A method to estimate the threshold of critical transitions in ecosystems

Background/Question/Methods

Evidences from various ecosystems, ranging from lakes to semi-arid ecosystems, suggest that gradually changing drivers can cause abrupt state shifts from one stable state to an alternative stable state. Such shifts, also called critical transitions, can often be irreversible and may result in significant ecological as well as economic losses. Recently, studies based on bifurcation theory and nonequilibrium phase transitions have devised early warning signals of abrupt ecological transitions. However, these signals cannot forecast the threshold value of the ecosystem state or the driver at which the transition will actually occur. In this work, we propose a method to analyze discrete spatial data such as data from satellite imagery to estimate this threshold. To do so, we employ a spatially explicit model of vegetation that includes local interactions such as facilitation between nearby vegetation and competition for resources (based on Kefi et. al. Nature 449 (2007)213-217| doi:10.1038/nature06111 and Lubeck J.Stat.Phys.123(1)(2006)193-221). These models exhibit transitions in ecological state from a vegetated to a bare state. We build mean-field theories of these spatial models and analyze the model near the critical point.

Results/Conclusions

We propose a method to calculate variance of the spatial data at different levels of coarse-graining (smoothing) of data. Our analytical arguments, based on the theory of non-equilibrium phase transitions, suggest that the spatial variance of the coarse-grained data should peak at the value of the state or the driver variable corresponding to a critical point. In contrast, the raw data (which is typically 0 or 1 at each location) exhibits peak of spatial variance at a mean spatial density of half, irrespective of the nature of local interactions. Results of our numerical simulations are consistent with the analytical theory. As an example, this method could be applied to satellite or other imagery based data of vegetation cover along a spatial gradient of rainfall or fire or related drivers. An important implication of our work is using such relatively easily available data, we may be able to estimate the threshold value of the driver or the state variable at which the critical point is expected. Our future work will involve testing the method with real data and to understand its strengths as well as limitations.