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Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

  • Odo Diekmann
  • Mats Gyllenberg
  • J. A. J. Metz
Article

Abstract

We consider the question of when delay systems, which are intrinsically infinite dimensional, can be represented by finite dimensional systems. Specifically, we give conditions for when all the information about the solutions of the delay system can be obtained from the solutions of a finite system of ordinary differential equations. For linear autonomous systems and linear systems with time-dependent input we give necessary and sufficient conditions and in the nonlinear case we give sufficient conditions. Most of our results for linear renewal and delay differential equations are known in different guises. The novelty lies in the approach which is tailored for applications to models of physiologically structured populations. Our results on linear systems with input and nonlinear systems are new.

Keywords

Linear chain trick Delay-differential equation Renewal equation Markov chain Physiologically structured populations Epidemic models 

Mathematics Subject Classification

34K17 93C23 92D25 

Notes

Acknowledgements

We thank Michael Mackey for turning our attention to the works [31] and [16]. Hans Metz’ work benefitted from the support from the “Chair Modélisation Mathématique et Biodiversité of Veolia Environnement-École Polytechnique-Muséum National d’Histoire Naturelle-Fondation X”.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Odo Diekmann
    • 1
  • Mats Gyllenberg
    • 2
  • J. A. J. Metz
    • 3
    • 4
    • 5
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  3. 3.Mathematical Institute and Institute of BiologyLeiden UniversityLeidenThe Netherlands
  4. 4.Evolution and Ecology ProgramInternational Institute for Applied Systems AnalysisLaxenburgAustria
  5. 5.Naturalis Biodiversity CenterLeidenThe Netherlands

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