Towards the prediction of critical transitions in spatially extended populations with cubical homology

TL;DR

This paper explores using topological data analysis, specifically Betti numbers, to predict critical transitions like extinction in spatially extended populations, offering a new approach to analyze complex spatial patterns.

Contribution

It introduces a novel method applying Betti numbers to characterize and track spatial patterns in population models for early warning of critical transitions.

Findings

Betti number time series show characteristic changes before extinction

Topological features correlate with system dynamics and transitions

Method applicable to spatial data analysis and model validation

Abstract

The prediction of critical transitions, such as extinction events, is vitally important to preserving vulnerable populations in the face of a rapidly changing climate and continuously increasing human resource usage. Predicting such events in spatially distributed populations is challenging because of the high dimensionality of the system and the complexity of the system dynamics. Here, we reduce the dimensionality of the problem by quantifying spatial patterns via Betti numbers ($\beta_0$ and $\beta_1$), which count particular topological features in a topological space. Spatial patterns representing regions occupied by the population are analyzed in a coupled patch population model with Ricker map growth and nearest-neighbors dispersal on a two-dimensional lattice. We illustrate how Betti numbers can be used to characterize spatial patterns by type, which in turn may be used to track spatiotemporal changes via Betti number time series and characterize asymptotic dynamics of the model parameter space. En route to a global extinction event, we find that the Betti number time series of a population exhibits characteristic changes. We hope these preliminary results will be used to aide in the prediction of critical transitions in spatially extended systems. Additional applications of this technique include analysis of spatial data (e.g., GIS) and model validation.