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Collocation Techniques for Structured Populations Modeled by Delay Equations

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Current Trends in Dynamical Systems in Biology and Natural Sciences

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 21))

Abstract

Collocation methods can be applied in different ways to delay models, e.g., to detect stability of equilibria, Hopf bifurcations and compute periodic solutions to name a few. On the one hand, piecewise polynomials can be used to approximate a periodic solution for some fixed values of the model parameters, possibly using an adaptive mesh. On the other hand, polynomial collocation can be used to reduce delay systems to systems of ordinary differential equations and established continuation tools are then applied to analyze stability and detect bifurcations. These techniques are particularly useful to treat realistic models describing structured populations, where delay differential equations are coupled with renewal equations and vital rates are given implicitly as solutions of external equations, which in turn change with model parameters. In this work we show how collocation can be used to improve the performance of continuation for such complex models and to compute periodic solutions of coupled problems.

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Acknowledgements

All the authors are members of the CDLab—Computational Dynamics Laboratory (cdlab.uniud.it) and of INdAM Research group GNCS. DB is supported by the project PSD_2015_2017_DIMA_PRID_2017_ZANOLIN “SIDIA – SIstemi DInamici e Applicazioni” (UNIUD).

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Authors and Affiliations

  1. Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

    Alessia Andò & Dimitri Breda

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  1. Alessia Andò
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  2. Dimitri Breda
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Correspondence to Dimitri Breda .

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Editors and Affiliations

  1. Department of Mathematics, University of Trento, Trento, Italy

    Maira Aguiar

  2. Department of Mathematics, University of Évora, Évora, Portugal

    Carlos Braumann

  3. Faculty of Science, VU University Amsterdam, Amsterdam, The Netherlands

    Bob W. Kooi

  4. Department of Mathematics, University of Trento, Trento, Italy

    Andrea Pugliese

  5. Department of Mathematics “CMAF-CIO”, University of Lisbon, Lisbon, Portugal

    Nico Stollenwerk

  6. Department of Mathematics “Giuseppe Peano”, University of Turin, Torino, Italy

    Ezio Venturino

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Andò, A., Breda, D. (2020). Collocation Techniques for Structured Populations Modeled by Delay Equations. In: Aguiar, M., Braumann, C., Kooi, B., Pugliese, A., Stollenwerk, N., Venturino, E. (eds) Current Trends in Dynamical Systems in Biology and Natural Sciences. SEMA SIMAI Springer Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-41120-6_3

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