🍩 Database of Original & Non-Theoretical Uses of Topology

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  1. Topological Descriptors for Coral Reef Resilience Using a Stochastic Spatial Model (2022)

    Robert A. McDonald, Rosanna Neuhausler, Martin Robinson, Laurel G. Larsen, Heather A. Harrington, Maria Bruna
    Abstract A complex interplay between species governs the evolution of spatial patterns in ecology. An open problem in the biological sciences is characterizing spatio-temporal data and understanding how changes at the local scale affect global dynamics/behavior. We present a toolkit of multiscale methods and use them to analyze coral reef resilience and dynamics.Here, we extend a well-studied temporal mathematical model of coral reef dynamics to include stochastic and spatial interactions and then generate data to study different ecological scenarios. We present descriptors to characterize patterns in heterogeneous spatio-temporal data surpassing spatially averaged measures. We apply these descriptors to simulated coral data and demonstrate the utility of two topological data analysis techniques--persistent homology and zigzag persistence--for characterizing the spatiotemporal evolution of reefs and generating insight into mechanisms of reef resilience. We show that the introduction of local competition between species leads to the appearance of coral clusters in the reef. Furthermore, we use our analyses to distinguish the temporal dynamics that stem from different initial configurations of coral, showing that the neighborhood composition of coral sites determines their long-term survival. Finally, we use zigzag persistence to quantify spatial behavior in the metastable regime as the level of fish grazing on algae varies and determine which spatial configurations protect coral from extinction in different environments.