International audienceThis note introduces a PDE-like Toy-Model that embeds several aspects of Territory Working. The long term goal is to build a software tool that behaves like a Territory, and in particular that incorporates its multi-scale-in-time-and-space nature, in order to make simulations, to explore scenarios, to foresee policy impacts, to help to make a decision when facing a change in the environment or in cultural behavior, {\it etc.}. The aim of this note is prove that the concept of building a model that couple Systemic Approach and PDE tools to achieve the evoked long term goal is possible

Frenod, Emmanuel

HAL-Université de Bretagne Occidentale

A PDE-like Toy-Model of Territory WorkingEmmanuel FrenodTo cite this version:Emmanuel Frenod. A PDE-like Toy-Model of Territory Working. 2013. <hal-00817522v2>HAL Id: hal-00817522https://hal.archives-ouvertes.fr/hal-00817522v2Submitted on 15 Oct 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A PDE-like Toy-Model of Territory Working ∗Emmanuel Frénod†Abstract - This note introduces a PDE-like Toy-Model that embeds several aspects of Territory Working. The longterm goal is to build a software tool that behaves like a Territory, and in particular that incorporates its multi-scale-in-time-and-space nature, in order to make simulations, to explore scenarios, to foresee policy impacts, to help to make a decisionwhen facing a change in the environment or in cultural behavior, etc.. The aim of this note is prove that the concept ofbuilding a model that couple Systemic Approach and PDE tools to achieve the evoked long term goal is possible.1 IntroductionThis note introduces a PDE-like Toy-Model that embeds several aspects of Territory Working. Thelong term goal is to build a software tool that behaves like a Territory, and in particular that incor-porates its multi-scale-in-time-and-space nature, in order to make simulations, to explore scenarios,to foresee policy impacts, to help to make a decision when facing a change in the environment orin cultural behavior, etc.. The aim of this note is prove that the concept of building a model thatcouple Systemic Approach and PDE tools to achieve the evoked long term goal is possible.Figure 1: Systemic Approach of Territory Working∗This work is supported by MGDIS and AMIES†Univ. Bretagne - Sud, UMR 6205, LMBA, F-56000 Vannes, France. emmanuel.frenod@univ-ubs.fr. http://web.univ-ubs.fr/lmam/frenod/index.html.12 Brief Systemic Approach of Territory WorkingBy Territory, we mean a Country or a Town or a set of Towns and Countries having a coherenceand being administrated by a common local government.The Woking of a Territory results from the interactions between several compartments. A repre-sentation of this Systemic Approach is given in Figure 1. Three compartments are considered. Thefirst one ("People") concerns the population, the second one ("Energy") concerns the energy ques-tions and the last one ("Enterprises") concerns the world of enterprises . Then, the compartmentsinfluence each other or themselves. This fact is symbolized by the arrows. For instance, the arrowwhich is the more at the top of the figure translates that when people are at work, they contributeto the production of wealth. In return, the arrow just below means that displacements are inducedby enterprise locations and the arrow in the left translates that an energy consumption is inducedby those displacements. The three arrows that point on the "Economy" compartment from itselfsymbolize that enterprises create wealth and that this created wealth is used by the enterprisesfor growing up and for efficiency improvement. The two last arrows, in the bottom, express thatenterprises consume energy to create wealth.Figure 2: Initial Population Density P0 (left) and Job-station Density E0 (right)3 The Toy-ModelIn this Toy-Model, all quantities are dimensionless. The simulations are done using Freefem++.All the simulations are done over the same time interval which is run by variable t ∈ [0, T ) for areal number T > 0. Yet as the characteristic times of the various phenomena are not the same,they are given with various steps.The geographic model of the Toy-Model is a disk D, provided with coordinates x, with a boundary∂D. The vector of norm 1, orthogonal to ∂D and pointing outside D is denoted µ.2Figure 3: Job-station Density E0 times Efficiency Indicator i0On this disk, several densities, which depend on time, are defined:• P = P(t, x) is the Population Density.• E = E(t, x) is the Job-station Density.• W =W(t, x) is the Wealth-that-goes-to-people Density.For those densities, initial values are defined: P0 = P0(x), which is drawn on the left of Figure2, and which is the Population Density when everybody is at home at night; E0 = E0(x) which isdrawn on the right of Figure 2; and, W0 =W0(x) = 1.An Efficiency Indicator i = i(t, x) is also defined on the disk. It ranges in [0, 1] and mea-sures the efficiency of the enterprises at every point of the disk; it is initialized at i0. The productof the initial Job-station Density E0 by the initial Efficiency Indicator i0 = i0(x) is given in Figure 3.From initial Population Density P0 and Job-station Density E two attractors are built. Thefirst one AE = AE(t, x) attracts the Population Density towards Job-stations. It is defined asAE(t, x) =∫DP0(x) dx∫DP0(x) dx+ α∫DE(t, x) dx(P0(x) + αE(t, x)) , (1)where α is a constant larger than 1. The second one AP = AP(x) makes the population to go backhome. It is defined asAP(x) = P0(x). (2)With those two attractors, we can simulate the daily motion of the population. For this, wefirstly adopt the convention that day length is 1 and we define two functions tm = tm(t) and3Figure 4: Displacements from home to work and back (Animation available on YouTube: http://youtu.be/LkPlVT-a4pg.)te = te(t) depending only on time and periodic of period 1. tm(t) > 0 for instants correspondingto the morning displacements, when people go to work and 0 otherwise; and, te(t) > 0 for instantscorresponding to the evening displacements, when people go back home and 0 otherwise. Secondlywe write the equation meaning that in the morning the Population Density is attracted by Job-station Attractor and in the evening by Home Attractor:∂P∂t(t, x) = tm(t) (AE(t, x)− P(t, x)) + te(t) (AP(x)− P(t, x)) , ∀x ∈ D,∀t ∈ (0,+∞), (3)The evolution of P is given by the movie in Figure 4. In this movie, we can see the alternation ofpeople displacements to their works (in the morning) and to their homes (in the evening). Becauseof Job-station distribution most of people converge everyday to work in a small region located nearthe center of the disk.This motion generates an energy consumption. It is modeled by a time density φ = φ(t), whichis constant per day and which value at any given day is the double integral over the disk of thedistances from home locations to Job-station locations weighted by the locations where PopulationDensity P increases and where it decreases.In other words, φ is a constant over every interval [n, n+ 1], n ∈ N with worthφn =∫D∫D|x− y| PnIncr(x)PnDecr(y) dxdy, (4)where PnIncr is the density of increasing population, where population increases, when comparingthe Population Density in the morning (before people go to work) and in the middle of the day(when workers are at their job-station). It is defined byPnIncr(x) = max(P(n+12, x)− P(n, x), 0). (5)4Figure 5: Evolution of Job-station Density E . (Animation available on YouTube: http://youtu.be/PorgK4q5Ds0.)In a similar way, PnDecr is the density of decreasing population, where population decreases, whencomparing the Population Density in the morning and in the middle of the day and is defined byPnDecr(x) = max(−P(n+12, x) + P(n, x), 0). (6)The fact that people is at work produces Wealth with a rate which depends on the product ofthe Population Density P by the Job-station Density E times the Efficiency Indicator i. To modelthis, we introduce a wealth production rate density ω = ω(t, x) which is constant per day and whichvalue at any given day, represented by interval [n, n+ 1], n ∈ N, isωn(x) = β0 P(n+ 12, x) E(n+ 12, x) i(n+12, x). (7)where β0 is a coefficient small in front of 1, meaning that the time scale of Wealth variation is largewhen compared to a day.This Wealth is split into three parts. The first one is allocated to enterprises’ growth. In otherwords, it is a source term of the ODE, set in every point of the disk, the Job-station Density E issolution to:∂E∂t(t, x) = β1 ω(t, x), ∀x ∈ D,∀t ∈ (0,+∞), (8)where β1 is a coefficient belonging to a triplet (β1, β2, β3) of positive coefficients such that β1 +β2 +β3 = 1.The second part (β2 ω(t, x)) contributes to improve enterprises’ efficiency. It contributes as asource term in the ODE, set in every point of the disk and which is such that its solution i alwaysranges in [0, 1]:∂i∂t(t, x) = i(t, x)(1− i(t, x)) (β2 ω(t, x)− β1 ω(t, x)) , ∀x ∈ D,∀t ∈ (0,+∞). (9)5The term β1 ω(t, x) in this equation models that when organization grows, if nothing is done, itsefficiency decreases.The result of the influence of Wealth on the Job-station Density E and on the product of theJob-station Density E times the Efficiency Indicator i are given in the movies of Figures 5 and 6.We see in those movies that, around the small region where most of people converges everyday towork, there is a strong increase of Job-station Density E and of enterprises’ Efficiency i.Figure 6: Evolution of Job-station Density E times Efficiency Indicator i. (Animation available onYouTube: http://youtu.be/B7gjmu7q8IU.)The third part of Wealth production will diffuse inside the population. It is then a source termin the diffusion equation W is solution to:∂W∂t(t, x)− ν∆W(t, x) = β3ω(t, x)− κφ(t), ∀x ∈ D,∀t ∈ (0,+∞), (10)∂W∂µ= FW , ∀x ∈ ∂D,∀t ∈ (0,+∞), (11)where ν is the diffusion coefficient of Wealth within the population on the territory. In this PDE,the energy consumption time density φ is also a source term but with an opposite action than thatof the Wealth production. This is because φ induces a consumption of Wealth. In this PDE thereis also a boundary condition on the disk border, translating the fact that there is a little wealththat enters the Territory through its border. The evolution of Wealth-that-goes-to-people Densityis given by the movie in Figure 7. We can see the diffusion of the Wealth-that-goes-to-people fromthe small region where most of people converges everyday to work towards the other parts of theTerritory.4 InterpretationClearly, this Toy-Model does not pretend to give realistic results. Nevertheless it seems that theproposed approach brings a way to couple several aspects of Territory Working.6Figure 7: Evolution of Wealth Density W. (Animation available on YouTube: http://youtu.be/05JNyznPxvw.)For instance, it can handle non-linearities. This is illustrated by the two pictures of Figure 8.The one on the left is the Population Density at midday of the first day and the one on the right isthe Population Density at midday of the last day. We can see that, in the small region close to thecenter of the disk, the density is higher in the right picture than in the left picture. This can beexplained as follow: Since people is working their, wealth is produced, which induces Job-stationnumber to increase. As a consequence, day after day, more and more people is coming to work inthis small region.The scales (regarding time and space) have no realistic meaning. Nevertheless, we can see thecapability of this kind of models to account for a wide variety of scales. For instance, regarding thespace scales, in the Toy-model, there is the size of the Territory (the disk) and there are the char-acteristic sizes of variation of Population-at-home Density, of Job-station Density and EnterpriseEfficiency. Beside this, this way of modeling allows us to connect several Territories through theircommon boundaries and then to consider a large number of connected Territories.Regarding the time scales, in the Toy-Model there are the characteristic time of Population motion,the characteristic time of wealth production and the one of wealth diffusion.7Figure 8: Population Density at midday of the first day (left) and of the last day (right)8

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