . . . . . . . . . . 27
3.3 Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Analytical Results for Fixation Time . . . . . . . . . . . . . . . . . . . . . 31
3.3.2 Simulation of Fixation Time . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Sojourn Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.5 Effective Rate of Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Location of the First Mutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Larger Graphs 45
4.1 Size Eight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 Removal of One and Two Links . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.2 Removal of Three Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Influence of the "Five Links" on Fixation Time . . . . . . . . . . . . . . . . . . . 48
5 Discussion 49
5.1 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References 55
 Rev. Method: -
 Identifiers: Other: Diss/12503
 Degree: Master

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