A numerical framework for computing steady states of structured population models and their stability.

Area of Science:

• Mathematical Biology
• Theoretical Ecology
• Computational Biology

Background:

• Structured population models are crucial for understanding biological systems, often represented by general evolution equations.
• Analytical solutions for steady states in these models are difficult to obtain, limiting theoretical analysis.
• Existing methods struggle with finding exact stationary solutions, necessitating new computational approaches.

Purpose of the Study:

• To develop a numerical framework for approximating stationary solutions of general evolution equations.
• To enable the computation of approximate existence and stability regions for steady states.
• To provide a tool for analyzing the asymptotic behavior of solutions in population dynamics.

Main Methods:

• Approximation of the infinitesimal generator using the Trotter-Kato Theorem on a finite-dimensional space.
• Reduction of evolution equations to systems of ordinary differential equations for numerical analysis.
• Application of the framework to linear and nonlinear population models, including coagulation-fragmentation equations.

Main Results:

• Demonstrated convergence of the numerical framework using a known linear Sinko-Streifer model.
• Successfully applied the framework to a nonlinear population balance equation, an extension of the Smoluchowski model.
• Showcased the framework's ability to provide insights into the theoretical stability of stationary solutions.

Conclusions:

• The developed numerical framework offers a robust method for approximating stationary solutions in structured population models.
• This approach facilitates the analysis of steady-state stability and asymptotic behavior in complex biological systems.
• An open-source Python program is available, promoting accessibility and further research in the field.