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

Area of Science:

• Mathematical Biology
• Numerical Analysis
• Dynamical Systems

Background:

• Population models with physiological structures are often described by hyperbolic partial differential equations (PDEs).
• The asymptotic stability of these models is linked to the spectrum of the infinitesimal generator.
• Approximating this spectrum is crucial for analyzing model behavior.

Purpose of the Study:

• To develop a general numerical method for approximating the spectrum of the infinitesimal generator of a linear population model.
• To reformulate the problem in a function space allowing for simpler boundary conditions.
• To provide a method applicable to models with two physiological structures.

Main Methods:

• Reformulation of the population model in the space of absolutely continuous functions (Carathéodory sense).
• Definition of the infinitesimal generator's domain with trivial boundary conditions.
• Discretization of the operator using bivariate collocation to form a finite-dimensional matrix.
• Approximation of eigenvalues and eigenfunctions.

Main Results:

• A general numerical method for approximating the spectrum of the infinitesimal generator is proposed.
• The method effectively discretizes the operator into a matrix.
• Test examples demonstrate the convergence of approximated eigenvalues and eigenfunctions.
• The convergence rate depends on the regularity of the model coefficients.

Conclusions:

• The proposed numerical method provides a viable approach for analyzing the asymptotic stability of linear population models.
• The technique simplifies boundary condition handling through function space reformulation.
• The method's accuracy is validated by numerical examples and its dependence on coefficient regularity is shown.