mathematical modeling for environmental...

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Mathematical Modeling for Environmental

Economics and Sustainable ManagementOutils mathématiques pour la décision en environnement

Master EDDEE

Luc Doyen,Michel De Lara

Doyen, DeLara Modeling for the Sustainable Management

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Trimester Mathematics of Bio-economics at IHP

Doyen, DeLara Modeling for the Sustainable Management

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Program

Monday 7, January 10h00-13h00 : IHP : Course Doyen-De Lara Introduction todecision modelling for sustainable management of natural resources

Monday 14, January 9h00-12h00 : ENGREF : Computer Session De Lara-Doyen

Introduction to the scientific software

Monday 21, January 9h00-12h00 : IHP Course Doyen-De Lara Controlleddynamics and equilibria

Monday 28, January 9h00-12h00 : ENGREF : Computer Session De Lara-DoyenEquilibria

Monday 4, February 9h00-12h00 : IHP Course Doyen-De Lara Optimality andsustainability

Monday 11, February 9h00-12h00 : ENGREF : Computer Session DeLara-Doyen Optimality

Monday 18, February 9h00-12h00 : IHP Course Doyen-De Lara Viability

Monday 25, February 9h00-12h00 : ENGREF : Computer Session DeLara-Doyen Viability

Doyen, DeLara Modeling for the Sustainable Management

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Some References

DeLara M.. & Doyen L., 2008, Sustainable Management of NaturalResources Mathematical Models and Methods, Springer.

Clark C.W., (1976), Mathematical Bio-economics : The Optimal

Management of Renewable Resource, J. Wiley & Sons, New York

Heal G. (1998) Valuing the Future, Economic Theory and Sustainability.Columbia University Press, New York.

Conrad J. M. 1999, Resource Economics, Cambridge University Press.

Doyen, DeLara Modeling for the Sustainable Management

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Outline

General issues

Deterministic models in discrete time

– General framework ; Examples– Equilibrium and stability.– Viability.– Inter-temporal optimality.

Models under uncertainty

– Robust approach.– Stochastic approach.

Partial information

Strategic interactions

Doyen, DeLara Modeling for the Sustainable Management

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Environmental issues

Management of exhaustible resources : oil, mining, ...

Pollution management : ex : C02

Management of renewable resources : Fisheries, wildlife,forest, agroecology

Doyen, DeLara Modeling for the Sustainable Management

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Sustainable management of biodiversity

Global changes in ecosystems

Alarming trends : MEA, IUCN, ...

Major concerns for ecosystem services

=⇒ bio-economic risk

Sustainable management

Doyen, DeLara Modeling for the Sustainable Management

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IPBES

"From IPCC"

To

Doyen, DeLara Modeling for the Sustainable Management

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Biodiversity scenarios

Cheung et al, Fish and Fisheries, 2009Doyen, DeLara Modeling for the Sustainable Management

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Bio-economic modeling

Doyen, DeLara Modeling for the Sustainable Management

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Bio-economic policies

Three major components :

Scientific knowledge :mechanisms, models, data.

Shared intertemporal objectives :ex : Johannesburg 2002 : MSY by 2015

Bio-economic instruments

standards : quotas, MPA, ...monetary : landing taxes, ITQ, ...informational : MSC,...

Doyen, DeLara Modeling for the Sustainable Management

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Important goals

Sustainability

– Reconciliation

Economy–Social-Ecology

– Intergenerational equity

– Intragenerational equity

Resilience

Precaution

Doyen, DeLara Modeling for the Sustainable Management

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Modeling requirements

Systemic approach

Dynamics

Uncertainties

Decision

Evaluation

Multi-criteria

– Ecology– Socio-Economy

Short and long term horizon

Doyen, DeLara Modeling for the Sustainable Management

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Decision approaches for sustainability

Equilibriumand stability :

Steady state,

MSY

Schaefer, 1954

IntertemporalOptimality

Present Value

CB, CE

Clark, 1976

Maximin criteria

....

Constraints, viabilityand risk

PVA

Beissinger, 2002

Viable control

TWA

SMS

Bishop, 1978

Doyen, DeLara Modeling for the Sustainable Management

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Control theory

Problem :Find controls to achieve various goals for states of a system

Three main ingredients :

– Controlled dynamics– Constraints– Criteria to optimize

Doyen, DeLara Modeling for the Sustainable Management

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Control theory in discrete time

Control theoryin discrete time

blanco

Economics Ecology Modelingdecision Life-cycle Simulationsunder meta-population

uncertainty

Doyen, DeLara Modeling for the Sustainable Management

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Examples for natural resource management

Stylized models :

– Exploitation of an exhaustible resource

– Management of a renewable resource

– Mitigation policies for CO2 emissions

Doyen, DeLara Modeling for the Sustainable Management

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Exploitation of an exhaustible resource

Doyen, DeLara Modeling for the Sustainable Management

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Management of a renewable resource

Doyen, DeLara Modeling for the Sustainable Management

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Mitigation policies for CO2 emissions

Doyen, DeLara Modeling for the Sustainable Management

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A controlled dynamics in discrete time

Difference equation :

{

xi(t + 1) = Fi(t, x(t), u(t)), t = 0, . . . , T (1)

where

– the state, x(t) = (x1(t), . . . , xn(t)) ∈ X = Rn ;– the control or decision, u(t) = (u1(t), . . . , un(t)) ∈ U = Rp ;– the time Horizon T

Doyen, DeLara Modeling for the Sustainable Management

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State constraints

The admissible states

x(t) ∈ A(t), t = 0, . . . , T − 1 , (2)

generally specified under inequality form

{

gj(t, x(t)) ≤ 0gk(t, x(t)) = 0 .

In many examples, a constant set :

x(t) ∈ A

Doyen, DeLara Modeling for the Sustainable Management

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Final state constraints

Final state reaches some target C ⊂ X :

x(T ) ∈ C (3)

Generally specified under inequality form

gj(x(T )) ≤ 0

gk(x(T )) = 0.

Doyen, DeLara Modeling for the Sustainable Management

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Decision or control constraints

The admissible decisions :

u(t) ∈ B(t, x(t)), t = 0, . . . , T − 1 (4)

generally specified under inequality form

{

gj(t, x(t), u(t)) ≤ 0gk(t, x(t), u(t)) = 0 .

Frequently, a constant set of feasible control

u(t) ∈ B

Doyen, DeLara Modeling for the Sustainable Management

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The feasible decisions and sustainability

The feasible strategy : u(.) = (u(0), u(1), . . . , u(T )) such that

– The constraints (3), (2) and (4)– where x(.) with the dynamics (1)

Feasibility set :

Tad =

{

(x(.), u(.))

∣

∣

∣

∣

(1), (3), (2), (4) hold true

}

Viability or invariance approach

A particular case : equilibria

Doyen, DeLara Modeling for the Sustainable Management

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Optimality

An inter-temporal criterion

π

(

x(0), x(1), . . . , x(T ), u(0), u(1), . . . , u(T − 1)

)

The optimal control problem :

π(x⋆(·), u⋆(·)) = max(x(.),u(.))∈Tad

π(x(·), u(·)).

where

– u⋆(·) = (u⋆(0), u⋆(1), . . .) optimal decision sequence,– x⋆(·) = (x⋆(0), x⋆(1), . . .) optimal state sequence.

Doyen, DeLara Modeling for the Sustainable Management

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Some criteria

Present value

π(x(·), u(·)) =

T−1∑

t=0

ρtL(x(t), u(t))

Maximin

π(x(·), u(·)) = mint=0,...,T−1

L(t, x(t), u(t)).

Greenπ(x(·), u(·)) = M(T , x(T ))

Chichilnisky

π(x , u) = θ

T−1∑

t=0

L(t, x(t), u(t)) + (1 − θ)M(T , x(T ))

Doyen, DeLara Modeling for the Sustainable Management

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A useful result

miny∈A

f (y) = − maxy∈A

−f (y)

Doyen, DeLara Modeling for the Sustainable Management

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Simulations can help

Given x0, compute outputs of specific strategies

u(t)u(t, x)

Example : scenarios of mitigation policies

Doyen, DeLara Modeling for the Sustainable Management

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SCILAB Code

t0=1990 ; T=2100 ; M0=354 ;alp=0.471 ; Minf=280 ; d=1/120 ;

EBAU0=7.2 ;g=0.01 ;

//

function y=EBAU(t) ; y=EBAU0*(1+g)^(t-t0); endfunction

//

function y=f(t,M,ab) ; y=M+alp*EBAU(t)*(1-ab)-d*(M-Minf) ; endfunction

//

ab=zeros(1,T) ; // for instance

x(t0)=M0 ;

for (t=t0 :1 :T)

x(t+1)=f(t,x(t),ab(t)) ;

end

//

plot2d(t0 :T+1,x(t0 :T+1),rect=[t0,0,T+1,1000])

Doyen, DeLara Modeling for the Sustainable Management

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The sensitivity problem

Pb : The outputs g(x(t), u(t)), L(x(t), u(t)), π depend on :

the parameters of the modelthe functional forms of the modelthe initial conditions

Doyen, DeLara Modeling for the Sustainable Management

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The sensitivity problem

Example : Different populationdynamics with r = 3, K = 10

Doyen, DeLara Modeling for the Sustainable Management

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What about complexity ?

More complex models :

Exploitation of an exhaustible resource with capital (DHS)

A trophic web and sustainable use values

An exploited metapopulation and PA management

SSVPA Model

Doyen, DeLara Modeling for the Sustainable Management

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Exploitation of an exhaustible resource with capital (DHS)

Doyen, DeLara Modeling for the Sustainable Management

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A trophic web and sustainable use values

Doyen, DeLara Modeling for the Sustainable Management

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Simulations can help

Cheung et al, Fish and Fisheries, 2009

Doyen, DeLara Modeling for the Sustainable Management

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Simulations can help : Fisheries in French Guiana

Cissé et al., Environment and Development Economics, 2013

Biodiversity

Seafood production

Doyen, DeLara Modeling for the Sustainable Management

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Simulations can help ! ! Farming land-use and birds

Mouysset et al., Ecological Economics, 2011

2008ւ ց

HQE2 2050 ENERGY 2050

Doyen, DeLara Modeling for the Sustainable Management

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Simulations can help ! ! Fisheries in Bay of Biscay

Doyen et al., Ecological Economics, 2012

Co-viability

SSB species 1 (Tons)

time(years)

2025202020152010

250

200

150

100

50

0

time(years)

SSB species 2 (Tons)

2025202020152010

40

35

30

25

20

15

10

5

0

20242020201620122008

3.5e+007

3.0e+007

2.5e+007

2.0e+007

1.5e+007

1.0e+007

5.0e+006

0.0e+0002028

profit fleet 2 (Keuro)

20242020201620122008

1.2e+007

1.0e+007

8.0e+006

6.0e+006

4.0e+006

2.0e+006

0.0e+0002028

Status Quo 2008

SSB species 1 (Tons)

time(years)

2025202020152010

250

200

150

100

50

0

time(years)

SSB species 2 (Tons)

2025202020152010

40

35

30

25

20

15

10

5

0

20242020201620122008

4.5e+007

4.0e+007

3.5e+007

3.0e+007

2.5e+007

2.0e+007

1.5e+007

1.0e+007

5.0e+006

0.0e+0002028 202820242020201620122008

8e+006

6e+006

4e+006

2e+006

0e+000

-2e+006

-4e+006

-6e+006

-8e+006

-1e+007

profit fleet 2 (Keuro)

NPV scenario

SSB species 1 (Tons)

time(years)

2025202020152010

250

200

150

100

50

0

time(years)

SSB species 2 (Tons)

2025202020152010

40

35

30

25

20

15

10

5

0

profit fleet 1 (Keuro)

202820242020201620122008

9e+007

8e+007

7e+007

6e+007

5e+007

4e+007

3e+007

2e+007

1e+007

0e+000

-1e+0072028202420202016

profit fleet 2 (Keuro)

20122008

9e+006

8e+006

7e+006

6e+006

5e+006

4e+006

3e+006

2e+006

1e+006

0e+000

Doyen, DeLara Modeling for the Sustainable Management

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But which synthetic indicators ? ? ?

Mouysset et al., Ecological indicators, 2011

Doyen, DeLara Modeling for the Sustainable Management

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Open loop or closed loop ?

Open-loop : time-dependent sequences :

u : t → u(t) .

Closed-loop ≈ feedback

u : (t, x) → u(t, x) .

– More robust– Dynamic programming methods.

Doyen, DeLara Modeling for the Sustainable Management

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Decision tree

Finite countable decisions(ex u ∈ U = {0, 1})

Given x0

Tree :

– nodes = states x(t)– edges = decisions u(t)

Evaluate the performance π onevery path

Doyen, DeLara Modeling for the Sustainable Management

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The curse of the dimensionality

A binary decision u ∈ {0, 1} on horizon T

=⇒ 2T possible sequences u(.)

On a computer :

– ram : 8 GBytes = 8(1 024)3 = 233 bytes– a double-precision real : 8 bytes = 23 bytes.

=⇒ 230 double-precision reals

Doyen, DeLara Modeling for the Sustainable Management