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Resilience and vulnerability from a stochastic controlled dynamical system perspective

Charles Rougé, Jean-Denis Mathias and Guillaume Deffuant

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The viability framework for resilience

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Example: The case of lake eutrophication(Carpenter et al., 1999)

Lake

(Phosphorus concentration P)Inflow Outflow

Algae

Phosphorus input L

Bounded!!! (by U>0)

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Deterministic viability: single trajectories

Event

Events

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Part I

Resilience of a stochastic controlled dynamical system

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Impact of uncertainty on the viability kernel

 

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Multiplicity of recovery trajectories

Events

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Resilience in a stochastic dynamical system

Recovery is defined by getting back to the stochastic viability kernel

Centrality of the probability of recovery after a given date: the Probability of resilience

No longer a unique measure of recovery but possibility to derive statistics.

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Resilience statistic:expected recovery date

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Resilience statistic: maximal recovery time (99% confidence)

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Resilience statistic: probability of resilience

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Part II

Vulnerability as a measure of future harm

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Harm: a value judgement on a state

Economic harm

Increases linearly as L decreases

Eco

logi

cal h

arm

Qua

drat

ic in

crea

se w

ith P

Threshold of harm

Properties

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Defining vulnerability

1) One associates harm values to a trajectory:Þ Sum of static harm values (cost criterion)Þ Crossing of a threshold (viability criterion)

2) Vulnerability is a statistic on the distribution of harm values:Þ Expected value of the costÞ Exit probability (crossing of a threshold) Þ Value-at-risk (e.g. worst 1%) of the cost

3) Interest in low-vulnerability kernels.

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Vulnerability as total costΤ=100

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Vulnerability as exit probability

Stochastic viability kernel!!!

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Part III

Towards a resilience-vulnerability framework

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Conceptual definitions

Resilience: capacity to keep or recover properties after a hazard, disturbance or change.

Probability of recovery at 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

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Combining resilience and vulnerability

Dynamic safety criterion

(or property of interest)

Low-vulnerability zone

Resilience: capacity to recover

Vulnerability: harm experienced

(equivalent to a restoration cost)

?

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The proposed framework

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Take home messages

Complimentarity of resilience and vulnerability

The notion of low-vulnerability kernel generalizes that of viability kernel.

Resilience is the ability to get back to this safety set after a disturbance or a change.

Vulnerability is a statistic based on the harm values associated to the possible trajectories.

Choice of the strategy dependent on the indicator.