Optimal Harvesting of a Semilinear Elliptic Logistic ... Harvesting of a ... Optimal Harvesting of a...

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


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


• 1920 - 1960: Establishment of Science• Ricker, Beverton and Holt, Leslie, Lotka and

Volterra, Thompson etc.• multi-species modeling,• age- and size-structure dynamics;


• 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)


• 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)


• The New Millenium• future models:

• habitat and spatial concerns• genetics• multispecies interactions• enviromental factors• effects of harvesting on the ecosystem• socioeconomic concerns


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.


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 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;




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;



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 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,


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.


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


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


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

)


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

)


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

)


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

)


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.


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


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 ... Harvesting of a ... Optimal Harvesting of a...

Documents

Transcript of Optimal Harvesting of a Semilinear Elliptic Logistic ... Harvesting of a ... Optimal Harvesting of a...