simulating heterogeneity in a consumption model linked to a water resource model: when is the linear...

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Simulation Modelling Practice and Theory 16 (2008) 65–75

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Simulating heterogeneity in a consumption model linked toa water resource model: When is the linear

approximation relevant?

Margaret Edwards a,c,*, Franc�ois Goreaud a, Nils Ferrand b

a LISC Cemagref, 24, av. des Landais, BP 50085, 63172 Aubiere, Franceb Cemagref-Irrigation-Montpellier, BP 5095, 361 rue J.-F. Breton 34033 Montpellier Cedex 1, France

c LIP6, 8 rue du Capitaine Scott, 75015 Paris, France

Received 21 September 2004; received in revised form 6 October 2007; accepted 10 October 2007Available online 4 November 2007

Abstract

Water resource represents an important stake for human development. Studies show that under the currently growingdemand this resource can become scarce, therefore, leading to the necessity of reducing unnecessary demand. Householdneeds often represent an important proportion of the overall demand, while individual costs are not high enough to leadthe consumer to restrict his demand to the essential. Therefore, besides pricing policies, evaluating the impact of variousincentives and information policies in reducing household demand represents a current stake in modelling for water man-agement support. One of the major difficulties for this is to account for local information ignoring spatial inter-dependenceof the resource and heterogeneity in the population.

In this paper, we tackle the possibility of simulating a composite population of water consumers and/or more than onewater resource by a linear approximation derived from simple configurations (a single consumer population linked to asingle resource). We model a reversible diffusion of careful consumption behaviours in a population influenced by publicinformation on the resource. Our results show that the adequacy of linear approximation depends on the choice of infor-mation provided to the consumers, and more particularly on the definition of a state of crisis. We discuss implications ofthis result from various points of view.� 2007 Elsevier B.V. All rights reserved.

Keywords: Linear approximation; Aggregation relevance; Water consumption; Information

1. Introduction

Water resource is crucial for human development and survival. However, under the currently growingdemand this resource risks to become scarce [20,11], therefore, leading to the necessity of finding alternativesolutions to the current withdrawals [12,23,15] and of reducing unnecessary demand [7,16,13]. Households’

1569-190X/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.simpat.2007.10.003

* Corresponding author. Address: LISC Cemagref, 24, av. des Landais, BP 50085, 63172 Aubiere, France. Tel.: +33 0662831756.E-mail addresses: [email protected], [email protected] (M. Edwards).

mailto:[email protected]

mailto:[email protected]

66 M. Edwards et al. / Simulation Modelling Practice and Theory 16 (2008) 65–75

needs can represent an important proportion of the overall demand. However, studies show that pricingpolicies are not always sufficient to reduce significantly the demand [22] since they apply uniformly to verydifferent levels of income and affect most significantly the poorer, whose consumption is already reduced toa minimum. Therefore, pricing policies are usefully complemented by incentives and informationcampaigns.

Assistance to such management choices is provided by modelling approaches, largely used to address envi-ronmental problems, particularly concerning resources [10,14,18], both as representation of the complex sys-tem considered and as tool for decision support and management [1,8,27,26,17].

It so appears with a simple model, that the success of information campaigns depends on the informationtransmitted to consumers about the state of the resource [9].

Therefore, a current stake for water consumption management support modelling is to evaluate (besidespricing policies) the impact of various incentives and information policies in reducing household demand[28,3,21]. One of the major difficulties for this is to account for heterogeneity in the population [21] and localinformation which does take into consideration spatial inter-dependence of the resource.

Simulating heterogeneity often requires refining the grain of description and, therefore, increasing simula-tion cost. For this reason it is important to understand which heterogeneity is important to take into account.For instance Picard and Franc [19] show that space-dependent individual based models and aggregated mod-els (regarding either spatial influence or description of the population) of forest dynamics lead to differentresults.

When heterogeneity can be approximated in a linear way and directly deduced from simulations corre-sponding to homogeneous configurations this represents a substantial gain in time. This means qualitativelythat heterogeneous elements do not interact. In this case simulating heterogeneity increases the precision of theresults but the improvement is quantitative and not qualitative.

Conversely when linear approximation is not justified for simulating heterogeneity this means that the het-erogeneous processes interact, leading to new phenomena which change qualitatively the evolution. Therefore,it seems particularly important in this case to study the effect of introducing heterogeneity, for which specificsimulations are required.

In this paper, we investigate whether it is possible to approximate in a linear way the evolution of more thanone population of consumers linked to one or more water resources from simple configurations where a homo-geneous population is linked to a single resource.

To do so, we link a simple resource model to a reversible behaviour diffusion model in reply to public infor-mation about the state of the resource, which we present in Section 2. We consider two kinds of informationabout the resource. For each of these we observe the results for four configurations, in which heterogeneity forthe population and for the resource are introduced separately (one population linked to one resource, twopopulations on the same resource, a homogeneous population dispatched on two resources, and finally twopopulations each linked to their own resource).

2. Presentation of the tested configurations and of the models used

2.1. General presentation of the model

The model we consider includes two sub-models which are linked: first an aggregate consumption modelwhich simulates the diffusion in the population of a careful withdrawal behaviour (versus a careless with-drawal behaviour) and second a model of the water resource. In this paper, in order to take into accountmutual influence we use a bi-directional link between these two sub-models: water consumption alters theresource available, which conversely conditions the consumption behaviours.

As concerns the resource, we used a monthly flow series (for the year 1991) in order to minimise climaticvariations; this series is repeated for every year of the simulation. Moreover, this flow series is based on amonthly mean of real measurements on the Orb river (Herault, France, upstream of Beziers city, see [2,6]).However, since these measurements integrate human withdrawals, we have added an estimate of the monthlywater withdrawals obtained from data for the Herault county. We finally obtained a ‘natural flow estimate’, towhich we then applied the different levels of water consumption.

M. Edwards et al. / Simulation Modelling Practice and Theory 16 (2008) 65–75 67

Our consumption model is initially inspired from the sociology of innovation diffusion [4,24,5,29,25]. Thesize of the population is supposed constant over time. We suppose that each individual can choose betweentwo kinds of water consumption behaviours: careful (A) or indifferent (B). The principle of the model is thatthe decision of an agent to change its behaviour relies on a trade-off between a social pressure (considering thebehaviour of its neighbours), and an intrinsic, personal interest. We use here an aggregate version of themodel, which is described in detail in Edwards et al. [9].

The choice of the kind of water consumption behaviour then leads to multiplying a basic need either by 1 or2, 5. The basic need is defined monthly following data collected for the Herault county and is supposed homo-geneous for a kind of population. More precisely we simulate two populations corresponding to differentkinds of water withdrawals: households and farmers, who have different basic needs for water and the evolu-tion of which is defined by different social parameters.

In our model, the evolution of behaviours depends on information available about the water resource,which we define by an indicator. We use different indicators reflecting different perceptions of the state ofthe resource. Alternatively in an engineering view of the analysis, it can be considered as a management choicewhere a principal agent chooses to provide specific information on the water status, in order to improve thissituation or to avoid crisis. The different forms of information lead to different kinds of reactivity and, there-fore, evolutions. The overall modelling process is a sequence of simulation and ‘‘what-if” prospective leadingto a public policy.

2.2. Indicators tested

Our indicators (information about the resource provided to the consumers) are based on a monthly averageof the flow. Indeed the month seems a more plausible interval than the day for re-evaluating one’s water con-sumption behaviour. Moreover this permits to avoid artificial daily variations due to meteorological condi-tions, the effects of which take a few days to spread in the geographical space we consider (the Orb river basin).

Our first indicator is the position to a threshold of the monthly average of the flow. This corresponds to anintuitive evaluation of drought for the current month. For instance restricting withdrawals will be moreimportant in summer at low flow than in winter. However an evident disadvantage of this indicator is tobe highly influenced by the period of the year (e.g. water is lower in summer than in winter whatever thewithdrawals).

Our second indicator is the yearly average of the first one (over the last 12 previous months). This indicatorcorresponds to a more general observation of the human influence on the resource, independently of theseason.

Table 1 summarizes the two indicators we used, with their mathematical formulas and interpretations.Lets us note:

r the threshold defined as a parameterwm the withdrawal for month m

qm the flow after withdrawal for month m

Both indicators are based on the flow after withdrawal and indicate the position to a threshold. Moreoverboth indicators provide the same possible values (�1, 1 and eventually 0). Their difference is, therefore, notnumerical but qualitative, by the way in which they define a state of crisis for the resource requiring a restric-tion of the withdrawals.

Table 1Indicators

Symbol Meaning Formula for month m

I1 Sign of the difference between the monthly flow after withdrawal and a threshold sign ðqm � rÞ? Is the flow under the alert threshold?

I2 Sign of the difference between the average over 12 months of the flow and a threshold sign 112

P11k¼0qm�k

� �� r

� �

? On an average over the last year is the flow under the alert threshold?


2.3. Tested configurations and parameter values

We chose to follow an incremental approach to help dissociate the possible influences of social versus spa-tial composition. The different configurations tested appear in Fig. 1.

To test how linearity can or not justify for approximating heterogeneity we test separately the introductionof heterogeneity in the population and in the resource.

We consider four kinds of configurations: homogeneous both for the population and the resource (C1), het-erogeneous for only either the population (noted C2) or the resource (noted C3), heterogeneous for both thepopulation and the resource (noted C4). Details about the parameters tested in the simulations appear inAnnex 1.

Heterogeneity for the population is introduced by considering two kinds of individuals: either quickly reac-tive to changes in the environment or slower reactive. Heterogeneity in the resource is obtained by separatingspatially the resource into two: upstream and downstream resources.

We seek to estimate whether the behaviour of a composite population (regarding either the kinds of indi-viduals or the resource where to withdraw) corresponds to the weighted composition of the behavioursobserved for homogeneous populations (regarding either of these aspects). Therefore for each configurationwe test different distributions of the population between the two kinds of individuals or on the two resources).

2.4. Variables observed

We observe the evolution over time of the percentage of careful consumers in the total population. Thisreflects the evolution of the system since there from can be deduced the total water consumption and the flowafter withdrawal.

We calculate synthetic values defining the distance of the simulation results to a linear approximation of theevolution computed from the reference evolutions (configuration 1); first we compute the average over time oftheir relative difference (obtained by subtracting one from the other the percentages of careful consumers atevery time step); second we calculate the correlation between the two time series.

From

1 to2 re

sour

ces

Populations with similar characteristicsand therefore equivalent

From1 to 2 populations

From

1 to2 re

sour

cesFrom

1 to 2 populations

1P, 1H

2P, 1H1P, 2H

2P, 2H

Fig. 1. Tested configurations: progressive introduction of heterogeneity for population and resource. P stands for population, H standsfor hydrological/water resource unit.


Moreover to make results clearer, we only retain for each configuration the most unfavourable values(observed over the sets of parameter tested). This means for the relative difference, the highest absolute value,and for the correlation, the furthest value from one, i.e. the lowest one.

3. Results for the first indicator: linear approximation of heterogeneity is possible

3.1. Reference evolutions for the first indicator

The reference evolutions of the behaviours (configuration 1) for this first indicator appear in Fig. 2. Theevolution appearing to be periodic, we have represented it over a single representative year (12 months) atthe end of the simulation. The time scale is the month.

The changes in the very reactive population are faster than in the less reactive one, as expected. However,the starting and ending of the indifference hollow seem to occur approximately at the same time for bothpopulations.

3.2. For the first indicator heterogeneity can be simulated in a linear way

The following graphs show for the first indicator the less favourable relative differences and correlation val-ues between the simulation results and the corresponding linear approximations for the differentconfigurations.

Whether for heterogeneity only in the population (configuration C2), only in the water resource (C3.1 andC3.2) or in both (C4.1 and C4.2), relative difference is all cases negligible (<0.05% whereas the maximum pos-sible difference is 100%) and the correlation, almost perfect (nearly equal to 1).

This shows that for this kind of indicator the heterogeneity either of the population or of the resource or ofboth together can be approximated in a linear way.

Let us now interpret these results in practical terms. This indicator is based upon the resource available forthe given month, the value of which varies greatly during the year. In fact the variation of the flow from onemonth to the other is generally greater than the maximum quantity of water withdrawn. Therefore, the with-drawals appear secondary in the value of the indicator, defined as the position of the flow after withdrawal tothe given threshold. The value of the indicator mainly reflects the period of the year [9].

Since the consumption choices of a population do not appear clearly in the value of the indicator, they can-not influence an other population withdrawing from the same resource (or from a dependent other resource).There is no indirect influence possible between the populations through the information given by the indicator.This is why composite configurations (introducing heterogeneity in a population either concerning the con-sumers’ reactivity or the resource available) lead to independent processes and a general additive evolution.

This means that the evolution of behaviours for composite configurations can be easily deduced from theevolution observed for homogeneous configurations. They do not require specific simulations, which repre-sents a significant gain in time. However, this means also that the processes at work do not interact. Simulat-ing heterogeneity improves quantitatively but not qualitatively the results (see Fig. 3).

1 resource 1 population

0

20

40

60

80

100

188

190

192

194

196

198

Time (months)

Perc

enta

ge o

f car

eful

be

havi

ours

Little reactive

Very reactive

Fig. 2. Reference evolutions for the first indicator.

Greatest relative differences observedfor the indicator 1

-1%

0%

1%

2%

3%

4%

5%

C2 C3.1 C3.2 C4.1 C4.2

indicator 1

Less satisfactory correlations observedfor the indicator 1

0

0.2

0.4

0.6

0.8

1

1.2

C2 C3.1 C3.2 C4.1 C4.2

indicator 1

Fig. 3. For the first indicator, less favourable relative differences (negligible) and correlations (close to 1) observed between the evolutionof behaviours for heterogeneous configurations and the corresponding linear approximations, show that the linear approximation isjustified.


4. Results for the second indicator

4.1. Reference evolutions

We have represented hereafter the reference evolutions for the second indicator (Fig. 4) over a representa-tive window of 12 months as for the first indicator. Contrary to the first indicator its evolution is not periodicbut irregularly oscillatory.

Here again the very reactive population seems to follow a careful behaviour over more time on the wholethan the little reactive one. However, the peaks of carefulness do not start and finish at the same moment for

1 resource 1 population

0

20

40

60

80

100

188

190

192

194

196

198

Time(months)

Perc

enta

ge o

f car

eful

be

havi

ours

Little reactive

Very reactive

Fig. 4. Reference evolutions for the second indicator.


both populations, as it was the case for the first indicator; on the contrary they even can be completely oppo-site. Therefore, for this indicator the behaviours do not seem to depend closely on the period of the year, con-trary to the first indicator we tested.

4.2. Heterogeneity of the population cannot be approximated in a linear way for the second indicator

4.2.1. Relative difference and correlation show that linear approximation is not justified

The following graphs show for the second indicator the less favourable relative differences and correlationvalues between the simulation results and the corresponding linear approximations for the different configu-rations (see Fig. 5).

For this second indicator we can see that the average relative difference between simulation results and lin-ear approximation is significant. It can attain 4% when introducing heterogeneity in the population (C2), 17%when introducing heterogeneity for the resource (C3.1 and C3.2), and 17% when introducing heterogeneityboth for the population and the resource (C4.1 and C4.2). Heterogeneity in the resource seems to have thegreatest effect.

The correlations prove also not to be satisfactory. When introducing heterogeneity in the population (C2)correlation can be expected to get as low as 0.26; it can get down to 0.24 when introducing heterogeneity forthe resource (C3.1 and C3.2) and 0.2 when introducing heterogeneity for both. Here again the effect of intro-ducing heterogeneity for the resource seems greater.

These results show that for this indicator, heterogeneity of the population and of the resource cannot beapproximated in a linear way and requires specific simulations. Moreover here introducing heterogeneitychanges not only quantitatively but also qualitatively the evolution.

Greatest relative differences observedfor the indicator 2

0%

10%

20%

C2 C3.1 C3.2 C4.1 C4.2

C2 C3.1 C3.2 C4.1 C4.2

indicator 2

Less satisfactory correlations observedfor the indicator 2

00.05

0.10.15

0.20.25

0.30.35

0.4

indicator 2

Fig. 5. For the second indicator, relative differences and correlations between the evolution of behaviours for heterogeneousconfigurations and the corresponding linear approximations show that the linear approximation is not justified.


4.2.2. Introducing heterogeneity can lead to qualitatively different evolutions

The following example shows that for a certain distribution of the population between the two resources(one upstream and the other downstream), the evolution of the behaviours can cease to be oscillatory (as itis for the homogeneous reference configurations) to become static.

When the population upstream represents only 25% of the whole population, the resource available to themon an average over the year (reference for this indicator) is more than sufficient and they are no longer boundto reduce their withdrawals. The resource available downstream is all that scarcer on the yearly average, lead-ing the individuals to permanently restrain their consumption (see Fig. 6).

When averaging over the period of the flow (as for this indicator), seasonal effects disappear, underliningthe global human influence of the resource. The value of the indicator is no longer mainly influenced by theperiod of the year (there are great variations of the flow from one month to another) but by the variations inthe consumption. We have shown in the case of a homogeneous configuration that the indicator about theresource, and more precisely the period over which to average the information, influenced significantly thedynamics [9].

In a heterogeneous configuration, this refined link between consumers and resource allows indirect interac-tions to appear between the populations. For this example the population downstream is less sensible to theseasonal variations of the flow than to the careless consumption of the population upstream. Since influencebetween heterogeneous populations exists, new phenomena occur and linearity does no longer suffice todescribe the introduction of heterogeneity.

4.2.3. Error on the resource level due to linear approximation

Our previous results show that for this second indicator the evolution of consumption behaviours for morethan one population and/or resource cannot be approximated in a linear way. We seek now to observe howimportant is the consequent error on the estimation of the resource level at the outlet. We compute in the sameway as for the populations, the difference between a simulated flow and its linear approximation by subtract-ing them at every time step and averaging this difference over the time steps, in absolute and relative values.

However, whereas changes can affect the whole population, a complete change in consumption behavioursonly concerns a fraction of the flow. Therefore, the differences observed between the simulated flows and theircorresponding linear approximation are lower.

When considering two resources the error due to inappropriate linear approximation of the dynamics canreach 2% (corresponding to 0.3 m3/s). For two populations the greatest error falls to 0.35% (0.05 m3/s). Het-erogeneity of the resource therefore seems to be more significant. Finally for two resources and two popula-tions the error can attain 2%.

Even though these differences smaller than 2% can seem minor from a management point of view, theyshould not be neglected, since they are only due to the approximation error. Moreover since errors mightadd, we can expect that introducing more than one resource and/or population might lead to even greaterdifferences.

2 resources, 1 populationlittle reactive

75% population downstream

0

20

40

60

80

100

176

181

186

191

196

Time

Perc

enta

ge o

f ca

refu

l be

havi

ours

upstreamdownstream

Fig. 6. Evolution of each (little reactive) sub-population (upstream and downstream) for different distributions of the whole population onthe two sub-basins at the end of the simulation (time in months).


5. Discussion–conclusion

This article shows that a particular care should be taken in water management to the choice of the infor-mation provided to the population. Indeed the importance of this choice previously observed in the case whereboth the population and the resource are supposed homogeneous [9] becomes even greater when simulatingheterogeneity either for the population or/and for the resource.

The purpose of this article was to evaluate whether it is possible to approximate in a linear way a compositepopulation of consumers linked to a composite resource in a linear way. The results show that the relevancy ofa linear approximation of heterogeneity depends on the link between sub-models and more particularly on theinformation about the resource available to the consumers. Indeed in some cases the choice of the informationavailable on the resource can allow an indirect influence between populations through the resource.

These results have practical implications for the modeller in terms of facility to take heterogeneity intoaccount. Indeed a previous work has led us to identify for homogeneous configurations a key property ofthe indicator of the resource defining two families of indicators which are, respectively, represented by the firstand the second indicator of this article [9]. The present results show that for indicators similar to the first onewe tested the evolution of a composite population and/or resource can be deduced from the evolutionsobserved in the homogeneous configurations. For the indicators similar to the second one, introducing heter-ogeneity leads to new phenomena and linear approximation is not appropriate. Specific simulations arerequired.

The results can also be interpreted in terms of ‘‘real world”. First the choice of the indicator determines thefeature to influence the evolution of the behaviours. For our first indicator the flow (closely linked to the per-iod of the year and the natural conditions) determines the necessity (or not) to reduce withdrawals. For thesecond indicator, the level of withdrawals observed plays a regulatory role. Therefore, different indicators per-mit to focus the water management on different features: here either environmental (for the first indicator), orsocial – in relation to environment (for the second). This focus conditions the solutions liable to be found.

Second the choice of the indicator, and, therefore, of the element defining the state of crisis, can either per-mit or prevent indirect interactions to appear between the populations. Firstly, from a practical point of view,when heterogeneity cannot be approximated in a linear way, simulating it becomes delicate because of the newphenomena which occur. In similar cases the results are submitted to some uncertainty since refining thedescription when nearing it to reality can change completely the evolution. This must be taken into accountwhen anticipating different management policies. Secondly, a kind of information which allows interactionbetween populations, can before leading to the effective solving of spatial inequalities concerning the availabil-ity of resource, raise unwanted social tensions, requiring specific measures. Knowing this can also influence thechoice of the information provided to the population.

Further possible developments include considering more complex configurations (with more than two dif-ferent populations and/or resources) and testing more complex models.

Annex 1. Detailed configurations and parameters tested

1.1. Configuration 1: 1 resource, 1 population

This is the reference configuration. We calculate the outcome of this configuration in two cases: first for apopulation quick to respond to a crisis announcement (i.e. very reactive) and second for a population slow torespond to a crisis announcement (i.e. little reactive). The parameters tested appear in Table 2. The thick linesseparate values leading to different simulations.

1.2. Configuration 2: 1 resource, 2 populations

We consider a single resource available at the same time for these two different populations (very or littlereactive), the proportions of which vary (0%, 25%, 50%, 75%, 100%) for a constant total number ofindividuals.

Table 2Parameter values tested for configuration 1

Responsiveness (cE) Proportions of the populations (%) River basin corresponding to the resource available

Configuration 1

Population Slow (cE = 0.5) 100 Upstream + downstreamQuick (cE = 1) 100 Upstream + downstream


Our purpose is to test whether the evolution of each population is or not similar to the reference ones; thisis, whether their evolutions are influenced indirectly through the resource by the other population. If they aresimilar to their references, this means we can easily deduce the evolution of a heterogeneous population (forindifferent proportions) from the reference evolutions, i.e. we can sensibly improve the accurateness of ourmodel at low cost.

We have detailed the parameter values we tested in the table hereafter (Table 3). (The thick lines separatevalues leading to different simulations.)

1.3. Configuration 3: 2 resources, 1 population

We now consider a single kind of population (either very or little reactive) distributed on two resources (thedownstream resource depending on the upstream one); each population withdraws and observes its ownresource. Here again we observe the outcome for various distributions of the population on both resources;the downstream population represents 0%, 25%, 50%, 75% and 100% of the total.

We try here to determine how taking into account a spatial heterogeneity of the resource affects the evo-lution of behaviours, and whether they vary much from the reference evolutions. We chose the thresholdsfor these two resources so as to induce evolutions similar to the reference when all the population is upstream(or respectively, downstream). The following parameter values have been tested (see Table 4).

1.4. Configuration 4: 2 resources, 2 populations

We now suppose that each of the two kinds of populations (quick or little reactive) is linked to either theupstream or downstream resource; we consider successively the quick upstream, then downstream (respec-tively, the slow downstream, then upstream). Moreover we let the proportions of both populations vary(0%, 25%, 50%, 75%, 100% of the total population is downstream (and for instance very reactive)). We showthe parameter values tested in Table 5.



Configuration 2

Population 1 0.5 0 25 50 75 100 Upstream + downstreamPopulation 2 1 100 75 50 25 0 Upstream + downstream



Configuration 3

Population 1 0.5 0 25 50 75 100 UpstreamPopulation 2 0.5 100 75 50 25 0 DownstreamPopulation 1 1 0 25 50 75 100 UpstreamPopulation 2 1 100 75 50 25 0 Downstream



Configuration 4

Population 1 0.5 0 25 50 75 100 UpstreamPopulation 2 1 100 75 50 25 0 DownstreamPopulation 1 1 0 25 50 75 100 UpstreamPopulation 2 0.5 100 75 50 25 0 Downstream


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simulating heterogeneity in a consumption model linked to a water resource model: when is the linear...

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