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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Bibliographic Details
Published in:Mathematics of control, signals, and systems signals, and systems, 2016-09, Vol.28 (3), p.1, Article 20
Main Authors: Obando, Germán, Poveda, Jorge I., Quijano, Nicanor
Format: Article
Language:English
Subjects:
Access control
Communications Engineering
Control
Dynamical systems
Engineering
Game theory
Mathematical models
Mathematics
Mathematics and Statistics
Mechatronics
Networks
Optimization
Original Article
Population
Robotics
Robust control
Systems Theory
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Summary: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.
ISSN:0932-4194
1435-568X
DOI:10.1007/s00498-016-0170-9