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Topological data analysis of biological aggregation models.

Chad M Topaz1, Lori Ziegelmeier1, Tom Halverson1

  • 1Department of Mathematics, Statistics, and Computer Science, Macalester College, Saint Paul, Minnesota, United States of America.

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Summary
This summary is machine-generated.

Topological data analysis reveals hidden structures in biological aggregations like bird flocks. This method uncovers dynamic events not detected by traditional order parameters.

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Area of Science:

  • Mathematical Biology
  • Complex Systems
  • Topological Data Analysis

Background:

  • Biological aggregations (e.g., flocks, schools, swarms) exhibit complex emergent behaviors.
  • Mathematical models like Vicsek and D'Orsogna simulate agent interactions (alignment, attraction, repulsion).
  • Traditional order parameters (e.g., polarization) quantify global aggregation behavior.

Purpose of the Study:

  • To apply topological data analysis (TDA) to mathematical models of biological aggregations.
  • To identify and characterize topological structures within agent movement data.
  • To compare TDA findings with conventional order parameters for enhanced understanding.

Main Methods:

  • Utilized numerical simulation output from Vicsek and D'Orsogna models.
  • Treated each simulation time frame as a point cloud in position-velocity space.
  • Calculated persistent homology and Betti numbers (counting components, circles, volumes) to analyze topological structure.
  • Developed a visualization for Betti numbers across time and persistence scales.

Main Results:

  • Topological analysis revealed distinct structural events and patterns within the simulated aggregations.
  • Betti numbers provided insights into the connectivity and spatial organization of agents.
  • Identified emergent phenomena and dynamic transitions not captured by standard order parameters.
  • Visualization effectively displayed topological changes over simulation time.

Conclusions:

  • TDA offers a powerful complementary approach to traditional methods for studying collective animal behavior.
  • Topological features provide novel metrics for characterizing the dynamics and structure of biological aggregations.
  • This approach enhances the understanding of complex systems by revealing hidden organizational principles.