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www.irstea.fr Pour mieux affirmer ses missions, le Cemagref devient Irstea Resilience and vulnerability from a stochastic controlled dynamical system perspective Charles Rougé, Jean-Denis Mathias and Guillaume Deffuant

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2 The viability framework for resilience

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3 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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4 Deterministic viability: single trajectories Event Events

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www.irstea.fr Pour mieux affirmer ses missions, le Cemagref devient Irstea Part I Resilience of a stochastic controlled dynamical system

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6 Impact of uncertainty on the viability kernel

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7 Multiplicity of recovery trajectories Events

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8 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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9 Resilience statistic:expected recovery date

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10 Resilience statistic: maximal recovery time (99% confidence)

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11 Resilience statistic: probability of resilience

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www.irstea.fr Pour mieux affirmer ses missions, le Cemagref devient Irstea Part II Vulnerability as a measure of future harm

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13 Harm: a value judgement on a state Economic harm Increases linearly as L decreases Ecological harm Quadratic increase with P Threshold of harm Properties

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14 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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15 Vulnerability as total cost Τ=100

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16 Vulnerability as exit probability Stochastic viability kernel!!!

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www.irstea.fr Pour mieux affirmer ses missions, le Cemagref devient Irstea Part III Towards a resilience-vulnerability framework

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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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19 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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20 The proposed framework

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21 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.

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