They are minimal models inspired to greater or lesser degree by biological observation, and their minimalism allows one to understand the basic behaviors necessary or sufficient for a particular type of aggregation phenomenon. The discrete and continuum models that we have mentioned here are largely phenomenological. An alternative modeling approach, also the subject of a rich literature, is to treat a sufficiently large population as a continuum and describe its dynamics with a partial integrodifferential equation as in, e.g. The common approach is to envision organisms as point particles with motion laws that are first or second order in time, and with behavioral rules that are some combination of self-propulsion and/or social alignment, attraction, and repulsion. Since then, hundreds of aggregation models have been created some of the most well-studied include. The forces in the model are social forces between two fish-namely attraction and repulsion-and are described by a simple functional form dependent on the distance between individuals, akin to gravitational or intermolecular forces in physics. Modeling of aggregations dates back (at least) to the 1950s with the seminal work of, which describes the motion of individual fish as particles obeying Newton’s law. Quantitative understanding of aggregations has been developed in part through mathematical modeling. Beyond the realm of biology, the understanding of biological aggregations has inspired applications from computer algorithms to robotic self-assembly. Beyond serving as examples of emergent pattern formation, organisms moving in groups can affect resource consumption, disease transmission, and at the longest spatiotemporal scales, evolution itself. Aggregations take on a vast array of morphologies: advancing fronts of running wildebeest, branched dendritic structures of bacteria, tornado-like vortices of swimming anchovy, and much more. Social interactions are behaviors like attraction, repulsion, and alignment, which are activated when one organism senses another via sight, sound, smell, touch, or perhaps some combination of senses. Social interactions between members can play a crucial role in the formation and behavior of these groups. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.Ĭompeting interests: The authors have declared that no competing interests exist.īiological aggregations are groups of organisms such as fish schools, bird flocks, insect swarms, and mammal herds. įunding: This work was supported by National Science Foundation grant DMS-1412674, to CT National Science Foundation grant DMS-1009633, to CT and Simons Foundation Grant 283311, to TH. Journal editors may view the submission using the following url. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are creditedĭata Availability: We have deposited our data with Dryad. Received: DecemAccepted: ApPublished: May 13, 2015Ĭopyright: © 2015 Topaz et al. The topological calculations reveal events and structure not captured by the order parameters.Ĭitation: Topaz CM, Ziegelmeier L, Halverson T (2015) Topological Data Analysis of Biological Aggregation Models. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. Each simulation time frame is a point cloud in position-velocity space. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms.
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