Numerical Analysis of Linearly Implicit Methods for Discontinuous Nonlinear Gurtin-MacCamy Model

Area of Science:

• Mathematical Biology
• Computational Science
• Ecology

Background:

• Population models often use continuous rates, but biological systems exhibit discontinuous mortality and fertility due to maturation.
• Age-structured models are crucial for understanding population dynamics, especially with varying life stages.

Purpose of the Study:

• To develop and analyze a novel numerical method for nonlinear age-structured population models with discontinuous rates.
• To investigate the convergence and stability properties of the numerical solutions for these models.

Main Methods:

• A novel numerical method using two-layer boundary conditions and linearly implicit methods on a special mesh was developed.
• Uniform boundedness analysis was employed to prove piecewise finite-time convergence.
• A numerical basic reproduction function was used to determine the existence and stability of endemic equilibria.

Main Results:

• The numerical method demonstrated piecewise finite-time convergence for models with discontinuous rates.
• The numerical basic reproduction function accurately predicted the existence of endemic equilibria, converging to the exact value with order 1 accuracy.
• Numerical processes approximated the global stability of disease-free and local stability of endemic equilibria for juvenile-adult models.

Conclusions:

• The developed numerical method is effective for analyzing age-structured population models with discontinuous rates.
• The method provides accurate approximations for population dynamics, equilibrium stability, and reproduction numbers.
• Numerical experiments validated the efficiency and accuracy of the proposed approach for ecological models.