The Spatial Covariation of Population Density and Agricultural Land Productivity in the Kanto District: A Mathematical Model

Abstract

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 35 km and showed a tendency to correlate negatively in the outside range of 35 km, except the non-agricultural city core and the mountainous area beyond 95 km. 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 35 km 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 fAwith 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),andhA(P) in(1) correspond to curves FAG, 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) isacomponent 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 fpbe the mean value of fp(r) with respect to overall r, we get p = fp(r)-fpand (d/dr)fP(r) < 0. Thus hA(p) becomes a function of r and (d/dr)hA(p(r) becomes positive. Let r0be 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 μp and agricultural land productivity uAwith azimuth angle in a specific distance belt. Both uPand uAare 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 UPand VPin Figure 7 respectively. The variable vP(θ) is a component representing long periodic variation; vP(θ + λ1) = vP(θ) for a constant λ1(>0). The variable wP(θ) is a component representing short periodic variation; wP(θ+λ2)= wP(θ) for a smaller constant λ2(>0) than λ1. The productivity uA(θ)at angle θ can be written in equation (5). Variables uA(θ) and vA(θ) in (5) correspond to curves UAand VAin Figure 7 respectively. The variable vA(θ) is acomponent representing long periodic variation and expressed as VA(θ)=k1VP(θ) using a constant k1(>0) and density vP(θ). The variable WAis a component representing short periodic variation and expressed as wA(θ)= -k2wP(θ) using a constant k2(>0) and density wP(θ). © 1991, The Human Geographical Society of Japan. All rights reserved.