Abstract
A variety of ecosystems exhibit spatial clustering devoid of characteristic sizes, also known as scale-free clustering. In physics, scale-free behaviour is known to arise when a system is at a critical point, which occurs at the edge of two phases of matter. Scale-free clustering in physics therefore indicates that a system is not resilient. Spatial ecological studies, however, posit that scale-free clustering arises away from critical points and is therefore an indicator of robustness. This inconsistency is troubling. Here, we synthesize the literature on cluster-size distributions together with analyses of a spatial ecological model that incorporates local birth, death and positive feedbacks. We argue that scale-free clustering in real ecological systems is driven by strong positive feedbacks. Using the model, we demonstrate that power-law relations may occur far away from, near or at the critical point of ecosystem collapse depending on the strength of local positive feedbacks. We therefore infer that clustering patterns are unrelated to critical points of ecosystem collapse. Power-law clustering, instead, indicates a different critical point, called a percolation point, that signifies the onset of spanning clusters in a landscape. Finally, we show that a collapse or a regime shift in an ecosystem is characterized by the emergence of scale-free spatial correlations in the system, reflected in a scale-free power-spectrum.
Glossary
- Regime shifts
- Qualitative changes in ecosystem state. These shifts can be abrupt or gradual functions of the underlying drivers.
- Phase transition
- A discontinuous change in a macroscopic property of a system. Some ecological relevant macroscopic quantities are canopy cover, biomass density, connectivity, etc.
- Critical point
- The value of a state variable (such as density) or the value of an environmental condition (such as rainfall) at which a system undergoes a phase transition.
- Criticality
- Characteristic features of a system at a critical point. These features are often independent of the system-specific details.
- Robust criticality
- The appearance of critical features over a broad range of conditions, and not only near the critical point.
- Scale-free
- Lacking a characteristic scale, or a quantity having infinite average value.
- Diverge
- Tending to infinity.
- Percolation
- Existence of a path (of sites in a particular state) from each edge of a system to all others.
- Resilience
- The amount of disturbance a system can withstand without transitioning to an alternate state.
- Stability
- The rate at which a system recovers from perturbations.
- Facilitation
- Interactions between individuals resulting in enhanced reproduction and/or reduced death rates.
- Spatial autocovariance function
- Covariance between states at two locations as a function of the distance between them.
- Power spectrum/Spectral density function
- Strength of fluctuations as a function of frequency; it is the Fourier transform of the autocovariance function.