Optimal Harvesting of aSemilinear Elliptic Logistic Fishery
ModelWandi Ding1, Suzanne Lenhart2
1University of Tennessee - Knoxville, [email protected] University of Tennessee - Knoxville & Oak Ridge National Laboratory
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.1/21
Outline• Development of Fishery Models• Motivation• The Model• Optimal Control Problems• Numerical Examples for J1
• Numerical Examples for J2
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.2/21
Development of Fishery Models[1]
• 1900 - 1920: First Efforts• F. I. Baranov: grandfather of fisheries
population dynamics• ICES (1902): International Council for the
Exploration of the Sea
[1] T. J. Quinn II, Ruminations on the Development and Future of Population Dynamics Models
in Fisheries, Natural Resource Modeling, 16:4, 2003
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.3/21
• 1920 - 1960: Establishment of Science• Ricker, Beverton and Holt, Leslie, Lotka and
Volterra, Thompson etc.• multi-species modeling,• age- and size-structure dynamics;
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.4/21
• 1960 - 1980: Deterministic Theory, StatisticalPractice• advances in age-structured models (Gulland,
Pope, Doubleday),
• improvements to surplus production (Pella,Tomlinson, Schnute, Fletcher, Hilborn) andstock recruitment models,
• bioeconomic models (Clark)• management control models (Hilborn,
Walters)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.5/21
• 1960 - 1980: Deterministic Theory, StatisticalPractice• advances in age-structured models (Gulland,
Pope, Doubleday),
• improvements to surplus production (Pella,Tomlinson, Schnute, Fletcher, Hilborn) andstock recruitment models,
• bioeconomic models (Clark)• management control models (Hilborn,
Walters)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.5/21
• 1960 - 1980: Deterministic Theory, StatisticalPractice• advances in age-structured models (Gulland,
Pope, Doubleday),
• improvements to surplus production (Pella,Tomlinson, Schnute, Fletcher, Hilborn) andstock recruitment models,
• bioeconomic models (Clark)• management control models (Hilborn,
Walters)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.5/21
• 1980-2000: The Golden Age• integration between mathematics and statistics
• Bayesian and time series methods(uncertainty)
• realistic modeling for:· age and size-structured population· spatial dynamics· harvesting strategies(stochasticity, time variation)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.6/21
• 1980-2000: The Golden Age• integration between mathematics and statistics
• Bayesian and time series methods(uncertainty)
• realistic modeling for:· age and size-structured population· spatial dynamics· harvesting strategies(stochasticity, time variation)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.6/21
• 1980-2000: The Golden Age• integration between mathematics and statistics
• Bayesian and time series methods(uncertainty)
• realistic modeling for:· age and size-structured population· spatial dynamics· harvesting strategies(stochasticity, time variation)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.6/21
• The New Millenium• future models:
• habitat and spatial concerns• genetics• multispecies interactions• enviromental factors• effects of harvesting on the ecosystem• socioeconomic concerns
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.7/21
MotivationNeubert(Ecology Letter, 2003) studied the fisherymanagement problem:Maximize the yield
J(h) =
∫
l
0
h(x)u(x) dx, 0 ≤ h(x) ≤ hmax
Subject to
−d2u
dx2= u(1 − u) − h(x)u, 0 < x < l,
u(0) = u(l) = 0.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.8/21
Neubert’s Results• No-take marine reserves are always part of an
optimal harvest designed to maximize yield;
• The sizes and locations of the optimal reservesdepend on a dimensionless length parameter;
• For small values of this parameter, the maximumyield is obtained by placing a large reserve in thecenter of the habitat;
• For large values of this parameter, the optimalharvesting strategy is a spatial “chatteringcontrol” with infinite sequences of reservesalternating with areas of intense fishing;
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.9/21
Neubert’s Results• No-take marine reserves are always part of an
optimal harvest designed to maximize yield;• The sizes and locations of the optimal reserves
depend on a dimensionless length parameter;
• For small values of this parameter, the maximumyield is obtained by placing a large reserve in thecenter of the habitat;
• For large values of this parameter, the optimalharvesting strategy is a spatial “chatteringcontrol” with infinite sequences of reservesalternating with areas of intense fishing;
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.9/21
Neubert’s Results• No-take marine reserves are always part of an
optimal harvest designed to maximize yield;• The sizes and locations of the optimal reserves
depend on a dimensionless length parameter;• For small values of this parameter, the maximum
yield is obtained by placing a large reserve in thecenter of the habitat;
• For large values of this parameter, the optimalharvesting strategy is a spatial “chatteringcontrol” with infinite sequences of reservesalternating with areas of intense fishing;
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.9/21
Neubert’s Results• No-take marine reserves are always part of an
optimal harvest designed to maximize yield;• The sizes and locations of the optimal reserves
depend on a dimensionless length parameter;• For small values of this parameter, the maximum
yield is obtained by placing a large reserve in thecenter of the habitat;
• For large values of this parameter, the optimalharvesting strategy is a spatial “chatteringcontrol” with infinite sequences of reservesalternating with areas of intense fishing;
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.9/21
Our Fishery Model
−∆u = ru(1 − u) − h(x)u, x ∈ Ω,
u = 0, x ∈ ∂Ω.
where u(x) is the fish density, r is the growth rate,
h(x) is the harvesting depending on the location of
fish, Ω ∈ Rn, smooth and bounded domain.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.10/21
Optimal Control ProblemsGoals:
• Maximizing the yield and minimizing the cost offishing.
J1(h) =
∫
Ω
h(x)u(x) dx −
∫
Ω
(B1 + B2h)h dx,
h ∈ U1.
• Maximizing the yield and minimizing thevariation of the fishing effort.
J2(h) =
∫
Ω
h(x)u(x) dx−A
∫
Ω
|∇h|2 dx, h ∈ U2,
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.11/21
Optimality System Istate equation
−∆u = ru(1 − u) − h(x)u, x ∈ Ω,
u = 0, x ∈ ∂Ω;
adjoint equation
−∆p − r(1 − 2u)p + hp = h, x ∈ Ω,
p = 0, x ∈ ∂Ω;
characterization of optimal control
h(x) = minmax0,u − pu − B1
2B2
, 1 − δ.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.12/21
Numerical Examples for J1: 1-D case, B2 effect
Set B1 = 0.1, vary B2 = 0.5, 1.25, 2.5, 5, 10
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
fish
den
sit
y f
or
J1(h
)B
2 = 10
B2 = 5
B2 = 2.5
B2 = 1.25
B2 = 0.5
B1 = 0.1
0 1 2 3 4 50
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
x
optim
al harv
esting for
J1(h
)
B2 = 10
B2 = 5
B2 = 2.5
B2 = 1.25
B2 = 0.5B
1 = 0.1
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.13/21
Numerical Examples for J1: 1-D case, small B2
Set B1 = 0, vary B2 = 0.1, 0.05, 0.01
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x
fish
den
sit
y f
or
J1(h
)
B2 = 0.1
B2 = 0.05
B2 = 0.01B
1 = 0
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
B2 = 0.01
B2 = 0.05
B2 = 0.1
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.14/21
Numerical Examples for J1: 2-D case
00.5
11.5
22.5
0
1
2
30
0.1
0.2
0.3
0.4
0.5
x−axis
B1 = 0, B
2 = 1, r = 5, L = 2.5
y−axis
fish
den
sit
y f
or
J1(h
)
00.5
11.5
22.5
0
1
2
30
0.05
0.1
0.15
0.2
x−axis
B1 = 0, B
2 = 1, r = 5, L = 2.5
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.15/21
Numerical Examples for J1: 2-D case, B1 effect
00.5
11.5
22.5
0
1
2
30
0.05
0.1
0.15
0.2
x−axis
B1 = 0.1, B
2 = 1, r = 5, L = 2.5
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
00.5
11.5
22.5
0
1
2
30
0.05
0.1
0.15
0.2
x−axis
B1 = 0, B
2 = 1, r = 5, L = 2.5
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.16/21
Numerical Examples for J1: 2-D case, domain
size effect
0
1
2
3
0
1
2
30
0.2
0.4
0.6
0.8
x−axis
B1 = 0, B
2 = 1, r = 5, L = 3
y−axis
fish
den
sit
y f
or
J1(h
)
0
1
2
3
0
1
2
30
0.1
0.2
0.3
0.4
x−axis
B1 = 0, B
2 = 1, r = 5, L = 3
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
00.5
11.5
22.5
0
1
2
30
0.1
0.2
0.3
0.4
0.5
x−axis
B1 = 0, B
2 = 1, r = 5, L = 2.5
y−axis
fish
den
sit
y f
or
J1(h
)
00.5
11.5
22.5
0
1
2
30
0.05
0.1
0.15
0.2
x−axis
B1 = 0, B
2 = 1, r = 5, L = 2.5
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.17/21
Numerical Examples for J1: 2-D case, small B2
00.5
11.5
22.5
0
1
2
30
0.1
0.2
0.3
0.4
x−axis
B1 = 0, B
2 = 0.05, r = 5, L = 2.5
y−axis
fish
den
sit
y f
or
J1(h
)
00.5
11.5
22.5
0
1
2
30
0.2
0.4
0.6
0.8
1
x−axis
B1 = 0, B
2 = 0.05, r = 5, L = 2.5
y−axis
op
tim
al h
arv
esti
ng
fo
r J
1(h
)
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.18/21
Optimality System IIstate equation
−∆u = ru(1 − u) − h(x)u, x ∈ Ω,
u = 0, x ∈ ∂Ω;
adjoint equation
−∆p − r(1 − 2u)p + hp = h, x ∈ Ω,
p = 0, x ∈ ∂Ω;
characterization of optimal control
minmax(pu − u − 2A∆h, h − (1 − δ)), h − 0 = 0.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.19/21
Numerical Examples for J2:
Vary A = 1, 2.5, 5, 10
0 1 2 3 4 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
fish
den
sit
y f
or
J2(h
)
A=2.5
A=1
A=5A=10
0 1 2 3 4 50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
x
op
tim
al h
arv
esti
ng
fo
r J
2(h
) A=1
A=5
A=10
A=2.5
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.20/21
Conclusion: in the long run• If we want to maximize yield and minimize cost
(J1),then increasing labor cost (B2) or fixed cost(B1) will decrease optimal harvesting;
• If we only want to maximize yield, then reserveis part of the optimal harvesting strategy;
• For J1, the optimal benefit inreases when domainsize increases;
• If we want to maximize yield and minimizevariation in fishing effort, then increasing (A)will reduce optimal harvesting.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.21/21
Conclusion: in the long run• If we want to maximize yield and minimize cost
(J1),then increasing labor cost (B2) or fixed cost(B1) will decrease optimal harvesting;
• If we only want to maximize yield, then reserveis part of the optimal harvesting strategy;
• For J1, the optimal benefit inreases when domainsize increases;
• If we want to maximize yield and minimizevariation in fishing effort, then increasing (A)will reduce optimal harvesting.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.21/21
Conclusion: in the long run• If we want to maximize yield and minimize cost
(J1),then increasing labor cost (B2) or fixed cost(B1) will decrease optimal harvesting;
• If we only want to maximize yield, then reserveis part of the optimal harvesting strategy;
• For J1, the optimal benefit inreases when domainsize increases;
• If we want to maximize yield and minimizevariation in fishing effort, then increasing (A)will reduce optimal harvesting.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.21/21
Conclusion: in the long run• If we want to maximize yield and minimize cost
(J1),then increasing labor cost (B2) or fixed cost(B1) will decrease optimal harvesting;
• If we only want to maximize yield, then reserveis part of the optimal harvesting strategy;
• For J1, the optimal benefit inreases when domainsize increases;
• If we want to maximize yield and minimizevariation in fishing effort, then increasing (A)will reduce optimal harvesting.
Optimal Harvesting of a Semilinear Elliptic Logistic Fishery Model – p.21/21