Skip to main content
SpringerLink
Log in

Finite Dimensional State Representation of Linear and Nonlinear Delay Systems

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987)

    MATH  Google Scholar 

  2. Barbour, A.D., Reinert, G.: Approximating the epidemic curve. Electron. J. Probab. 18, 1–30 (2013)

    Article  MathSciNet  Google Scholar 

  3. Breda, D., Diekmann, O., Gyllenberg, M., Scarabel, F., Vermiglio, R.: Pseudospectral discretization of nonlinear delay equations: new prospects for numerical bifurcation analysis. SIAM J. Appl. Dyn. Syst. 15, 1–23 (2016)

    Article  MathSciNet  Google Scholar 

  4. Calsina, À., Saldaña, J.: A model of physiologically structured population dynamics with a nonlinear individual growth rate. J. Math. Biol. 33, 335–364 (1995)

    Article  MathSciNet  Google Scholar 

  5. de Roos, A.M.: Numerical methods for structured population models: the Escalator Boxcar train. Numer. Methods Partial Differ. Equ. 4, 173–195 (1988)

    Article  MathSciNet  Google Scholar 

  6. de Roos, A.M., Persson, L.: Population and Community Ecology of Ontogenetic Development. Princeton University Press, Princeton (2013)

    Google Scholar 

  7. Diekmann, O., Gyllenberg, M.: Equations with infinite delay: blending the abstract and the concrete. J. Differ. Equ. 252, 819–851 (2012)

    Article  MathSciNet  Google Scholar 

  8. Diekmann, O., Metz, H.: Exploring linear chain trickery for physiologically structured populations. In: Bij het afscheid van Prof. Dr. H.A. Lauwerier, TW in Beeld, CWI, Amsterdam, pp. 73–84 (1988)

  9. Diekmann, O., Getto, Ph, Gyllenberg, M.: Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal. 39, 1023–1069 (2007)

    Article  MathSciNet  Google Scholar 

  10. Diekmann, O., Gyllenberg, M., Metz, J.A.J.: Finite dimensional state representation of physiologically structured population models (in preparation)

  11. Diekmann, O., Gyllenberg, M., Metz, J.A.J.: Steady-state analysis of structured population models. Theor. Popul. Biol. 63, 309–338 (2003)

    Article  Google Scholar 

  12. Diekmann, O., Heesterbeek, J.A.P., Roberts, M.G.: The construction of next-generation matrices for compartmental epidemic models. J. R. Soc. Interface 7, 873–885 (2010)

    Article  Google Scholar 

  13. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.-O.: Delay Equations. Functional-, Complex-, and Nonlinear Analysis. Springer, New York (1995)

    MATH  Google Scholar 

  14. Diekmann, O., Gyllenberg, M., Metz, J.A.J., Nakaoka, S., de Roos, A.M.: Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example. J. Math. Biol. 61, 277–318 (2010)

    Article  MathSciNet  Google Scholar 

  15. Diekmann, O., Gyllenberg, M., Huang, H., Kirkilionis, M., Metz, J.A.J., Thieme, H.R.: On the formulation and analysis of general deterministic structured population models. II. Nonlinear theory. J. Math. Biol. 43, 157–189 (2001)

    MATH  Google Scholar 

  16. Fargue, D.: Réductibilité des systèmes héréditaires à des systèmes dynamiques (régis par des équations différentielles ou aux dérivées partielles). C. R. Acad. Sci. Paris Sér. B 277, 471–473 (1973)

  17. Farina, L., Rinaldi, S.: Positive Linear Systems. Theory and Applications. Wiley, New York (2000)

    Book  Google Scholar 

  18. Gripenberg, G., Londen, S.-O., Staffans, O.: Volterra Integral and Functional Equations. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  19. Gurtin, M.E., MacCamy, R.C.: Non-linear age-dependent population dynamics. Arch. Ration. Mech. Anal. 54, 281–300 (1974)

    Article  MathSciNet  Google Scholar 

  20. Gyllenberg, M.: Nonlinear age-dependent population dynamics in continuously propagated bacterial cultures. Math. Biosci. 62, 45–74 (1982)

    Article  MathSciNet  Google Scholar 

  21. Gyllenberg, M.: Stability of a nonlinear age-dependent population model containing a control variable. SIAM J. Appl. Math. 43, 1418–1438 (1983)

    Article  MathSciNet  Google Scholar 

  22. Gyllenberg, M.: Mathematical aspects of physiologically structured populations: the contributions of JAJ Metz. J. Biol. Dyn. 1, 3–44 (2007)

    Article  MathSciNet  Google Scholar 

  23. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 115, 700–721 (1927)

    Article  Google Scholar 

  24. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society, Zürich (2006)

    Book  Google Scholar 

  25. Leung, K.Y., Diekmann, O.: Dangerous connections: on binding site models of infectious disease dynamics. J. Math. Biol. 74, 619–671 (2017)

    Article  MathSciNet  Google Scholar 

  26. MacDonald, M.: Time Lags in Biological Models. Lecture Notes in Biomathematics, vol. 27. Springer, Berlin (1978)

    Chapter  Google Scholar 

  27. MacDonald, M.: Biological Delay Systems. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  28. Metz, J.A.J., Diekmann, O.: Exact finite dimensional representations of models for physiologically structured populations. I. The abstract foundations of linear chain trickery. In: Goldstein, J.A., Kappel, F., Schappacher, W. (eds.) Differential Equations with Applications in Biology, Physics and Engineering. Lecture Notes in Pure & Applied Mathematics, vol. 133, pp. 269–289. Marcel Dekker, New York (1991)

  29. Miller, J.C.: A note on a paper by Erik Volz: SIR dynamics in random networks. J. Math. Biol. 62, 349–358 (2011)

    Article  MathSciNet  Google Scholar 

  30. Miller, J.C., Volz, E.M.: Incorporating disease and population structure into models of SIR disease in contact networks. In: PLoS ONE [E], vol 8, no. 8, Public Library of Science, San Francisco, pp. 1–14 (2013)

    Article  Google Scholar 

  31. Vogel, T.: Théorie des Systèmes Évolutifs. Gauthier-Villars, Paris (1965)

    MATH  Google Scholar 

  32. Zadeh, L.A., Desoer, C.A.: Linear System Theory: The State Space Approach. McGraw-Hill, New York (1963)

    MATH  Google Scholar 

  33. Zadeh, L.A., Polak, E.: System Theory. McGraw-Hill, New York (1969)

    Google Scholar 

Download references

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”.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Utrecht, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands

    Odo Diekmann

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Mats Gyllenberg

  3. Mathematical Institute and Institute of Biology, Leiden University, 2333 CA, Leiden, The Netherlands

    J. A. J. Metz

  4. Evolution and Ecology Program, International Institute for Applied Systems Analysis, 2361, Laxenburg, Austria

    J. A. J. Metz

  5. Naturalis Biodiversity Center, 2333 CR, Leiden, The Netherlands

    J. A. J. Metz

Authors
  1. Odo Diekmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mats Gyllenberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. A. J. Metz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mats Gyllenberg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Diekmann, O., Gyllenberg, M. & Metz, J.A.J. Finite Dimensional State Representation of Linear and Nonlinear Delay Systems. J Dyn Diff Equat 30, 1439–1467 (2018). https://doi.org/10.1007/s10884-017-9611-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-017-9611-5

Keywords

Mathematics Subject Classification

Search

Navigation