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Lake eutrophication: using resilience evaluation tocompute sustainable policies
Laetitia Chapel Sophie Martin Guillaume Deffuant
Laboratory of Engineering for Complex Systems (LISC)Cemagref
10th International Conference on Environmental Science andTechnology - September 2007
Outline
1 Lake eutrophication
2 Viability kernel
3 Resilience value computation
4 Summary
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 2 / 19
Lake eutrophicationDefinition
Oligotrophic lake
clear water
low input nutrient
high economic value
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 3 / 19
Lake eutrophicationDefinition
Eutrophic lake
turbid water
high input nutrient
low economic value
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 4 / 19
Lake eutrophicationDefinition
Phosphorus P in the lake
is the most critical nutrient
used by the farmers in form of fertilizer or animal feedsupplements
excess P accumulates in the soil and is transported to thelakes
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 5 / 19
Lake eutrophicationSimplified model in 3 dimensions (L, P, M)
x ′(t) =
L′(t) = u, u ∈ [−VL; +VL]P ′(t) = −(s + h)P(t) + L(t) + rM(t)f (P(t))M ′(t) = −kM(t) + sP(t)− rM(t)f (P(t))
(1)
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 6 / 19
Lake eutrophicationProperty of interest
Property of interest
the lake must remain in an oligotrophic state (populationpoint of view)P ∈ [0;Pmax ]
the profitability of the farmers activities must be ensuredL ∈ [Lmin; Lmax ]
We evaluate the resilience of this property of interest
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 7 / 19
Outline
1 Lake eutrophication
2 Viability kernel
3 Resilience value computation
4 Summary
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 8 / 19
Viability kernel
Aim: define levels of P, M and L that are compatible with theobjective to maintain the property of interestConce
ntr
atio
nofphosp
hor
us
(P)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Viab(K) K
Eutrophic lakeN
o re
ntab
ility
for t
he fa
rme r
s
M=0
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 allowsstaying in the viability constraint setConce
ntr
atio
nofphosp
hor
us
(P)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Viab(K) KNo
rent
abili
ty fo
r the
farm
e rs
Eutrophic lake
u=-VL
u=0 u=+VL
M=0
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 theproperty of interest can be maintainedConce
ntr
atio
nofphosp
hor
us
(P)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Viab(K) K
Eutrophic lakeN
o re
ntab
ility
for t
he fa
rme r
s
M=0
Input of phosphorus in the lake (L)
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 11 / 19
Outline
1 Lake eutrophication
2 Viability kernel
3 Resilience value computation
4 Summary
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 12 / 19
Resilience value computationDefinition
Resilience: capacity of the system to maintain its property ofinterest in spite of disturbance
Martin proposed a mathematical interpretation of resilience
S. MartinThe cost of restoration as a way of defining resilience: a viabilityapproach applied to a model of lake eutrophication.Ecology and Society, 9(2), 2004.
• Resilience: inverse of the cost of restoration of the property ofinterest
• Based on the viability theory framework
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 13 / 19
Resilience value computationCost function
Viability kernel is the 0-level of the cost function
Conce
ntr
atio
nofphosp
hor
us
(P)
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
Viab(K) :Null Cost K
Eutrophic lake: Ecological costN
o re
ntab
ility
: Eco
nom
ic c
ost
Outside Viab(K):Management cost M=0
Input of phosphorus in the lake (L)
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Resilience value computationMethods
Algorithm to compute resilience values
Approximating viability kernel algorithm can be used tocompute 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
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Resilience value computationRestoration costs
Starting from a non-viable state, the system is doomed toleave K and we look for policies that bring back the systeminside Viab(K )
Conce
ntr
atio
nof(P
) 4
0
M=1
0 1
Input of phosphorus in the lake (L)
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 16 / 19
Resilience value computationResilience
Inverse of the cost to restore the property of interest, lost dueto exogenous disturbances
Maximal disturbance: jump of magnitude P = 0.5Conce
ntr
atio
nof(P
) 4
0
M=1
0 1
Input of phosphorus in the lake (L)
Chapel, Martin & Deffuant Lake eutrophication: using resilience evaluation 17 / 19
Outline
1 Lake eutrophication
2 Viability kernel
3 Resilience value computation
4 Summary
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 ofthe approach
Resilience values allow to define sustainable policies, with theminimal cost
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