Turing-Hopf patterns on growing domains: The torus and the sphere

Area of Science:

• Mathematical Biology
• Reaction-Diffusion Systems
• Computational Neuroscience

Background:

• Reaction-diffusion models, like the FitzHugh-Nagumo model, are crucial for understanding biological pattern formation.
• Investigating these models on curved and growing domains is essential for realistic biological system simulations.
• Previous studies often focused on static or flat domains, limiting applicability to dynamic biological structures.

Purpose of the Study:

• To investigate spatial and spatio-temporal patterns in the FitzHugh-Nagumo model on growing curved domains (torus and sphere).
• To analyze the influence of domain growth and curvature on pattern selection and stability.
• To compare predictions from linear theory with numerical simulations, especially in regimes of multistability and Hopf bifurcations.

Main Methods:

• Analytical computation of bifurcation boundaries for the homogeneous steady state on monostable systems.
• Numerical simulations to observe pattern formation on static and exponentially growing domains.
• Investigation of Turing, Turing-Hopf bifurcations, and pattern selection influenced by geometry and growth.

Main Results:

• Identified Turing and Turing-Hopf bifurcations, alongside additional patterning due to multistability.
• Demonstrated that domain growth and curvature significantly impact pattern selection on spheres and tori.
• Found parameter regimes where linear theory predicts pattern types but not nonlinear structure, and regimes where it fails, particularly in Hopf bifurcations.

Conclusions:

• Domain growth and curvature are critical factors in reaction-diffusion pattern formation, extending beyond simple instability modification.
• Multistability plays a key role in pattern selection and can lead to deviations from linear theory predictions.
• The study highlights the limitations of linear theory in complex geometric and dynamic scenarios, emphasizing the need for nonlinear analysis.