Wildebeest herds on rolling hills: Flocking on arbitrary curved surfaces

Area of Science:

• Physics
• Soft Matter Physics
• Biological Physics

Background:

• Active matter physics studies collective behavior in systems like animal herds and cytoskeletal filaments.
• The Toner-Tu flocking theory provides a continuum framework for active matter.
• Analyzing active matter on complex, curved biological surfaces presents significant challenges for existing analytical methods.

Purpose of the Study:

• To develop and validate a finite element method formulation of the Toner-Tu flocking theory for arbitrary curved surfaces.
• To enable the study of active matter collective behavior in complex, biologically relevant geometries.
• To overcome the analytical intractability of active matter problems on curved surfaces.

Main Methods:

• Formulation of the Toner-Tu flocking theory using the finite element method.
• Numerical implementation and testing on channel flow, cylinders, and spheres with comparisons to analytical solutions.
• Application to surfaces with arbitrary curvature to explore herding dynamics.

Main Results:

• Successful implementation of the finite element method for Toner-Tu theory on curved surfaces.
• Validation of the numerical approach against analytical solutions in simpler geometries.
• Demonstration of the ability to study complex herding behaviors on arbitrary landscapes.

Conclusions:

• The finite element method provides a versatile and powerful tool for studying active matter on curved surfaces.
• This approach extends the applicability of Toner-Tu theory to realistic biological environments.
• The framework facilitates future research integrating theory with experimental data on collective motion in complex geometries.