1. Criticality and disturbance in spatial ecological systems Mercedes Pascual & Frederic Guichard TREE, February 2005 dominant spatial scaledominant patch sizedominant size phase transitionthresholdthreshold behavior self-organizedpercolationpercolation-type transition connectednessresilienceresilience inversely correlated with connectedness long-range correlationslong-range correlations arising from short-range or local interactions long-range correlations in the distribution of organisms statistically correlated over large scalesshort-range facilitation long-range inhibitionpower lawpower-law scaling scale invariancescaling holdsbroad scaling regionparameter space fine-tuning of parameter spaceforcing parameter sensitive to parameter changeshigh sensitivity is low resilience emergent propertyS-I and S-I-R modelexploiter-victim model Ising modelperturbationadaptive cycle of ecosystems spatial stochastic modellattice modelgrid-based model robustdisturbance lifespan2-state, 3-state system (or more) separation of time scalesdouble separation of time scalesfast vs. slow process well-mixed disturbance and recovery vs. distributed disturbance and local recovery environmental gradients

2. What is criticality? Classical physics/statistical mechanics system poised at a phase transition, e.g., gas-liquid local interactions, but long-range correlation (long-range order) appears power-law distribution and scale invariance? Self-organized What is self-organization? “…the development of a system from an unpatterned to a patterned state without the intervention of an external control.” --Richards 2002 “…the spontaneous emergence of new structures and new forms of behavior in open systems far from equilibrium, characterized by internal feedback loops and described mathematically by nonlinear equations.” --Capra 1996 Self-organized criticality “…the internal dynamics of a system organises itself into a pervasive critical state.” --Richards 2002 “…the system takes itself to the critical state through its own dynamics and a slow external forcing…” --Pascual and Guichard 2005 Benard cells/convection

3. Pascual and Guichard: Ecological systems Threshold behavior near state shift Spatial pattern – patchiness with power-law distribution & scale invariance – arising from local interactions Disturbance & recovery systems “Can spatial patterns be useful indicators of the proximity of a system to catastrophic change?” Three types of criticality for D&R systems: classical, self-organized, and robust Spatial stochastic (lattice/grid) models & empirical studies

4. Classical well-mixed disturbance well-mixed recovery process? 2-state model disturbance (spread?) rate >> recovery rate power-law pattern only for narrow range (combo) of D&R rates small change in D or R rate = “collapse” Self-Organized distributed disturbance well-mixed recovery 3-state model: disturbed state disturbance spread rate >> recovery rate >> disturbance intro rate power-law pattern for wide range of D&R values, as long as “separated”…? large, intermittent temporal fluctuations but no collapse Robust distributed disturbance local recovery process 3-state model power-law pattern for wide range of D&R values no collapse; no large, intermittent temporal fluctuations

6. Criticality Type Dist. TypeRecovery Type Dist. Intro Rate Dist. Rate Recovery Rate Spatial power-law distribution Temporal power-law distribution ClassicalWell-mixedWell- mixed? ??FastSlowYesNo? ??Well-mixedLocal?? Self- Organized DistributedWell- mixed SlowerFastSlowYesYes? RobustDistributedLocal??FastFast?YesNo?

7. Conclusions Spatial pattern alone is not enough to characterize a system as exhibiting threshold behavior (i.e., potential state shift) Disturbance and recovery processes/rates must be considered Framework of possible types of criticality: future work should explore the validity of these in more detail, in both models and natural systems

8. --from Turner et al. 1993 Questions Do systems self-organize to a state that confers stability, resilience, or adaptability? Does the idea of criticality(-ies) lead us any closer to generality? How does this overlap with or extend Turner et al. 1993? What does this mean for IDH? What are the implications for restoration?

9. Artwork by Elaine Wiesenfeld (from Bak, How Nature Works)