This study examined how population density and agricultural land productivity in a metropolitan area varied spatially by discussing the case of the Kanto district. A mathematical model was presented for describing the relationship of these two values. Population density values were calculated from grid-cell data of the population census of 1980. Values of agricultural land productivity were represented by agricultural income per hectare estimated from grid-cell data of the agricultural census of 1980. Figure 3 shows distribution of these values.
We attempted to find the complicated spatial patterns of these values by analyzing the covariation of them in two aspects: the covariation with distance from the city center (Fig. 4) and with azimuth angle in distance belts (Fig. 5).
The facts that we found are summarized as follows:
The covariation with distance showed a tendency to correlate positively in the inside range of 35km and showed a tendency to correlate negatively in the outside range of 35km, except the non-agricultural city core and the mountainous area beyond 95km.
The variation with azimuth angle, both of density and of productivity, consisted of long waves and short waves. The long waves of density and productivity showed similar curves. By contrast, the short waves of them showed inverse curves. This inverse relationship was weaker in the outside range of 35km than the inside.
A mathematical model was then presented on the basis of these facts. This model consists of the following two parts:
1) The covariation with distance
The following formulation describes the covariation of population density
fP and agricultural land productivity
fA with distance. Equation (1) shows the productivity
fA (
r, p) at a place where distance from the edge of the city center is
r and deviation of density is
p. Variables
fA (
r, p),
gA (
r), and
hA (
p) in (1) correspond to curves
FA,
G, and
H in Figure 6 respectively. The variable
gA (
r) is a component declining exponentially with distance
r; gA (
r)>0 and (
d/dr) gA (
r)<0 for
r>0. The variable
hA (
r) is a component increasing with decreasing deviation
p; (d/dp) hA (
p)<0;
hA (
p)≈0 at
p=0. Let
fP(
r) be the population density at distance
r and
fP be the mean value of
fP(
r) with respect to overall
r, we get
p=
fP(
r)-
fP and (
d/dr)fP(
r)<0. Thus
hA (
p) becomes a function of
r and (
d/dr)hA(
p(r)) becomes positive. Let
r0 be a constant distance, the relationship between
gA (
r) and
hA (
p) is given by inequalities (2) and (3).
2) The covariation with azimuth angle
The following formulation describes the covariation of population density up and agricultural land productivity
uA, with azimuth angle in a specific distance belt Both
uP and
uA are standardized values with respect to overall angle in the distance belt. The density
uP(θ) at angle θ can be written in equation (4). Variables
uP(θ) and
vP(θ) in (4) correspond to curves
UP and
VP in Figure 7 respectively. The variable
vP(θ) is a component representing long periodic variation;
vP(θ+λ
1)=
vP(θ) for a constant λ
1(>0)
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