Trends in Ecology & Evolution
Criticality and disturbance in spatial ecological systems
Introduction
Ecological systems can exhibit threshold behavior with sudden shifts between states 1, 2. These shifts are characterized by important changes in population abundance and species diversity, and their prediction has become an important focus of ecological science. Threshold behavior originates from the sensitivity of ecological systems to parameters controlling their dynamics, and therefore implies a high sensitivity to environmental perturbations. One theoretical concept of relevance to this type of dynamics is criticality (see Glossary): in its classical form, criticality comprises a drastic shift in state following only slight changes in an underlying process. Mathematical models of wind-disturbed forests predict, for example, the rapid collapse of the area covered by trees when gap cover increases only slightly. Similarly, small changes in predator efficiency [3] or wave force [4] lead to the collapse of the mussel bed in models of the intertidal. Of particular relevance when evaluating the occurrence of these phenomena in nature are the spatial patterns that develop near these critical points of transition. Patchiness develops from local interactions in the complete absence of an underlying blueprint, and exhibits the signature of a complete lack of characteristic, or dominant, spatial scale. In particular, the size distribution of gaps, or that of clusters of trees or mussels, exhibits power-law behavior, with patches of all sizes present and no dominant size. Such patterns have been described for intertidal mussel beds [5] and for wind-disturbed [6] and fire-disturbed [7] forests in nature, within the practical limits of a finite range of scales.
Thus, the concept of criticality lies at the interface of two important subjects in ecology, threshold behavior and patchiness. One central question follows: can spatial patterns be useful indicators of the proximity of a system to catastrophic change (see also [8])?
Most ecological examples of criticality pertain to natural and model systems that incorporate disturbance, whether abiotic, such as fires in forests and wave action in the rocky intertidal zone, or biotic, such as disease and predation 5, 9, 10. Disturbance, broadly defined to include biological and physical processes, is fundamental to ecological dynamics, affecting population persistence and species coexistence in dynamic habitats 11, 12, 13, 14, 15. The dynamics of ecological systems at any level of organization typically encompass processes of growth and inhibition that are characteristic of disturbance and recovery. These processes are found, for example, in exploiter–victim (or antagonistic) interactions, in the destruction and recovery of landscapes, and in the adaptive cycle of ecosystems [16].
Although mathematical models of complex systems that incorporate processes of disturbance and recovery are known to exhibit different classes of criticality, the specific ecological mechanisms underlying these differences are not sufficiently recognized. Small differences in the local nature of disturbance and recovery, and in the relative temporal scale of these processes, lead to striking differences in dynamical properties, including the sensitivity to perturbations and the size distribution of temporal change. We argue here that observable properties of patchiness in critical systems, such as spatial power laws and connectedness, are not sufficient to identify these differences. Instead, properties of the processes of disturbance and recovery themselves must be considered to infer properties of the temporal dynamics, and in particular, the possibility of threshold behavior.
Here, we review the different types of critical behavior currently known in ecological systems that incorporate disturbance, including predation, disease and abiotic processes. We use examples from spatial stochastic models, and the systems in nature for which they were proposed, to illustrate three general classes of behavior regarding criticality and associated patterns of spatial self-organization, and to establish both the specific ecological processes underlying these differences and their dynamical consequences*. We rely specifically on lattice (or grid) based models 17, 18, in which the local nature of recovery and disturbance is easily represented. These systems have been used extensively in population and community ecology to consider individual-based representations. They are also relevant to landscape ecology as implementations of habitat destruction and restoration, and further provide a starting point for spatial extensions of the concept of adaptive cycles developed by Holling and Gunderson to address ecosystem resilience [16]. In this concept, as an ecosystem develops, it also becomes increasingly interconnected among its components. Connectedness is inversely related to resilience and reaches a threshold, driving the system to a large-scale collapse and providing opportunities for growth and innovation [16]. We discuss the generality of this inverse relationship in systems with disturbance.
Section snippets
The three classes of criticality
Different types of disturbed systems can be distinguished that differ significantly in their sensitivity to parameter changes and therefore, in their response to external perturbations and environmental change. There are three types of criticality that we consider here: (i) classical criticality; (ii) self-organized (SOC); and (iii) ‘robust’ criticality. Box 1 illustrates classical criticality in the context of disturbed systems, such as wind-disturbed forests. At the critical point,
Well-mixed disturbance
When is criticality in disturbed systems indicative of low resilience, specifically, of the possibility of a sudden state shift for small parameter changes? To address this question, we begin with an example. Wind-disturbed tropical forests have been described as critical systems 6, 23, 24, 25, 26, and viewed as a landscape of trees and gaps corresponding to occupied and unoccupied sites. In real forests, a scale-invariant distribution in tree cover has been observed, suggesting the existence
Distributed disturbance with well-mixed recovery
Criticality and scale invariance are not always associated with high sensitivity to parameter values and can instead be associated with large and unpredictable temporal fluctuations. An example of this type of behavior is given by the dynamics of a childhood disease. Observed patterns for past outbreaks of measles in small islands were shown to occur intermittently in time, with the size of these events lacking a characteristic size [9]. Instead, the size distribution is a power law, implying
Distributed disturbance and local recovery
Is criticality always associated with drastic changes, either in the form of the large temporal fluctuations of self-organized criticality or the high sensitivity to parameters of classical phase transitions? To address this question, we begin with an example of wave disturbance in intertidal mussel beds 5, 34.
In marine rocky intertidal zones, mussels are often the dominant competitors for space, a key limiting resource. Gap formation resulting from wave disturbance and recovery through mussel
Some evolutionary considerations
Evolutionary processes have also been invoked to explain the ubiquity of criticality in biological systems showing phase transitions 20, 37. Rand et al. [38] and Keeling [39] studied a three-state epidemic model in which the local spread of infected individuals (transmissibility) is allowed to mutate. They showed that transmissibility evolves to its critical value associated with scale invariance. Similarly, in models of epidemics where a separation of temporal scales between transmission and
Conclusions
Criticality has been an appealing ecological concept from two different perspectives: first, as an explanation for scale-invariant patterns in nature, and second, as a mechanism underlying drastic change, in the form of either large unpredictable temporal fluctuations (SOC) or sudden state shifts driven by small perturbations (classical phase transitions). We have argued that these two perspectives are not necessarily linked: properties of the spatial patterns, including power laws and high
Acknowledgements
We thank Manojit Roy for discussions on the subject and Andy Dobson for comments on an earlier version of the article. We are pleased to acknowledge support from the James S. McDonnell Foundation through a Centennial Fellowship in Global and Complex Systems to M.P, and from the Fonds Québécois de la Recherche sur la Nature et les Technologies (FQRNT) through a grant to F.G.
Glossary
- Connectedness:
- two sites are connected if they belong to the same patch (or cluster). In lattice models, a patch can be defined by starting at a given site and adding all the sites among its neighbors that are occupied by that same species or state (e.g. mussel bed). The neighborhood itself is defined by the local rules of interaction. One measure of connectedness is the size of the largest patch relative to the total size of the system.
- Criticality:
- in statistical mechanics, the state of a system
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