Resilience and vulnerability from a
stochastic controlled dynamical
system perspective
Charles Rougé, Jean-Denis Mathias and Guillaume Deffuant
Pour mieux
affirmer
ses missions,
le Cemagref
devient Irstea
www.irstea.fr
2
The viability framework for resilience
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Example: The case of lake eutrophication
(Carpenter et al., 1999)
Phosphorus
input L
Inflow
Bounded!!! (by U>0)
Lake
(Phosphorus concentration P)
Algae
Outflow
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Deterministic viability: single trajectories
Event
Events
Part I
Resilience of a stochastic
controlled dynamical system
Pour mieux
affirmer
ses missions,
le Cemagref
devient Irstea
www.irstea.fr
6
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
Part II
Vulnerability as a measure of
future harm
Pour mieux
affirmer
ses missions,
le Cemagref
devient Irstea
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13
Ecological harm
Quadratic increase with P
Harm: a value judgement on a state
Threshold of harm
Properties
Economic harm
Increases linearly as L decreases
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Defining vulnerability
1) One associates harm values to a trajectory:
 Sum of static harm values (cost criterion)
 Crossing of a threshold (viability criterion)
2) Vulnerability is 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
3) Interest in low-vulnerability kernels.
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Vulnerability as total cost
Τ=100
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Vulnerability as exit probability
Stochastic viability
kernel!!!
Part III
Towards a resilience-vulnerability
framework
Pour mieux
affirmer
ses missions,
le Cemagref
devient Irstea
www.irstea.fr
18
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
Resilience: capacity to recover
?
Dynamic safety criterion
(or property of interest)
Low-vulnerability zone
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.