Lake eutrophication: using resilience evaluation to compute sustainable policies Laetitia Chapel Sophie Martin Guillaume Deffuant Laboratory of Engineering for Complex Systems (LISC) Cemagref 10th International Conference on Environmental Science and Technology - September 2007
Outline Lake eutrophication 1 Viability kernel 2 Resilience value computation 3 Summary 4 Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 2 / 19
Lake eutrophication Definition Oligotrophic lake clear water low input nutrient high economic value Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 3 / 19
Lake eutrophication Definition Eutrophic lake turbid water high input nutrient low economic value Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 4 / 19
Lake eutrophication Definition Phosphorus P in the lake is the most critical nutrient used by the farmers in form of fertilizer or animal feed supplements excess P accumulates in the soil and is transported to the lakes Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 5 / 19
Lake eutrophication Simplified model in 3 dimensions ( L , P , M ) L ′ ( t ) = u , u ∈ [ − VL ; + VL ] x ′ ( t ) = P ′ ( t ) = − ( s + h ) P ( t ) + L ( t ) + rM ( t ) f ( P ( t )) (1) M ′ ( t ) = − kM ( t ) + sP ( t ) − rM ( t ) f ( P ( t )) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 6 / 19
Lake eutrophication Property of interest Property of interest the lake must remain in an oligotrophic state (population point of view) P ∈ [0; P max ] the profitability of the farmers activities must be ensured L ∈ [ L min ; L max ] We evaluate the resilience of this property of interest Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 7 / 19
Outline Lake eutrophication 1 Viability kernel 2 Resilience value computation 3 Summary 4 Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 8 / 19
Viability kernel Aim: define levels of P , M and L that are compatible with the objective to maintain the property of interest Concentration of phosphorus (P) 0,8 0,7 Eutrophic lake 0,6 No rentability for the farmers 0,5 0,4 M=0 0,3 0,2 0,1 Viab(K) K 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 9 / 19
Viability kernel Viable state: there exists at least one evolution which allows staying in the viability constraint set Concentration of phosphorus (P) 0,8 0,7 Eutrophic lake 0,6 No rentability for the farmers 0,5 u=0 u=+VL 0,4 M=0 0,3 0,2 u=-VL 0,1 Viab(K) K 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 10 / 19
Viability kernel Viability kernel: set of all viable states = states for which the property of interest can be maintained Concentration of phosphorus (P) 0,8 0,7 Eutrophic lake 0,6 No rentability for the farmers 0,5 0,4 M=0 0,3 0,2 0,1 Viab(K) K 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 11 / 19
Outline Lake eutrophication 1 Viability kernel 2 Resilience value computation 3 Summary 4 Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 12 / 19
Resilience value computation Definition Resilience: capacity of the system to maintain its property of interest in spite of disturbance Martin proposed a mathematical interpretation of resilience S. Martin The cost of restoration as a way of defining resilience: a viability approach applied to a model of lake eutrophication . Ecology and Society, 9(2), 2004. • Resilience: inverse of the cost of restoration of the property of interest • Based on the viability theory framework Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 13 / 19
Resilience value computation Cost function Viability kernel is the 0-level of the cost function Concentration of phosphorus (P) 0,8 0,7 Eutrophic lake: Ecological cost 0,6 No rentability: Economic cost 0,5 0,4 Outside Viab(K) : M=0 Management cost 0,3 0,2 Viab(K) : 0,1 K Null Cost 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 14 / 19
Resilience value computation Methods Algorithm to compute resilience values Approximating viability kernel algorithm can be used to compute resilience values Use a classification method: Support Vector Machines We propose a new algorithm that • deals with more realistic systems • allow to introduce uncertainties on the parameters Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 15 / 19
Resilience value computation Restoration costs Starting from a non-viable state, the system is doomed to leave K and we look for policies that bring back the system inside Viab ( K ) 4 Concentration of (P) M=1 0 0 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 16 / 19
Resilience value computation Resilience Inverse of the cost to restore the property of interest, lost due to exogenous disturbances Maximal disturbance: jump of magnitude P = 0 . 5 4 Concentration of (P) M=1 0 0 1 Input of phosphorus in the lake (L) Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 17 / 19
Outline Lake eutrophication 1 Viability kernel 2 Resilience value computation 3 Summary 4 Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 18 / 19
Summary Resilience can be defined thanks to viability theory We propose a new algorithm that enhances the potential of the approach Resilience values allow to define sustainable policies, with the minimal cost Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 19 / 19
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