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Did mathematical modelling really matter for fishery policy purposes ?

Nadia Raïssi

Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA)

Premier Congrés Franco‘Marocain de Mathématiques Appliquées 19 Avril 2018

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Motivation

◮ Sustainability Management stocks’s conservation necessity of the "pur et dur" rebuilding program for overexploited capture fishery resources

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Motivation

◮ Sustainability Management stocks’s conservation necessity of the "pur et dur" rebuilding program for overexploited capture fishery resources ◮ Bioeconomic Modelling Decision Support ◭

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

conceptual framework

1931 with Hotelling works = natural resource economics.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

conceptual framework

1931 with Hotelling works = natural resource economics. 1955 static analysis Scott and Gordon MSY

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

conceptual framework

1931 with Hotelling works = natural resource economics. 1955 static analysis Scott and Gordon MSY 1975 Clark and Munro brought the theories capital and investment into fisheries economics; dynamical analysis.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Gordon 1956 "The conservation ...requires a dynamic formulation. . . for use at a future date . . . it is necessary to arrive at an optimum, which is a catch per unit of time, and one must reach this objective through consideration of the interaction between the rate of catch, the dynamics of fish populations, and the economic time- preference schedule of the community or the interest rate on invested capital. This is a very complicated problem and I suspect that we will have to look to the mathematical economists for assistance in clarifying it"

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

1975 Bioeconomic management school

created by C.Clark

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Clark’s Fundamental Fishery Model FFM

Resource dynamic

˙ X = F(X) − h(X) X(0) = X0 0 ≤ h ≤ hmax (1)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Clark’s Fundamental Fishery Model FFM

Resource dynamic

˙ X = F(X) − h(X) X(0) = X0 0 ≤ h ≤ hmax (1)

Income

Max ∞ e−δt(p − c(X))h(X)dt

• .

(2)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

MRAP’solution

bionomic equilibrium

F ′(X ⋆) − c′(X ⋆)(F(X ⋆) p − c(X ⋆) = δ. (3) capital theory

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

MRAP’solution

bionomic equilibrium

F ′(X ⋆) − c′(X ⋆)(F(X ⋆) p − c(X ⋆) = δ. (3) capital theory

MRAP

h(X) =    hmax, if X > X ⋆; F(X ⋆), if X = X ⋆; 0, if X < X ⋆. (4) investment theory

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

key’s indicators in Morocco

≃ 15% GDP 7% Annual growth turnover; 10% exportation 10% population lives on this activity.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Limits

the demand for harvested fish and the supply of fishing effort are perfectly elastic.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Limits

the demand for harvested fish and the supply of fishing effort are perfectly elastic. the produced capital and the human capital employed in the fishery are perfectly malleable.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Limits

the demand for harvested fish and the supply of fishing effort are perfectly elastic. the produced capital and the human capital employed in the fishery are perfectly malleable. research perspectives "...non-malleable human capital—an issue of particular importance to developing coastal states" "...intra-EEZ management of fisheries and the application of game theory." 2017 Survey Colin W. Clark and Gordon R. Munro, (University of British Columbia) Capital Theory and

the Economics of Fisheries: Implications for Policy Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 1/3

ecosystem approach

Fleet Distribution between different areas no long individual fishery assets, but species portfolio as a whole.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 1/3

ecosystem approach

Fleet Distribution between different areas no long individual fishery assets, but species portfolio as a whole.

Protected Marine Areas

Artificial Coral Reef (in the Mediterranean Sea)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 1/3

ecosystem approach

Fleet Distribution between different areas no long individual fishery assets, but species portfolio as a whole.

Protected Marine Areas

Artificial Coral Reef (in the Mediterranean Sea)

Pricing Policy

Investment-quasi malleability of produced capital variable prices-relaxing the perfect elasticity of the demand

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 2/3

Several fleets

Nadia RAÏSSI. Features of bioeconomics models for the optimal management of a fishery exploited by two different fleets. Natural Resource Modeling, 2001. C Sanogo, S BenMiled, and N Raissi. Viability Analysis of Multi-fishery. Acta Biotheor., 60(1-2):189–207, jun 2012. Chata sanogo Thesis. Different fleets competition study 2015. Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 2/3

Several fleets

Nadia RAÏSSI. Features of bioeconomics models for the optimal management of a fishery exploited by two different fleets. Natural Resource Modeling, 2001. C Sanogo, S BenMiled, and N Raissi. Viability Analysis of Multi-fishery. Acta Biotheor., 60(1-2):189–207, jun 2012. Chata sanogo Thesis. Different fleets competition study 2015.

Several areas

Rachid MCHICH, Pierre AUGER, My Lhassan HBID, and Nadia RAÏSSI. Optimal spatial distribution of a bioeconomical fishing model on several zones: Allee effect. International Journal of Ecological Economics and Statistics (IJEES), 2007. Rachid Mchich, Najib Charouki, Pierre Auger, Nadia Raissi and Omar Ettahiri,(2006), Optimal spatial distribution of the fishing effort in a multifishing zone model,Ecological Modelling, 197(3/4), pp.274-280. Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 3/3

Protected Marine Area

Dubey B.,Chandra P. and Sinha P.(2003),A model for fishery resource with reserve area, Nonlinear Analysis:Real world Appl.4(2003),pp.625-637. Dominique Ami, Pierre Cartigny and Alain Rapaport(2005),Can marine protected areas enhance both economic and biological situations?,C.R.Biologies,328,pp.357-366 Chakib Jerry and Nadia Raissi (2008),Impact and effectiveness of marine protected area on economic sustainability,BIOMAT2008,RubemP.Mondaini editor,World Scientific, ISBN :978-981-4271-81-3,Pages182-191,2008. Mounir Jerry, Pierre Cartigny and Alain Rapaport (2010),The study of the viability domain for a fishing problem with reserve,Nonlinear Analysis:Real world Applications,11,pp.720,734. Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

FFM’s Development 3/3

Protected Marine Area

Dubey B.,Chandra P. and Sinha P.(2003),A model for fishery resource with reserve area, Nonlinear Analysis:Real world Appl.4(2003),pp.625-637. Dominique Ami, Pierre Cartigny and Alain Rapaport(2005),Can marine protected areas enhance both economic and biological situations?,C.R.Biologies,328,pp.357-366 Chakib Jerry and Nadia Raissi (2008),Impact and effectiveness of marine protected area on economic sustainability,BIOMAT2008,RubemP.Mondaini editor,World Scientific, ISBN :978-981-4271-81-3,Pages182-191,2008. Mounir Jerry, Pierre Cartigny and Alain Rapaport (2010),The study of the viability domain for a fishing problem with reserve,Nonlinear Analysis:Real world Applications,11,pp.720,734.

Pricing

Clark, C. W., F. H. Clarke, and G. Munro. 1979. “The Optimal Management of Renewable Resource Stocks: Problems of Irreversible Investment.” Econometrica 47(1):25–47. Chakib Jerry and Nadia Raissi (2010),Can management measures ensure the biological and economical stabilizability of a fishing model?,Math.and Comp. Modell.,51,516-526, 2010. Pierre Auger, Rachid Mchich, Nadia Raïssi, and Bob W. Kooi. Effects of market price on the dynamics of a spatial fishery model: Over-exploited fishery/traditional fishery. Ecological Complexity, 7(1):13-20. Chakib Jerry and Nadia Raissi (2012) Optimal exploitation for commercial fishing problem; Acta Biotheoretica vol 60 pp 209-223 Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Methods of Dynamic Optimization

Hamilton Jacobi approach

Several level optimal control problem Differential Games; Nash equilibrium

Dynamical Systems

Variable aggregation

Viability Theory

Viable constraints Viability Kernel

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Several Fleets

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

several fleets

The model

Profit issues        Ri(E) = ∞ e−δit (piqix − ci)Ei) dt ˙ X = F(X) − (q1E1 + q2E2)X X(0) = X0 (5) Control Constraints E min

i

≤ Ei(t) ≤ E max

i

, ∀t ≥ 0; i = 1, 2. (6) State Constraints 0 ≤ X(t) ≤ K, ∀t ≥ 0; K = carryingcapacity. (7)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Principal-Agent analysis

1987 F. Clarke and G.Munro studied the access right of a foreign fleet 2001 N.R. generalized : foreign + domestic fleet

tax and subsidies

Principal Revenue RP(r, m, E) = ∞ e−δPt (riqix(t) + mi)Ei(t)) dt (8) tax and subsidies pi = p0

i − ri

(9) ci = c0

i + mi; i = 1, 2.

(10) Revenue Agreement Ri ≥ Li; i = 1, 2. (11)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Bilevel dynamic optimization

First step: ˆ E1(., r, m), ˆ E2(., r, m), ˆ x(., r, m)

(PA(r, m))                                      max

(r,m,E) R2(r2, m2, E2)

R2 ≥ L2 0 ≤ E2 ≤ E max

2

t ≥ 0 0 ≤ x ≤ K t ≥ 0 r2 ≤ p0

2

− c0

2 ≤ m2

(E1, x)              max

(r,m,E) R1(r1, m1, E1)

R1 ≥ L1 0 ≤ E1 ≤ E max

1

t ≥ 0 ˙ X = F(X) − (q1E1 + q2E2)X X(0) = X0 (12)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Taxes choice

Second step: (r,m) sol’n

(Pp)      max

(r,m) Rp(r, m, ˆ

x(r, m), ˆ E(r, m)) Ri ≥ Li ri ≤ p0

i

− c0

i ≤ mi

(13)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

collapse of less efficient fleet

bionomic equilibrium

F ′(x∗

i ) +

c∗

i F(x∗ i )

x∗

i (piqix∗ i − c∗ i ) = δi,

i = 1, 2 (14)

Solution

If x∗

2 < x∗ 1

ˆ E(x) =        (0, 0), if x < x∗

2

(E max

1

, E max

2

), if x > x∗

1

(0, E max

2

), if x∗

2 < x < x∗ 1

(0, E ∗

2 ),

if x = x∗

2.

(15)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Fleet coexistence: Differential games

Nash equilibrium

R1( ˆ E1, ˆ E2) ≥ R1(E1, ˆ E2) ∀ feasible E1 R2( ˆ E1, ˆ E2) ≥ R2( ˆ E1, E2) ∀ feasible E2 (16)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Fleet coexistence: Differential games

Nash equilibrium

R1( ˆ E1, ˆ E2) ≥ R1(E1, ˆ E2) ∀ feasible E1 R2( ˆ E1, ˆ E2) ≥ R2( ˆ E1, E2) ∀ feasible E2 (16)

Candidate

ˆ E =    (E max

1

, E max

2

)ifx > x∗ (E ∗

1 , E ∗ 2 )ifx = x∗

(0, 0)ifx < x∗ (17)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

E ∗

j = F(x)

qjx − δi piqix − ciln(x) qj(piqix − ci) , i, j = 1, 2; i = j. (18)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

E ∗

j = F(x)

qjx − δi piqix − ciln(x) qj(piqix − ci) , i, j = 1, 2; i = j. (18) Nash equilibrium x∗ solution of F(x∗) = q1E ∗

1 (x∗) + q2E ∗ 2 (x∗)

(19)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Optimality’s Proof : Gold Road of Caratheodory

First step: Perturbe initial conditions of OC problem → family of CO problems

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Optimality’s Proof : Gold Road of Caratheodory

First step: Perturbe initial conditions of OC problem → family of CO problems Second step: Identify Nash equilibrium candidate for each problem

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Optimality’s Proof : Gold Road of Caratheodory

First step: Perturbe initial conditions of OC problem → family of CO problems Second step: Identify Nash equilibrium candidate for each problem Third step: Build conjecture about optimality of family of perturbed problems

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Optimality’s Proof : Gold Road of Caratheodory

First step: Perturbe initial conditions of OC problem → family of CO problems Second step: Identify Nash equilibrium candidate for each problem Third step: Build conjecture about optimality of family of perturbed problems Fourth step: Use this conjecture in order to compute value function

Value functions

Vi(τ, x) = ∞

τ

e−δit(piqi ˆ x(t) − ci) ˆ Ei(t)dt (20)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Optimality’s Proof : Gold Road of Caratheodory

First step: Perturbe initial conditions of OC problem → family of CO problems Second step: Identify Nash equilibrium candidate for each problem Third step: Build conjecture about optimality of family of perturbed problems Fourth step: Use this conjecture in order to compute value function

Value functions

Vi(τ, x) = ∞

τ

e−δit(piqi ˆ x(t) − ci) ˆ Ei(t)dt (20) Fifth and last step: verification H.J. equation and confirmation of conjecture

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Hamilton Jacobi equation

Hi : R × R − → R; i, j = 1, 2i = j defined by : Hi(X, λ) = Max{λ(F(X) − (qiEi + qj ˆ Ej)X) + (piqiX − ci)Ei); E min

i

≤ Ei ≤ E max

i

}.

Theorem

Assume that Vi is C 1 and Hi is continuous then Vi is the unique solution

• f Hamilton Jacobi equation

Vt(t, x) + Hi(x, Vx(t, x)) = 0. V (0, x0) = ˆ V

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Fleet Distribution

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Several Areas 1/2

Aggregation approach

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Several Areas 1/2

Aggregation approach Najib Charouki et al.

                           ε ˙ n1 = (˜ kn2 − kn1) + ε

• r1n1(1 −

n1 K1 ) − qE1n1

• ε ˙

n2 = (kn1 + ˆ kn3 − (˜ k + ¯ k)n2) + ε

• r2n2(1 −

n2 K2 ) − qE2n2

• ε ˙

n3 = (¯ kn2 − ˆ kn3) + ε

• r3n3(1 −

n3 K3 ) − qE3n3 ε ˙ E1 = ( ˜ mE2 − mE1) + ε(bn1 − c)E1 ε ˙ E2 = (mE1 + ˆ mE3 − ( ˜ m + ¯ m)E2) + ε(bn2 − c)E2 ε ˙ E3 = ( ¯ mE2 − ˆ mE3) + ε(bn3 − c)E3 (21) b = pa Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Several Areas 2/2

aggregated model

• ˙

n = rn(1 − n K ) − (˜ aEn) ˙ E = (˜ apn − c)E (22) ˜ a is related to spatial distrubition This aggregated model is a good approximation of the global model if ǫ is small enough therefore it exists structurally stable equilibrium. (n∗, E ∗) = c p˜ a; r ˜ a(1 − c pK ˜ a)

• (23)

arising measure ˜ aopt sol’n max

x≥ c

pK

E(x) = r x (1 − c pKx ) (24)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

viability approach

J.P.AUBIN

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

closed loop control    ˙ x(t) = F(x(t), u(t)) u(t) ∈ U(x(t))a.e. x(0) = x0 (25) viability constraint ∀t ≥ 0, x(t) ∈ K (26) viable set if for all x ∈ BondK , there is a tangent direction along which the path enter into K (or stay on its boundary), then K is said viable,

• therwise we sketch the biggest subset viable in K, viability kernel

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

protected area 1/4

the model

˙ X1 = F1(X1) + λ(X2 − X1), ˙ X2 = F2(X2) − λ(X2 − X1) − qE2X2, X(0) = X 0, (27) E2(·) belong to [E min

2

, E max

2

]. If K1, K2 are the carrying capacities of the two areas, then the total density is X = ρX1 + (1 − ρ)X2, where K = K1 + K2 is the carrying capacities of the whole domain and ρ = K1 K .

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

protected area 2/4

constraints

X(t) ≥ c and X2(t)E2(t) ≥ α2, ∀t ≥ 0, (28) Let c1 = c ρ, c2 = c 1 − ρ, and define

• i. C1 := {X ≥ c}, D := BondC1,
• ii. C2 := {X / X2E max

2

≥ α2},

• iii. Ni := null − clines,
• iv. X ∗(E2) equilibrium point for giving E2 away from the origin,
• vii. (X E2

1 , α2/E2) is the intersection of N2(E2) with the line X2 = α2/E2,

• viii. (c1 − c1

c2 X E2 2 , X E2 2 ) is the intersection of N2(E2) with the line D.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Protected Area 3/4

hypothesis

H1 α2 ≤ E max

2

, H2 λ + qE min

2

≥ F ′

2(0),

H3 λ ≥ F ′

1(0),

H4 ρ ≤ 1/2. H1 = ⇒C2 = ∅ H3-H4 = ⇒N1 and N2(E2) do not cross the axes away from the origin, whatever is E2 ∈ [E min

2

, E max

2

].

main result

Under hypotheses H1, H2, H3 and H4, when X ∗(E min

2

) ∈ C1 and E = ∅,

• ne has

Viab(C1 ∩ C2) =

• X | X1 ≥ ˜

X1(X2), X2 ∈ [ ˜ X2, 1]

• .

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

protected area 4/4

Figure : Viability kernels with intersection of L1 and L2.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Pricing 1/4

The model

As, in S.Svizerro and C.Jerry et al.:

Problem

(S)        ˙ P = ε( D(P) K − qEX), P(0) = P0 > 0, ˙ X = F(X) − qEX, X(0) = X0, 0 < E min ≤ E ≤ E max and F(X) = rX(1 − X)

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Pricing 1/4

The model

As, in S.Svizerro and C.Jerry et al.:

Problem

(S)        ˙ P = ε( D(P) K − qEX), P(0) = P0 > 0, ˙ X = F(X) − qEX, X(0) = X0, 0 < E min ≤ E ≤ E max and F(X) = rX(1 − X) ◮ Sustainability constraint: P ¯ ≤ P(t) ≤ ¯ P

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Pricing 1/4

The model

As, in S.Svizerro and C.Jerry et al.:

Problem

(S)        ˙ P = ε( D(P) K − qEX), P(0) = P0 > 0, ˙ X = F(X) − qEX, X(0) = X0, 0 < E min ≤ E ≤ E max and F(X) = rX(1 − X) ◮ Sustainability constraint: P ¯ ≤ P(t) ≤ ¯ P Where, ◮ ¯ P guarantees a consumption level throughout time. ◮ P ¯ guarantees a global positive net benefit in the sector.

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

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Viable stationary point

Hypothesis:

H0: r ≥ qE max. H1: lim

p→∞ D(P) = 0 and lim p→0 D(P) = +∞. Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

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Viability kernel

Consider the following set: W =

• P∗/∃ X ∗ where (P∗, X ∗) is an equilibrium point of (S) and P

¯ ≤ P∗ ≤ ¯ P

• .

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

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Viability kernel

Consider the following set: W =

• P∗/∃ X ∗ where (P∗, X ∗) is an equilibrium point of (S) and P

¯ ≤ P∗ ≤ ¯ P

• .

Proposition

Under hypothesis H0, one has Case 1. If W = ∅ then Viab(Z) = ∅. Case 2. If W = ∅: Viab(Z) =   (P, X)

• P

¯

′(t) ≤ P ≤ ¯

P if X ¯ ≤ X ≤ ¯ X ′ P ¯ ≤ P ≤ ¯ P if ¯ X ≤ X ≤ X ¯ P ¯ ≤ P ≤ ¯ P′(t) if X ¯

′ ≤ X ≤ ¯

X   

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

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Viability kernel

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Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

Conclusion H.J. vs Viability

Hamilton Jacobi Viability Optimal Control controled and constrainted system ˆ u optimal Set E viability kernel Compute value function compute viable control Hamilton Jacobi equation E c is not viable

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41

At last

Did mathematical modelling really matter for fishery policy purposes, or are they of theoretical interest only?

Nadia Raïssi ( Mohammed V University in Rabat Faculty of Sciences Laboratory Mathematical Analysis and Applications (LAMA) Did mathematical modelling really matter for fishery policy purposes ? Premier Congrés Franco‘Marocain de Mathématiques / 41