MotivationThe ModelNumerical ExampleSummaryReferencesFishing in a Shallow Lake: Exploring a ClassicFishery Model in a Habitat with Shallow LakeDynamics.John Janmaat11Department of EconomicsUniversity of British Columbia - Okanagan CampusWEAI Paci�c Rim Conference 2011John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesHarvest E�ort and Habitat Damage.Biological resource models focus on e�ort - stock relationship.Biological resources exist within habitats.Harvesting e�ort can cause habitat damage.Harvesting e�ort can choose between multiple equilibria.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesHarvest E�ort and Habitat Damage.Biological resource models focus on e�ort - stock relationship.Biological resources exist within habitats.Harvesting e�ort can cause habitat damage.Harvesting e�ort can choose between multiple equilibria.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesHarvest E�ort and Habitat Damage.Biological resource models focus on e�ort - stock relationship.Biological resources exist within habitats.Harvesting e�ort can cause habitat damage.Harvesting e�ort can choose between multiple equilibria.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesHarvest E�ort and Habitat Damage.Biological resource models focus on e�ort - stock relationship.Biological resources exist within habitats.Harvesting e�ort can cause habitat damage.Harvesting e�ort can choose between multiple equilibria.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesExamples.Direct habitat impact: substrate destructionBottom dragging [Collie et al., 2000]Direct habitat impact: habitat destructionNursery destruction [Barbier and Strand, 1998, Barbier, 2003]Indirect habitat impact: food web reorganizationKeystone species [Batabyal, 2002]Reorganizing interspeci�c competitionSelection between multiple equilibria [Sche�er et al., 1997,Hare and Mantua, 2000, Muradian, 2001, Rosser Jr., 2001,Folke et al., 2004]John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesExamples.Direct habitat impact: substrate destructionBottom dragging [Collie et al., 2000]Direct habitat impact: habitat destructionNursery destruction [Barbier and Strand, 1998, Barbier, 2003]Indirect habitat impact: food web reorganizationKeystone species [Batabyal, 2002]Reorganizing interspeci�c competitionSelection between multiple equilibria [Sche�er et al., 1997,Hare and Mantua, 2000, Muradian, 2001, Rosser Jr., 2001,Folke et al., 2004]John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Basic ProblemExamplesExamples.Direct habitat impact: substrate destructionBottom dragging [Collie et al., 2000]Direct habitat impact: habitat destructionNursery destruction [Barbier and Strand, 1998, Barbier, 2003]Indirect habitat impact: food web reorganizationKeystone species [Batabyal, 2002]Reorganizing interspeci�c competitionSelection between multiple equilibria [Sche�er et al., 1997,Hare and Mantua, 2000, Muradian, 2001, Rosser Jr., 2001,Folke et al., 2004]John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsSystem Dynamics.Habitat State Dynamicsz =−r(z)+d(y ,z)r(z): habitat recovery rate (positive)d(y ,z): habitat damaged(0,z) = 0, r(z)> 0∀z > 0, habitat recovers absent e�ort.Steady states where r(z) = d(y ,z).John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsSystem Dynamics.Stock Dynamics x = g(x ,z)−h(x ,y)g(x ,z): stock growth rateh(x ,y): harvesth(x ,0) = 0, stock moves to x : g(x ,z) = 0, carrying capacity.∂g(x ,z)/∂z < 0, more habitat damage, lower carryingcapacity. John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsSystem Dynamics.Response Correspondenced(yss(x ,z),z) = r(z)yss(x ,z): e�ort consistent with x = 0 and z = 0.implicit relationship between x and z such that x = 0 andz = 0.requires correct choice of e�ort.Analogous to requiring x = 0 for classic static optimum.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsObjectives.Pro�t p(h(x ,y))h(x ,y)− c(y)p(h): harvest pricec(y): e�ort costWelfare∫ h(x ,y)0 p(τ)dτ − c(y)If p(h) = p, pro�t equals welfare.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsObjectives.Pro�t p(h(x ,y))h(x ,y)− c(y)p(h): harvest pricec(y): e�ort costWelfare∫ h(x ,y)0 p(τ)dτ − c(y)If p(h) = p, pro�t equals welfare.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsObjectives.Pro�t p(h(x ,y))h(x ,y)− c(y)p(h): harvest pricec(y): e�ort costWelfare∫ h(x ,y)0 p(τ)dτ − c(y)If p(h) = p, pro�t equals welfare.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsOpen Access.Zero Pro�t p(g(x ,z))g(x ,z) = c(yss(x ,z))Steady states with zero pro�t,Solutions must also lie in response correspondence.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsStatic OptimumSteady State Welfaremax∫ h(xss (y ,zss (y)),y)0 p(τ)dτ − c(y)xss(y ,z): stock consistent with x = 0 and z = 0, given y and z .zss(y): habitat damage consistent with z = 0, given y .Maximize steady state welfare,Solutions must also lie in response correspondence.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic OptimumHamiltonianH=

∫ h(x ,y)0 p(τ)dτ−c(y)+πx [g(x ,z)−h(x ,y)]+πz [−r(z)+d(y ,z)]Optimal path to steady state,Steady states must also lie in response correspondence.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic OptimumDynamic System x = g(x ,z)−h(x ,yFOC )z = −r(z)+d(yFOC ,z)πx = πx [ρ − (gx −hx)]−p(h)hxπz = πz [ρ − (−rz +dz)]−πxgzyFOC ⇒ y : ∂H/∂y = 0, gx = ∂g(x ,z)/∂x , etc.Optimal path to steady state,John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπx Hotelling Form

ρ =πxπx + p(h)hx

πx +(gx −hx)πx Steady State

πx = p(h)hx/ [ρ − (gx −hx)]Habitat in�uence through gxJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπx Hotelling Form

ρ =πxπx + p(h)hx

πx +(gx −hx)πx Steady State

πx = p(h)hx/ [ρ − (gx −hx)]Habitat in�uence through gxJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπx Hotelling Form

ρ =πxπx + p(h)hx

πx +(gx −hx)πx Steady State

πx = p(h)hx/ [ρ − (gx −hx)]Habitat in�uence through gxJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπz Hotelling Form

ρ =πzπz + πxgz

πz +(−rz+dz)πz Steady State

πz = πxgz/[ρ − (−rz+dz)]Value of habitat health from gzJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπz Hotelling Form

ρ =πzπz + πxgz

πz +(−rz+dz)πz Steady State

πz = πxgz/[ρ − (−rz+dz)]Value of habitat health from gzJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences System DynamicsObjectivesSolutionsDynamic Optimumπz Hotelling Form

ρ =πzπz + πxgz

πz +(−rz+dz)πz Steady State

πz = πxgz/[ρ − (−rz+dz)]Value of habitat health from gzJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyHabitat Damage DynamicsShallow Lake Dynamicsz =my −αz+θz2/(φ2+ z2)my : e�ort impact−αz : recovery of habitat healthθz2/(φ2+ z2): inertia of habitat damageSpecialization of shallow lake dynamics developed byCarpenter and Ludwig 1999, Sche�er 2004 and studied byBrock and De Zeeuw [2002], Brock and Starrett [2003], Mäleret al. [2003], Wagener [2003]John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyStock DynamicsStock Dynamics x = rx [1− x(z +1)]−qxyfrom x = rx(1− x/K (z))−qxy with K (z) = 1/(z +1).Habitat damage reduces carrying capacity.

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyObjectivePrice of Harvest p(h) = A−BhCost of E�ort c(y) = cyDownward sloping linear demand,Constant marginal cost of e�ort.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyObjectivePrice of Harvest p(h) = A−BhCost of E�ort c(y) = cyDownward sloping linear demand,Constant marginal cost of e�ort.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyObjectivePrice of Harvest p(h) = A−BhCost of E�ort c(y) = cyDownward sloping linear demand,Constant marginal cost of e�ort.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyObjectivePrice of Harvest p(h) = A−BhCost of E�ort c(y) = cyDownward sloping linear demand,Constant marginal cost of e�ort.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyParametersStock Habitat EconomicGrowth r = 0.05 Impact m = 1 Discount ρ = 0.05Catch q = 0.5 Recovery α = 0.52 Intercept A= 4.8Inertia θ = 1 Slope B = 231Scale φ = 1 Cost c = 0.3Chosen to illustrate cases where multiple equilibria exist.Existence of multiple equilibria very sensitive to α .John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyLocating SolutionsStock x

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

Stock x

Hab

itat D

amag

e z

c) FOC and Response

FOC OAFOC DOFOC SOResponse

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyStock Growth

0.0 0.2 0.4 0.6 0.8 1.0

0.00

00.

004

0.00

80.

012

Stock x

Har

vest

h

a) Stock Growth

OASO

DOMax

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyRevenue and Cost

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.02

0.04

0.06

Effort y

Rev

enue

/ C

ost

b) Revenue and Cost

SurplusTRTCJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicySupply and Demand

0.000 0.004 0.008 0.012

01

23

45

Harvest h

Pric

e

d) Supply and Demand

SODOOA

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyLocating Solutions

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

Stock x

Hab

itat D

amag

e z

c) Response and Conditional FOC

Response, α = 0.5076Response, α = 0.6198Cond. FOC, α = 0.5076Cond. FOC, α = 0.6198

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyPaths when α = 0.52040.

00.

51.

01.

52.

0

Hab

itat D

amag

e z

a) α = 0.5204

ResponseCond. FOCP. AxisNew Fishery

DOSOOANew Fishery

c) Response and Conditional FOCJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyPaths when α = 0.5108b) α = 0.5108

ResponseCond. FOCP. AxisNew Fishery

DOSOOANew Fishery

d) Principle AxesJohn Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyPrinciple Axes

1.0 0.0 0.2 0.4 0.6 0.8 1.0

Stock x

d) Principle Axes

α = 0.5076α = 0.5236α = 0.5397α = 0.6198

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyImpact of Changing r

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.5

1.0

1.5

2.0

Instantaneous Growth Rate (r)

Hab

itat D

amag

e (z

)

a) Instantaneous Growth (r)

OASO

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyImpact of Changing α

0.5 0.3 0.4 0.5 0.6 0.7

b) Habitat Recovery (α)

Habitat Recovery (α)

all λ realsome λ imagMax SS profit

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyOutline1 MotivationThe Basic ProblemExamples2 The ModelSystem DynamicsObjectivesSolutions3 Numerical ExampleFunctional FormsResultsPolicy John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling E�ortE�ort tax or lumpsum taxcannot singly pick somesteady states.Combine lumpsum taxand e�ort subsidy.E�ort quota must be tiedto habitat health.

0.00 0.02 0.04 0.06 0.08 0.100.

000.

020.

040.

06

Effort y

Rev

enue

/ C

ost

b) Revenue and Cost

SurplusTRTC

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling E�ortE�ort tax or lumpsum taxcannot singly pick somesteady states.Combine lumpsum taxand e�ort subsidy.E�ort quota must be tiedto habitat health.

0.00 0.02 0.04 0.06 0.08 0.100.

000.

020.

040.

06

Effort y

Rev

enue

/ C

ost

b) Revenue and Cost

SurplusTRTC

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling E�ortE�ort tax or lumpsum taxcannot singly pick somesteady states.Combine lumpsum taxand e�ort subsidy.E�ort quota must be tiedto habitat health.

0.00 0.02 0.04 0.06 0.08 0.100.

000.

020.

040.

06

Effort y

Rev

enue

/ C

ost

b) Revenue and Cost

SurplusTRTC

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling HarvestHarvest tax alone may notreach optimal steadystate.Harvest level dependenttax rotates demand curve,eliminates multipleequilibria.Harvest quota must betied to habitat health. 0.000 0.004 0.008 0.012

01

23

45

Harvest h

Pric

e

d) Supply and Demand

SODOOA

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling HarvestHarvest tax alone may notreach optimal steadystate.Harvest level dependenttax rotates demand curve,eliminates multipleequilibria.Harvest quota must betied to habitat health. 0.000 0.004 0.008 0.012

01

23

45

Harvest h

Pric

e

d) Supply and Demand

SODOOA

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferences Functional FormsResultsPolicyControlling HarvestHarvest tax alone may notreach optimal steadystate.Harvest level dependenttax rotates demand curve,eliminates multipleequilibria.Harvest quota must betied to habitat health. 0.000 0.004 0.008 0.012

01

23

45

Harvest h

Pric

e

d) Supply and Demand

SODOOA

John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesSummaryReal �sh live in habitats.Human actions that impact on habitat should be in analysis.Dynamics messy,Optimal path may cross in stock-damage space,One shot policies (constant tax, �xed quota) unlikely to attainor sustain optimum steady state.Optimal policy likely contingent on state variables.Further workDetailed analysis of policy options,Uncertainty (habitat damage, stock, price and cost shocks),Spatial interactions.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesSummaryReal �sh live in habitats.Human actions that impact on habitat should be in analysis.Dynamics messy,Optimal path may cross in stock-damage space,One shot policies (constant tax, �xed quota) unlikely to attainor sustain optimum steady state.Optimal policy likely contingent on state variables.Further workDetailed analysis of policy options,Uncertainty (habitat damage, stock, price and cost shocks),Spatial interactions.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesSummaryReal �sh live in habitats.Human actions that impact on habitat should be in analysis.Dynamics messy,Optimal path may cross in stock-damage space,One shot policies (constant tax, �xed quota) unlikely to attainor sustain optimum steady state.Optimal policy likely contingent on state variables.Further workDetailed analysis of policy options,Uncertainty (habitat damage, stock, price and cost shocks),Spatial interactions.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesSummaryReal �sh live in habitats.Human actions that impact on habitat should be in analysis.Dynamics messy,Optimal path may cross in stock-damage space,One shot policies (constant tax, �xed quota) unlikely to attainor sustain optimum steady state.Optimal policy likely contingent on state variables.Further workDetailed analysis of policy options,Uncertainty (habitat damage, stock, price and cost shocks),Spatial interactions.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesReferences IEdward B. Barbier. Habitat-�shery linkages and mangrove loss inThailand. Contemporary Economic Policy, 21(1):59�77, 2003.Edward B. Barbier and Ivar Strand. Valuing mangrove-�sherylinkages: A case study of Campeche, Mexico. Environmental andResource Economics, 12:151�166, 1998.Amitrajeet A. Batabyal. Human actions, the survival of keystonespecies, and the resilience of ecological-economic systems.Resources Policy, 28:153�157, 2002.W.A. Brock and A. De Zeeuw. The repeated lake game. EconomicLetters, 76:109�114, 2002.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesReferences IIW.A. Brock and D. Starrett. Managing systems with non-convexpositive feedback. Environmental and Resource Economics, 26:575�602, 2003.S.R. Carpenter and W.A Ludwig, D.and Brock. Management ofeutropication for lakes subjects to potentially irreversible change.Ecological Applications, 9:751�771, 1999.Jeremy S. Collie, Stephen J. Hall, Michel J. Kaiser, and Ian R.Poiner. A quantitative analysis of �shing impacts on shelf-seabenthos. Journal of Animal Ecology, 69:785�798, 2000.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesReferences IIICarl Folke, Steve Carpenter, Brian Walker, Marten Sche�er,Thomas Elmqvist, Lance Gunderson, and C.S. Holling. Regimeshifts, resilience, and biodiversity in ecosystem management.Annual Review of Ecology, Evolution and Systematics, 35:557�81, 2004. doi: 10.1146/annurev.ecolsys.35.021103.105711.S.R. Hare and N.J. Mantua. Empirical evidence for North Paci�cregime shifts in 1977 and 1989. Progress in Oceanography, 47:103�145, 2000.K.G. Mäler, A. Xepadadeas, and A. De Zeeuw. The economics ofshallow lakes. Environmental and Resource Economics, 26:603�624, 2003.Roland Muradian. Ecological thresholds: a survey. EcologicalEconomics, 38:7�24, 2001.John Janmaat Fishing in a Shallow Lake

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MotivationThe ModelNumerical ExampleSummaryReferencesReferences IVJ. Barkley Rosser Jr. Complex ecological�economic dynamics andenvironmental policy. Ecological Economics, 37:23�37, 2001.M. Sche�er, S. Carpenter, J.A. Foley, C. Folke, and B. Walker.Catastropich shifts in ecosystems. Nature, 413:591�596, 1997.Marten Sche�er. Ecology of Shallow Lakes. Kluwer AcademicPublishers, 2004.F. O. O. Wagener. Skiba points and heteroclinic bifurcations, withapplications to the shallow lake system. Journal of EconomicDynamics and Control, 27:1533�1561, 2003.John Janmaat Fishing in a Shallow Lake