Ferreira EM, Neves AGM. Fixation probabilities for the Moran process with three or more strategies: general and coupling results.
J Math Biol 2020;
81:277-314. [PMID:
32476038 DOI:
10.1007/s00285-020-01510-0]
[Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/24/2018] [Revised: 12/19/2019] [Indexed: 10/24/2022]
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. Contrary 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 Núñez Rodríguez and Neves (J Math Biol 73:1665-1690, 2016). 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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