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Resilience and Vulnerability in Stochastic Controlled Systems

Explore resilience and vulnerability from a dynamic systems perspective. Understand stochastic viability frameworks and recovery trajectories in controlled dynamical systems. Analyze harm values, exit probabilities, and cost distributions to assess vulnerability. The proposed framework emphasizes the interplay between resilience and vulnerability to define low-vulnerability zones.

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Resilience and Vulnerability in Stochastic Controlled Systems

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  1. Resilience and vulnerability from a stochastic controlled dynamical system perspective Charles Rougé, Jean-Denis Mathias and Guillaume Deffuant

  2. The viabilityframework for resilience

  3. Example: The case of lake eutrophication (Carpenter et al., 1999) Phosphorus input L Bounded!!! (by U>0) Lake (Phosphorus concentration P) Outflow Inflow Algae

  4. Deterministicviability: single trajectories Event Events

  5. Part I Resilience of a stochastic controlled dynamical system

  6. Impact of uncertainty on the viabilitykernel

  7. Multiplicity of recoverytrajectories Events

  8. Resilience in a stochasticdynamical system Recoveryisdefined by getting back to the stochasticviabilitykernel Centrality of the probability of recoveryafter a given date: the Probability of resilience No longer a unique measure of recovery but possibility to derivestatistics.

  9. Resiliencestatistic:expectedrecovery date

  10. Resiliencestatistic: maximal recovery time (99% confidence)

  11. Resiliencestatistic: probability of resilience

  12. Part II Vulnerability as a measure of future harm

  13. Harm: a value judgement on a state Threshold of harm Properties Ecologicalharm Quadraticincreasewith P Economicharm Increaseslinearly as L decreases

  14. Definingvulnerability One associatesharm values to a trajectory: • Sum of staticharm values (costcriterion) • Crossing of a threshold (viabilitycriterion) Vulnerabilityis a statistic on the distribution of harm values: • Expected value of the cost • Exit probability (crossing of a threshold) • Value-at-risk (e.g. worst 1%) of the cost Interest in low-vulnerabilitykernels.

  15. Vulnerability as total cost Τ=100

  16. Vulnerability as exit probability Stochasticviabilitykernel!!!

  17. Part III Towards a resilience-vulnerability framework

  18. Conceptualdefinitions Resilience:capacity to keep or recoverpropertiesafter a hazard, disturbance or change. • Probability of recoveryat date t • Statistic on a recovery time distribution Vulnerability: a measure of future harm(Hinkel, 2011). • Statistic on an exit probability • Statistic on a cost distribution

  19. Combiningresilience and vulnerability Resilience: capacity to recover Vulnerability: harmexperienced (equivalent to a restorationcost) ? Dynamicsafetycriterion (or property of interest) Low-vulnerability zone

  20. The proposedframework

  21. Take home messages Complimentarityof resilience and vulnerability The notion of low-vulnerabilitykernelgeneralizesthat of viabilitykernel. Resilienceis the ability to get back to thissafety set after a disturbance or a change. Vulnerabilityis a statisticbased on the harm values associated to the possible trajectories. Choice of the strategydependent on the indicator.

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