Replicator dynamics under perturbations and time delays
Along with the appearance of new optimization and control problems, novel paradigms emerge. A large number of them are based on behavioral ecology, where population dynamics play an important role. One of the most known models of population dynamics is the replicator equation, whose applications in...
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| Published in: | Mathematics of control, signals, and systems Vol. 28; no. 3; p. 1 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
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01-09-2016
Springer Nature B.V |
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| Abstract | Along with the appearance of new optimization and control problems, novel paradigms emerge. A large number of them are based on behavioral ecology, where population dynamics play an important role. One of the most known models of population dynamics is the replicator equation, whose applications in optimization and control have increased in recent years. This fact motivates the study of the replicator dynamics’ properties that are related to the implementation of this method for solving optimization and control problems. This paper addresses implementation issues of the replicator equation in engineering problems. We show by means of the Lyapunov theory that the replicator dynamics model is robust under perturbations that make the state to leave the simplex (among other reasons, this phenomenon can emerge due to numerical errors of the solver employed to obtain the replicator dynamic’s response). A refinement of these results is obtained by introducing a novel robust dynamical system inspired by the replicator equation that allows to control and optimize plants under arbitrary initial conditions on the positive orthant. Finally, we characterize stability bounds of the replicator dynamics model in problems that involve
N
strategies that are subject to time delays. We illustrate our results via simulations. |
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| AbstractList | Along with the appearance of new optimization and control problems, novel paradigms emerge. A large number of them are based on behavioral ecology, where population dynamics play an important role. One of the most known models of population dynamics is the replicator equation, whose applications in optimization and control have increased in recent years. This fact motivates the study of the replicator dynamics' properties that are related to the implementation of this method for solving optimization and control problems. This paper addresses implementation issues of the replicator equation in engineering problems. We show by means of the Lyapunov theory that the replicator dynamics model is robust under perturbations that make the state to leave the simplex (among other reasons, this phenomenon can emerge due to numerical errors of the solver employed to obtain the replicator dynamic's response). A refinement of these results is obtained by introducing a novel robust dynamical system inspired by the replicator equation that allows to control and optimize plants under arbitrary initial conditions on the positive orthant. Finally, we characterize stability bounds of the replicator dynamics model in problems that involve N strategies that are subject to time delays. We illustrate our results via simulations. Along with the appearance of new optimization and control problems, novel paradigms emerge. A large number of them are based on behavioral ecology, where population dynamics play an important role. One of the most known models of population dynamics is the replicator equation, whose applications in optimization and control have increased in recent years. This fact motivates the study of the replicator dynamics’ properties that are related to the implementation of this method for solving optimization and control problems. This paper addresses implementation issues of the replicator equation in engineering problems. We show by means of the Lyapunov theory that the replicator dynamics model is robust under perturbations that make the state to leave the simplex (among other reasons, this phenomenon can emerge due to numerical errors of the solver employed to obtain the replicator dynamic’s response). A refinement of these results is obtained by introducing a novel robust dynamical system inspired by the replicator equation that allows to control and optimize plants under arbitrary initial conditions on the positive orthant. Finally, we characterize stability bounds of the replicator dynamics model in problems that involve N strategies that are subject to time delays. We illustrate our results via simulations. |
| ArticleNumber | 20 |
| Author | Poveda, Jorge I. Obando, Germán Quijano, Nicanor |
| Author_xml | – sequence: 1 givenname: Germán orcidid: 0000-0003-4271-4944 surname: Obando fullname: Obando, Germán email: ge-oband@uniandes.edu.co organization: Department of Electrical and Electronics Engineering, Universidad de los Andes – sequence: 2 givenname: Jorge I. surname: Poveda fullname: Poveda, Jorge I. organization: Department of Electrical and Computer Engineering, University of California – sequence: 3 givenname: Nicanor surname: Quijano fullname: Quijano, Nicanor organization: Department of Electrical and Electronics Engineering, Universidad de los Andes |
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| CitedBy_id | crossref_primary_10_1016_j_physa_2019_01_045 crossref_primary_10_1016_j_physa_2022_127927 crossref_primary_10_1016_j_amc_2021_126204 crossref_primary_10_1007_s11107_020_00906_8 crossref_primary_10_1016_j_jde_2021_05_043 crossref_primary_10_1016_j_jfranklin_2018_10_016 crossref_primary_10_1016_j_spa_2021_09_007 crossref_primary_10_3390_w16121696 crossref_primary_10_4995_riai_2024_21215 crossref_primary_10_1016_j_amc_2023_128228 crossref_primary_10_1016_j_ijepes_2020_106407 crossref_primary_10_3390_math8122120 |
| Cites_doi | 10.1109/MCS.2010.939135 10.1007/s11235-010-9307-1 10.1016/j.jtbi.2004.06.012 10.1109/TSMCB.2006.880134 10.1086/423827 10.1109/TIE.2011.2107714 10.1017/CBO9780511806292 10.1016/j.arcontrol.2014.09.002 10.1109/TSMC.2016.2523934 10.1109/TCST.2013.2252175 10.3390/g4040561 10.1016/0040-5809(84)90023-6 10.1007/BF02124750 10.1038/246015a0 10.1016/j.automatica.2015.05.002 10.1017/CBO9781139173179 10.1109/TSMCB.2009.2034631 10.1080/00207179.2010.501389 10.1006/jtbi.1997.0427 10.1016/j.automatica.2016.02.004 10.1016/S0167-6911(99)00039-0 10.1109/72.883403 10.1016/0025-5564(78)90077-9 10.1137/120878537 10.1088/1748-3182/6/1/016007 10.1016/j.jtbi.2012.01.001 10.1109/CDC.2014.7040053 10.1515/crll.1955.194.66 10.1007/978-1-4612-0039-0 10.1142/7759 10.1090/memo/0211 10.2200/S00330ED1V01Y201101CNT009 10.1109/CDC.2012.6426134 10.3182/20120914-2-US-4030.00019 10.1109/CDC.2012.6426277 |
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| Keywords | Time-delay systems Replicator dynamics Singular perturbation Evolutionary game theory |
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| Title | Replicator dynamics under perturbations and time delays |
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