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Replicator dynamics under perturbations and time delays

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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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References

  1. Maynard Smith J, Price G (1973) The logic of animal conflict. Nature 246:15–18

    Article  Google Scholar 

  2. Maynard Smith J (1982) Evolution and the theory of games. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  3. Thomas B (1984) Evolutionary stability: states and strategies. Theor Popul Biol 26(1):49–67

    Article  MathSciNet  MATH  Google Scholar 

  4. Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40(1):145–156

    Article  MathSciNet  MATH  Google Scholar 

  5. Weibull J (1997) Evolutionary game theory. The MIT press, Cambridge, MA, USA

  6. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  7. Sandholm W (2010) Population games and evolutionary dynamics. The MIT press, Cambridge, MA, USA

  8. Cressman R, Křivan V, Garay J (2004) Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments. Am Nat 164(4):473–489

    Article  Google Scholar 

  9. Fox MJ, Shamma JS (2013) Population games, stable games, and passivity. Games 4(4):561–583

    Article  MathSciNet  MATH  Google Scholar 

  10. Pais D, Caicedo-Núnez CH, Leonard NE (2012) Hopf bifurcations and limit cycles in evolutionary network dynamics. SIAM J Appl Dyn Syst 11(4):1754–1784

    Article  MathSciNet  MATH  Google Scholar 

  11. Leonard N (2014) Multi-agent system dynamics: bifurcation and behavior of animal groups. Annu Rev Control 38(4):171–183

    Article  MathSciNet  Google Scholar 

  12. Barreiro-Gomez J, Obando G, Quijano N (2016) Distributed population dynamics: optimization and control applications. IEEE Trans Syst Man Cybern Syst 99:1–11. doi:10.1109/TSMC.2016.2523934

    Article  Google Scholar 

  13. Barreiro-Gomez J, Quijano N, Ocampo-Martinez C (2016) Constrained distributed optimization: a population dynamics approach. Automatica 69:101–116

    Article  MathSciNet  MATH  Google Scholar 

  14. Obando G, Pantoja A, Quijano N (2014) Building temperature control based on population dynamics. IEEE Trans Control Syst Technol 22(1):404–412

    Article  Google Scholar 

  15. Ramírez-Llanos E, Quijano N (2010) A population dynamics approach for the water distribution problem. Int J Control 83(9):1947–1964

    Article  MathSciNet  MATH  Google Scholar 

  16. Bomze I, Pelillo M, Stix V (2000) Approximating the maximum weight clique using replicator dynamics. IEEE Trans Neural Netw 11(6):1228–1241

    Article  Google Scholar 

  17. Tembine H, Altman E, El-Azouzi R, Hayel Y (2010) Evolutionary games in wireless networks. IEEE Trans Syst Man Cybern Part B Cybern 40(3):634–646

    Article  Google Scholar 

  18. Tembine H, Altman E, El-Azouzi R, Hayel Y (2011) Bio-inspired delayed evolutionary game dynamics with networking applications. Telecommun Syst 47(1–2):137–152

    Article  MATH  Google Scholar 

  19. Poveda J, Quijano N (2012) Dynamic bandwidth allocation in wireless networks using a shahshahani gradient based extremum seeking control. In: Proceedings of the 2012 6th international conference on network games, control and optimization (NetGCooP), IEEE, pp 44–50

  20. Pantoja A, Quijano N (2011) A population dynamics approach for the dispatch of distributed generators. IEEE Trans Ind Electron 58:4559–4567

    Article  Google Scholar 

  21. Poveda JI, Quijano N (2015) Shahshahani gradient-like extremum seeking. Automatica 58:51–59

    Article  MathSciNet  MATH  Google Scholar 

  22. Pantoja A, Quijano N (2012) Distributed optimization using population dynamics with a local replicator equation. In: Proceedings of the 51st IEEE conference on decision and control, IEEE, pp 3790–3795

  23. Yi T, Zuwang W (1997) Effect of time delay and evolutionarily stable strategy. J Theor Biol 187(1):111–116

    Article  Google Scholar 

  24. Alboszta J, Miekisz J (2004) Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay. J Theor Biol 231(2):175–179

    Article  MathSciNet  Google Scholar 

  25. Menache I, Ozdaglar A (2011) Network games: theory, models, and dynamics, vol 4. Morgan & Claypool Publishers, San Rafael, CA, USA

  26. Shahshahani S (1979) A new mathematical framework for the study of linkage and selection. Memoirs of the American Mathematical Society, Providence, RI, USA

  27. Poveda J, Quijano N (2012) A shahshahani gradient based extremum seeking scheme. In: Proceedings of the 51st IEEE conference on decision and control, IEEE, pp 5104–5109

  28. Quijano N, Passino KM (2007) The ideal free distribution: theory and engineering application. IEEE Trans Syst Man Cybern Part B Cybern 37(1):154–165

    Article  Google Scholar 

  29. Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  30. Pantoja A, Quijano N, Leirens S (2011) A bioinspired approach for a multizone temperature control system. Bioinspir Biomim 6(1):016007

    Article  Google Scholar 

  31. Teel AR, Peuteman J, Aeyels D (1999) Semi-global practical asymptotic stability and averaging. Syst Control Lett 37(5):329–334

    Article  MathSciNet  MATH  Google Scholar 

  32. Lobry C, Sari T (2005) Singular perturbation methods in control theory. Contrôle non linéaire et Applications 4(64):155–182

    MATH  Google Scholar 

  33. Barreiro-Gomez J, Quijano N, Ocampo-Martínez C (2014) Constrained distributed optimization based on population dynamics. In: Proceedings of the 53rd IEEE conference on decision and control, IEEE, pp 4260–4265

  34. Iijima R (2012) On delayed discrete evolutionary dynamics. J Theor Biol 300:1–6

    Article  MathSciNet  Google Scholar 

  35. Chen C-T (1998) Linear system theory and design. Oxford University Press Inc, Oxford

    Google Scholar 

  36. Corless RM, Gonnet GH, Hare DE, Jeffrey DJ, Knuth DE (1996) On the lambert w function. Adv Comput Math 5(1):329–359

    Article  MathSciNet  MATH  Google Scholar 

  37. Yi S, Nelson PW, G U (2010) Time-delay systems: analysis and control using the Lambert W function. World Scientific, Singapore

  38. Gu K, Kharitonov V, Chen J (2003) Stability of time-delay systems. Birkhauser, Cambridge, MA, USA

  39. Sipahi R, Niculescu S, Abdallah C, Michiels W, Gu K (2011) Stability and stabilization of systems with time delay. IEEE Control Syst Mag 31(1):38–65

    Article  MathSciNet  Google Scholar 

  40. Engelborghs K, Luzyanina T, Samaey G (2001) Dde-biftool v. 2.00: a matlab package for bifurcation analysis of delay differential equations. tech. rep., Department of Computer Science, Katholieke Universiteit Leuven, Belgium

  41. Wright EM (1955) A non-linear difference-differential equation. J. Reine Angew. Math 194(1):66–87

    MathSciNet  MATH  Google Scholar 

  42. Giraldo J, Quijano N (2012) Delay independent evolutionary dynamics for resource allocation with asynchronous distributed sensors. In: Proceedings of the 3rd IFAC workshop on distributed estimation and control in networked systems, pp 121–126

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Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Universidad de los Andes, Bogotá, Colombia

    Germán Obando & Nicanor Quijano

  2. Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, USA

    Jorge I. Poveda

Authors
  1. Germán Obando
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  2. Jorge I. Poveda
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  3. Nicanor Quijano
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Corresponding author

Correspondence to Germán Obando.

Additional information

Research supported in part by ALTERNAR project, BPIN 20130001000089, Acuerdo 005 de 2013, OCAD-Fondo de CTel SGR, Colombia; and in part by Programa de becas doctorales Colciencias–Colfuturo Convocatoria 528 and Colciencias programa Jóvenes Investigadores 2012.

This work was done while J. I. Poveda was with the Department of Electrical and Electronics Engineering, Universidad de los Andes, Bogotá, Colombia.

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Obando, G., Poveda, J.I. & Quijano, N. Replicator dynamics under perturbations and time delays. Math. Control Signals Syst. 28, 20 (2016). https://doi.org/10.1007/s00498-016-0170-9

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  • DOI: https://doi.org/10.1007/s00498-016-0170-9

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