  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.
