Computer simulation of population dynamics inside the urban environment

In this paper using a mathematical model of the so-called “space-dynamic” approach we investigate the problem of development and temporal dynamics of different urban population groups. For simplicity we consider an interaction of only two population groups inside a single urban area with axial symmetry. This problem can be described qualitatively by a system of two non-stationary nonlinear differential equations of the diffusion type with boundary conditions of the third type. The results of numerical simulations show that with a suitable choice of the diffusion coefficients and interaction functions between different population groups we can receive different scenarios of population dynamics: from complete displacement of one population group by another (originally more “aggressive”) to the “peaceful” situation of co-existence of them together.


Introduction
In recent years models of spatial economy have been widely used for describing different urban processes. Processes of urbanization along with urban consolidation have been observed for a long time in both developed and developing countries, but modern cities are more and more characterized by decentralization tendencies. At present we can frequently observe heterogeneous rather than homogeneous urban formations. The examples of urban formations of both established and still establishing forms may be such metropolises as Tokyo, New York, Mexico City, Moscow, Shanghai, Singapore etc. It is well known that distribution of different population groups according to living areas represents quite complicated function of spatial variables and time period. The given work presents computerized results which were obtained by means of "soft" mathematical model ( [1]; Arnold) of interaction of two groups of urban population with non-linear spatial diffusion taken into account. 1 To whom any correspondence should be addressed. For example, urban population may be classified by their income level (the "rich" and the "poor"), according to their level of education ("more educated" and "less educated") and by a number of other factors. In a number of economically and socially developed countries coexistence of population groups belonging to different social layers brings about serious social problems. Therefore, at present such processes are intensively studied from different viewpoints including mathematical models within spatial economy.
Interaction of similar population groups within a single urban agglomeration may be described at qualitative level using non-standard equations in partial derivatives of the diffusion type. For simplicity and visibility the given work considers only two groups of population characterized by density of distribution x y of the fixed dimensional area D at a given period of time t . The system of non-linear differential equations describing time dynamics of these population groups looks formal as: where elliptical differential operator of the second order 1,2 s = , describe "interaction" of different population groups living in the same area. Quite realistic seems the following type of these functions: In this work we consider the simple area as a two dimensional area D , though this is not a necessary restriction, but it leads to a better understanding of the processes under study, namely rectangle x y x l y l = £ £´£ £ on the plane of variables ( , ) x y . At the boundary D G = ¶ are given heterogeneous boundary conditions (boundary conditions of the 3rd type): where . n ¶ ¶ shows a derivative along the single outer normal at the points of boundary G .
It is necessary to add original conditions (the initial Cauchy conditions for equations (1) and boundary conditions (3)) and, as a result, we have the following boundary value problem consisting of two nonlinear differential diffusion (parabolic) equations: This system of equations (4)

Computer simulation results
Suppose that the relations between two groups of population, chosen by us, may be both "friendly" and "neutral" or even "unfriendly". A similar approach was proposed earlier in [4;Zhang].
As initial distribution density for both interacting population groups let's take "symmetrical" functions, so that the whole urban area was partially occupied by this or that population group (Fig. 1). It may be shown how urban structure for different parameters of the modeled system will change. For example, when relations between two groups of the population are of "aggressive" character and the obtained calculation results (Fig. 2, a-d) show that both population groups equally fill the given area, but due to "hostility" one group (the more "aggressive" one) dislodges the other one as a result, and, therefore, the urban structure of the population becomes practically homogeneous. If coefficients 1 0 d < and 2 0 d = , diffusion coefficients 1 K and 2 K depend only on function u , one can see the following picture of spatial-time diffusion of the population (Fig. 3, a-f). This example can be classified according to educational level when less educated people tend to live in areas with more educated population. It can also be said that areas with more favourable facilities are preferable for life which defines evident population migration from one area to another. As an example, suppose that interaction between population groups can be classified as purely competitive. Diffusion coefficients in equations are considered to have sedate dependence. Under such supposition one population group dominates over the other, dislodging it outside the city borders (Fig.  4, a-f).

Conclusions
The present work considers the mathematical model aimed at describing the mechanism of interaction between different population groups leaving in the same urban area. This problem is formulated in terms of non-linear differential equations of the diffusion type. It is no use applying the so-called "hard" model in such kind of mathematical simulation. At present we can speak only about "soft" mathematical modeling. The term "soft" mathematical model implies only the parameters and factors that are essential or available. Having a confirmed and verified mathematical model one can proceed to solve purely applied problems.
In his time (to be more exactly in 1997) an outstanding Russian mathematician, a member of the Russian Academy of Sciences Vladimir I. Arnold noted that "… success comes not so much with application of ready-made recipes ("hard" models), but rather with mathematical approach to the problems of the real world phenomena" [1; Arnold].

Appendix
To confirm the accuracy of the difference schemes and the convergence rate of the numerical solutions we made the corresponding analysis using precise analytical solutions of non-linear differential equations of the diffusion type.
The similar analysis was made on the example of the system of non-linear equations of "reactiondiffusion" type [5;Polyanin & Zaitsev]: where parameter 1 n = defines problem with the axial symmetry and parameter 2 n = defines problem with the central symmetry. In literature function u it is customary to call 'activator' and the second function w , which leads to slowing down the described diffusion process, is called 'inhibitor'.
x t q q = satisfies the linear partial derivatives equation: Let's multiply the first equation of system (A.1) by ' b ' and add it to the second one, multiplied by