Persistent Homology Classifies Parameter Dependence of Patterns in Turing Systems.

Area of Science:

• Complex Systems
• Chemical Systems
• Biological Systems

Background:

• Self-organising systems exhibit complex pattern formation.
• Reaction-diffusion systems are key models for studying these patterns.
• Understanding parameter dependence is crucial for predicting system behavior.

Purpose of the Study:

• To apply topological data analysis (TDA) to study self-organising models.
• To investigate if topological summaries can capture parameter dependence in pattern topology.
• To explore TDA for refining parameter estimation in self-organising systems.

Main Methods:

• Utilized topological data analysis (TDA).
• Examined homology of sublevel sets of solutions to Turing reaction-diffusion systems.
• Applied a topological clustering algorithm to specific systems (chlorite-iodide-malonic acid and Schnakenberg).

Main Results:

• Demonstrated that topological summaries can capture parameter dependence.
• Showcased a topological clustering algorithm's ability to reveal how pattern topology varies with parameters.
• Illustrated findings using the chlorite-iodide-malonic acid and Schnakenberg reaction-diffusion systems.

Conclusions:

• Topological data analysis offers a powerful approach to characterizing pattern topology in reaction-diffusion systems.
• The developed topological clustering method can effectively link pattern topology to system parameters.
• This TDA application holds promise for improving parameter estimation in self-organising systems.