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On the asymptotic behaviour of the solutions to the replicator equation
Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Oxford University Press
2011
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3107970/ https://www.ncbi.nlm.nih.gov/pubmed/20435663 http://dx.doi.org/10.1093/imammb/dqq006 |
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author | Karev, Georgy P. Novozhilov, Artem S. Berezovskaya, Faina S. |
author_facet | Karev, Georgy P. Novozhilov, Artem S. Berezovskaya, Faina S. |
author_sort | Karev, Georgy P. |
collection | PubMed |
description | Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of the replicator equation should be transformed; in particular, the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form. The original n-dimensional problem is reduced to the analysis of asymptotic behaviour of the solutions to the so-called escort system, which in some important cases can be of significantly smaller dimension than the original system. The Newton diagram methods are applied to study the asymptotic behaviour of the solutions to the escort system, when interaction matrix has Rank 1 or 2. A general replicator equation with the interaction matrix of Rank 1 is fully analysed; the conditions are provided when the asymptotic state is a polymorphic equilibrium. As an example of the system with the interaction matrix of Rank 2, we consider the problem from Adams & Sornborger (2007, Analysis of a certain class of replicator equations. J. Math. Biol., 54, 357–384), for which we show, for an arbitrary dimension of the system and under some suitable conditions, that generically one globally stable equilibrium exits on the 1-skeleton of the simplex. |
format | Online Article Text |
id | pubmed-3107970 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2011 |
publisher | Oxford University Press |
record_format | MEDLINE/PubMed |
spelling | pubmed-31079702011-06-07 On the asymptotic behaviour of the solutions to the replicator equation Karev, Georgy P. Novozhilov, Artem S. Berezovskaya, Faina S. Math Med Biol Articles Selection systems and the corresponding replicator equations model the evolution of replicators with a high level of abstraction. In this paper, we apply novel methods of analysis of selection systems to the replicator equations. To be suitable for the suggested algorithm, the interaction matrix of the replicator equation should be transformed; in particular, the standard singular value decomposition allows us to rewrite the replicator equation in a convenient form. The original n-dimensional problem is reduced to the analysis of asymptotic behaviour of the solutions to the so-called escort system, which in some important cases can be of significantly smaller dimension than the original system. The Newton diagram methods are applied to study the asymptotic behaviour of the solutions to the escort system, when interaction matrix has Rank 1 or 2. A general replicator equation with the interaction matrix of Rank 1 is fully analysed; the conditions are provided when the asymptotic state is a polymorphic equilibrium. As an example of the system with the interaction matrix of Rank 2, we consider the problem from Adams & Sornborger (2007, Analysis of a certain class of replicator equations. J. Math. Biol., 54, 357–384), for which we show, for an arbitrary dimension of the system and under some suitable conditions, that generically one globally stable equilibrium exits on the 1-skeleton of the simplex. Oxford University Press 2011-06 2010-04-30 /pmc/articles/PMC3107970/ /pubmed/20435663 http://dx.doi.org/10.1093/imammb/dqq006 Text en © The Author 2010. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org. This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.5), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited. |
spellingShingle | Articles Karev, Georgy P. Novozhilov, Artem S. Berezovskaya, Faina S. On the asymptotic behaviour of the solutions to the replicator equation |
title | On the asymptotic behaviour of the solutions to the replicator equation |
title_full | On the asymptotic behaviour of the solutions to the replicator equation |
title_fullStr | On the asymptotic behaviour of the solutions to the replicator equation |
title_full_unstemmed | On the asymptotic behaviour of the solutions to the replicator equation |
title_short | On the asymptotic behaviour of the solutions to the replicator equation |
title_sort | on the asymptotic behaviour of the solutions to the replicator equation |
topic | Articles |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3107970/ https://www.ncbi.nlm.nih.gov/pubmed/20435663 http://dx.doi.org/10.1093/imammb/dqq006 |
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