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Propagating fronts, chaos and multistability in a cell replication model.

Rebecca Crabb¹, Michael C. Mackey, Alejandro D. Rey

¹Department of Applied Mathematics, University of Washington, Seattle, Washington 98195Center for Nonlinear Dynamics and Departments of Physiology, Physics, and Mathematics, McGill University, 3655 Drummond, Room 1124, Montreal, Quebec, H3G 1Y6, CanadaDepartment of Chemical Engineering, McGill University, 3455 University Street, Montreal, Quebec, H3A 2A7, Canada.

Chaos (Woodbury, N.Y.)

|September 1, 1996

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This study analyzes a cell population model with delays, revealing that increased spatio-temporal delays lead to complex spatial patterns and irregular wave propagation in cell dynamics.

Area of Science:

Mathematical Biology
Computational Science
Biophysics

Background:

Cell population dynamics are often modeled using partial differential equations.
Incorporating delays due to cell cycle length is crucial for realistic models.
Previous models often lack the complexity arising from nonlinear, nonlocal, and delayed terms.

Purpose of the Study:

To numerically analyze a hyperbolic partial differential equation model for cell population dynamics.
To investigate the impact of delayed arguments in time and maturation variables.
To explore the complex behaviors arising from nonlinear, nonlocal reaction terms.

Main Methods:

Numerical solutions of a hyperbolic partial differential equation with delayed arguments.
Analysis of a model balancing linear convection with nonlinear, nonlocal, and delayed reaction terms.

Related Experiment Videos

Validation of numerical results using different computational methods.

Main Results:

The nonlinear, nonlocal birth term generates soliton-like or front solutions.
Increasing nonlocality and temporal delays create fine periodic structures on solution fronts.
Hopf bifurcations and period doublings leading to apparent chaos were observed.
Spatial chaos emerged in the time-maturation plane for specific parameter values.

Conclusions:

The model exhibits rich phenomenology, including soliton propagation and chaotic behavior.
Spatio-temporal delays significantly influence population dynamics, leading to complexity.
Findings are relevant for understanding biological systems modeled by hyperbolic delayed partial differential equations.