Investigation of Clustering in Turing Patterns to Describe the Spatial Relations of Slums

Investigation of Clustering in Turing Patterns to Describe the Spatial Relations of Slums Jakob Hartig 1, John Friesen 1∗ ID and Peter F. Pelz 1 ID 1 Chair of Fluid Systems, Technische Universität Darmstadt, Otto-Berndt Str. 2, 64289 Darmstadt, Germany * Correspondence: john.friesen@fst.tu-darmstadt.de; Tel.: +49-6151-16-27100 Academic Editor: name Version August 17, 2020 submitted to Abstract: Worldwide, about one in eight people live in slums. Empirical studies based on satellite 1 data have identified that the size distributions of this type of settlement are similar in different cities 2 of the Global South. Based on these results, we developed a model describing the formation of 3 slums with a Turing mechanism, in which patterns are created by diffusion-driven instability and the 4 inherent characteristic length of the system is independent of boundary conditions. We examine the 5 model in this paper by critically reflecting its assumptions, comparing them with recent empirical 6 observations and discussing possible adjustments and future extensions based on new methods of 7 identifying pattern formation mechanisms. 8

basic assumptions and conclusions (sec. 2). In a further step, we discuss in detail the  Therefore, the population is divided into two groups "rich" and "poor" on the basis of income 60 and the behaviour and interaction of both groups is described by two coupled partial differential 61 equations 62 ∂ u 1 ∂ t = UR f 1 (u 1 , u 2 ) + D 1 ∆ u 1 , where u i describes the density fields of both population groups (i = 1 represent "rich", i = 2 63 represent "poor"), U the popultion density and D i the diffusion coefficients. R has the dimenson of a 64 rate and f i are the coupled reaction rates.

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These equations describe the change of the concentration of inhabitants as the sum of dwellers 66 moving into, being born or dying within the city (reaction term) and moving accross the city borders 67 (diffusion term).

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The equations are brought into a dimensionless form by using the following parameters t := R t, x j := x j √ R/D 1 , u i := u i / U leading to Thus, instead of describing the behavior of the different inhabitants of the city separately (as is 69 done, for example, in agent-based models [20]), their behavior is averaged and described by means of 70 partial differential equations. The authors interprete the behavior of the two groups by using two basic 71 phenomena: short distance migration and long distance migration.
While the short distance migration, which is represented by the diffusion term, depends only on the concentration of the respective population group at a specific location, the interaction of the two 74 groups is represented in the coupled reaction terms, called long distance migration.

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The basic idea of the model is that a stable state of long-distance migration is assumed, in which 76 the interaction between the two social groups leads to an even distribution. This corresponds to 77 the analogy from the model of Turing, where the system is stable in the absence of diffusion. This 78 stable initial state can be described by certain conditions, which are expressed in a special form of 79 the Jacobian, called behavior matrix in the model. For detailed derivations and necessary as well as 80 sufficient conditions for the formation of patterns please refer to the standard literature, e.g. [21].

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A necessary condition for a stable state is a negative trace and a positive determinant of the Jacobian a ij := To fulfill this conditions, the Jacobian has to have one of the following four proporties However, due to different migration behaviour of the two groups, described by the diffusion 87 terms in the equation, the system can be transformed from a stable to an unstable state in which the 88 faster diffusion or mobility of the more wealthier population group leads to a regular pattern formation 89 (cf. Figure 3).

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While other models usually take a people-centred view, the model of Pelz et al. [18] represents an 91 empirically observed quantity, independent, of the city, country and culture the morphological slum 92 are found in. In the model, this is the resultof the simple interaction of two groups, whose behaviour 93 can be represented by partial differential equations and thus analytically investigated.

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Like every model, the presented model [18] is based on a number of assumptions, which we 95 examine in more detail below. In a further step, we compare these assumptions in detail with current 96 results from empirical research on morphological slums and with possible approaches to extend the 97 model.  In the literature it is repeatedly mentioned that simply dichotomous classifications can be 106 problematic and perceived as judgmental, because people, groups of people or whole states are 107 often reduced to a single characteristic and categorized according to that characteristic [26].

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Furthermore, the question arises how to distinguish between the two population groups. That is, 109 from which border on the population is classified as poor, from which as rich. This discussion is well 110 known in poverty research, and leads to distinctions between relative and absolute poverty [27]. While 111 relative poverty affects those parts of the population whose wealth is in a certain lower percentile of the wealth distribution, absolute poverty is defined by the fact that the respective person or group has 113 less than a certain amount of money available per day.

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In the model [18] the population is divided into two groups based on the respective income of the 115 individuals. Since only two groups were studied, the model implies that the behavioural characteristics to such a liquid in the described model is. Also, self-reinforcing effects in an activator-substrate model 126 assume an interaction between both morphogens, which then produce new activator morphogens.

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However, it is difficult to derive an analogy to this behavior in the proposed model, since the formal 128 dwellers would be the substrate for the morphological slum dwellers.

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Lastly, it should be mentioned that easy allocation to two-part groups in terms of income is often   Zero-flux boundary conditions were chosen. This assumption is based on the fact that the 147 population traveling into the city can be mapped in the reaction term via long-distance migration, while 148 within the city only short-distance migration takes place.

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The limitation of the system means that only certain wavelengths can be formed in one [21]. If   With regard to the model presented [18] it must be stated that the assumption that human 166 behaviour can be modelled by means of random walk is possible, but that the approaches to the 167 mobility of the different groups described there should be refined.   However, while these size distributions show a wide variance (Figure 1, (d)), size distributions of 183 concentration peaks of Turing patterns are usually very narrow (Figure 1, (b)).

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The formation of Turing patterns in an RD system is based on the interaction of two morphogens, 189 whose temporal development of the concentration curves are interrelated, as can be seen in Figure   190 1, A. Due to the coupling in the differential equations, an increase in the activator concentration is Although studies have already shown that morphological slums can also shrink and thus split up, it is 202 not usually the case that slums move.
203 Figure 3. Qualitative illustration of a activator substrate model with clusters as initial condition and homogeneous parameters. Initial condition (a) t = 0, (b) t = 0.16t = t end , (c) t = t = 0.27t end and end (d) t = t end .

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In recent studies, however, it is repeatedly stated that the criteria for Turing patterns can be 216 extended in different ways [41]. For example, the interaction could be extended from two to more 217 social groups to address the problem of dichotomous views. show that with piecewise constant parameters patterns occur which are restricted to a part of the 230 considered area. However, this behavior is due to the fact that the diffusion coefficient on one side 231 is greater than than the critical diffusion coefficient for a homogeneous area. Furthermore, spatially 232 dependent diffusion coefficients were investigated and it was shown that the wavelengths imprinted 233 on the system from outside can interact with those intrinsically present in the system [47].

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The strong topographic and spatially varying social structures in cities of the Global South 235 could therefore be represented in an adapted model by spatially and possibly temporally dependent parameters. Here, the cross-diffusion, which has been neglected up to now, could be included in the 237 model in order to be able to reject the strong assumption of the independent mobilities of the two 238 social groups. The influence of cross-diffusion on the Turing mechanism was also investigated [48,49].

Parameter identification 240
If it is assumed that the formation of morphological slums is based on a mechanism that can be 241 described by partial differential equations, the question of (i) the underlying reaction kinetics and ( By analysing both, the model and empirical findings on morphological slums, we found 258 differences in the spatial relations and development of both patterns. Using spatially and temporally 259 homogeneous parameters, it is possible to map a characteristic size of morphological slums, but this 260 also results in a spatially homogeneous pattern, which contradicts the empirical observations. These