2020/03/11 by Evandro P. de Souza, A. Neves, de Souza, Evandro P. +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · Social Sciences · #Biology #Birth–death process #Broom #Combinatorics #Complex Network Analysis Techniques #Demography #Ecology #Evolutionary Game Theory and Cooperation #Fixation (population genetics) #Graph #Mathematics #Opinion Dynamics and Social Influence #Population #Statistics #Vertex (graph theory) #math.PR #msc:91A22 #msc:92D15 #physics.bio-ph #q-bio.PE
paper · pdf · doi:10.48550/arxiv.2003.05462
published in arXiv (Cornell University) (Cornell University) · 20 pages, 4 figures
openalex publication_date 2020/03/11 · arxiv created 2020/03/18 · arxiv updated 2020/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Broom and Rychtář [Proc. R. Soc. A (2008) 464, 2609--2627] found an exact solution for the fixation probabilities of the Moran process for a structured population, in which the interaction structure among individuals is given by the so-called star graph, i.e. one central vertex and n leaves, the leaves connecting only to the center. We generalize on their solution by allowing individuals' fitnesses to depend on the population frequency, and also by allowing a possible change in the order of reproduction and death draws. In their cited paper, Broom and Rychtář considered the birth-death (BD) process, in which at each time step an individual is first drawn for reproduction and then an individual is selected for death. In the death-birth (DB) process, the order of the draws is reversed. It may be seen that the order of the draws makes a big difference in the fixation probabilities. Our solution method applies to both the BD and the DB cases. As expected, the exact formulae for the fixation probabilities are complicated. We will also illustrate them with some examples and provide results on the asymptotic behavior of the fixation probabilities when the number n of leaves in the graph tends to infinity.