Time Delayed Dynamic Model of Renewable
Resource and Population
Akio Matsumoto1
Department of Economics
Chuo University

Mami Suzuki2
Department of Management
Aichi Gakusen University

Yasuhisa Saito3
Department of Systems Engineering
Shizuoka University
August 30, 2005

1 742-1,

Higashi-Nakano, Hachioji, 192-0393, Japan. akiom@tamacc.chuou.ac.jp; +81-426-74-3351(tel); +81-426-74-3425 (fax).
2 1, Shiotori, Ohike-Machi, Toyoda, 471-8532, Japan. m-suzuki@gakusen.ac.jp;
81-565-35-1313 (tel), 81-565-35-1677 (fax).
3 3-5-2, Johoku, Hamamatsu, 432-8561, Japan. y-saito@sys.eng.shizuoka.ac.jp.

Abstract
This paper reconsiders a "Ricardo-Malthus" dynamic model of population
and renewable resource developed by Brander and Taylor [1998]. To thisend, it sheds light on a delay in production, a key feature of long-run evolution in a pre-industrial economic society. It is clear that the delay has aconsiderable eﬀect on long-run dynamics of natural resource and population
This notwithstanding, many previous works relate to a case in which production is instantaneous. As such, the purpose of this paper is to investigate
the eﬀect caused by the delay in production. Our analysis shows that there
is a critical value of the delay with which a delayed version of an otherwise
stable system becomes unstable. In addition, it numerically shows that thecritical value is negatively related to the size of the carrying capacity and theexogenous net birth rate.

Acknowledgement
This is a revised version of the paper with the same title, which was presented at the international conference on Dynamical Systems Theory and Its
Applications to Biology and Environmental Science held at Shizuoka University, Shizuoka, Japan, March 15 - 17, 2004. We are grateful for commentsfrom Bob Sacker and participants of the conference. We appreciate the financial supports from the Japan Ministry of Education, Culture, Sports, Scienceand Technology (Grand-in-Aid for Scientific Research (C) 15540217 for the
first two authors and Grand-in-Aid for JSPS Fellows 00000472 for the third
author). Needless to say, remaining errors are our responsibility.

1

Introduction

This study reconstructs an economic dynamic model for a small island basedon the Easter Island study by Brander and Taylor [1998] (BT henceforth).
Introducing a delay in production into the BT model, it deals with analyticalas well as numerical representations of evolutions of a small island economy,
evolutions of which are characterized by a two-fold process of transformation;
environmental degradation and population growth. Its main purpose is toinvestigate eﬀects caused by the delay in production on an adjustment process
in population and natural resource stock.
Archeological studies have suggested that many Pacific islands followed
similar evolutionary patterns of natural resource and population dynamics,
that is, rapid population growth, resource degradation, economic decline andthen population collapse. BT reconsider archeological and anthropologicalevidences of Easter Island, as an example, from an economic point of view. In
particular, BT present a general equilibrium model of renewable resource andpopulation dynamics and seek to explain the rise and fall of Easter Island for
1400 years between the 4th century and the middle of the 18th century. Theybring to light on economic conditions under which the small island economy
can survive or perish. Their findings indicate that an economic model linking resource and population dynamics may explain not only the sources ofpast historical evolutions discovered in these small islands but also a possibility of sustainable growth of our world economy in which rapid-increasing
population and rapid-degrading environment become serious problems. The
analysis in the BT model has been extended in various directions. Dalton
and Coats [2000] examine the impact of market institutions and diﬀerent
property-rights structures. Reuveny and Decker [2000] numerically consider
how technological progress and population management reform aﬀect thelong-run dynamics of Easter Island. Matsumoto [2002] reconstructs the BT
continuous model in discrete steps and shows that the modified model cangenerate various dynamics ranging from simple dynamics to complex dynamics involving chaos.
In the existing literature, however, not much has been revealed with respect to a "history" of Easter Island. It has been believed that a small groupof Polynesians arrived the island around 400 A.D., deforestation occurred
around 1000 A.D., most of statures were carved during 1000-1400 A.D., andso forth.1 Based on this "conventional wisdom," BT as well as other researchers attempt to reproduce a dynamic pattern of natural resource andpopulation. However, the "wisdom" is still one of possible hypotheses and1

See section 1 of Brander and Taylor [1998] for more details.

1not fully confirmed yet. In particular, according to Intoh [2000], recent reconsiderations of archeological evidences on the island imply indeterminacyof arrival time of Polynesians. It can only be estimated that a small groupsettled between 410 and 1270 A.D. This new finding is inconsistent with thetraditional wisdom. In other words, we may have a diﬀerent history (as well
as diﬀerent evolution pattern) of Easter Island, even though the available
historical evidences are the same. It is thus imperative to construct a model
for small islands that can generate various patterns of dynamics in order todeal with such ambiguous characteristics of archeological evidence.
In this study, we extend the BT model to include a delay or lag in production with the following reason. Agricultural production may play anessential part of economic activity in a pre-industrial economic society. It is
well-known that an important characteristic of such an agricultural production is the significant time lag between the time at which producers maketheir decisions to plant seeds in the fields and the time when they actuallygather crops from the fields. It is thus natural to raise a question: how thedelayed production aﬀects an evolutionary pattern of small island economy?
This study is organized as follows. Section 2 constructs a simple economic
dynamics model of an small island based on BT’s Easter Island analysis.
Section 3 analytically as well as numerically examines eﬀects caused by adelay in production on evolutions of natural resource and population. Section
4 provides summary and concluding remarks.

2

Basic Model for Small Islands

Since our analysis is based on BT’s dynamic model of renewable resourceand population, we recapitulate the basic part of their model in this section.
See BT’s paper for more details.
The model describes the dynamics of an economy with two goods andthree types of economic agents (two producers and one consumer). The
harvest of the renewable resource is called agricultural good and some othergood called manufactured good. The model functions as follows. At time t,
the stock of natural resource S(t) and the size of population L(t) are given.2
Producers determine their demands for labor and supplies of goods so as tomaximize their profits. A manufactured producer supplies the manufactured
good produced with constant returns to scale using only labor. Since, by
choice of units, one unit of manufactured goods can be produced by one unitof labor, the total supply of manufactured good M S is determined by the2

For a time being, t is suppressed for the notational simpliciy.

2demand for labor LD
M

M S = LD
M.

(1)

An agricultural producer supplies the agricultural good carried out according
to the Shaefer harvesting production function,
H S = αSLD
H,

(2)where H S is the harvest supplied, α is a positive constant indicating theharvesting eﬃciency and LD
H is the labor demand in resource harvesting. A
representative consumer is endowed with one unit of labor and is assumed
to have a Cobb-Douglas utility function,u(h, m) = hβ m1−β ,(3)where h and m are individual consumptions of the agricultural good and of
manufactured good, and β ∈ (0, 1) is a positive constant reflecting preferenceof the agricultural good. Each consumer supplies one unit of labor anddemands both goods so as to maximize his utility subject to the budget
constraint, ph + m = w where p is the price of the agricultural good, w is thewage rate, and the price of the manufactured good is normalized to 1 as it
is treated as a numeraire. The usual utility maximization procedure yields
optimal demands for both goods, hd = wβand md = (1 − β)L where d ispattached to a variable indicating individual demand. The total number of
population is L so that the total demands are
H D = Lhd and M D = Lmd .

(4)

Prices are adjusted to establish temporary equilibrium in each of three markets; agricultural good market (H D = H S ), manufactured good market
D(M D = M S ) and labor market (LDM + LH = L) in which labor force isassumed to be equal to the population. It can be verified that the fixedproportion of the total population is employed in the agricultural section,SLD
H = βL and thus the resource harvest is H = αβSL at the temporary
equilibrium state. After finishing transactions in each market, new values ofnatural resource and the size of population are determined at the next instant of time. With these new values, the process repeats until the stationary
state is attained.
Dynamics of temporary equilibrium is described as follows. A change inthe stock at time t is determined by the natural growth rate G(S) minus theharvest rate,dS(t)= G(S(t)) − H S (t).
(5)
dt
3

For the analytical simplicity, the logistic functional form for G is assumed,
G(S) = r(1 − S/K)S where K is the maximum possible size for the resource
stock, r is an intrinsic growth rate of natural resource, and both are positive constants. A change of population depends on a diﬀerence between anunderlying birth rate and death rate. b denotes the exogenously-determinedbirth rate, and d the endogenously-determined death rate. It is assumed
that the net rate, denoted as c = b − d, is negative. Following the formulation of Malthusian population dynamics, it is further assumed that theendogenously-determined birth rate depends on the economic activities, thatis, per capita consumption of the agricultural good increases fertility and/orS
decreases mortality. Let φ HL be a fertility function where φ is positive constant. Then the population growth rate is
1 dL(t)H S (t)= (c + φ),L(t) dtL(t)(6)

where the first factor is the exogenous net birth rate, and the second is theendogenous birth rate. Substituting the logistic function into the natural
growth rate and the optimal harvest into the fertility function yields the
dynamics process of the natural resource and population,
µ
¶

dS(t)
S(t)


= r(1 −
) − αβL(t) S(t),

 dt
K
(7)



 dL(t) = (c + αβφS(t)) L(t).
dtThis is a two-dimensional dynamic system of diﬀerential equations and isa variant of the Lotka-Volterra perdator-prey model in which the human isthe predator and the resource stock is the prey. The system has a steadystate if dS= 0 and dL= 0 hold. We can solve the last simultaneous systemdt
dt
for stocks of the natural resource and population size.3 Coordinates of aninterior solution are
µ
¶
c
r
c
e
e
S =−
and L =
1+
.
(8)
αβφ
αβ
αβφ

BT find the following result about the stability of the interior steady point:
3

The system can have a corner solution depending on values of parameters. However,the corner solution with L = 0 means an extinction of human, which is not interesting.
Thus, in this study, we focus only on the interior solution.

4

Theorem 1 (Proposition 4 (iii) of Brander and Taylor) An interior steadystate (Le > 0 and S e > 0) is a spiral node with cyclical convergence if−cr+ 4(−c − αβφK) < 0
αβφKand an improper node allowing monotonic convergence if not.
Using the parameterization for the basic model for which BT provide a
detailed justification,4 we perform two simulations, which are reproductionsof BT’s numerical examples. Figure 1 illustrates time series for population
size (i.e., a mountain-shaped curve) and resource stocks (i.e., a downward
sloping concave-convex curve) when the intrinsic growth rate of renewable
resource is low, r = 0.04. Figure 2 also illustrate time series for the same
variables when the rate is roughly 9 times higher, r = 0.35. In these figures, one period represents one decade and the horizontal axis shows 140
periods. The initial period corresponds to the year 400 A.D., when the firstindigenous people are said to have arrived on the island, and the last period
corresponds to sometimes of the 18th century when European first arrived
on the island, after which the substantial changes in environments make thedynamic model, (7), hold no longer.
Figure 1 shows oscillatory dynamics and appears to replicate what is
known of Easter Island history. The island was settled by a Polynesian groupabout 400 A.D and was covered with great palm trees at this time. For the
first 300 years, the populations size is small, and the resource has little degradation. However, soon after, the population size begins to increase rapidly,
and the resource begins to decline correspondingly. The heyday of Easter
Island is supposed to be between 1100 and 1300: the population reaches its
maximum (about 10,000) and the statue carving is intensive. After reachingits peak of the population size, the island entered into a declining period
and then disappeared from the history; the palm forest was entirely gone by
1400, carving ceased by 1500 and violent internecine conflict appeared.
4

BT use the following parameters’ values for their simulations: L0 = 40 (initial humanpopulatoin), S0 = 12, 000 (initial stock of the renewable resource), K = 12, 000 (carrying
capacity), α = 0.00001 (harvesting eﬃcienty), β = 0.4 (preference for agricultural good),r = 0.04 (intrinsic growth rate of the renewable resource), φ = 4 (fertility rate), c = −0.1
(intrinsic net birth rate).

5

S,L
12000

8000

4000

4c

9c

11c

14c 15c18c

TimeFigure 1.Slow Growth Rate of Natural Resource (r = 0.01)
According to Theorem 1, the model can generate monotonic behavior forsome combinations of parameters values, which may explain monotonic evolutions observed on some Polynesian islands. In the second simulation wechange the growth rate of the natural resource from the lower value (r = 0.04)
to the higher value (r = 0.35) and use the same values of any other parameters as in the first simulation. As illustrated in Figure 2, the simulationshows a entirely diﬀerent dynamics; a smooth adjustment converging to thestationary state. Comparing these two figures, we observe that an island witha slow-growing resource base will exhibit overshooting and collapse while anisland with a rapid growing resource exhibits a near-monotonic adjustmentof population and resource stocks towards steady state.
S,L
40000

20000
12000

4c

9c

11c

14c 15c18c

TimeFigure 2. Rapid Growth Rate of Natural Resource (r = 0.35)

63Application Model with Time Delay

In this section, we investigate the hypothetical role that a delay might has
played in disturbing monotonic dynamics and disturbing further an oscillatory dynamics. In order to see this disturbing or destabilizing eﬀect, we
introduce a delay in production into the basic model, taking account of thefact that it takes some time for agricultural good from seeding to harvesting,H(t) = αβL(t)S(t − τ ).

(9)

τ is a delay in production. The current harvesting depends on the current
amount of labor and the stock of natural resource at time t − τ . Substituting(9) into the basic system (7) generates the dynamical system with the delay
in production, τ ,

dS(t)

S(t)


 dt = rS(t)(1 − K ) − αβL(t)S(t − τ )
(10)


dL(t)


= L(t)(b − d + φαβS(t − τ )).
dt
It can be verified that the delayed system has the same equilibrium point
as the basic system. To investigate the stability of the delayed system, wefirst make a coordinate transformation such that a new system is centeredat the equilibrium point (S e , Le ) and then linearize the resultant system atthe origin to derive its characteristic equation.
Let S = S − S e and L = L − Le . Then the centered system is reduced to

dS(t)
e


)S(t) − αβS e L(t) − Kr S(t)2= r(1 − 2S

K

dt


−αβLe S(t − τ ) − αβL(t)S(t − τ )
(11)





 dL(t) = (Le + L(t))φαβS(t − τ ).
dt
¶
µ
S(t)
= Ceλt where C ∈ C2 and λ ∈ C. Comparing the linear terms,
PutL(t)we have¶¶µµer(1 − 2S
−αβLe 0) −αβS e
λt
λt
K
Ce +
Ceλ(t−τ ) .
Cλe =φαβLe 000In consequence, for (10), we get the characteristic equation of the linearized
system around (S e , Le ),2S e) + αβLe e−λτ }λ + (αβ)2 φS e Le e−λτ = 0,
λ + {−r(1 −K
27and substitute (8) into it to haveλ2 + {−(r +rc2cr) + (r +)e−λτ }λ + (αβ)2 φS e Le e−λτ = 0.
φαβK
φαβK

(12)

When τ = 0, (12) becomes
λ2 −cr
λ + (αβ)2 φS e Le = 0.
αβφK

(13)

Since c < 0 and (αβ)2 φS e Le > 0, all the characteristic roots of (13) havenegative real parts, by which, in this case, the equilibrium point (S e , Le ) is
locally asymptotically stable for (10).
For the sake of notational simplify, we introduce new variables, p and q,
defined asrc
p=r+
and q = (αβ)2 φS e Le > 0.
αβφK
Substituting the new variables into (12), we can rewrite the characteristic
equation with delay as
λ2 + (−(2p − r) + pe−λτ )λ + qe−λτ = 0.

(14)

Substituting λ = iy into (14) gives
py sin yτ + q cos yτ = y 2 ,py cos yτ − q sin yτ = (2p − r)y.

(15)
(16)

Squaring and adding (15) and (16) yields
y 4 + (3p2 − 4pr + r2 )y 2 − q2 = 0.

(17)

Let Y = y 2 ≥ 0 where the direction of inequality is due to y ∈ R. By the way,
if Y = 0, we have q = 0 which is a contradiction. Thus we are concernedwith Y > 0 and have roots y0 of (17),y0 = ±

Ã−(3p2 − 4pr + r2 ) +! 12p(3p2 − 4pr + r2 )2 + 4q2
.
2

(18)

From (15) and (16),cos(y0 τ ) =y02 (q + 2p2 − pr)y0 {py02 − q(2p − r)}
and
sin(y
τ
)
=
,
0
p2 y02 + q2
p2 y02 + q2

which imply that there is a τ 0 such thatτ0 =y 2 (q + 2p2 − pr)|y0 |{py02 − q(2p − r)}1
1
arcsin 0 2
arccos=. (19)|y0 |q + p2 y02|y0 |p2 y02 + q2

Then we have the following theorem.

8

Theorem 2 The delayed dynamical system (10) is unstable if τ > τ 0 whereτ0 =y 2 (q + 2p2 − pr)|y0 | {py02 − q(2p − r)}1
1
arcsin 0 2
arccos=,|y0 |q + p2 y02|y0 |q 2 + p2 y02
p=r+rc> 0, q = (αβ)2 φS e Le > 0
αβφK

and
y0 = ±

Ã−(3p2 − 4pr + r2 ) +! 12p(3p2 − 4pr + r2 )2 + 4q 2.
2

Proof. See Appendix.
To the best of our knowledge, a nonlinear dynamical system with timedelay does not have an analytical solution.5 Nevertheless, it is possible toexamine its dynamical behavior by simulating the system numerically. Since
we have performed two simulations without delay in the last section, we
conduct simulations with production delay and then compare the resultswith time delay with the one without it. By doing so, we can detect eﬀects
on long-run dynamics of population and natural stocks caused by the delay
in production.
Figure 3 presents simulation results that occur when the delay production
is introduced into the first example, ceteris paribus. Real lines show timeseries of population and natural resource obtained in the current simulation
while dotted lines are reproductions of simulation results depicted in Figure 1.
With delay in production, the populations and the resource stock generates
more volatile fluctuations relative to the ones without delay. Population
reaches a much higher peak and much lower trough, jumping to 17,500 andfalling quickly down to near zero while the natural resource declines more
rapidly and gets closer to zero stock. These numerical simulations indicate
the destabilizing eﬀect caused delay in production in the oscillatory case.
5

It is possible to construct an analytical solution when a nonlinear dynamical system
is discrete. See Suzuki [1996, 2000].

9St ,Lt
17500

12500

7500

2500
4c

9c

11c

14c 15c18c

TimeFigure 3. Production Delay in Fluctuating Economy
Figure 4 presents simulations that occur when the delay production is
introduced into the second example, ceteris paribus. As in Figure 3, reallines show the simulation results with the delay in production and dottedlines the ones without it, which reproduce the results depicted in Figure 2.
Comparing these results, we observe firstly the earlier and much more volatilefluctuations in population as well as natural resource; secondly the almost
exhaustion of natural resource and the much more severe falls in population
at the beginning of 18th century. This numerical simulations again indicate
the destabilizing eﬀect of the delay even on otherwise monotonic dynamics.
S,L
50000

30000

10000

4c

9c

11c

14c 15c18c

TimeFigure 4. Production Delay in Stable Economy
In these numerical simulations, we adopt the parameterization used by
BT for which the critical value of the delay is τ 0 = 10.1652.6 This is indeed alarge delay, but we should not be too much concerned about this. As seen in(19), the critical value depends on any other parameters. We should thus be6

See Appendix for calculations.

10

content with the existence of the critical value, trusting that it might be considerately small under slightly diﬀerent parameter specifications. Although it
may be possible to derive analytically its dependency on parameters, we can
expect the computations to become messy and thus confirm it numerically.
Figure 5(A) illustrates the relationship between the delay and the maximum
size of the natural resource, ceteris paribus. It displays the downward sloping
borderline between stable region and unstable region. It is observed that thecritical value of τ gets smaller as the size of K becomes larger. Figure 5(B)illustrates the relationship between the delay and the exogenous net birthrate. We can find the same property that τ 0 is negatively related to b − d.
A combination of larger K and smaller b − d in absolute value can lead tosmaller τ 0 .
t

t

10

5
8

Unstable Region

4

Unstable Region

63
4

Stable Region22Stable Region

1
K
14000 16000 18000 20000 22000 24000- 0.15(A) the size of resource stock- 0.1- 0.050b -d(B) the net birth rate

Figure 5. Downward sloping τ 0 curves.

4

Concluding Remarks

This study has investigated the long-run dynamics of renewable natural resource and population introducing a delay in agricultural production activities. It first confirms that the basic mode without delay exhibits stable dynamics. Then it analytically derives a critical value of the delay in production
for which the loss of stability occurs. Subsequent two numerical simulations
show the destabilizing eﬀect caused by the delay in production on evolutionsof the small island economy. It makes fluctuations of the resource stock andthe population much more widely over time when the growth rate of thenatural resource is low (see Figure 3) and it also generates fluctuations evenin a constantly-growing economy with the higher growth rate (see Figure 4).
11

Appendix
To prove Theorem 2, we apply the method used in Saito [2002. 116-122],
which makes it possible to check the simplicity of a characteristic root on theimaginary axis without tedious calculations. Let
P (λ, τ ) = λ2 + (−(2p − r) + pe−λτ )λ + qe−λτ ,and τ n = τ 0 + 2πn (n = 0, 1, 2, · · · ). Then, P (iy0 , τ n ) = 0 and we obtain thefollowing from (14),∂P (iy0 , τ n )= iy0 [−y02 − (2p − r)iy0 ],∂τ∂P (iy0 , τ n )= 2iy0 − (2p − r) + [p − τ n (q + piy0 )]e−iy0 τ n .
∂λClearly, ∂P (iy∂τ0 ,τ n ) 6= 0. We now consider the following value:K=

p2 y04 + 2q 2 y02 + (2p − r)2 q 2.
[((2p − r)y0 )2 + y04 ][q 2 + (py0 )2 ]We get K > 0. Furthermore, from (14),¶¸· µp
2iy0 − (2p − r)+signK = sign Re−iy0 (−y02 − (2p − r)iy0 ) iy0 (q + piy0 )¶¸· µpe−iy0 τ n2iy0 − (2p − r)τn+= sign Re−−iy0 (−y02 − (2p − r)iy0 ) iy0 (q + piy0 )e−iy0 τ n iy0¶¸· µ2iy0 − (2p − r) + pe−iy0 τ n
τn= sign Re−−iy0 (−y02 − (2p − r)iy0 )iy0" Ã ∂P (iy ,τ ) !#0

= sign Ren∂λ− ∂P (iy0 ,τ n )∂τ.

0 ,τ n )6= 0 and, by the well-known implicit function
Hence, we can obtain ∂P (iy∂λtheorem, we have!#" Ã ∂P (iy ,τ ) !#
" Ã ¯
0 n
dλ ¯¯
∂τ
=
sign
Re
−
sign Re∂P (iy0 ,τ n )dτ ¯λ=iy0 ,τ =τ n Ã ∂λ! ∂P (iy0 ,τ n ) −1 − ∂P (iy∂τ0 ,τ n )= sign Re∂λ= signK > 0.
12

This implies that (S e , Le ) becomes unstable if τ > τ 0 holds (see, for example
Kuang [1993]).
Under the current parameter specification in which K = 12000, c = −0.1,
r = 0.04, φ = 4, α = 0.00001 and β = 0.4, we have p = 0.0191667, q =0.00191667 and y0 = 0.0459087. Then solving either arcsin[τ 0 y0 ] = 0.449917
or arccos[0.466673] = τ 0 y0 for τ 0 yields
τ 0 = 10.1652.
If we take τ > 10.1652, then the dynamical system (10) becomes unstable asshown in Figure 3 and Figure 4.

References[1] Brander, J. A., and Taylor, N. S., "The Simple Economics of Easter
Island: A Ricardo-Malthus model of Renewable Resource Use," American
Economic Review, 88, 119-138, 1998.
[2] Dalton, T. R., and Coats, R.M., "Could Institutional Reform Have Saved
Easter Island?" Journal of Evolutionary Economics, 10, 489-505, 2000.
[3] Intoh, M., "Prehistorical Oceania," in Oceania History, ed., M. Yamamoto, 17-45, 2000, Yamakawa.
[4] Kuang, Y., Delay Diﬀerential Equations with Application in Population
Dynamics, Academic Press, New York, 1993.
[5] Matsumoto, A., "Economic Dynamic Model for Small Islands," Discrete
Dynamics in Nature and Society, 7, 121-132, 2002.
[6] Reuveny. R., and Decker, C. S., "Easter Island: Historical Anecdote or
Warning for the Future," Ecological Economics, 35, 271-287, 2000.
[7] Saito, Y., "The Necessary and Suﬃcient Condition for Global Stabilityof a Lotka-Volterra Cooperative or Competition System with Delays,"
Journal of Mathematical Analysis and Applications, 268, 109-124, 2002.
[8] Suzuki, M., "On Some Diﬀerential Equations in Economic Model," Mathematical Japonica, 43, 129-134, 1996.
[9] Suzuki, M., "Diﬀerence Equation for a Population Model," Discrete Dynamics in Nature and Society, 5, 9-18, 2000.

13