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A pseudospectral method for investigating the stability of linear population models with two physiological structures

  • Received: 30 September 2022 Revised: 30 November 2022 Accepted: 07 December 2022 Published: 26 December 2022
  • The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.

    Citation: Alessia Andò, Simone De Reggi, Davide Liessi, Francesca Scarabel. A pseudospectral method for investigating the stability of linear population models with two physiological structures[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4493-4515. doi: 10.3934/mbe.2023208

    Related Papers:

  • The asymptotic stability of the null equilibrium of a linear population model with two physiological structures formulated as a first-order hyperbolic PDE is determined by the spectrum of its infinitesimal generator. In this paper, we propose a general numerical method to approximate this spectrum. In particular, we first reformulate the problem in the space of absolutely continuous functions in the sense of Carathéodory, so that the domain of the corresponding infinitesimal generator is defined by trivial boundary conditions. Via bivariate collocation, we discretize the reformulated operator as a finite-dimensional matrix, which can be used to approximate the spectrum of the original infinitesimal generator. Finally, we provide test examples illustrating the converging behavior of the approximated eigenvalues and eigenfunctions, and its dependence on the regularity of the model coefficients.



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    [1] F. Bekkal Brikci, J. Clairambault, B. Ribba, B. Perthame, An age-and-cyclin-structured cell population model for healthy and tumoral tissues, J. Math. Biol., 57 (2008), 91–110. https://doi.org/10.1007/s00285-007-0147-x doi: 10.1007/s00285-007-0147-x
    [2] J. Dyson, R. Villella-Bressan, G. Webb, A nonlinear age and maturity structured model of population dynamics, J. Math. Anal. Appl., 242 (2000), 93–104. https://doi.org/10.1006/jmaa.1999.6656 doi: 10.1006/jmaa.1999.6656
    [3] K. E. Howard, A size and maturity structured model of cell dwarfism exhibiting chaotic behavior, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 3001–3013. https://doi.org/10.1142/S0218127403008363 doi: 10.1142/S0218127403008363
    [4] P. Magal, S. Ruan, Structured Population Models in Biology and Epidemiology, no. 1936 in Lecture Notes in Math., Springer, Berlin, Heidelberg, 2008. https://doi.org/10.1007/978-3-540-78273-5
    [5] J. A. J. Metz, O. Diekmann, The Dynamics of Physiologically Structured Populations, no. 68 in Lect. Notes Biomath., Springer, Berlin, Heidelberg, 1986. https://doi.org/10.1007/978-3-662-13159-6
    [6] H. Inaba, Endemic threshold analysis for the Kermack–McKendrick reinfection model, Josai Math. Monogr., 9 (2016), 105–133. https://doi.org/10.20566/13447777_9_105 doi: 10.20566/13447777_9_105
    [7] J. W. Sinko, W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910–918. https://doi.org/10.2307/1934533 doi: 10.2307/1934533
    [8] H. Kang, X. Huo, S. Ruan, On first-order hyperbolic partial differential equations with two internal variables modeling population dynamics of two physiological structures, Ann. Mat. Pura Appl., 200 (2021), 403–452. https://doi.org/10.1007/s10231-020-01001-5 doi: 10.1007/s10231-020-01001-5
    [9] G. F. Webb, Dynamics of populations structured by internal variables, Math. Z., 189 (1985), 319–335. https://doi.org/10.1007/BF01164156 doi: 10.1007/BF01164156
    [10] D. Breda, O. Diekmann, M. Gyllenberg, F. Scarabel, R. Vermiglio, Pseudospectral discretization of nonlinear delay equations: New prospects for numerical bifurcation analysis, SIAM J. Appl. Dyn. Syst., 15 (2016), 1–23. https://doi.org/10.1137/15M1040931 doi: 10.1137/15M1040931
    [11] D. Breda, O. Diekmann, S. Maset, R. Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7 (2013), 4–20. https://doi.org/10.1080/17513758.2013.789562 doi: 10.1080/17513758.2013.789562
    [12] D. Breda, P. Getto, J. Sánchez Sanz, R. Vermiglio, Computing the eigenvalues of realistic Daphnia models by pseudospectral methods, SIAM J. Sci. Comput., 37 (2015), A2607–A2629. https://doi.org/10.1137/15M1016710 doi: 10.1137/15M1016710
    [13] D. Breda, S. Maset, R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482–495. https://doi.org/10.1137/030601600 doi: 10.1137/030601600
    [14] D. Breda, C. Cusulin, M. Iannelli, S. Maset, R. Vermiglio, Stability analysis of age-structured population equations by pseudospectral differencing methods, J. Math. Biol., 54 (2007), 701–720. https://doi.org/10.1007/s00285-006-0064-4 doi: 10.1007/s00285-006-0064-4
    [15] D. Breda, M. Iannelli, S. Maset, R. Vermiglio, Stability analysis of the Gurtin–MacCamy model, SIAM J. Numer. Anal., 46 (2008), 980–995. https://doi.org/10.1137/070685658 doi: 10.1137/070685658
    [16] F. Scarabel, D. Breda, O. Diekmann, M. Gyllenberg, R. Vermiglio, Numerical bifurcation analysis of physiologically structured population models via pseudospectral approximation, Vietnam J. Math., 49 (2021), 37–67. https://doi.org/10.1007/s10013-020-00421-3 doi: 10.1007/s10013-020-00421-3
    [17] L. N. Trefethen, Approximation Theory and Approximation Practice, Society for Industrial and Applied Mathematics, Philadelphia, 2013.
    [18] D. Breda, S. Maset, R. Vermiglio, Stability of Linear Delay Differential Equations, SpringerBriefs Control Autom. Robot., Springer, New York, 2015. https://doi.org/10.1007/978-1-4939-2107-2
    [19] D. Breda, S. De Reggi, F. Scarabel, R. Vermiglio, J. Wu, Bivariate collocation for computing ${R}_0$ in epidemic models with two structures, Comput. Math. Appl., 116. https://doi.org/10.1016/j.camwa.2021.10.026
    [20] D. Breda, F. Florian, J. Ripoll, R. Vermiglio, Efficient numerical computation of the basic reproduction number for structured populations, J. Comput. Appl. Math., 384 (2021), 113165. https://doi.org/10.1016/j.cam.2020.113165 doi: 10.1016/j.cam.2020.113165
    [21] D. Breda, T. Kuniya, J. Ripoll, R. Vermiglio, Collocation of next-generation operators for computing the basic reproduction number of structured populations, J. Sci. Comput., 85 (2020), 40. https://doi.org/10.1007/s10915-020-01339-1 doi: 10.1007/s10915-020-01339-1
    [22] D. Breda, D. Liessi, Approximation of eigenvalues of evolution operators for linear renewal equations, SIAM J. Numer. Anal., 56 (2018), 1456–1481. https://doi.org/10.1137/17M1140534 doi: 10.1137/17M1140534
    [23] D. Breda, D. Liessi, Approximation of eigenvalues of evolution operators for linear coupled renewal and retarded functional differential equations, Ric. Mat., 69 (2020), 457–481. https://doi.org/10.1007/s11587-020-00513-9 doi: 10.1007/s11587-020-00513-9
    [24] D. Breda, S. Maset, R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations, SIAM J. Numer. Anal., 50 (2012), 1456–1483. https://doi.org/10.1137/100815505 doi: 10.1137/100815505
    [25] F. Scarabel, O. Diekmann, R. Vermiglio, Numerical bifurcation analysis of renewal equations via pseudospectral approximation, J. Comput. Appl. Math., 397 (2021), 113611. https://doi.org/10.1016/j.cam.2021.113611 doi: 10.1016/j.cam.2021.113611
    [26] F. Scarabel, R. Vermiglio, Equations with infinite delay: Pseudospectral approximation of characteristic roots in an abstract framework, in preparation.
    [27] E. Sinestrari, G. F. Webb, Nonlinear hyperbolic systems with nonlocal boundary conditions, J. Math. Anal. Appl., 121 (1987), 449–464. https://doi.org/10.1016/0022-247X(87)90255-1 doi: 10.1016/0022-247X(87)90255-1
    [28] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori e Stampatori, Pisa, 1995.
    [29] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. https://doi.org/10.1007/978-981-10-0188-8
    [30] P. P. J. E. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, B. de Pagter, One-Parameter Semigroups, no. 5 in CWI Monogr., North-Holland Publishing Company, Netherlands, 1987.
    [31] K. J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, no. 194 in Grad. Texts in Math., Springer, New York, 2000. https://doi.org/10.1007/b97696
    [32] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, H.-O. Walther, Delay Equations, no. 110 in Appl. Math. Sci., Springer, New York, 1995. https://doi.org/10.1007/978-1-4612-4206-2
    [33] J. Šremr, Absolutely continuous functions of two variables in the sense of Carathéodory, Electron. J. Differential Equations, 2010 (2010), 154, 1–11.
    [34] L. N. Trefethen, Spectral Methods in MATLAB, Software Environ. Tools, Society for Industrial and Applied Mathematics, Philadelphia, 2000. https://doi.org/10.1137/1.9780898719598
    [35] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edition, Dover, Mineola, NY, 2001, reprint of the Springer, Berlin, 1989 edition.
    [36] M. H. Schultz, ${L}^\infty$-multivariate approximation theory, SIAM J. Numer. Anal., 6 (1969), 161–183. https://doi.org/10.1137/0706017 doi: 10.1137/0706017
    [37] C. W. Clenshaw, A. R. Curtis, A method for numerical integration on an automatic computer, SIAM J. Numer. Anal., 2 (1960), 197–205. https://doi.org/10.1007/BF01386223 doi: 10.1007/BF01386223
    [38] L. N. Trefethen, Is Gauss quadrature better than Clenshaw–Curtis?, SIAM Rev., 50 (2008), 67–87. https://doi.org/10.1137/060659831 doi: 10.1137/060659831
    [39] A. M. de Roos, O. Diekmann, P. Getto, M. A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, Bull. Math. Biol., 72 (2010), 259–297. https://doi.org/10.1007/s11538-009-9445-3 doi: 10.1007/s11538-009-9445-3
    [40] C. Barril, À. Calsina, O. Diekmann, J. Z. Farkas, On the formulation of size-structured consumer resource models (with special attention for the principle of linearised stability), arXiv: 2111.09678.
    [41] À. Calsina, O. Diekmann, J. Z. Farkas, Structured populations with distributed recruitment: From PDE to delay formulation, Math. Methods Appl. Sci., 39 (2016), 5175–5191. https://doi.org/10.1002/mma.3898 doi: 10.1002/mma.3898
    [42] O. Diekmann, M. Gyllenberg, J. A. J. Metz, S. Nakaoka, A. M. de Roos, Daphnia revisited: Local stability and bifurcation theory for physiologically structured population models explained by way of an example, J. Math. Biol., 61 (2010), 277–318. https://doi.org/10.1007/s00285-009-0299-y doi: 10.1007/s00285-009-0299-y
    [43] E. Franco, O. Diekmann, M. Gyllenberg, Modelling physiologically structured populations: Renewal equations and partial differential equations, arXiv: 2201.05323.
    [44] E. Franco, M. Gyllenberg, O. Diekmann, One dimensional reduction of a renewal equation for a measure-valued function of time describing population dynamics, Acta Appl. Math., 175 (2021), 12. https://doi.org/10.1007/s10440-021-00440-3 doi: 10.1007/s10440-021-00440-3
    [45] O. Diekmann, F. Scarabel, R. Vermiglio, Pseudospectral discretization of delay differential equations in sun-star formulation: Results and conjectures, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2575–2602. https://doi.org/10.3934/dcdss.2020196 doi: 10.3934/dcdss.2020196
    [46] M. Gyllenberg, F. Scarabel, R. Vermiglio, Equations with infinite delay: Numerical bifurcation analysis via pseudospectral discretization, Appl. Math. Comput., 333 (2018), 490–505. https://doi.org/10.1016/j.amc.2018.03.104 doi: 10.1016/j.amc.2018.03.104
    [47] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, no. 89 in Monogr. Textb. Pure Appl. Math., Marcel Dekker, Inc., New York, 1985.
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