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

Alessia Andò; Simone De Reggi; Davide Liessi; Francesca Scarabel; Alessia Andò; Simone De Reggi; Davide Liessi; Francesca Scarabel

doi:10.3934/mbe.2023208

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 4493-4515. doi: 10.3934/mbe.2023208

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

1.
Area of Mathematics, Gran Sasso Science Institute, viale F. Crispi 7, 67100 L'Aquila, Italy
2.
Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy
3.
Department of Mathematics, The University of Manchester, Oxford Rd, M13 9PL, Manchester, United Kingdom
4.
Joint UNIversities Pandemic and Epidemiological Research, United Kingdom
5.
CDLab – Computational Dynamics Laboratory, University of Udine, Italy

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.
- bivariate collocation,
- infinitesimal generator,
- partial differential equations,
- stability of equilibria,
- physiologically structured populations
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

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Abstract

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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