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Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations

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Abstract

Recently, systems of coupled renewal and retarded functional differential equations have begun to play a central role in complex and realistic models of population dynamics. In view of studying the local asymptotic stability of equilibria and (mainly) periodic solutions, we propose a pseudospectral collocation method to approximate the eigenvalues of the evolution operators of linear coupled equations, providing rigorous error and convergence analyses and numerical tests. The method combines the ideas of the analogous techniques developed separately for renewal equations and for retarded functional differential equations. Coupling them is not trivial, due to the different state spaces of the two classes of equations, as well as to their different regularization properties.

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Notes

  1. Different choices are possible for the norm of a product space: as an example, the norm defined as \(\max \{\Vert x \Vert _{X}, \Vert y \Vert _{Y}\}\) is equivalent to (1).

  2. Proposition 1 and 2 assume that the integrand function u is integrable: in this case, being u continuous, the assumptions can probably be relaxed.

References

  1. Arino, O., van den Driessche, P.: Time delays in epidemic models: modeling and numerical considerations. In: Arino, O., Hbid, M.L., Ait Dads, E. (eds.) Delay Differential Equations and Applications, no. 205 in NATO Science Series II: Mathematics, Physics and Chemistry. Springer, Dordrecht (2006). https://doi.org/10.1007/1-4020-3647-7

    Chapter  Google Scholar 

  2. Breda, D., Diekmann, O., de Graaf, W.F., Pugliese, A., Vermiglio, R.: On the formulation of epidemic models (an appraisal of Kermack and McKendrick). J. Biol. Dyn. 6(sup2), 103–117 (2012). https://doi.org/10.1080/17513758.2012.716454

    Article  MathSciNet  MATH  Google Scholar 

  3. Breda, D., Diekmann, O., Maset, S., Vermiglio, R.: A numerical approach for investigating the stability of equilibria for structured population models. J. Biol. Dyn. 7(sup1), 4–20 (2013). https://doi.org/10.1080/17513758.2013.789562

    Article  MathSciNet  MATH  Google Scholar 

  4. Breda, D., Getto, P., Sánchez Sanz, J., Vermiglio, R.: Computing the eigenvalues of realistic Daphnia models by pseudospectral methods. SIAM J. Sci. Comput. 37(6), A2607–A2629 (2015). https://doi.org/10.1137/15M1016710

    Article  MathSciNet  MATH  Google Scholar 

  5. Breda, D., Liessi, D.: Floquet theory and stability of periodic solutions of renewal equations. J. Dynam. Diff. Eqn. (2020). https://doi.org/10.1007/s10884-020-09826-7

  6. Breda, D., Liessi, D.: Approximation of eigenvalues of evolution operators for linear renewal equations. SIAM J. Numer. Anal. 56(3), 1456–1481 (2018). https://doi.org/10.1137/17M1140534

    Article  MathSciNet  MATH  Google Scholar 

  7. Breda, D., Maset, S., Vermiglio, R.: Approximation of eigenvalues of evolution operators for linear retarded functional differential equations. SIAM J. Numer. Anal. 50(3), 1456–1483 (2012). https://doi.org/10.1137/100815505

    Article  MathSciNet  MATH  Google Scholar 

  8. Breda, D., Maset, S., Vermiglio, R.: Stability of Linear Delay Differential Equations: A Numerical Approach with MATLAB. Springer Briefs Control Autom. Robot. Springer, New York (2015). https://doi.org/10.1007/978-1-4939-2107-2

    Book  MATH  Google Scholar 

  9. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). https://doi.org/10.1007/978-0-387-70914-7

    Book  MATH  Google Scholar 

  10. Chatelin, F.: Spectral Approximation of Linear Operators. No. 65 in Classics Appl. Math. Society for Industrial and Applied Mathematics, Philadelphia (2011). https://doi.org/10.1137/1.9781611970678

  11. Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. No. 21 in Texts Appl. Math. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-4224-6

  12. Diekmann, O., Getto, P., Gyllenberg, M.: Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars. SIAM J. Math. Anal. 39(4), 1023–1069 (2008). https://doi.org/10.1137/060659211

    Article  MathSciNet  MATH  Google Scholar 

  13. Diekmann, O., van Gils, S.A., Verduyn Lunel, S.M., Walther, H.O.: Delay Equations: Functional-, Complex- and Nonlinear Analysis. No. 110 in Appl. Math. Sci. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-4206-2

  14. Diekmann, O., Gyllenberg, M.: Equations with infinite delay: blending the abstract and the concrete. J. Differ. Equ. 252(2), 819–851 (2012). https://doi.org/10.1016/j.jde.2011.09.038

    Article  MathSciNet  MATH  Google Scholar 

  15. 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(2), 277–318 (2010). https://doi.org/10.1007/s00285-009-0299-y

    Article  MathSciNet  MATH  Google Scholar 

  16. Erneux, T.: Applied Delay Differential Equations. No. 3 in Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York (2009). https://doi.org/10.1007/978-0-387-74372-1

  17. Feller, W.: On the integral equation of renewal theory. Ann. Math. Stat. 12(3), 243–267 (1941). https://doi.org/10.1214/aoms/1177731708

    Article  MathSciNet  MATH  Google Scholar 

  18. Fridman, E.: Introduction to Time-Delay Systems: Analysis and Control. Systems Control Found. Appl. Birkhäuser, Basel (2014). https://doi.org/10.1007/978-3-319-09393-2

  19. Gripenberg, G., Londen, S.O., Staffans, O.: Volterra Integral and Functional Equations. No. 34 in Encyclopedia Math. Appl. Cambridge University Press, Cambridge (1990)

  20. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-Delay Systems. Control Eng. Birkhäuser, Boston, MA (2003). https://doi.org/10.1007/978-1-4612-0039-0

  21. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. No. 99 in Appl. Math. Sci. Springer, New York (1993). https://doi.org/10.1007/978-1-4612-4342-7

  22. Iannelli, M.: Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori e Stampatori, Pisa (1995)

    Google Scholar 

  23. Iannelli, M., Pugliese, A.: An Introduction to Mathematical Population Dynamics: Along the Trail of Volterra and Lotka. No. 79 in UNITEXT – La Matematica per il 3+2. Springer, Cham, Switzerland (2014). https://doi.org/10.1007/978-3-319-03026-5

  24. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A Math. Phys. Charact. 115(772), 700–721 (1927). https://doi.org/10.1098/rspa.1927.0118

    Article  MATH  Google Scholar 

  25. Kolmanovskii, V., Myshkis, A.: Introduction to the Theory and Applications of Functional Differential Equations. No. 463 in Math. Appl. Kluwer Academic Publishers, Dordrecht (1999). https://doi.org/10.1007/978-94-017-1965-0

  26. Krylov, V.I.: Convergence of algebraic interpolation with respect to roots of Čebyšev’s polynomial for absolutely continuous functions and functions of bounded variation. Dokl. Akad. nauk SSSR 107(3), 362–365 (1956). In Russian

    MathSciNet  MATH  Google Scholar 

  27. Kuang, Y.: Delay Differential Equations: With Applications in Population Dynamics. No. 191 in Math. Sci. Eng. Academic Press, San Diego (1993)

  28. Liessi, D.: Pseudospectral methods for the stability of periodic solutions of delay models. Ph.D. thesis (2018)

  29. Lythgoe, K.A., Pellis, L., Fraser, C.: Is HIV short-sighted? Insights from a multistrain nested model. Evolution 67(10), 2769–2782 (2013). https://doi.org/10.1111/evo.12166

    Article  Google Scholar 

  30. MacDonald, N.: Time Lags in Biological Models. No. 27 in Lecture Notes in Biomathematics. Springer, Berlin, Heidelberg (1978). https://doi.org/10.1007/978-3-642-93107-9

  31. MacDonald, N.: Biological Delay Systems: Linear Stability Theory. No. 8 in Cambridge Studies in Mathematical Biology. Cambridge University Press, Cambridge (1989)

  32. Metz, J.A.J., Diekmann, O. (eds.): The Dynamics of Physiologically Structured Populations. No. 68 in Lecture Notes in Biomathematics. Springer, Berlin, Heidelberg (1986). https://doi.org/10.1007/978-3-662-13159-6

  33. Michiels, W., Niculescu, S.I.: Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach, second edition. No. 27 in Adv. Des. Control. Society for Industrial and Applied Mathematics, Philadelphia (2014). https://doi.org/10.1137/1.9781611973631

  34. Rivlin, T.J.: An Introduction to the Approximation of Functions. Blaisdell Book in Numerical Analysis and Computer Science. Blaisdell Publishing Company, Waltham (1969)

    Google Scholar 

  35. Royden, H.L., Fitzpatrick, P.M.: Real Analysis, 4th edn. Prentice Hall, Upper Saddle River (2010)

    MATH  Google Scholar 

  36. Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts Appl. Math. Springer, New York (2011). https://doi.org/10.1007/978-1-4419-7646-8

  37. Stépán, G.: Retarded Dynamical Systems: Stability and Characteristic Functions. No. 210 in Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow (1989)

  38. Trefethen, L.N.: Spectral Methods in MATLAB. Software Environ. Tools. Society for Industrial and Applied Mathematics, Philadelphia (2000). https://doi.org/10.1137/1.9780898719598

  39. Walther, H.O.: Topics in delay differential equations. Jahresber. Dtsch. Math.-Ver. 116(2), 87–114 (2014). https://doi.org/10.1365/s13291-014-0086-6

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

All the authors are members of the Gruppo Nazionale di Calcolo Scientifico (GNCS) of the Istituto Nazionale di Alta Matematica “Francesco Severi” (INdAM). They were supported by the INdAM GNCS project “Approssimazione numerica di problemi di evoluzione: aspetti deterministici e stocastici” (2018) and by the project PSD_2015_2017_DIMA_PRID_2017_ZANOLIN “SIDIA – SIstemi DInamici e Applicazioni” of the Department of Mathematics, Computer Science and Physics of the University of Udine. D.L. was supported also by the PhD course in Computer Science, Mathematics and Physics of the Department of Mathematics, Computer Science and Physics of the University of Udine.

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  1. CDLab – Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Via delle scienze 206, 33100, Udine, UD, Italy

    Dimitri Breda & Davide Liessi

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  1. Dimitri Breda
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Breda, D., Liessi, D. Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations. Ricerche mat 69, 457–481 (2020). https://doi.org/10.1007/s11587-020-00513-9

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