Criticality and disturbance in spatial ecological systems

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Criticality and disturbance in spatial ecological
systems Mercedes Pascual Frederic
Guichard TREE, February 2005
dominant spatial scale dominant patch
size dominant size phase transition threshold thr
eshold behaviorself-organized percolation percol
ation-type transition connectedness resilience r
esilience inversely correlated with
connectednesslong-range correlations long-range
correlations arising from short-range or local
interactions long-range correlations in the
distribution of organisms statistically
correlated over large scales short-range
facilitation long-range inhibition power
law power-law scaling scale invariance scaling
holds broad scaling region parameter
space fine-tuning of parameter space forcing
parameter sensitive to parameter changes high
sensitivity is low resilience emergent
property S-I and S-I-R model exploiter-victim
modelIsing model perturbation adaptive cycle of
ecosystems spatial stochastic model lattice
model grid-based modelrobust disturbance
lifespan 2-state, 3-state system (or
more) separation of time scales double
separation of time scales fast vs. slow
processwell-mixed disturbance and recovery vs.
distributed disturbance and local
recovery environmental gradients
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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

Benard cells/convection
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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 DR systems
  classical, self-organized, and robust
• Spatial stochastic (lattice/grid) models
  empirical studies

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• Classical
• well-mixed disturbance
• well-mixed recovery process?
• 2-state model
• disturbance (spread?) rate gtgt
• recovery rate
• power-law pattern only for narrow
• range (combo) of DR rates
• small change in D or R rate collapse
• Self-Organized
• distributed disturbance
• well-mixed recovery
• 3-state model disturbed state
• disturbance spread rate gtgt
• recovery rate gtgt
• disturbance intro rate
• power-law pattern for wide range of
• DR values, as long as separated?
• large, intermittent temporal fluctuations

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• 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

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• 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?

--from Turner et al. 1993
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Artwork by Elaine Wiesenfeld (from Bak, How
Nature Works)