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Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation

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Abstract

Physiologically structured population models are typically formulated as a partial differential equation of transport type for the density, with a boundary condition describing the birth of new individuals. Here we develop numerical bifurcation methods by combining pseudospectral approximate reduction to a finite dimensional system with the use of established tools for ODE. A key preparatory step is to view the density as the derivative of the cumulative distribution. To demonstrate the potential of the approach, we consider two classes of models: a size-structured model for waterfleas (Daphnia) and a maturity-structured model for cell proliferation. Using the package MatCont, we compute numerical bifurcation diagrams, like steady-state stability regions in a two-parameter plane and parametrized branches of equilibria and periodic solutions. Our rather positive conclusion is that a rather low dimension may yield a rather accurate diagram! In addition we show numerically that, for the two models considered here, equilibria of the approximating system converge to the true equilibrium as the dimension of the approximating system increases; this last result is also proved theoretically under some regularity conditions on the model ingredients.

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Acknowledgments

The research of FS is funded by NSERC-Sanofi Industrial Research Chair. DB, FS and RV are members of the INdAM Research group GNCS. MG was supported by the Centre of Excellence in Analysis and Dynamics Research, Academy of Finland.

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Authors and Affiliations

  1. LIAM–Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    Francesca Scarabel

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

    Francesca Scarabel, Dimitri Breda & Rossana Vermiglio

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

    Odo Diekmann

  4. Department of Mathematics and Statistics, University of Helsinki, P.O. 68 (Pietari Kalmin katu 5), 00014, Helsinki, Finland

    Mats Gyllenberg

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  1. Francesca Scarabel
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  2. Dimitri Breda
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  3. Odo Diekmann
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  4. Mats Gyllenberg
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  5. Rossana Vermiglio
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Correspondence to Francesca Scarabel.

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Scarabel, F., Breda, D., Diekmann, O. et al. Numerical Bifurcation Analysis of Physiologically Structured Population Models via Pseudospectral Approximation. Vietnam J. Math. 49, 37–67 (2021). https://doi.org/10.1007/s10013-020-00421-3

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  • DOI: https://doi.org/10.1007/s10013-020-00421-3

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