Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1033–1065

# Fixation probabilities for the Moran process in evolutionary games with two strategies: graph shapes and large population asymptotics

• Evandro P. de Souza
• Eliza M. Ferreira
• Armando G. M. Neves
Article

## Abstract

This paper is based on the complete classification of evolutionary scenarios for the Moran process with two strategies given by Taylor et al. (Bull Math Biol 66(6):1621–1644, 2004. ). Their classification is based on whether each strategy is a Nash equilibrium and whether the fixation probability for a single individual of each strategy is larger or smaller than its value for neutral evolution. We improve on this analysis by showing that each evolutionary scenario is characterized by a definite graph shape for the fixation probability function. A second class of results deals with the behavior of the fixation probability when the population size tends to infinity. We develop asymptotic formulae that approximate the fixation probability in this limit and conclude that some of the evolutionary scenarios cannot exist when the population size is large.

## Keywords

Markov chains Asymptotic analysis Birth death processes

## Mathematics Subject Classification

91A22 92D15 60J20

## Notes

### Acknowledgements

We thank Max O. Souza for early discussions and encouragement for writing this paper.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Evandro P. de Souza
• 1
• Eliza M. Ferreira
• 2
• Armando G. M. Neves
• 1
Email author
1. 1.Departamento de MatemáticaUniversidade Federal de Minas GeraisBelo HorizonteBrazil
2. 2.Departamento de Ciências ExatasUniversidade Federal de LavrasLavrasBrazil