Travelling Waves in a PDE-ODE Coupled Model of Cellulolytic Biofilms with Nonlinear Diffusion

Area of Science:

• Nonlinear dynamics
• Mathematical biology
• Partial differential equations

Background:

• Travelling wave (TW) solutions are crucial in modeling phenomena like biofilm growth.
• Previous studies observed stable TW solutions numerically and experimentally in cellulolytic biofilm growth.
• However, analytical justification for these TW solutions in degenerate nonlinear systems was lacking.

Purpose of the Study:

• To provide an analytical justification for the existence and stability of travelling wave solutions.
• To analyze nonlinear systems involving an ODE coupled to a degenerate PDE.
• To investigate models where the diffusion coefficient varies with the solution's value.

Main Methods:

• Utilized the travelling wave ansatz and a first integral to simplify the system.
• Reduced the complex system to an autonomous dynamical system with two unknowns.
• Analyzed the system in its phase-plane to determine solution properties.

Main Results:

• Proved the existence of a unique travelling wave solution with a sharp front and diffusive tail.
• Demonstrated that the travelling wave moves at a constant speed.
• Established the linear stability of the travelling wave in two space dimensions under specific initial conditions.

Conclusions:

• The analytical results confirm and justify previous numerical and experimental observations of stable travelling waves.
• The study provides a theoretical foundation for understanding travelling wave phenomena in degenerate nonlinear systems.
• Numerical simulations validated the theoretical predictions regarding the existence, stability, and parameter dependence of these travelling waves.