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A Reduced Dynamic Model for
Evaluating the Impact of Man on the Environment
L.Gardini, H.Sedehi, R.Serra
TEMA S.p.A.
ABSTRACT
A representation of socio-economic systems using reduced models allows a
"qualitative" type of analysis to be carried out. It is often the case,
especially in long term processes, that the main interest is directed
towards the asymptotic behaviour of the solutions as a function of the
initial state and to evaluating the properties of stability of stationary
states. In this article, after a short outline of the procedure and
methodology adopted, we describe the application of these techniques in
the construction and use of a dynamic model for the design of a tourist
village. The model, which mainly deals with the impact of man on the
environment, serves to evaluate the social and economic effects of the
construction of a tourist centre in a natural environment which must be
conserved.
INTRODUCTION
A typical feature of socio-economic modelling as distinct from physical
modelling, is the lack of natural laws which may be relied upon for
rigorous foundations (leaving aside for the time being subtle questions
about what the laws of nature really are). This leads to uncertainties
about the form of the basic model equations. Further uncertainties
regard parameter values and the initial conditions.
These well-known observations, which are common to almost all
socio-economic models (with the exception of a few rather trivial
examples) lead to the conclusion that families of models, rather than
individual ones, have to be considered.
This situation is not dramatic, as long as one is dealing with
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structurally stable models. However, most models do not seem to belong
to the set of structurally stable systems. The well known phenomena of
sensitive dependence on initial conditions, bifurcations, chaos (Allen et
al., 1982; Nicolis and Prigogine, 1977; Haken, 1977, 1983; Cvitanovic,
1984; Serra et al., in press) invite caution in the use of such models.
This does not mean, of course, that models are useless. It means,
however, that in order for them to be useful rather than misleading, they
must be carefully handled and their results properly interpreted. It is
clear that such observations could make the penetration of system
dynamics into industrial environments more difficult. It is always
easier to sell a reliable tool, rather than problems. However, a growing
number of managers and staff assistants are realizing the need for a
better understanding of the dynamic behaviour of the systems they are
working with. A key step in this direction has been the observation, on
several occasions, of counter-intuitive transient behaviour of complex
systems by J. Forrester and co-workers (Forrester, 1968). There are
also examples of counterintuitive dynamic behaviour in the asymptotic
regime, which are worthy of study.
A question naturally rises about the "correct" modelling level. Recent
developments in hard and soft sciences has led to the so-called "science
of complexity" (Jantsch, 1980), (Capra,1982). Here, it will suffice to
say that, in the science of complexity, there is no single, privileged
viewpoint, or modelling level. The need is rather that of integrating
information coming from different perspectives, which are all partial and
incomplete. System dynamics can provide a field of application for this
approach, since different models are needed for answering different
questions about the same physical system. A major requirement is
non-contradiction among the different models, while it seems unrealistic,
in general, to require that they all be deduced by an overall global
model via projection techniques.
We discuss here a model for a tourist settlement in a natural
environment. This problem has been studied by our group for a tourist
village in Sardinia, and a model was described in a preceding system
dynamics conference (Sedehi et al., 1983). The model considered was
rather a detailed one, describing the interaction between different
natural species, different kinds of tourists, the structures available in
the village and their ageing, and the economic-financial subsystem. The
main goal of the model. described here is a study of the long term
behaviour of the system.
In this case it seems particularly interesting to analyse possible
complex asymptotic behaviours. However, such an analysis is much better
performed in models with only a few variables. We present here a
simplified version of the model, and discuss its behaviour.
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The need for reduced descriptions has been stressed previously (Serra et
al., 1984), and we will not go into the details here. The basic idea is,
that since our brain can work with a limited number of variables, a
projection of some kind is always required, even if we use a complicated
model with a lot of interacting dynamic variables. This projection is
often implicitly done by the system dynamist, but in some cases one can
devise formal projection techniques. This is the case when there are
variables whose dynamics differ considerably, and the corresponding
projection techniques are known as adiabatic elimination procedures.
Another point deserves consideration. In the case of the tourist village
it is natural to consider a discrete map rather than a continuous dynamic
system, because the main cooperative phenomenon is the transmission of
information among tourists. The tourists who visited the village one
year will express their opinion, positive or negative, to their friends,
and this will influence the number of visitors in the following year.
There is a natural clock, and the natural unit of time is the year. An
analogous situation exists for natural species, if their reproduction
cycle is annual. We are thus led to a model of the type:
N(t+1) = f£(N(t), other variables) (1)
rather than
N(t) = £(N(t), other variables) (2)
It is well known that complicated dynamic behaviours (e.g.,chaos) are
much more common in discrete maps than in continuous systems.
It is to be stressed, in this respect, that although the conceptual
framework of system dynamics is continuous in time, most applications
integrate the dynamic system with the simple Euler algorithm, thus
transforming it into a discrete map of the type of Eq. (1).
MODEL DESCRIPTION
We refer the reader to Sedehi et al. (1983) for a detailed discussion of
the model. Its main features are summarized below.
The number of visitors at time tiNg » is determined by two factors:
advertising and the number of visitors in the preceding years and their
satisfaction. Satisfaction, in turn, is determined by the state of the
environment and by crowding.
Let us start from the environment, which will be represented by M
interacting species. We choose a simple standard model in population
biology (Goel et al., 1971), i.e.:
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Ay SA; ~ Reet 4 ai AS AAR S23 NA (3)
4 Ts 2 = y
ey
The last term takes into account the interaction between man and the j-th
species, and may be positive or negative (for instance, tourist
settlements often cause a reduction in animal or vegetable species but
might lead to an increase in parasitic species which grow on waste). The
coefficients Cym may also take positive or negative values according to
the competitive or cooperative nature of the interaction between the j-th
and k-th species. The coefficients will vanish if there is no
interaction.
Now, in order to simplify the preceding model, let us suppose that the
dynamics of a single species, say the first, is much slower than the
dynamics of the other species interacting with it. Then we can apply the
so-called direct adiabatic elimination procedure, which has been
described elsewhere (Serra et al., 1984). The procedure consists of
setting
Ay=° gs 2)-.-)M (a)
The fast variables, according to this approximation, instantaneously
adjust themselves to their equilibrium values, corresponding to a given
value of the slow variable. The equilibrium values of the fast variables
turn out to be linear functions of A, and N:
A500 = 4 -%y As - pyN (5)
The time evolution of the slow species is therefore given by:
A, = KiA4 — KAT -— ky AWN
=i =
K, = z, [+ Yo % 45]
4
etla-g-
Ke T, [ 34 L 43% (6)
Kz =t]a,-
. [~ do eak
A
We thus reach the following important conclusion: if the time scale of a
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species is widely separated from the time scale of the other interacting
species, we can use a Verhulst model for the slow species alone, instead
of the whole set of Lotka-Volterra-Verhulst equations.
From now on we will suppose that these conditions are met, and we will
therefore consider only one species, which will be regarded as
representative of the state of the natural environment. We will now
introduce three models for tourist-environment interaction, whose
behaviour will be discussed in the next section.
Let us first of all consider the simplest model, which will be called
"model 1c" - c meaning continuous:
As tA-7A- BNA
N= (s-4)N+ Pu
(7)
If the satisfaction coefficient s>1, then we have a positive tendency to
the growth in the number of visitors, N. Another (constant) term which
contributes to the growth in the number of tourists is advertisig (Pu).
We will suppose that the effect of man on the A species is negative,
therefore 0.
The form of the satisfaction coefficient is given below (Eq.10). We
suppose that the quantities A and N are normalized. The former is
normalized by dividing the original equation by the equilibrium number of
type A individuals in a condition where tourists are absent. This
explains why the coefficients of the linear and quadratic terms are
equal: it is a consequence of normalization. The tourist number is
normalized by dividing the original equation by the maximum number of
tourists which can be accommodated in the village.
As discussed in the introduction, it is more correct to consider the
analogous discrete model which will be called Model 1d:
Alt+4) = (44+ 0) Alt) - TARLY- BN (t) Alt)
N(t+4) = 6(t) N(t) + Pu (8)
This model considers a homogeneous tourist population, whereas it is
known that people differ strongly with respect to destructiveness towards
nature and sensitivity to nature and crowding. We will therefore
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investigate the differences between the previous model and another model,
whose discrete version (Model 2d) is given below:
Altes) = (A+ At) - TAME) - BA) N1(L)— B, AG) N2t)
Na(t+4)= 5,(t) N4(t) + Pu, (9)
N2(t+4)= S(t) N2(t) + Pu 2
The corresponding continuous model (2c) can be written in a
straightforward way. An Ni type of tourist may be considered as being
very sensitive and very careful towards the environment, with N2 tourists
being very sensitive towards overcrowding.
The satisfaction coefficient s is the sum of two contributions, one
coming from the state of the environment and the other from crowding.
The two contributions are weighted by a coefficient k €4
S(t) = Smax [ Kf (A(t)) + A-K) g(N (td) J (10)
f(A), which measures the satisfaction due to the natural environment, is
assumed to have a logistic shape, while g(N), sensitivity to the presence
of other people, should be belishaped (indeed, people generally do not
like to be too lonely and do not like overcrowding). Two simple such
functions are the following:
A
F(A) = A 7 (a1)
g(N)= exp | -4 [ Ney | (22)
In the following section we will discuss some of the dynamic behaviour
characterising the above models.
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DYNAMIC BEHAVIOUR OF THE MODEL
Asymptotic behaviour will first be discussed, followed by a short account
of transient phenomena as reduced models are particularly interesting for
long-term analysis.
The discussion begins with Model 1, considering in particular the dynamic
features of its discrete version (Eq. 8), and briefly mentioning those
of the continuous model (Eq. 7). Taking first the case where Pu=0,
{i.e., mo advertising) attention will be focused on the cooperative
phenomena due to the spread of information about tourists' satisfaction.
Starting from a "reasonable" set of parameter values, (see Note on
Parameters), the following behaviour was observed as the parameter was
changed. There was always an equilibrium point P* (N=0, A=1), stable for
0<@<2, which corresponded to the unperturbed situation (no tourists,
completely natural environment). The basin of attraction of this point
was, however, small (the basin of attraction is the set of initial
conditions which lead the system to the given attractor in the asymptotic
time limit). There were two further equilibrium points, one of which was
always unstable. The third equilibrium point was stable for sufficiently
low values of & (fig.1) and its basin of attraction much wider (in the
phase plane) than the other one, including "realistic" values of the
population N,
i
A(O) = 0.8 N(O)=0.1 T= 0.6
Figure 1
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As @ is increased this point becomes unstable, and a supercritical Hopf
bifurcation takes place: a stable limit cycle appears, which includes
the (now unstable) equilibrium point. The term "limit cycle" is correct
for the continuous model, while for the discrete model it would be more
proper to speak of quasiperiodic behaviour. The phase plane plot of this
cyclic attractor is shown in fig. 2 for the discrete case (Eq.8). The
analogous one for the continuous model (Eq.7) is qualitatively the same,
in shape and dimensions (not reported here).
IODIC ORBIT
A(O) = 0.8 N(O)=0.1 ®= 0.7
Figure 2
This indicates a close relationship between the two models, which
persists for greater values of the parameter. In fact, on increasing T,
another bifurcation occurred in both models: the unstable equilibrium
point became stable via a subcritical Hopf bifurcation (the stable limit
cycle dimensions diminished around the unstable equilibrium point and
reduced towards it giving a stable equilibrium point). The asymptotic
behaviour is qualitatively the same for the continuous and discrete
models: orbits tending towards the equilibrium points.
On increasing %, the equilibrium point always remains stable in the
continuous case, while in the discrete case another threshold value is
found, after which quasi periodic behaviour is observed.
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This phenomenon is due to the different kinds of bifurcations for
continuous models and maps. We note here that in the bifurcations
mentioned above the eigenvalues relating to the equilibrium. point cross
the unitary circle in the complex plane with positive real parts for the
map and cross the imaginary axis for the continuous model, while in this
last bifurcation the eigenvalues cross the unitary circle with the
negative real parts in the map having no analogue in the continuous model
where a complex eigenvalue with a negative real part gives a stable
equilibrium point.
After another threshold, a strictly periodic behaviour is observed, with
period three (fig. 3). In the phase plane the attractor is composed of
three distinct points, between which the system's representative point
continuously jumps.
A(O) = 0.8 N(O) = 0.1 T= 2.1
Figure 3
After another threshold, the period doubles, and a six year cycle is
found, which is followed by a twelve year cycle, and so on, leading to
chaos (fig. 4). This sequence of bifurcations reminds one of the
well-known Feigenbaum route to chaos (Cvitanovic, 1984; Guckenheimer and
Holmes, 1983).
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A(O) = 0.7) N(O) = 0.4 W= 2.5
Figure 4
The series of threshold values has not yet been analysed in sufficient
detail to categorically state that it follows Feigenbaum's rule.
However, a first estimate of such a sequence suggests a convergence rate
quite similar to the well-known value for flip bifurcations of minimal
maps of the interval.
On further increasing % (%>2.71) the chaotic behaviour drastically
changes. In all examples considered, the tourist population, after a
chaotic transient, tends to zero, only the chaotic behaviour of the first
variable A surviving.
In an attempt to explain this, it is to be noted that the equilibrium
point, say P*, stable for 6<2 with a "small" basin of attraction
becomes unstable via a flip-bifurcation at wW=2 and two new stable
equilibrium points appear near P*. At about T =2.5 they become unstable
and 4 stable equilibria appear. On further increasing 8, a full
sequence of period doubling bifurcations occurs reaching a chaotic
behaviour in A, with the variable N=0. This sequence of bifurcation
values is different from the previous one and thus, two distinct routes
to chaos take place on increasing 8 and, at high values, the latter
seems to dominate.
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Variations in p while keeping @ fixed were also tried. Table 1 shows
the model's asymptotic behaviour for a fixed value of &. As £
increases, we pass from a stable point to a three year cycle, to a quasi
periodic behaviour, and back again to a stable point. In general, it
seems that, by varying $ » we can obtain a set of dynamic behaviours,
whose "upper limit of complexity" is determined by the value of %.
Value Type of Attractor
0.4 < 8 < 0.5 Stable equilibrium point
0.55< §< 0.7 Stable periodic orbit of period 3
0.80< 8< 1.1 Quasiperiodic orbit
Table 1 Ta2
A low value of the advertising variable (Puz.1) is then introduced. In
this case there is only one equilibrium point for the model (Eq.8) and
the tourist population value, say N., increases with increasing %& (and
is greater in value than the corresponding one for the previous case,
Pu=0, as expected). The dynamic behaviour is similar to that discussed
above. Some differences detected at high @-values are worth noting. As
before the range of ®-values over which there is chaotic behaviour is
quite wide (here about 2.3-2.75) but on further increasing T we now
observe a phenomenon which we shall call "reorganization". In fact,
periodic orbits of period 9 and 3 are observed. However, the transient
to reach this "order" has chaotic dynamics and persists over a long
period which increases with increasing T.
Let us now consider model 2d, where interaction between two different
kinds of tourist is taken into account. Let us again increase & while
keeping the other parameters fixed (see Note on Parameters). We consider
first the no advertising case: at sufficiently low J values (fig. 5),
we have a stable fixed point which corresponds to the complete extinction
of one of the two kinds of tourists (N1=0).
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375
A(O) = 0.8 N1(0) = 0.7 N2(0) =0.1 TH 1.5
Figure 5
As @W reaches a threshold value, we observe two kinds of dynamic
behaviour, completely different in character. Until the population Nl
reaches a characteristic value, ni* say, (which may depend on the A
value), i.e. with an initial condition of Nl < ni* the orbits tend
towards the equilibrium point with Ni=0 and the second population
dominates, while with N1 2 Nni* the dynamics is reversed; now in fact, it
is the second population which tends to zero and a quasiperiodic
behaviour survives between Nl and A. (fig. 6 and fig. 7).
On further increasing &, the characteristic value Ni* decreases and, as
above, two different dynamics are detected: either Nl —* 0 and (N2,A)
tend to a periodic orbit of period three, or N2 —™ O and (N1,A) tend to
a quasiperiodic orbit. These dynamics can be explained as follows:
depending on which of the two populations dominates, the asymptotic
behaviour (which is reached after only a few iterations) is the same as
that of model 1d (Eq.8) with the corresponding Tandp values.
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A(O) = 0.8 N1(0) = 0.7 N2(0) = 0.1 T= 1.8
Figure 6
QUAGIPERIGOIC ORBIT
se.
Te eecenegenreree™
A(O) = 0.8 N1(O0) = 0.7 N2(0) = 0.1 T=1.8
Figure 7
Thus, on further increasing @, the dynamics of the surviving population
Ni is a quasiperiodic orbit of equilibrium point, while for the
dominating population N2, a chaotic behaviour is soon reached (fig. 8),
as expected.
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--
A(O) = 0.6 N1(O0) = 0.2 N2(0)=0.2 T= 2.5
Figure 8
An interesting behaviour is detected at high values of J. At 0 =2.8
neither of the two populations dominates, both tending towards zero and
the dynamic of A is chaotic. Again, the dynamics of the bidimensional
model (Eq.8) can explain this phenomenon.
Finally, we consider the same model 2d in the case where a constant level
of advertising is allowed. The main difference from the previous case is
that now we no longer have the extinction of one of the two populations.
Both populations survive due to the constant flow via advertising
channels. However, the dynamic behaviours are quite similar to those
observed in the other models. Let us briefly outline this qualitative
behaviour at various @ values. Also in this case we have first (for
small @ values) a stable equilibrium point with population N2 prevailing
over Nl. On increasing 87, a first bifurcation occurs and at @=2 a
periodic orbit of period three is detected.
In two of these states N2 prevails (but the difference between the two
populations is less than before) and in the third one Nl prevails
(although by only a small amount). On increasing the value of Wa full
sequence of period doubling bifurcations takes place and a chaotic
behaviour is observed for high values (22.4).
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CONCLUSIONS
A brief comment will be made on the asymptotic analysis carried out,
although still incomplete. First of all, how relevant is asymptotic
analysis? In the first place, it can be seen that a typical transient
lasts for 10-30 years. Asymptotic analysis is therefore relevant for
long-term planning. There are, however, some exceptions, discussed in
the previous section, where a chaotic transient lasts so long that the
more regular asymptotic behaviour becomes apparent only after centuries:
it is clear that in this case asymptotic analysis is of no direct
practical use.
Second, we remark that a major goal of simulation models is to understand
how things will go, rather than actually predict accurate values. So
asymptotic analysis can tell us if corrective actions are to be taken,
such as more advertising, for instance.
The use of reduced models requires caution. We have seen formal
projection techniques at work in the case of widely separated time
scales. However, we often find that such formal techniques are
inapplicable, as in the case of the two different populations of
tourists. We have shown that the frequently adopted procedure of taking
average values can be misleading. Indeed, if one population dominates
there is no sense in averaging its characteristic parameters with those
of an extinguishing population, for asymptotic analysis. We have also
seen an example of a delicate relationship between dynamic behaviour and
the dominance relation between two groups of tourists, which requires
further analysis. Let us also remark the importance of initial
conditions in strongly cooperative models such as those considered.
We have also shown an illustrative example of a well-known property,
namely that discrete-time models give rise to a much wider set of dynamic
behaviour than continuous-time models. System dynamists usually think in
terms of continuous models, but there are instances (such as the present
one) where it is better to use discrete maps. This, however, does not
conflict with the basic principles of the system dynamics method.
A further point which needs be mentioned is the extreme sharpness of the
thresholds encountered. The most striking phenomenon discussed, namely
the change in dynamic behaviour accompanied by a change of prevailing
population takes place over a very narrow range. This sharpness can be
observed with respect both to the parameter values and to the initial
conditions.
Lastly, it is tc be remarked that the "counterintuitive behaviour of
complex systems" is a concept which refers not only to their transients,
but also to their asymptotic patterns.
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NOTE ON PARAMETERS
For all simulations a set of fixed parameters was chosen which are as
follows:
Model 1d (A-N)
Smax=1.8 , g=10 , R=1000 ,w=0.5 ,6=0.25 , k=0.5 , ( =0.8 , PusO
Model 2d (A-N1,N2)
Smax=1.8 , q=10 , R=1000 pao.5 s6=0.25 , k1=0.7 , k2=0.3 »P1=0.6 +P 2=1
Pu=0 :
The variable parameter in both models ,mentioned above, for all results
reported is %& . Together with the initial conditions of the
variables the value of %& is noted under each figure.
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