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

TL;DR

This paper introduces a pseudospectral method using bivariate collocation to analyze the stability of linear population models with two physiological structures, providing a practical approach to approximate spectral properties.

Contribution

It reformulates the hyperbolic PDE model in an absolutely continuous function space and discretizes it to efficiently approximate eigenvalues and eigenfunctions.

Findings

Converging eigenvalue approximations demonstrated

Method's accuracy depends on model coefficient regularity

Provides a practical spectral analysis tool for structured population models

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. We propose an equivalent reformulation of the problem in the space of absolutely continuous functions in the sense of Carath\'eodory, 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.