Fixation probabilities for the Moran process with three or more strategies: general and coupling results

Eliza M. Ferreira; Armando G. M. Neves

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arXiv · 1811.09552

Fixation probabilities for the Moran process with three or more strategies: general and coupling results

Eliza M. Ferreira·Armando G. M. Neves

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Abstract

We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrarily to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth-death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by N\'u\~nez Rodr\'iguez and Neves (J. Math. Biol. (2016) 73:1665--1690). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

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Eliza M. Ferreira, Armando G. M. Neves. 2018-11-23. Fixation probabilities for the Moran process with three or more strategies: general and coupling results. https://arxiv.org/abs/1811.09552

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