Optimal Management of Spatial Economic / Ecological Models Lecture prepared for the EAERE Summer School

Venice 2008

Anastasios XepapadeasAthens University of Economics and Business

and Beijer Fellow

Economic and ecological systems evolve in time and space. Spatial patterns could refer, for example, to:

Stripes or spots on animal coats Ripples in sandy desserts Vegetation patterns in arid grazing systemsResource concentration Geographical patterns of production activitiesUrban concentrationsEmergence of commercial districts within citiesNorth - South dualism

How the leopard got its spots . . .

Ripples in sandy desserts

Vegetation patterns

Chlorophyll concentrations in oceans

North -

South

Urban concentration

Edinburgh

In this lecture I will cover the following topics:

1.

Modelling spatial movements through short-

and long-range spatial diffusion

and systems of reaction-diffusion equations

2.

Modelling of economic/ecological systems under diffusion

3.

Emergence of pattern formation in reaction-diffusion economic/ecological models through the Turing mechanism

4.

Optimal control of partial differential equations in economic/ecological models under spatial diffusion

5.

Emergence of pattern formation in the optimal control of reaction-diffusion systems of economic/ecological models through the mechanism of optimal diffusion-induced instability

6.

Issues of regulation in reaction- diffusion economic/ecological systems

1. Modelling Spatial Movements

Modelling Diffusion: Local Effects – Short Range Diffusion

Let ( )tzx , , denote the concentration of a biological or economic variable at time 0≥t at the spatial point Z∈z , where space is assume to be one dimensional and modelled by a line segment. Let ( )tz,φ denote the flow of ‘material’ such as animals, commodities, or capital, past z at time .t We assume that this flux is proportional to the gradient of the concentration of the material, or

( ) ( )z

tzxDtz x ∂∂

−=,,φ

where xD is the diffusion coefficient and the minus sign indicates that material moves from high levels of concentration to low levels of concentration. Under this diffusion assumption, the evolution of the material's stock in a small interval zΔ is defined as:

( ) ( ) ( ) ( )dstsFtzztzdstsxdtd zz

z

zz

z,,,, ∫∫

++++−=

ΔΔΔφφ (1)

where ( )tzF , , is a source or a growth function for the material in question.

Dividing (1) by zΔ and taking limits as 0→zΔ the evolution of the material is determined as:

( ) ( ) ( ).,,, tzFz

tzt

tzx+

∂∂

−=∂

∂ φ

Using the definition of diffusion of flux we obtain the basic diffusion equation ( ) ( ) ( ) ( ) ( ) .,, where,,,,

2

222

ztzxtzxtzxDtzF

ttzx

x ∂∂

=∇∇+=∂

∂

If the source term represents logistic population growth ( ) ( ) ( )( ),,,, tzrxstzxtzF −= where s is intrinsic growth rate and rs / is the environment's carrying capacity, then we obtain the Fisher equation:

( ) ( ) ( ) ( ).,,1,, 2 tzxDs

tzrxtzsxt

tzxx∇+⎟

⎠⎞

⎜⎝⎛ −=

∂∂

The Fisher equation can be generalized to several interacting species or activities. With two interacting species ),( yx and no cross diffusion, we obtain:

( )

( ) yDyxFty

xDyxFtx

y

x

22

21

,

,

∇+=∂∂

∇+=∂∂

(2)

System (2) is referred to as a reaction - diffusion system or as an interacting population diffusion system.

Modelling Diffusion: Nonlocal Effects -

Long Range Diffusion

Long range effects can be modelled by integral equations. The change of x at spatial point z and time t can be represented by:

( ) ( )( ) ( ) ( )∫+∞

∞−

−+=∂

∂ ','',, dztzxzzwtzxFt

tzx (3)

( )'zzw − is the kernel function, which quantifies the effect of ( )tzx ,' on ( )tzx , α(m)

Positive Spatial Spillovers(Activation)

Negative Spatial Spillovers(Inhibition)

m=z’-z

z z’

2. Economic/ecological systems under diffusion

Private Optimization Management Problems

Let ( ) ( ) ( )( )ztxztxzt ,,,, 21=x and ( ) ( ) ( )( ),,,...,,, 1 ztuztuzt m=u ,1≥m denote the vectors of state and control variables respectively, at time ),0[ ∞∈t and spatial point [ ].,0 Lz∈ The reaction-diffusion system can be written as:

( ) ( ) ( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( ) circlea is space the, ,,0,

given,0,,0conditionsboundary with

,,,,,,,

,,,,,,,

21

2

22

2122

2

21

2111

2

1

ttLttzxzx

zztxDztztxztxf

tztx

zztxDztztxztxf

tztx

x

x

∀==

∂∂

+=∂

∂∂

∂+=

∂∂

xxx

u

u

Assume that an economic agent is located at each spatial point .z Each agent has a benefit function ( ) ( )( )ztztU ,,, ux defined over the state and the control variables . The benefit function is assumed to be increasing and strictly concave in the controls. Each economic agent considers herself/himself to be small in relation to the spatiotemporal evolution of the state variables and thus chooses controls to maximize an objective at each instant of time for the given spatial site, by treating the values of the state variables as exogenous parameters. Thus each agent ignores the impacts of his/her actions on other sites. However, these impacts emerge because of the diffusion of the state variables and this is the source of a diffusion induced spatial externality.

( ) ( ) ( )( ) mjztztUtzujuj ,...,1,,,,maxarg,0 == ux

( ) ( )( ) mjzthtzu jj ,...,1,,, 00 == x

( ) ( ) ( )( ) . allfor 0,ˆ,,:,ˆ sztztUtz =uxu

( ) ( )( ) mjzthtzu jj ,...,1,,ˆ,ˆ == x

3. The Emergence of Patterns

Pattern formation and Management in Reaction- Diffusion Ecosystems: The Turing mechanism in POMP

To analyze pattern formation we define first a spatially homogeneous or flat steady state (FSS) which is defined for ,0

21== xx DD as:

( ) ( )( )( )( )

( ) ( ) ( )( )xxxh

xh

xhx

001

0

0002

012

0002

011

02

01

0

,...,

0,,

0,,:,

mhh

xxf

xxfxx

=

=

==

Let ( ) ( ) ( )( ) ( ) ( )( )′′

=−− txtxxtxxtxt 21022

011 ,,=x denote deviations

around this FSS and define the linearization.

Linearize around the FSS

( ) ( ) ( )( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛==

2221

1211,,2

1

bbbb

JttJt P

dttxd

dttxd

tP

t xxx

where the elements of the Jacobian matrix, evaluated at the FSS, are defined as:

2

2

12

222

1

2

11

221

2

1

12

112

1

1

11

111

,

,

xu

uf

xfb

xu

uf

xfb

xu

uf

xfb

xu

uf

xfb

j

j

m

j

j

j

m

j

j

j

m

j

j

j

m

j

∂

∂

∂∂

+∂∂

=∂

∂

∂∂

+∂∂

=

∂

∂

∂∂

+∂∂

=∂

∂

∂∂

+∂∂

=

∑∑

∑∑

==

==

Assume that tr 02211 <+= bbJ P and .0det 21122211 >−= bbbbJ P This implies that the FSS is locally stable to spatially homogeneous perturbations.

To analyze pattern formations we proceed by considering the linearization of the full reaction diffusion system, which is:

( ) ( ) ( ) ( )( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=+=

∂

∂∂

∂

2

1

22

2

21

2

00

,,,,,,,

,

x

x

zztx

zztx

zzzzP

t DD

DztztDztJzt xxxx

Spatial patterns emerge if the FSS is unstable to spatially heterogeneous perturbations which take the form of spatially varying solutions, defined as:

( ) ( ) ,...2,1,2,2,1,cos, ±±==== ∑ nLnkikzecztx t

ikk

iπσ

,/2 Lnk π= and πnLk 2//1 = is a measure of the wave-like pattern. k is called the wavenumber and k/1 is proportional to the wavelength nLk //2: == πωω . while σ is the eigenvalue which determines temporal growth and ,ikc 2,1=i are constants determined by initial conditions and the eigenspace of σ .

After substituting the spatially heterogeneous solutions into the full linearization, the linearization becomes

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

== 22221

122

11

2

1,,,kDbb

bkDbJztJzt

x

xLLt xx

Since tr ,0222211 21

<−−+= kDkDbbJ xxL destabilization of the FSS under

spatially heterogeneous perturbations requires that the following dispersion relationship satisfy:

( ) ( ) 0det,0detdet 22211

421221

><++−== PPxxxx

L JJkDbDbkDDkJ φ

where 0det >PJ by the stability assumption about the FSS. Instability of the FSS implies that at least one eigenvalue of LJ becomes positive. The instability requirement will be satisfied if there exist wave numbers 1k and 2k such that ( ) 02 <kφ for ( ),, 2

221

2 kkk ∈ which implies that ( ) 02 >kσ for ( )., 2

221

2 kkk ∈ This in turn requires that: (i) 2mink which

corresponds to the wavenumber which maximizes ( )2kφ is positive and,

(ii) ( ) 02min <kφ or

( )0det

4,0

221

21

21

21

211221122 <+

+−>

+ P

xx

xx

xx

xx JDD

DbDbDD

DbDb

The dispersion relationship

( )

( )( )

( ) { }α

δδ

,0for 0 given,0,),0,(

0,2

2

==∇=∇

>∇+−=∂∂

∇+−−=∂∂

zExzEzx

EDEACpqxEtE

xDqExrxsxtx

E

x

Examples: Biomass/Effort

Linearization around a spatially homogeneous steady state ( ) :, ∗∗ Ex

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=∗

∗

EExx

J =www ,&

( ) .2221

1211⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

−−−

=∗∗∗

∗∗

′ aaaa

EACEpqEqxrx

Jδδ

Linearization of the full system:

.0

0,

//

,2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂∂∂

=

∇=

E

xt

t

DD

DtEtx

DJ

w

www +

Spatially heterogeneous perturbations:

( ) . ,...,2,1 ,cos, ±±=⎟⎠⎞

⎜⎝⎛=∑ n

aznectz t

kk

πλw

Possible spatiotemporal evolution in the neighbourhood of the FSS

( )

( ) .cosexp,

,cosexp,

2

2

22

2

2

azt

aEtzE

ak

azt

axtzx

E

x

ππλε

πππλε

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+∼

⎟⎠⎞

⎜⎝⎛=⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+∼

∗

∗

Substituting the spatially heterogeneous perturbation into the full linearization the eigenvalue ( ),kλ as a function of the wavenumber, is obtained as the roots of

( ) ( )[ ] ( )( ) ( ) ( ).

02

112242

22211

22

JDetkaDaDkDDkh

khaakDD

ExEx

Ex

++−=

=++−++ λλ

If ( ) 02 >kλ for ( )22

21

2 ,kkk ∈ , then the FSS is instable to spatially heterogeneous perturbations.

Example: Management of Semi-arid systems

( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ttWLtWtW

ttPLtPtPzWzP

ztWDztWrztPztWVRztPFztWztPDztTHPbztPztWGztP

zzWWt

zzPt

∀==

∀==

+−−=+−−=

,,0, ,,0,

given,0,,0,,,,,,,,

,,,,,,

( ) ( ) ( )( ) ( ) ( ) 0,,,,,,

0,,0,1,, 1

>=+=>≥+== +

uuWPPWVRPRPFdPPdPbgWPPWG

ζβζβδηδη

( ) ( )[ ] ( )[ ] αα −= 1,,, ztEztPztTH

P: Plant biomass, W : soil water, E : effort, TH : total harvesting

Profits

Profit-maximizing effort-harvesting, Property rights

Open Access

( ) ( ) ( ) ( ) ( )aA

apcztAPztTHztPztE

a−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

=== 100 ,1

,,,,,,1

γγγ

( ) ( ) ( ) ( ) aApcztPAztHTztPztE

a−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛=== 1ˆˆ,ˆ,,ˆ,ˆ,,ˆ,ˆ

1

γγγ

( )[ ] ( )[ ] ( ).,,, 1 ztcEztEztPp −−αα

Turing diffusion induced instability for the semi arid system

The stability of a FSS for the semi arid system requires

( )( )( ) ( )

( )( )( ) ( )( )( )

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛

+−−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

>−−+−−=<−

<+−−=>−

−

−

−

−

WWW

WPPF

WF

WP

ruPuWRgPPdPgW

ggff

J

uWRruPdPgWJuWR

ruPPdPgWJdPgW

00

100

00100

0

0100

100

0det0

:conditionst Determinan0tr

0:conditions Trace

ζδη

ζδηζ

δηδη

ηη

η

η

η

Pattern formation through the Turing mechanism requires:

( ) ( )

( ) ( )( )[ ] )0det4

21000

0100

<+⎟⎟

⎠

⎞−++−−

+>−

−

−

P

WP

WPW

WW

P

JDD

DdPgWDruP

ruPDDdPgW

δη

δη

η

η

The last condition is equivalent to having the dispersion relationship being negative for a certain range of wavenumbers k .

For these wavenumbers one eigenvalue of the Jacobian matrix

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

=WWW

WPPPS Dkgg

fDkfJ 2

2

becomes positive. Thus the FSS is unstable to spatial perturbations and a spatial pattern is formed around the FSS.

The growing spatial instability for the plant biomass and soil water is proportional to ( ) ( )3/2cosexp 2

min zk πσ and is given approximately by

( )

( ) 11.0,3

2cos)114757.0exp(327.87,

99.0,3

2cos)114757.0exp(053.44,

22

11

−=⎟⎠⎞

⎜⎝⎛+∼

=⎟⎠⎞

⎜⎝⎛+∼

vztvBztP

vztvBztP

w

p

π

π

where 2,1, =jv j is the first and the second component of the eigenvector which corresponds to the eigenvalue ( ) 114757.02

min =kσ and sB′ are determined by initial conditions.

A numerical example

Spatial Instability around the FSS

Spatially heterogeneous steady state

( ) ( )( ) ( )

( ) ( )( ) ( )2

22

02

21

20

1

2

1

,0

,0

zzxDzzf

zzxDzzf

x

x

∂∂

+=

∂∂

+=

hx

hx

Long term spatiotemporal evolution

4. Optimal control of partial differential equations

The Social Optimization Management Problem

The Maximum Principle

Let ( ) ( )ztuztx ,,, be the scalar state and control variables respectively at time t and spatial point .z Let ( ) ( )( )ztuztxf ,,, be a net benefit function satisfying standard concavity assumptions and consider the following infinite horizon optimal control problem:

( ){ }( ) ( )( )

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) circle a is space the, ,,,(conditionsboundary spatial with

given,,,,,,,s.t.

,,,max

10

02

2

0,

1

0

ttxztxztx

ztxz

ztxDztuztxgt

ztx

dtdzztuztxfe tz

zztu

∀==

∂∂

+=∂

∂

−∞

∫∫ ρ

Maximum Principle under Diffusion: Necessary Conditions (MPD-NC) Let ( )ztuu ,∗∗ = be a choice of instrument that solves problem (Objectve)-(circle1)

and let ( )ztxx ,∗∗ = be the associated path for the state variable. Then there exists a function ( )ztq , such that for each t and z : 1) ( )ztuu ,∗∗ = maximizes the generalized current value Hamiltonian function

( ) ( )( )

( ) ( ) ( ) ( )( ) ( )⎥⎦

⎤⎢⎣

⎡∂

∂++

=

2

2 ,,,,,,

,,,,~

zztxDztuztxgztquxf

ztquztxxH

or for interior solutions: ( ) 0, =∂∂

+∂∂

ugztq

uf

2) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )2

2

2

2

2

2

,,,,,

,,,

,,,

zztxDztuztxg

tztx

zztqD

xgztq

xfztq

zztqD

xHztq

tztq

∂∂

+=∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂+

∂∂

+∂∂

−

=∂

∂−

∂∂

−=∂

∂

∗

ρ

ρ

evaluated at ( ) ( )( ),,,, ztqztxuu ∗∗ = ( ) ( )uxqguxfH ,, +=

3) The following limiting intertemporal transversality condition holds

( ) ( ) zdzzTxzTqez

z

T

T allfor 0,,lim 1

0

=∫−

∞→

ρ

4) The following spatial transversality condition holds for all dates t :

( ) ( )10 ,, ztqztq =

Basic reference for the result: DERZKO, N., SETHI, P. and THOMPSON, G. (1984), Necessary and Sufficient Conditions for Optimal Control of Quasilinear Partial Differential Systems, Journal of Optimization Theory and Applications, 43, 89-101. Also, BROCK, W. and XEPAPADEAS, A. (2006), "Diffusion-Induced Instability and Pattern Formation in Infinite Horizon Recursive Optimal Control, The Beijer International Institute of Ecological Economics, Swedish Academy of Sciences Discussion Paper, 205/2006, also available at SSRN: http://ssrn.com/abstract=895682. Provide a heuristic derivation of necessary and sufficient conditions for this problem using a variational approach based on Kamien and Schwartz (1991).

Two state variable problemThe Hamiltonian is:

( ) ( ) ( )( ) ( ) ( )( )

( ) ( ) ( )( ) ( )⎥⎦

⎤⎢⎣

⎡∂

∂++

=

∑=

2

2

2,1

,,,,,

,,,,,,,,

zztxDztztfztp

ztztUztztzt

ixii

ii

ux

uxpuxH

which is a generalization of the ‘flat’ Hamiltonian function ( ) ( )uxux ,,

2,1ii

i

fpUH ∑=

+=

For interior solutions ( )ztu j ,∗ is defined by ( ) ( )( ) ( ) ( ) ( )( ) mj

uztztfztp

uztztU

u j

ii

ijj

,...1,0,,,,,,,2,1

==∂

∂+

∂∂

=∂∂ ∑

=

uxuxH

Then ( ) ( ) ( )( ) mjztztgztu jj ,...1,,,,, == ∗∗ px . The costate variables need to satisfy:

( ) ( ) ( ) ( )( ) ( ) 2,1,,,,,,,,2

2

=∂

∂−−=

∂∂ ∗ i

zztpDztztztH

tztp i

xxi

iixgpxρ

Temporal and spatial transversality conditions: ( ) ( )

( ) ( ) 2,1,,0,

2,1,0,,lim0

==

==∫−

∞→

iLtptp

idzzTxzTpe

ii

ii

LT

T

ρ

Modified Hamiltonian Dynamic System

zzxxt

zzxxt

zzxpt

zzxpt

pDHpp

pDHpp

xDHx

xDHx

222

111

22

11

22

11

22

11

−−=

−−=

+=

+=

ρ

ρ

Reference for the two state variable case:

Brock, W. and A. Xepapadeas, (2008), “Pattern Formation, Spatial Externalities and Regulation in Coupled Economic-Ecological Systems,” Beijer Discussion Papers Series, 214, http://www.beijer.kva.se/discussions.asp.

Available also at SSRN, http://ssrn.com/abstract=1144071

5. Emergence of pattern formation through optimal diffusion-induced

instability

We approach this problem by examining how diffusion, regarded as a perturbation, affects the steady state of an optimal control problem without spatial considerations. This is the special case of the control problem with .0=D From optimal control theory, with

,0=D we know that under appropriate concavity assumptions, if a steady state defined as ( )∗∗ qx , such that ,0=x& 0=q& exists, then this steady state will have the local saddle point property or it will be unstable (e.g. Kurz, 1968; Levhari and Liviatan, 1972). Saddle point stability is a concept of conditional stability and implies that for the general n -dimensional problem there exists an n -dimensional locally stable manifold such that if the state-costate dynamical system starts on this manifold in the neighborhood of the steady state, it remains on it for all time and converges to the saddle point steady state. Brock and Scheinkman (1976), Cass and Shell (1976), and Rockafellar (1976) extended this local stability concept to global asymptotic stability (GAS) by introducing a Hamiltonian curvature assumption.

By definition, when D=0 the steady state with the saddle point property is spatially homogeneous or a Flat Optimal Steady State (FOSS). If diffusion destabilizes the stable manifold of this flat optimal steady state, in the sense that the only negative eigenvalue becomes positive, then the result might be the emergence of a regular stable patterned distribution for the state, costate and control variables across the spatial domain of the optimal control problem. The Turing mechanism for diffusion induced instability shown before does not however include optimal control considerations. This might be a new mechanism of diffusion-induced instability of optimal control, the ODI mechanism, which could be used to study pattern formation in optimal control problems.

A Linear Quadratic Approximation

( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )( ) ( ) tLtxtx

Lzzzxzx

GSz

ztxDztGuztSxt

ztxNABBA

dzdtztuxztNxztuBztxAe tL

ztu

allfor ,0,],0[ circlea in,given,0

0,,,,, s.t.

0,0,,

,,,2

,2

max

0

2

2

2

22

00,

=∈=

>∂

∂+−=

∂∂

>−>

⎥⎦⎤

⎢⎣⎡ +−−−∞

∫∫ρ

ρ

where, by a slight abuse of notation, ( ),, ztx ( )ztu , are now deviations from the FOSS values ( ),, ∗∗ ux and the costate variable ( )ztq , associated with the Hamiltonian of the original problem is measured in deviations from the FOSS value, ∗q .

The Jacobian of the MHDS at the flat optimal steady state ( )∗∗ qx , is defined as:

( ) ( ) ( )( ) ( ) ⎥

⎦

⎤⎢⎣

⎡

+−−−

=⎥⎦

⎤⎢⎣

⎡

−−= ∗∗∗∗

∗∗∗∗∗∗

BGN

BN

BG

BGN

qxxx

qqxq

FAF

qxHqxHqxHqxH

qxJρρ 2

2

,,,,

, 00

000

The eigenvalues of ( ),,0 ∗∗ qxJ for ,0>ρ are either positive or they have opposite signs.

Theorem Assume that in the linear quadratic approximation with ,0=D the optimal flat steady state ( )∗∗ qx , associated with the Jacobian matrix

( )∗∗ qxJ ,0 has the local saddle point property. Then if ( )

( ) ( )∗∗∗∗

∗∗

−>

>

qxHqxH

qxH

qqxx

xq

,,4

,2

002

0

ρ

ρ

there is a ,0>D such that the negative eigenvalue of the linearization

⎟⎟⎠

⎞⎜⎜⎝

⎛−

==D

DDDJ zzt 0

0~,~0 www +

( ) ( )( )∗∗ −− qztqxztx ,,,=w becomes positive. That is, both eigenvalues of the Jacobian matrix in (linearTheorem1) have positive real parts. Thus diffusion locally destabilizes the optimal flat steady state.

We look for solutions of the form ( ) ( )zeczt kt

kk

Ww λ∑=,

( ) ( ) ( ) ,...,2,1 ,2,sincos 21 ±±==+= nL

znkkkz nnkπAAW

where nA are arbitrary constants. This solution satisfies circle boundary conditions at 0=z and .Lz = The eigenvalue is ,/2 Lnk π= and

πnLk 2//1 = is a measure of the wave-like pattern. The eigenvalue k is called the wavenumber and k/1 is proportional to the wavelength

.//2: nLk == πωω ( )zkW is the eigenfunction corresponding to the wavenumber .k Substituting the candidate solutions into the linearization, and canceling ,teλ we obtain for each k or equivalently each ,n that .20

kkk DkJ WWW −=λ Since we require non-trivial solutions for ,kW λ must solve

0~ 20 =+− kDJIλ

Then the eigenvalue ( )kλ as a function of the wavenumber is obtained as

the roots of ( )

( ) ( ) 020422

22

det20

JkHDkDkhkh

xq +−+−=

=+−

ρ

ρλλ

State -

Costate

Paths around the FOSS

The solution of the flat LQ control problem takes the form ( )( ) *

222211

*122111

21

21

qevCevCtq

xevCevCtxtt

tt

++=

++=λλ

λλ

For a saddle point FOSS let 0,0 21 <> λλ . To satisfy transversallity conditions at infinity and stay on the stable manifold we set 01 =C . When the FOSS is destabilized by spatially heterogeneous perturbations and ( ) 02

2 >kλ for some k, then the state-cotate paths around the FOSS will take the form:

( )

( ) ∗

∗

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∼

=+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∼

qL

zL

ztL

Cztq

kL

xL

zL

ztL

Cztx

πππλ

ππππλ

2sin2cos4exp,

4,2sin2cos4exp,

2

2

2'

2

22

2

2

2

2'

2

Spatially Heterogeneous Optimal Steady State

The HOSS will satisfy

( ) ( ) ( )( )

( ) ( )( ) 2

20

2

20

,0

,0

zxDzqzxH

zqDzqzxHzq

q

x

∂∂

+=

∂∂

−−= ρ

with the appropriate spatial boundary conditions. Setting ,zxv ∂∂= ,z

qu ∂∂=

we obtain the first-order system

( )

( )( ) uzqqxHq

Dzu

vzxqxH

Dzv

x

q

=∂∂

−=∂∂

=∂∂

−=∂∂

,,1

,,1

0

0

ρ

An economic intuition behind ‘optimal’ spatial heterogeneity

The LQ approximation suggests that ODI occurs, provided that the circle [ ]L,0 is large enough to accommodate the potentially unstable nodes, when the discount rate on the future is larger than a critical value. This ODI emerges as the relative marginal benefits that are associated with the optimal control of a state variable in space-time change between a spatially homogeneous and a spatially heterogeneous solution, as the strengh of diffusion across sizes increases and the size of the space increases. Under certain ciscomstances the optimal discounted benefits of the spatially heterogeneous system could exceed the optimal discounted benefits of the spatially homogeneous system.

Two-state variable problems

Linearization of the MHS around the FOSS) ( )( )

( )( )

( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛

2

1

00

,

,,

,,

,,

2

0

0

x

x

zz

zz

t

t

DD

DHIH

HHJ

ztzt

Dztzt

Jztzt

xpxx

pppx

px

px

px

ρ

where xppxxxpp HHHH =,, are ( )22× matrices of second derivatives of the Hamiltonian.

Consider spatially heterogeneous perturbations of the FOSS of the form

( ) ( ) ( ) ( ) ,...2,1,2,cos,,cos, ±±==== ∑∑ nLnkkzecztpkzecztx tp

ikk

itx

ikk

iπσσ

( ) 3212

3

2

2

2,

2,1,

2121

2121 KKKkKHHHH

K

ikDHH

HkDHK

pxxx

ppxp

xpxxx

ppxxpi

iiiii

iiiii

++=−

=

=+−−

−=

ρ

( )[ ][ ]

( ) ( )

( )22

2

22

02

2,1

42

2,1

0

3210

)2(

2,1,0for2

2

2

2

1

21212121

21

kJDkHIH

HIDkHJ

KkHDkDDkK

HHHH

HHHHK

iDDKKKK

SS

pxxi

xx

ppxxpxxp

ppxxxppxi

xx

iii

iiiiiiii

=⎟⎟⎠

⎞⎜⎜⎝

⎛

+−−−

=

+⎥⎦

⎤⎢⎣

⎡−++−=

+−+

+−=

===++=

∑

∑

=

=

xpxx

pppx

ρ

ρ

ρ

Theorem Assume that with ,021== xx DD the FOSS ( )∗∗∗∗

2121 ,,, ppxx associated with

the Jacobian matrix 0J has the local saddle point property with either two positive and two negative real roots, or with complex roots with two of them having negative real parts. Then there is a ( ) 0,

21>xx DD and wave numbers ( ) 0, 21 >∈ kkk such that,

if: (a) [ ]

( )[ ]

( )( ) ( )22

02

2,1

2,1

2/det0

04

)2(

02

)2(

2

2

2

1

2

2

2

1

KkJ

KDD

HD

DDHD

S

xx

pxxi

xx

pxxi

iii

iii

≤<

>++

−∑

>+

−∑

=

=

ρ

ρ

then all the eigenvalues of the linearization matrix) of the linearized system are real and positive. (b)

( ) 0det 2 <kJ S then the linearization matrix has one negative real eigenvalue, while all the other eigenvalues have positive real parts. (c)

( )( ) ( ) ( )2/2/det

0det4222

22

KKkJ

kJKS

S

ρ+<

<−

then all the eigenvalues of the linearization are complex with positive real parts. In all cases above the optimal dynamics associated with the reaction-diffusion system are unstable in the neighborhood of the FOSS, in the time-space domain.

Patterned state and costate paths around the FOSS can be approximated as: ( )( ) ( )[ ] ( ) ( )[ ] ( )

Lnkkzkkzk

ztzt

n

n

nn

n

n

πσσ 2,cosexpcosexp,, 2

442

33

2

1

2

1

=+∼⎟⎟⎠

⎞⎜⎜⎝

⎛ ∑∑ ccpx

The HOSS will satisfy the system of second-order differential equations in the space variable ,z defined by the MHDS for ,02121 ==== tttt ppxx or,

′′

′

−−=

+=

zz

zz

DH

DH

pp0

x0

x

p

ρ

This second order ( )44× system can either be solved numerically with appropriate boundary conditions, or can be transformed to an ( )88× first-order system by the transformation ,z∂

∂= xX z∂∂= pP and then solved given the spatial boundary

conditions on the circle.

Heterogeneous Optimal Steady State

Example: The Shallow Lake Problem with Spatial Diffusion of Phosphorus

The spatial optimal management shallow lake problem can be stated as:

( )( )( ) ( )( )[ ]

( ) ( ) ( ) ( )( ) ( )

( ) ( )Ltxtxz

tzxDztxhztbxztat

tzxts

dzdtztxCztaBe tL

zta

,0,

,,,,,..

,,max

2

2

00,

=∂

∂++−=

∂∂

−−∞

∫∫ ρ

i ( )ix ∗∗ λ, Eigenvalues 0det J Stability

1 ( )019.16,571.0 −

00812.0,13542.0 − 00109.0−

Saddle Point

2 ( )350.16,583.0 −

00897.0,11833.0 00106.0 Unstable

3 ( )157.2,445.2 − 49719.0,62449.0 − 31049.0−

Saddle Point

Table 1: FOSS for the Shallow Lake

Table 2: Emergence of Spatial Heterogeneity in the Shallow lake Diffusion-induced instability conditions Value at ( )ix ∗∗ λ, (T2) and (T3) 1=i 3=i ( )[ ] 022 0 >−+−=− ∗′

ρρλ ix xhbH 091633.0 00648.1−

( )( ) 02

22

400

4 >⎥⎦

⎤⎢⎣

⎡ −+=+∗∗

∗′′

λλρ

λλρ cxh

xxiHH

00100.0 057241.0−

Example: Optimal resource management with stock effects

We assume that net harvesting benefits at each point in space-time can be represented by a concave net benefit function ( ).qExQ The optimal harvesting problem in space-time is then defined as:

( )( ) ( )( )

( ) ( )( ) ( ) ( ) ( )

( ) ( ) pastingsmooth and ,0,

,,,,, s.t.

,,max

2

2

0,

Ltxtxz

ztxDztEztqxztxft

ztx

dzdtztxztqEQe t

ztE

=∂

∂+−=

∂∂

−∞

∫∫ ρ

Z

The generalized current value Hamiltonian for this problem is defined as:

( ) ( ) ( )⎥⎦

⎤⎢⎣

⎡∂

∂+−+= 2

2 ,~z

ztxDqExxfqExQH μ

Model Specification

• Standard logistic model, ( ) ( ),1 Kxrxxf −= where r is the intrinsic

growth rate and K the carrying capacity, with 08.0=r and 400000=K . • Quadratic benefit function ( ) ( ) ,2/1 2hhAhQ βα −+= • Harvesting ,qExh =

( )

( ) ( )( ) ( )[ ] ( )( )[ ] μμμρμ

μμμμρμ

μμ

μ

μμμ

=−−=

−−+−=

+−=

<=∂∂

<−=∂∂

=+

=⇒=⇒=

′′

′′

′′

′′′′

′′

QDxf

orDxEqQxqExf

DxxxqExfxxQ

ExE

xE

dEdxQxdEQ

xEEQxqxqQ

zz

zz

zz

using

,,

,

01,0

,

&

&

&

Spatially Heterogeneous OSS

The HOSS is characterized by:

( ) ( )[ ]

( )[ ]μρμ

μ

xfDz

xfxxEDz

x

′

−=∂∂

−=∂∂

1

,1

2

2

2

2

Making the substitutions ( ),zvz

x =∂∂ ( )zwz =∂

∂μ and using the circle boundary conditions and the appropriate transversality conditions ( ) ( ) ,750000 == Lxx ( ) ( ) ,31.76920 == Lμμ ( ) ( ),0 Lvv = ( ) ( ),0 Lww = we solve the system by multiple

shooting. figure 7 shows the spatial paths for x and μ at the HOSS for .2π=L The U curve is the spatial path for the biomass stock, while the lower curve, which has a very flat inverted U shape, is the spatial path for the resource's user cost.

6. Optimal diffusion-induced instability and regulation

Temporal Steady State: Optimality Conditions POMP -

SOMP

•Use the spatially heterogeneous shadow values for the state variables to design optimal decentralized price instruments. Taxes with spatiotemporal structure

•

Use optimal spatiotemporal controls to design quantity instruments

•Limits, quotas, possible tradable, with spatiotemporal structure

∂Uxz,uz∂uj

0 , j 1, . . . ,m

∂Uxz,uz∂uj

∑i1,2

piz∂f ixz,uz

∂uj 0

We define the optimal quantity instrument to be a value for a limit or a quota on private controls such that the steady state of a reaction diffusion system, with private agents maximizing private profits, is the same as the socially optimal steady state for the reaction diffusion system.

We define the optimal price instrument to be a value for a linear tax on private controls such that the steady state of a reaction diffusion system, with private agents maximizing private profits, is the same as the socially optimal steady state for the reaction diffusion system.

Quantity and Price Instruments

Example: A Semiarid System The Socially Optimal Steady State

The Hamiltonian for the SOMP is ( )( )[ ]

( )[ ]zzWW

zzP

WDWruWPRPPDTHPPdgWPcEEpP

+−−+++−+−+−= −

ζβμδλ ηαα 11H

Quantity Regulation

Price Regulation