Stability analysis of a state-dependent delay differential equation for cell maturation: analytical and numerical...

Area of Science:

• Mathematical Biology
• Cellular Dynamics
• Dynamical Systems

Background:

• Stem cell maturation is a complex process crucial for tissue regeneration and development.
• Understanding the dynamics of cell maturation requires sophisticated mathematical modeling.
• Previous models often simplified maturation rates or delays, limiting biological insights.

Purpose of the Study:

• To develop and validate a mathematical model for stem cell maturation incorporating state-dependent delays.
• To investigate the impact of maturation rate regulation on the stability of cellular equilibria.
• To establish a numerical framework for analyzing nonlinear delay-differential equations in biological systems.

Main Methods:

• Formulation of a differential equation model with state-dependent delay to describe cell maturity.
• Application of the principle of linearized stability for analytical insights.
• Extension of the pseudospectral discretization technique for approximating state-dependent delay equations.
• Numerical analysis of characteristic equations and stability boundaries in parameter planes.

Main Results:

• Demonstrated the applicability of linearized stability for the proposed model.
• Successfully adapted pseudospectral methods for nonlinear state-dependent delay equations, a novel application.
• Identified significant qualitative changes in stability boundaries based on model parameters.
• Validated numerical methods against fixed-delay equation cases.

Conclusions:

• The developed mathematical and numerical approaches provide a robust framework for studying stem cell maturation dynamics.
• The findings highlight how regulatory mechanisms in maturation rates can drastically alter system stability.
• This work offers a powerful tool for analyzing complex biological systems with delays, with potential implications for regenerative medicine and disease modeling.