Reaction-diffusion Model Describing the Morphogenesis of Urban

Systems in the US

John Friesen, Ruben Tessmann and Peter F. Pelz

Chair of Fluid Systems, Technische Universit

¨

at Darmstadt, Otto-Berndt-Str. 2, Darmstadt, Germany

Keywords:

Urbanization, Reaction-diffusion Models, Urban Development, US.

Abstract:

Urbanization is currently one of the greatest challenges facing mankind. In order to be able to anticipate

the rapid changes in urban structures, models are in need, mapping the morphological development of these

structures. In this paper we present an urban development model that is based on reaction diffusion equations

and can be interpreted sociologically. We apply this model to the urban development of US-American cities

and show that this very simple model can already map basic characteristics of urban development.

1 INTRODUCTION

More than every second person on earth lives in a

city. According to UN estimates, this share will rise

to almost two thirds by 2050 (United Nations, 2016).

These rapid changes lead to major challenges for the

infrastructure systems (energy, water, wastewater) of

urban areas (Kraas et al., 2016). In order to develop

adequat solution stragetegies to face this problem it

is necessary to anticipate the development of cities.

Therefore, for many years efforts have been made to

understand, model and simulate the development of

urban structures. Very common are statistically, cel-

lular automata and agent based models. A compre-

hensive summary of these different approaches can be

found in (Benenson and Torrens, 2004). The question

is, what the central mechanisms of this structure for-

mation are and how they can be simulated.

While in earlier studies it was repeatedly assumed

that urban systems behave scale free, more recent

studies have shown that urban systems do indeed con-

tain typical variables, as shown for example in the

work of (Gonz

´

alez-Val et al., 2015) for cities within

countries, or of (Friesen et al., 2018) for slums within

cities. Assuming that urban systems are free of scales,

simulation models are also developed that lead to

scale free systems. This can be seen for example in

the development of models for fractal morphogenesis

(Frankhauser, 1998). However, if urban systems have

typical scales, it is advantageous to use models that

take this circumstance into account.

Therefore, in this paper we use an analogy from

another scientiﬁc discipline to model urban change.

Structural changes can not only be identiﬁed in ur-

ban structures, but are also observed in other systems,

for example in biology. These processes of morpho-

genesis were fundamentally described in a mathemat-

ical way in 1952 by Alan Turing, who was able to

show how processes of pattern formation in biolog-

ical systems can be described using relatively sim-

ple reaction-diffusion equations (Turing, 1952). He

showed, that when certain conditions are satisﬁed,

diffusion can lead to instability of the system.

In this paper we use a framework based on the

work of Turing for describing the structural devel-

opment of cities with a system of reaction-diffusion

(RD) equations. Although RD equations have already

been used to describe urban processes (Schweitzer

and Steinbink, 2002), no stability analyses of the

equations have been performed. An exeption is a re-

cent paper of (Pelz et al., 2019) using instability ef-

fects to explain the formation of slums, a speciﬁc ur-

ban class. Also worth mentioning in this context is the

work of Paul Krugman, who also relates the spatial

distribution of industry to the work of Turing (Krug-

man, 1996). However, the equations presented in the

last mentioned paper describe economic and not soci-

ological aspects.

We take up this idea and ask the research ques-

tion whether and to what extend it is possible to pre-

dict the structural development of cities with reaction-

diffusion equations. For this purpose, we use Census

data from cities in the USA from the years 2000 and

2010, showing particularly strong economic growth.

After ﬁrst recapitulating a stability analysis of the

equations from literature, we interpret the equations

88

Friesen, J., Tessmann, R. and Pelz, P.

Reaction-diffusion Model Describing the Morphogenesis of Urban Systems in the US.

DOI: 10.5220/0007711300880096

In Proceedings of the 5th International Conference on Geographical Information Systems Theory, Applications and Management (GISTAM 2019), pages 88-96

ISBN: 978-989-758-371-1

Copyright

c

2019 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

sociologically and examine their suitability for de-

scribing structural changes in the cities. To do this,

we conduct a simulation study varying four different

parameters on two cities to ﬁnd the parameter com-

binitation describing the structural change within the

cities the best. We then simulate the structal change in

a thrid city by using the best parameters found before

and analyse the results.

In this paper we ﬁrst describe the reaction-

diffusion model, the sociological processes it de-

scribes, the data used and then our approach (sec. 2).

We then present our results (sec. 3), discuss them

(sec. 4) and ﬁnally summarize the work (sec. 5).

2 METHODS AND DATA

Our approach is as follows: First, we brieﬂy present

the mathematical principles of the RD equations.

Building on this, we explain the sociological pro-

cesses that can be mapped with these equations and

discuss the assumptions we made within our setup.

2.1 Reaction-diffusion Model

The Reaction-Diffusion Model which shall be used

for this simulation study is presented by Gray and

Scott ﬁrst in (Gray and Scott, 1983; Gray and Scott,

1984), which is why we shall refer to it as the

“Reaction-Diffusion Model by Gray/Scott” or short

the “Gray/Scott-model”. The basic model as de-

scribed in (Gray and Scott, 1983) can be expressed

as follows:

∂v

∂t

= D

v

∆v + R[uv

2

−(F +k)v]

∂u

∂t

= D

u

∆u + R[−uv

2

+ F(1 −u)]

(1)

v and u are (dimensionless) concentrations of two

chemicals V and U. Here V is the autocatalytic

activator, whereas U represents the substrate in the

activator-depleted substrate scheme. D

v

and D

u

are

the diffusion constants of V and U. F and k are two

more constants, which have an outstanding effect on

the emergence of the outcoming patterns. We added

the reaction rate R, which is set to one in the original

papers, but is needed for proper dimensional analysis.

As a ﬁrst step Eq. 1 shall be transformed to a non-

dimensional representation. Therefore we introduce

the dimensonsless parameters

d :=

D

u

D

v

;γ :=

RL

2

D

v

;t

∗

:=

D

v

t

L

2

;x

∗

:=

x

L

(2)

with the time t and the typical length L and of

the system. The non-dimensional reaction-diffusion

equations are then:

∂v

∂t

∗

= ∆v + γ[uv

2

−(F +k)v]

∂u

∂t

∗

= d∆u + γ[−uv

2

+ F(1 −u)]

(3)

We use the common zero-ﬂux boundary conditions

((n · ∇)). For the stability analysis we follow the

structure of the derivations by (Gray and Scott, 1984).

When investigating the steady state (i.e. ∂/(∂t

∗

) = 0),

one can ﬁnd three steady states. The ﬁrst is the

trivial solution and shall be called “red state” (in-

dex R) in accordance with (Gray and Scott, 1984):

(v

R

,u

R

) = (0,1). Furthermore, one can derive the

”blue state” (index ”B”)

(v

B

,u

B

) = (

F

2(F + k)

(1 +

√

p),

1

2

(1 −

√

p)) (4)

and the “intermediate state” (index “I”)

(v

I

,u

I

) = (

F

2(F + k)

(1 −

√

p),

1

2

(1 +

√

p)) (5)

when the discriminator p is:

p := 1 −

4(F + k)

2

F

> 0. (6)

A further analysis of the steady states shows, that the

red state is always stable, the intermediate state is al-

ways unstable and the blue state can be stable, but

does not have to be. Thus the blue state is the only

steady state which can lead to diffusion driven insta-

bility, the so called Turing instability. The necessary

and sufﬁcient condition for Turing instability can be

derived to:

2

p

d detA

B

< a

11

d + a

22

(7)

with the Jacobian

A

B

=

a

11

a

12

a

21

a

22

(8)

Eq. 7 and Eq. 8 deliver the stability map (Fig. 1) of

the Gray/Scott-model for different values of d. Every

(F,k)-combination to the right of the Saddle-Node-

Bifurcation curve leads to a system which only leads

towards the red state. Every (F,k)-combiantion that

is located between the Saddle-Node-Bifurcation and

the respective instability curve for a given value of d

leads towards Turing patterns, whereas every (F,k)-

combination to the left of the instability curve can ei-

ther lead to a stable blue state or a red state solution,

depending on the initial conditions.

Further examination of the stability of the steady

states leads to the implicit formula that is the neces-

sary and sufﬁcient condition for Turing instability in

the Gray-Scott-model:

Reaction-diffusion Model Describing the Morphogenesis of Urban Systems in the US

89

Figure 1: Stability Map of the Gray/Scott-model for differ-

ent values of d. The region right to the curve is the red state

(R). The grey regions are the Intermediate states according

to the different diffusion numbers (I). The region left to the

intermediate regions is the blue state (B).

2

v

u

u

u

u

u

u

u

t

d

"

(F + k)

F

2(F + k)

(1 +

√

p)

2

+ F

−

F

2(F + k)

(1 +

√

p)

2

(−2(F + k))

#

< (F + k)d −

F

2(F + k)

(1 +

√

p)

2

+ F

(9)

2.2 Methodology

A simulation study is chosen to test the capability of

reaction diffusion equations to predict human move-

ment patterns within cities. To reach the highest data

availability, the USA was chosen to provide city data.

One can get data down to the level of so called “Cen-

sus Blocks”, which vary in size but typically cover an

area of about 10 000 m

2

. At this level of resolution,

the accessible data is limited to:

• Population count,

• Age and sex,

• Marital status and

• Ethnicity.

This paper connects the natural science in form of

reaction-diffusion-model by Gray-Scott with certain

sociological phenomena (Figure 2). A broad litera-

ture analysis led to two phenomena that were regu-

larly stated as typical for (fast) growing cities. These

are segregation (S. Fowler, 2016; Farrell, 2016; Alba

and Logan, 1993; England et al., 1988; Koch, 2011;

Schulz et al., 2006) and gentriﬁcation (Timberlake

and Johns-Wolfe, 2017; Hwang and Sampson, 2014;

Janoschka et al., 2014; Lees, 2012; Smith, 2002;

Smith, 2005). Segregation in this context means a

separation of people and groups with similar social

(religious, ethnical, class-speciﬁc, a.o.) backgrounds

from groups with other social backgrounds to avoid

reciprocal contact. The Gray/Scott-model shows sim-

ilar behavior in being able to produce stripe and bub-

ble patterns out of a simulated “mixed ﬂuid bowl”.

Figure 2: Sociological interpretation of the Gray/Scott

Model.

The gentriﬁcation in contrast to the segregation

means that a wealthier population is slowly squeezing

the former – poorer – population out of their neigh-

borhood or district. This squeeze-out is happening

due to a valorization of these neighborhoods by ren-

ovations and redevelopment of houses, as well as a

different business structure being attracted by a more

solvent clientele, up to a point where the former pop-

ulation cannot or does not want to afford to live there

anymore. Looking at the Gray/Scott-model, this can

be incorporated if you see the richer population as

the activator (V ) and the poorer population as the in-

hibitor (U). The Gray/Scott-model contains the di-

minisher term −(F +k)v and the reﬁll term F(1 −u).

Given above made assumption, the rich population is

diminished more likely in areas where there is already

a high concentration of rich people v ≈ 1). At the

same time some of the poorer population has to move

out of their neighborhoods into close-by districts (in-

dicated by the higher diffusion of the inhibitor [d > 1])

or further away (indicated by the reﬁll term). This

shows that wealth would be an excellent morphogene

for the Gray/Scott-model. Unfortunately – as stated

above – at the highest resolution, no wealth data is

available. Therefore ethnicity data was chosen, as it

shows the highest correlation to wealth of the accessi-

ble data (Shapiro et al., 2013; Tessmann and Friesen,

2017).

For the simulation study different cities have to

be chosen. As megacities in developing countries are

particularly interesting in their fast growth and fast

changes in both population and infrastructure, cities

were chosen that behave similarly to those megacities

of developing countries. Four criteria can be identi-

ﬁed, that characterize such cities: An already large

number of inhabitants, a fast growth of population, a

high inequality and strong economic growth.

Especially strong economic growth is seen to be

a key factor in attracting rural population to migrate

GISTAM 2019 - 5th International Conference on Geographical Information Systems Theory, Applications and Management

90

Table 1: Variation of non-dimensional parameters in the simulation study.

Parameters Values

d d

−

= 2 d

0

= 6 d

+

= 10 d

++

= 100

F F

min

= 0 F

max

= 0.25

k k

min

= 0 k

max

= 0.65

γ γ

−

=

(

(588.31 +

1961.7

d

)L

2

for 2 ≤d ≤ 10

280.82L

2

for d = 100

γ

+

=

(

(58831 +

196170

d

)L

2

for 2 ≤d ≤ 10

28082L

2

for d = 100

towards those mega cities. Economic growth also has

the potential to change city layouts when e.g. resi-

dential areas are replacing industrial zones due to the

inﬂux of inhabitants. Therefore, the US sample set

was chosen based on economic growth, i.e. the GDP

growth between the census years 2000 and 2010 of all

Metropolitan Statistical Areas (MSA). The cities with

the highest GDP growth rates were:

1. Midland/Odessa, TX

2. Victoria, TX

3. Bismarck, ND.

For these cities/metropolitan areas all available shape-

ﬁles and available population data was gathered from

the respective sources at the highest available res-

olution. As simulation software a combination of

FlexPDE for the calculation of the reaction-diffusion-

equations and MATLAB for the preparation and eval-

uation of FlexPDE results was used. For the simula-

tion various pre-assessments had to be performed to

deﬁne the variation of the non-dimensional parame-

ters. L is deﬁned to be the square root of the city

area in m

2

and the other parameters were deﬁned like

shown in Table 1.

While d and γ are simulated as stated in Table 1, F

and d are simulated with a higher resolution. In total

11 F steps of 0.025 between F

min

and F

max

and 14 k

steps of 0.005 between k

min

and k

max

are simulated.

This results in 4

d

∗11

F

∗14

k

∗2

γ

= 1232 simulations

per city.

The values u and v are calculated by determining

the number of inhabitants N per block. u := N

u

/N at

a given location then corresponds to the percentage of

all non-white inhabitants N

u

relative to the total num-

ber of inhabitants N in that block. Accordingly, v is

then deﬁned as v := N

v

/N, where N

v

is the number of

white residents.

Deﬁning the nondimensional simulation time

t∗

max

is crucial to the simulation experiment as it

needs to be equivalent to the 10 years of change that

the census data represent. As it is – per deﬁnition

– not independent of γ, two nondimensional simu-

lation times – one per γ simulated – were deﬁned

based on pre-simulations to: t

∗

max,Simγ−

= 5 ∗10

−6

and

t

∗

max,Simγ+

= 5 ∗10

−8

. Furthermore a quality measure

QF =

B

max

∑

i=1

|AC

2010,i

−AC

Sim,i

| (10)

has to be deﬁned to evaluate the results of the sim-

ulation experiments, with AC

2010,i

the activator con-

centration at point i in 2010. B

max

is the number of

discrete simulation points used for the respective sim-

ulated city. The smaller the QF value is, the better is

the result of the simulation. In the experiment Mid-

land/Odessa, TX and Bismarck, ND shall be simu-

lated to ﬁnd the best factor combination and identify

main and interdependency effects of the individual

factors. Then, to test a linear dependency, Victoria,

TX, the city with the 2nd fastest GDP growth is tested

with the interpolated factor combination also.

2.3 Assumptions

For the simulations, some further general, simplify-

ing assumptions are made in order to obtain a simpli-

ﬁed city model, which can then provide insights into

the suitability of the reaction-diffusion equations for

predictions of city morphology by means of the sim-

ulation. First, it should be assumed that the cities to

be simulated represent homogeneous levels, i.e. the

entire urban area can in principle be developed within

its boundaries. So undevelopable geographies (e.g.

rivers or steep mountains) are neglected as well as de-

liberately undeveloped areas such as roads, parking

lots, parks and other means of transport (e.g. trams,

etc.). Besides that, only ”concentrations” of people

are considered. This means that ultimately no con-

crete statements can be made about the total number

of people living in a block, but only the proportion

of a certain ethnic group (or, if a lower resolution was

chosen, the proportion of rich/poor population). How-

ever, since the core of the model is an interplay be-

tween two social groups, this should be regarded as a

reasonable simpliﬁcation.

3 RESULTS

In this section we present the results of our study.

First, we present the two cities we trained on (Mid-

Reaction-diffusion Model Describing the Morphogenesis of Urban Systems in the US

91

land/Odessa, TX and Bismarck, ND). We show the

effects, the different parameters have on the qual-

ity function and interpret the outcoming simulations

qualitivly. Then we present the result for the city (Vic-

toria, TX) we gained using the best parameters from

the two cities simulated before.

3.1 Midland

The minimal QF value is 105,881 at a combination of

(γ,d,F, k) = (−,10, 0.075,0.005). The maximal QF

value is 188,349 is the result of (-, 2, 0.25, 0.065).

Furthermore, the main and interdependency effects

(derived from a regression model) on the QF value

are shown in Table 2. Here γ,d ,F and k are normed,

so that γ

∗

,d

∗

,F

∗

,k

∗

∈ [−1, 1]. The regression model

based on Table 2, can be used to estimate the effects

of the parameters γ,d,F and k on the QF value.

If minimized within the set boundaries, the regres-

sion model leads to an optimized factor combination

of (γ

∗

,d

∗

,F

∗

,k

∗

) = (−, +,−,−), which means that

small γ,F and k and large d values seem to lead to

better results for Midland/Odessa, TX. The graphical

results of the simulation based on the factor combina-

tions is shown in Figure 3.

Figure 3: Representation of the empirical and simulation

data for Midland/Odessa. The simulation was conducted

with (γ,d, F, k) = (−,10,0.007, 0.005).

Looking at Midland/Odessa’s graphical results, it

can be seen that there have been signiﬁcant demo-

graphic changes between 2000 and 2010. In the west

and southeast of the simulation area completely new

structures were created. These fundamental structural

changes could be mapped by the simulation model,

even if larger deviations can still be detected.

3.2 Bismarck

The minimal QF value is 154,584 at a combination of

(γ,d,F, k) = (+,100, 0,0). The maximal QF value is

287,979 is the result of (-, 2, 0, 0.05). Furthermore,

the main and interdependency effects (derived from a

regression model) on the QF value are as presented in

Table 2. Once again γ,d,F and k are normed, so that

γ

∗

,d

∗

,F

∗

,k

∗

∈ [−1,1]. The regression model based

Table 2, can be used to estimate the effects of the pa-

rameters γ,d,F and k on the QF value.

If minimized within the set boundaries, the regres-

sion model leads to an optimized factor combination

of (γ

∗

,d

∗

,F

∗

,k

∗

) = (+, +,−,−), which means that

small F and k and large γ and d values seem to lead to

better results for Bismarck, ND.

Figure 4: Representation of the empirical and simulation

data for Bismarck. The simulation was conducted with

(γ,d, F, k) = (+,100,0, 0).

The so far presented results lead to the various ef-

fects on the QF factor for the two simulated cities pre-

sented in Table 2.

If the results just described are summarized, the

best combinations found can be entered in the stabil-

ity map (Figure 1). This Figure 5 shows that the best

results are obtained with small values of k and F, as

well as large values for d. The only difference is γ.

Figure 5: Stability Map of the Gray/Scott-model with the

best factor combination for both cities investigated (Mid-

land/Odessa, TX and Bismarck, ND).

3.3 Victoria

As described above, to test the linear dependency of

the factor effects, Victoria, TX, which had the 2nd

strongest GDP growth of all US cities, shall be tested

with an altered simulation scenario. On the one hand,

the “regular” test plan described in the methodology

GISTAM 2019 - 5th International Conference on Geographical Information Systems Theory, Applications and Management

92

Table 2: Effects.

Midland/Odessa

Main effect γ

∗

d

∗

F

∗

k

∗

-8726 -6680 21317 13909

Interdependency effect 1st O. γ

∗

d

∗

γ

∗

F

∗

γ

∗

k

∗

D

∗

F

∗

D

∗

k

∗

F

∗

k

∗

5853 -6729 -7823 -3600 -4554 732

Inter-dependency effect 2nd & 3rd O. γ

∗

d

∗

F

∗

γ

∗

d

∗

k

∗

γ

∗

F

∗

k

∗

d

∗

F

∗

k

∗

γ

∗

d

∗

F

∗

k

∗

-2318 455 1336 392 343

Bismarck

Main effect γ

∗

d

∗

F

∗

k

∗

-35605 -24505 8684 19681

Interdependency effect 1st O. γ

∗

d

∗

γ

∗

F

∗

γ

∗

k

∗

D

∗

F

∗

D

∗

k

∗

F

∗

k

∗

8377 5952 -6703 2056 -4127 3903

Inter-dependency effect 2nd & 3rd O. γ

∗

d

∗

F

∗

γ

∗

d

∗

k

∗

γ

∗

F

∗

k

∗

d

∗

F

∗

k

∗

γ

∗

d

∗

F

∗

k

∗

-2362 -1393 -4388 -2107 1983

Table 3: Overview to factor combinations, that promise low QF values for Midland/Odessa, TX and Bismarck, ND. The “best

factor combination” is the one that delivered the lowest QF for the respective city. The factor combinations in the last column

are based on a GRG (generalized reduced gradient method) minimization of the respective regression model when respecting

all constraints.

Best factor combination Main Effects with interdependency effects

γ

∗

d

∗

F

∗

k

∗

γ

∗

d

∗

F

∗

k

∗

γ

∗

d

∗

F

∗

k

∗

Midland/ Odessa - + - - + + - - - + - -

Bismarck + + - - + + 0 - + + - -

and used for Midland/Odessa, TX and Bismarck, ND,

was simpliﬁed to test less combinations of F and k.

These were now only varied in three steps each. On

the other hand, an additional factor combination was

tested, that lies – with regards to the factor combina-

tion – in the middle of the optimized solutions found

for Midland/Odessa, TX and Bismarck, ND, which

are displayed in the table above.

Figure 6: Simulation for Victoria, TX with γ

0

= 10 ∗γ

−

for

variations of the factor d. F and k are changed equally from

small (−) to medium (0) to large (+). (−) is F = k = 0,

(0) is F = 0.125 and k = 0.03 and (+) is F = 0.25 and

k = 0.065.

As the results for the two cities are consistent

for d,F and k, but not for γ,γ

0

= 10γ

−

was de-

ﬁned as the middle between γ

−

and γ

+

. Con-

sequently, if the factor effects were linear depen-

dent across the tested cities, one would expect the

best factor combination for Victoria, TX to be at

(γ

∗

,d

∗

,F

∗

,k

∗

) = (0,+,−,−). The results for the Vic-

toria, TX simulation can be seen in Figure 6.

Figure 7: Representation of the empirical and simulation

data for Victoria. The simulation was conducted with

(γ,d, F, k) = (10γ

−

,100,0, 0).

The best QF value for Victoria, TX was found

with the factor combination that was indicated by the

linearity test, i.e. with small values for F and k, a

large value for d and the middle value of γ, γ

0

.

It is interesting to note that both the formation of

a kind of coridor, in the east of the area under con-

Reaction-diffusion Model Describing the Morphogenesis of Urban Systems in the US

93

sideration, and the splitting off of a small ”island”

with a high concentration of non-white population at

the northern edge of the area was represented by the

model in the simulation and can also be seen in the

empirical data (Figure 7).

4 DISCUSSION

An analysis of the above results shows that small val-

ues for F and k lead to the best quality values.

For F, a slight tendency towards small values can

be observed, which in terms of the reaction-diffusion

equations means that the inhibitor or substrate U - i.e.

in the sociological analogy the non-white population

- is ”ﬁlled up” at a low rate. This means that few

non-white population ”react” into the system, which

can be interpreted in such a way that there is little in-

ﬂux of non-white population groups from outside the

system boundaries under consideration. The factor k

stands for the difference between the ﬁlling rate for

the inhibitor/substrate and the reduction of the activa-

tor. If k is zero, the activator in the system decreases

at the same rate as the inhibitor/substrate is replen-

ished. The factor d, which represents the ratio of

the diffusion coefﬁcient of the inhibitor or substrate

U to that of the activator V , gives a similar picture

to the factor γ. For most of the cities investigated, it

seems advantageous to use a large value for d for the

simulation of urban morphology development. This

means that the diffusion coefﬁcient of the inhibitor

or substrate - i.e. in the sociological analogy used

here the diffusion coefﬁcient of the non-white pop-

ulation - is signiﬁcantly higher than the coefﬁcient of

the white population. This means that in relation to

the real cities, the non-white population must have a

higher migration rate in order to better reﬂect real de-

velopments. This means that non-white households,

i.e. households that tend to be poorer, have to move

more often. In principle, this is very much in line

with the sociological phenomena of segregation and

gentriﬁcation, since poorer households are displaced

from their previous residential locations by gentriﬁca-

tion and segregation and therefore have to move more

frequently.

This interpretation is to be understood qualita-

tively and should be investigated in further studies

with extended data material.

The simulations carried out depict a total of 10

years of urban development, as the best data avail-

ability in the USA is available for the census years

2000 and 2010. The population data from the year

2000 serve as starting values for the simulations. The

boundary conditions correspond to the usual zero ﬂux

conditions for the Gray-Scott model, i.e. there is

no ﬂow across system boundaries. In later investi-

gations it is therefore necessary to check more pre-

cisely whether the selected boundary conditions have

a signiﬁcant inﬂuence on the quality of the results and

to what extent a different simulation time can have

a positive or negative inﬂuence on the results. The

evaluation of the result quality of a simulation was

carried out using a so-called quality function, which

forms the absolute difference between the reference

solutions from the census data of 2010 and the simu-

lation result at each simulation grid point. These sim-

ulation grid points depend on the individual geome-

try of the city under investigation, but always have

a distance of 0.001. The results of the simulation

study suggest that the investigated factor combina-

tions are only applicable to a very limited extent for

the prediction of urban structures. Thus, the initial

values in for out of six cities are a better prediction

for the reference solution than the result of the simu-

lation with the best factor combination in each case.

On the other hand, in two cases, for Midland/Odessa,

TX and Victoria, TX, the simulation result of the best

factor combination is a much better estimate of the

reference solution than the corresponding initial val-

ues. It is noticeable that the changes from 2000 to

2010 in Midland/Odessa, TX are particularly large

compared to the other cities investigated. This may

suggest that the reaction-diffusion equations are bet-

ter suited to approximate large changes than to depict

the smallest change in detail. This in turn would mean

that a lower data resolution than that available in the

USA would sufﬁce. A more detailed mapping of city

structures may be possible with the reaction-diffusion

equations, but then a local optimization of the factors

is very likely necessary. At present, however, it re-

mains questionable whether the Gray/Scott equations

can be used to map the patterns.

Other quality values should also be examined in

further investigations, since the quality measure pre-

sented here is only an integral quantity and does not

say anything about which areas in the area under con-

sideration the deviations between simulation and em-

piricism are large and which they are small.

Finally, it remains to be seen that good factor com-

binations can actually be derived to a limited extent

from other, similar cities. This was investigated for

two cities. To what extent future developments can

be estimated by a historical determination of the best

factor combination has to be examined in further in-

vestigations.

GISTAM 2019 - 5th International Conference on Geographical Information Systems Theory, Applications and Management

94

5 CONCLUSIONS

In this paper we have investigated a Gray/Scott

reaction-diffusion equation model to simulate the de-

velopment of economically strong cities in the USA.

We transferred the model known from chemistry and

biology to the development of urban systems. We

show that reaction-diffusion equations can be inter-

preted sociologically and we describe fundamental

structural changes in a third city by training the equa-

tions on two cities.

A more detailed mapping of urban structures may

be possible with the reaction-diffusion equations, but

then a local optimization of the factors will be nec-

essary. At the moment, however, it remains ques-

tionable whether the patterns can be mapped with the

Gray-Scott equations investigated here. Finally, it re-

mains to be seen that good factor combinations can

actually be derived to a limited extent from other, sim-

ilar cities. This was investigated for two cities. To

what extent future developments can be estimated by

a historical determination of the best factor combina-

tion is to be examined in further investigations.

Finally, it should be noted that the approach pre-

sented here is not intended to show that simple math-

ematical equations can fully describe complex phe-

nomena such as urbanisation. Rather, it addresses

the question of whether (i) there are fundamental pro-

cesses that dominate urbanization, while other pro-

cesses may be neglected. On the other hand, (ii) it

is examined whether these processes can be mapped

with simple equations.

ACKNOWLEDGEMENTS

We would like to thank the KSB Foundation for pro-

viding funds for the research.

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