Convergence of generalized urn models to non-equilibrium attractors
| Autor: | Mathieu Faure, Sebastian J. Schreiber |
|---|---|
| Přispěvatelé: | Groupement de Recherche en Économie Quantitative d'Aix-Marseille (GREQAM), École Centrale de Marseille (ECM)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), École des hautes études en sciences sociales (EHESS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2014 |
| Předmět: |
Statistics and Probability
education.field_of_study Markov chain Applied Mathematics Population Probability (math.PR) Ode [SHS.ECO]Humanities and Social Sciences/Economics and Finance 60Jxx 37C70 92D25 [SHS]Humanities and Social Sciences Combinatorics Distribution (mathematics) Modeling and Simulation Ordinary differential equation Replicator equation Attractor FOS: Mathematics Applied mathematics Quantitative Biology::Populations and Evolution Limit (mathematics) education Economie quantitative Mathematics - Probability Mathematics |
| Zdroj: | Stochastic Processes and their Applications Stochastic Processes and their Applications, Elsevier, 2015, 125 (8), pp.3053--3074 Stochastic Processes and their Applications, 2015, 125 (8), pp.3053--3074. ⟨10.1016/j.spa.2015.02.011⟩ |
| ISSN: | 0304-4149 1879-209X |
| Popis: | Generalized Polya urn models have been used to model the establishment dynamics of a small founding population consisting of k different genotypes or strategies. As population sizes get large, these population processes are well-approximated by a mean limit ordinary differential equation whose state space is the k simplex. We prove that if this mean limit ODE has an attractor at which the temporal averages of the population growth rate is positive, then there is a positive probability of the population not going extinct (i.e. growing without bound) and its distribution converging to the attractor. Conversely, when the temporal averages of the population growth rate is negative along this attractor, the population distribution does not converge to the attractor. For the stochastic analog of the replicator equations which can exhibit non-equilibrium dynamics, we show that verifying the conditions for convergence and non-convergence reduces to a simple algebraic problem. We also apply these results to selection-mutation dynamics to illustrate convergence to periodic solutions of these population genetics models with positive probability. 29 pages, 2 figures |
| Databáze: | OpenAIRE |
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