space in unified models of economy and ecology or... space: the final frontier a. xepapadeas*...
TRANSCRIPT
Space in Unified Models of Economy and Ecology
or . . . ? Space: The final frontier
A. Xepapadeas*University of Crete, Department of Economics
*Research presented in this lecture has been conducted jointly with William Brock.
• Economics studies how human societies use scarce resources to produce commodities and distribute them among their members.
• Ecology is the study of living species, such as animals, plants and micro-organisms and the relations among themselves and their natural environment. An ecosystem includes these species and their nonliving environment, their interactions and evolution in time and space.
• Human economies and natural ecosystems are inexorably linked, but economic models and ecological models are not usually linked. Ecological economics aims at providing this link.
Economic and ecological systems evolve in time and space. Interactions take place among units occupying distinct spatial points. Thus geographical patterns of production activities, urban concentrations, or species concentrations occur. My purpose is:
• to discuss approaches for modeling, in a meaningful way, economic and ecological processes evolving in space time.
• to examine mechanism under which a spatially homogenous state – a flat landscape – acquires a spatial pattern.
• to examine how this pattern evolves in space-time.
• The emergence of spatial patterns in economics has received relatively little systematic analysis, with the notable exception of the new economic geography.
• Spatial patterns in human economies are profound . . .
Spatial Patterns
Spatial Analysis in Economics
• However spatial analysis was not given sufficient attention until the early 1990s
• Main ideas in location theory rely on economies of scale that enforce geographical concentrations
• Inability of earlier research to work with tractable models of imperfect competition which is implied by unexhausted economies of scale.
• In a homogeneous environment with transportation costs but no returns to scale, spatial patterns of economic activity cannot emerge. Economic activity should spread evenly across space to minimize transportation costs.
• Need for increasing returns to generate spatial patterns
• Need to model imperfect competition• Increasing returns/Imperfect competition
– New Industrial Organization– New Trade Theory – New Growth Theory – New Economic Geography
The Racetrack Economy*• Many regions equally spaced around the
circumference of a circle• Transportation takes place around the
circumferences• From a spatially homogeneous – flat – initial
distribution of manufacturing activities, a perturbation generates a spatial structure. Manufacturing is concentrated in two regions.
• What is the mechanism that generates this spatial pattern?
*M. Fujita, P. Krugman and A. Venables, The Spatial Economy, MIT Press 2001.
Paul Krugman, “Space: The Final Frontier” Journal of Economic Perspectives, Vol. 12, 2, 1998, pp. 161-174
Spatial Analysis in Ecology• Pattern formation and the emergence of
spatial patterns have received relatively more attention in ecology.
• Morphogenesis is the study of patterns and form, e.g.:– Mammalian coat patterns– Butterfly wing patterns
• Spatial patterns in resources.• Spatial patterns of species.• The concept of diffusion has been used in
ecological modeling to explain spatial pattern formation in ecological systems.
Modelling Diffusion• Biological resources tend to disperse in space
and time under forces promoting "spreading" or "concentrating" (Okubo, 2001); these processes along with intra and inter species interactions induce the formation of spatial patterns.
• Economic activities also tend to disperse in space and time. Flows of capital, labour, commodities, resources
• Spatial issues in economic-ecological problems: – resource management in patchy environments
(Sanchirico and Wilen 1999, 2001; Sanchirico 2004; Brock and Xepapadeas 2002)
– the study of control models for interacting species (Lenhart and Bhat 1992, Lenhart et al. 1999)
– the control of surface contamination in water bodies (Bhat et al. 1999)
• A central concept in modelling the dispersal of biological or economic resources is that of diffusion.
• Diffusion is defined as a process where the microscopic irregular movement of particles such as cells, bacteria, chemicals, animals, or commodities, results in some macroscopic regular motion of the group (Okubo and Levin 2001; Murray 1993, 2003).
• Diffusion is based on random walk models, which when coupled with population growth equations or capital accumulation equations lead to general reaction-diffusion systems.
Let tzx , denote the concentration of a biological or economic
variable at time t 0 at the spatial point .z Space is assume to be a line. We assume that the flux of ‘material’ animals, commodities, is proportional to the gradient of the concentration of the material , or
z
tzxDtz x
,,
where xD is the diffusion coefficient and the minus sign indicates that material moves from high levels of concentration to low levels or concentration Under this diffusion assumption the evolution of the material’s stock in a small interval z is defined as
dstsFtzztzdstsxdt
d zz
x
zz
z,,,,
(1)
where txF , is a growth function for the material in question Dividing (1) by z and taking limits as ,0z the evolution of the material is determined as:
tzFx
tz
t
tzx,
,,
Using the definition of diffusion we obtain the basic diffusion equation
2
2 ,,
,
x
tzxDtzF
t
tzxx
• In general a diffusion process in an ecosystem tends to produce a uniform population density, that is spatial homogeneity. Thus it might be expected that diffusion would "stabilize" ecosystems where species disperse and humans intervene through harvesting.
• There is however one exception known as diffusion induced instability, or diffusive instability (Okubo et al. 2001). Alan Turing (1952) suggested that under certain conditions reaction-diffusion systems can generate spatially heterogeneous patterns. This is the so-called Turing mechanism for generating diffusion instability.
Emergence of Spatial Patterns
• We examine conditions under which the Turing mechanism induces diffusive driven instability and creates heterogeneous spatial patterns in Economic/Ecological models.
• This is a different approach to the one most commonly used to address spatial issues, which is the use of metapopulation models in discrete patchy environments with dispersal among patches.
• Thus the Turing mechanism can be used to uncover conditions which generate spatial heterogeneity in models where ecological variables interact with economic variables. When spatial heterogeneity emerges, the concentration of variables of interest (e.g. resource stock and level of harvesting effort), in a steady state, are different in different locations of a given spatial domain. Once the mechanism is uncovered, the impact of regulation in promoting or eliminating spatial heterogeneity can also be analyzed.
A Bioeconomic ModelThe movement of biomass and effort in time and space can be described by the following reaction diffusion system
2
22
2
2
,
,0for 0 given,0,),0,(
0,
z
x
z
xx
zExzEzx
EDEACpqxEt
E
xDqExrxsxt
x
E
x
where EAC is the average cost curve, assumed to be U-shaped. By (zero flux) it is assumed that there is no external biomass or effort input on the boundary of the spatial domain.
The spatially homogeneous system for biomass evolution is:
0,
EACpqxEE
qExrxsxx
where a steady state 0, Ex for the spatial homogeneous system
is determined as the solution of .0Ex Linearizing around a steady state ,, Ex the linearized spatial homogeneous system can be written as
EE
xxJ www ,
where the linearization matrix J around a steady state is defined as
2221
1211
aa
aa
EACEpqE
qxrxJ
The Turing Mechanism
• The Turing mechanism implies that the spatially homogeneous steady state can be destabilized by a spatial perturbation depending on
• Condition for diffusive instability
xE DD /
02 2/12/1211222112211 xExE DDaaaaDaDa
Linearizing the full system and we obtain
E
xt
t
D
DD
tE
tx
DJ
0
0,
/
/
,2
w
www
The solution for the biomass and effort takes the form
a
zt
aEtzE
a
zt
axtzx
E
x
cosexp,
cosexp,
2
2
2
2
Since ,02
2 a as t increases the deviation from the spatial
homogeneous solution does not die out and could eventually be transformed into a spatial pattern which is like a single cosine mode. Then the growing solution approaches, as ,t a cosine like spatial pattern, which implies spatial heterogeneity of the steady state. Figure 1, draws on Murray (2003 Vol II, pp. 94-95) to represent one possible spatial pattern for ., tzx Shaded areas represent spatial biomass
concentration above ,x while non shaded areas represent spatial biomass
Capital Accumulation and Pollution Accumulation*
* Based on current research of S. Levin and A. Xepapadeas
We assume that pollution and capital diffuse along a segment 0,L. Pollution accumulation evolves in time and space according to:
Px,tt
gx,Kx,t xPx,t DP2Px,tx2
#
Capital stock evolves in time and space and according to:
Kx,tt
sxAKx,tfKx,t x,Px,tKx,t
DK2Kx,tx2
#
If the determinant of the linearization matrix around the equilibrium point
J sa AKa1 P PK
gK
does not vanish then an equilibrium exists for the spatially homogeneous system
If 1 then the a11 element of the linearization matrix is positive. In this case if
sa AKa1 P trJ 0
|J| 0 the linearization matrix has two real negative eigenvalues and the equilibrium is stable.
• The spatially homogeneous capital-pollution steady state can be destabilized by diffusive or Turing instability if:
• In this case we have a spatially heterogeneous pattern of capital accumulation (economic activity) and pollution accumulation
• But we require α11>1, increasing returns
02 2/12/1211222112211 KPKP DDaaaaDaDa
Control of BioinvasionsWe consider that a "propagule" of an invasive species (e.g. insects) of size x0,0 x0 is released at the origin of a one dimensional space of infinite length.
xt
sx Dx 2xz2
The invasive species moves in time and space as in figure 2.
Assume that we apply effort (spraying) Ez,t at time t and size z and by doing that qEz,txz,t insects are removed when xz,t are present. Then the evolution of the species is given by the Skellam equation
xt
s qEx Dx 2xz2
#
Spraying reduces the growth of the invasive species as shown in figure 2.
-1000
-500
0
500
1000
200
400
600
800
1000
0
1106
2106
3106
4106
5106
-1000
-500
0
500
1000
t
z
x
Figure 2
To make the model more realistic assume that the removal of the insects generates benefit equal to pzqEz,txz,t, where pz is the value associated with insect removal. Spraying has a time invariant average cost
ACE c0z 12c1zEz,t #
Assume that spraying moves fast across sites so that costs and benefits are equated and the p, c0 and c1 are spatially homogeneous. In this case the evolution of the invasive species is given by the Fisher equation
xt
ŝx1 âx Dx 2xz2
ŝ s 2qc0c1
, r 2q2pc1
, â rŝ
#
It can be seen that a wave solution exists for the evolution of the invasive species with minimum wave speed ,
cmin 2 s 2qc0
c1Dx
1/2
In the wave front solution the maximum concentration of the invasive species is
1â c1s 2qc0
2q2p #
Thus maximum concentration is declining in p the value associated with the removal of the invasive species.
Conclusions• This paper develops methods of analyzing
spatial dynamical ecological/economic systems.
• In particular the Turing mechanism for diffusive instability is adopted to bioeconomic problems
• The potential power of the method is shown in the analysis of spatial pattern formation in– A resource management problem– A spatial growth under pollution
accumulation problem– A bioinvasion control problem
Regulation Issues• In the resource management
problem and the growth pollution problem, spatial pattern creation is possible.– Spatial pattern creation may have
welfare implications regarding the spatial distribution of welfare.
– Regulation can eliminate spatial patterns and induce spatial homogeneity.
• Control of Bioinvasions– When benefits and costs are equated
across sites, the invasion could take the form of a travelling wave.
– Regulation affects the wave's speed and the spatially homogeneous stable carrying-capacity biomass of the invasive species.
We are living in a spatially heterogeneous world.. The modeling approach presented here might help in gaining some new insights into the interrelations between ecological systems and human economies and might provide a basis for more efficient regulation of environmental externalities in the space – time continuum