The entropy mainly characterizes the dispersion degree of indicators. The smaller the entropy value indicates that the greater the dispersion of the indicator, the greater the weight of the indicator for the comprehensive evaluation; conversely, the smaller the weight of the indicator for the comprehensive evaluation. The scientific setting of the weights of each indicator plays a pivotal role in the accurate calculation of the economic agglomeration index. This paper adopts this objective weighting method, which effectively avoids the bias of subjective artificial factors on the weighting of economic agglomeration indicators in the YRD region. Finally, it uses composite weighting to calculate the economic agglomeration index of the YRD region from 2000 to 2019. For detailed calculation steps, see Reference (Xue and Zeng, 2016).

Allometric growth is a classical scalar concept in biology and ecology, reflecting the geometric measure of the relationship between the local and the whole of an organism (Li et al., 2009). Allometric growth relation is the constant proportion between the relative growth rate of one part of a system and the relative growth rate of the whole system or another part of the system. This paper measures the allometric relationship between the population and land elements in urban systems. The different research objects divide into longitudinal allometry and transverse allometry. Longitudinal allometric growth measures the allometric relationship between geographic units in time series. Transverse allometric growth explores the allometric relationship of different geographical units on a time scale. The calculation formula of the allometric growth model is (Yin and Yang, 2021): A t = a P t b ;   A k = a P k b (1)

Taking the logarithm of both sides of Eq. 1, we can get: ln ⁡ A t = ln ⁡   a + b   ln P t ;   ln ⁡ A k = ln ⁡   a + b   ln P k (2)

Where A _t is the urban population at time t, P _t is the urban land use situation at time t, a is the proportion coefficient, and b is the scale coefficient, also known as the allometric growth coefficient. Studies by many scholars showed that when the allometric growth coefficient b is between 2/3 and 1, that is, when the average value is about 0.85, population and land are regarded as basically coordinated and equal speed growth. When b > 0.85, it is positive allometric growth, and the growth rate of the population is lower than that of land. When b < 0.85, it is negative allometric growth, and the growth rate of the population is greater than that of land (Li et al., 2009; Sun et al., 2011; Ma et al., 2018; Yin and Yang, 2021). Positive and negative allometric growth is divided into three levels, and the classification criteria and characteristics are shown in Table 2.

The trend surface uses spatial data to fit a mathematical surface that illustrates the spatial trends in the distribution patterns of observations of geographic elements over a sizeable spatial span (Xu, 2002). This paper uses trend surface analysis to abstract the overall pattern of divergent trends of economic agglomeration in the YRD region in different years.

Based on the geographical location of each city and the 20-year panel data of the economic agglomeration index, using spatial interpolation to obtain the multi-year trend surface of the geographical distribution. Based on the trend surface definition, the formula is (Li et al., 2013): Z i x i , y i = T i x i , y i + ε i (3) T i x i , y i = β 0 + β 1 x + β 2 y + β 3 x 2 + β 4 y 2 + β 5 x y (4)

Where Z _i (x _i , y _i ) is the geographic location of the city i, (x _i , y _i ) is the plane space coordinates, T _i (x _i ,y _i ) is the trend function, ε _i is the residual value, β ₀ is the constant term, β ₁ -β ₅ are the coefficients of the variables in the trend function.

Using Global spatial autocorrelation (Global Moran’s I) to analyze the spatial correlation of economic agglomeration between study regions, i.e., it contains spatially positive, spatially negative, or randomly distributed. The calculation formula is as follows (Liu et al., 2019): I = n ∑ i = 1 n ∑ j = 1 n w i j x i − x ¯ ( x j − x ¯ ) ∑ i = 1 n ∑ j = 1 n w i j × ∑ i = 1 n x i − x ¯ 2 (5) where, n is the number of study units, x _i and x _j are the economic agglomeration indices of city i and city j, x ¯ is the average of the economic agglomeration index of all cities, w _ij is the regional spatial weight matrix, and the regional spatial adjacency is 1 and non-adjacency is 0. Moran’s I > 0 is a positive spatial correlation, and Moran’s I <0 is a negative spatial correlation.

Using Pearson correlation coefficient to Explore the relationship between population-land and economic agglomeration. The calculation formula is as follows (Sun, 2014): Q X Y = ∑ i = 1 n X i − X ¯ Y i − Y ¯ ∑ i = 1 n X i − X ¯ 2 ∑ i = 1 n Y i − Y ¯ 2 (6) where, Q _XY is the correlation coefficient, and its value range Q _XY ∈[−1,1], Q _XY > 0 is a positive correlation, Q _XY < 0 is a negative correlation, Q _XY = −1 and Q _XY = 1 are absolute correlation, i.e., linear relationship, X ¯ and Y ¯ are the mean values, n is the number of samples.

A double-log linear fit of population and land in the YRD region from 2000 to 2019 by an allometric growth model. It obtains the results of the population-land allometric growth models for the YRD region as a whole and the three regions of the Shanghai metropolitan area, core area, and central area (Table 3). The results show the variation of each region’s scaling coefficient (Figure 2).

The scalar coefficient for the YRD region fluctuates in the range of 0.6052–1.0112 from 2000 to 2019. During the study period, the YRD region gradually evolves from negative allometric I to positive allometric growth Ⅱ stage. The fitting effect of each year is good, and the goodness of fit of the fitting equation is above 0.65. According to the change of the scale coefficient, there is a general fluctuating upward trend during the study period, but it showed a slight downward trend in 2001, 2007, and 2014. The scale coefficient’s value and the type of allometric growth division can be divided into three periods: 2000–2011, 2012–2017, and 2018–2019. From 2000 to 2011, the scale coefficient fluctuated from 0.6052 to 0.8247, which is in the negative allometric growth I stage of a weakly expanding population. From 2012 to 2017, the scale coefficient fluctuated continuously above 0.85, increasing from 0.8556 to 0.9113, and showed a sharp upward trend from 0.8501 to 0.9737, increasing by 14.5% in 2014–2015. This period is in the positive allometric growth I stage. That is, the land-relative growth rate is slightly higher than the population-relative growth rate of the bare coordination stage. In 2018–2019, the scale coefficient was higher than one but with a slight decrease, and it is in the positive allometric growth II stage of weak land expansion. Overall, the three periods can be combined into two more periods: a negative allometric period of population expansion from 2000 to 2011 and a favorable allometric period of land expansion from 2012 to 2019, i.e., the YRD region as a whole is in a transition stage from population expansion to land expansion during the study period.

Further comparison is to explore the population-land allometric growth relationship at different scales in the YRD (Figure 2). The goodness of fit for the Shanghai metropolitan area is above 0.81, the core area is above 0.76, and the central area is above 0.70, indicating that the allometric growth model fits well in each region. The scale coefficients of the Shanghai metropolitan area and the core area have similar trends. The years from 2000 to 2006 were in a fluctuating upward trend, rising from 0.7313 to 1.0408 and 0.8156 to 1.1533, respectively, with the years from 2005 to 2006 in a sharp upward trend, increasing by 29.1% and 23.7%, respectively; the years 2007–2019 were in a relatively stable stage. From 2000 to 2019, the scale coefficient of the central area was in a continuous fluctuating upward trend, which increased from 0.6552 to 1.076, increasing by 64.2% during the study period. In addition, the core area, Shanghai metropolitan area, and central area have crossed the 0.85 demarcation line since 2002, and urban development has gradually transformed from population expansion to land expansion.

Combined with the typology of allometric growth (Table 3), all scale ranges from 2000 to 2019 transition from negative to positive allometric growth. Shanghai metropolitan area was in the negative allometric growth I stage of weak population expansion from 2000 to 2006 but fluctuated in 2004 and 2006. From 2007 to 2019, the average annual population and land development rate is at the positive allometric growth I stage, indicating that the urban population growth and land expansion in Shanghai metropolitan area are reasonable, and their growth rates are the same. With the expansion of the research scope and scale, the allometric growth types of the core and central areas are similar, showing the development transition of “population expansion, population-land coordination, and land expansion,” but there is a certain lag in the central area. The core area was in the negative allometric growth I stage of weak population expansion from 2000 to 2001, the positive allometric growth I stage of population-land coordination from 2002 to 2005, and the positive allometric growth II stage of land expansion from 2006 to 2019. The central area was a negative allometric growth I stage from 2000 to 2010 (with fluctuations in 2006), a positive allometric growth I stage from 2011 to 2014, and a positive allometric growth II stage from 2015 to 2019.

The mean scale coefficients of the Shanghai metropolitan area, core area, central area, and YRD from 2000 to 2019 were 0.9083, 1.0113, 0.8668, and 0.8022, respectively; the standard deviations of the scaling coefficients are 0.0906, 0.0919, 0.1379, and 0.1319, respectively. With the continuous expansion of the scale of the research scope, the allometric growth pattern of core area > Shanghai metropolitan area > central area > YRD as a whole was presented, and the scale coefficient of the central area changed the most dramatically, while the scale coefficient of Shanghai metropolitan area was the most stable. Generally, the regions of different scales are transitioning from the negative allometric growth I stage of weak population expansion to positive allometric growth. Shanghai metropolitan area is gradually transformed into the population and land coordination type, and the core area, central area, and the YRD as a whole is gradually transformed into the weak land expansion type.

From the above, divide the allometric growth in the YRD region from 2000 to 2019 into two major stages to further explore the population-land allometric growth relationship among different cities. Calculating the transverse allometric growth fitting model for 2000–2011 and 2012–2019 and visualizing its scale coefficients (Figure 3).

From 2000 to 2011, the YRD region was dominated by positive allometric growth, with 23 vibrant allometric cities accounting for 56.1% of all cities. There were 18 hostile allometric cities, accounting for 43.9% of all cities. Vivacious allometric cities are mainly distributed in Shanghai, Zhejiang, southern Jiangsu, and central Anhui, among which 7, 12, and 19 cities in Shanghai metropolitan area, core area, and central area are positive allometric growth, accounting for about 77.8%, 75%, and 70.4%, respectively. Specifically, there were 14 positive allometric growth III stage cities with substantial land expansion during the study period, accounting for 60.9% of the vibrant allometric growth cities. It is mainly distributed in Zhejiang province, Nanjing, Zhenjiang, Changzhou in Jiangsu Province, and Tongling, Chizhou, and Huaibei in Anhui Province. Only one city, Ma ‘anshan city, Anhui Province, was in the positive allometric growth I stage of population-land coordination. The remaining eight cities were in the positive allometric growth II stage of weak land expansion. Negative allometric growth was dominated by hostile allometric III growth cities with shrinking populations and land, accounting for 88.9% of hostile allometric growth cities, mainly concentrated in northern Jiangsu, western and southern Anhui. Only two cities, Zhoushan City, Zhejiang Province, and Bozhou City, Anhui Province, were in the negative allometric growth II stage with substantial population expansion. The goodness of fit of the allometric growth model for each city is generally high, and only four cities, Nantong, Suzhou, Lu’an, and Bozhou, have a goodness of fit below 0.40.

From 2011 to 2019, the YRD region still showed positive allometric growth. There are 37 vibrant allometric cities, accounting for 90.2% of all cities. It is mainly concentrated in Shanghai, Jiangsu, southeastern Zhejiang, and most areas of Anhui. Nine positive allometric growth II stage cities, accounting for 24.3% of the total number of vibrant allometric cities, distributed in central and southern Anhui and northern Zhejiang. There were 28 positive allometric growth III stage cities, accounting for 75.7% of the total vivacious allometric cities, distributed in Shanghai, Jiangsu, southern Zhejiang, northern and southern Anhui, and other regions. There were four hostile allometric cities, accounting for 9.8% of all cities. Among them, there was only one negative allometric growth I stage city, Xuzhou. There were three negative allometric growth III stage cities, Yancheng, Lu’an, and Anqing. The overall fitting effect is good from the perspective of the goodness of fit. Only four cities Yancheng, Taizhou, Huainan, and Anqing have a goodness of fit below 0.40.

According to the different categories of population-land allometric growth in the two time periods of 2000–2011 and 2012–2019, they are classified into five types of evolution. Namely, stepwise upward (positive and negative allometric velocity levels fluctuate upward by one level), cross-level upward (positive and negative allometric velocity levels fluctuate upward by more than one level), stepwise downward (positive and negative allometric velocity levels fluctuate downward by one level), cross-level downward (positive and negative allometric velocity levels fluctuate upward by more than one level), and smooth transition (positive and negative allometric velocity levels do not change). Specifically, there were 22 cities with upward changes (stepwise upward and cross-level upward), accounting for 53.7% of all cities. After the change, 20 cities turned into positive allometric growth III stage cities, and the number of cities with upward change was large and widely distributed. Twelve cities, mainly distributed in southwestern Zhejiang, southern Jiangsu, and western Anhui, have maintained a smooth transition, accounting for 29.2% of all cities. There were seven cities with downward changes (stepwise downward and cross-level downward), including Xuzhou in Jiangsu Province, Hangzhou in Zhejiang Province, Ningbo and Quzhou in Zhejiang Province, and Huaibei, Tongling, and Chizhou in Anhui Province, accounting for 17.1% of all cities.

In general, from 2000 to 2019, the YRD region was dominated by land expansion, where the relative growth rate of land exceeded the population. Moreover, the degree of land expansion increased continuously from 2012 to 2019. Over time, the rate of land expansion in the YRD keeps a high level and continues to improve, and land expansion has become the main focus of urban development.

Using the entropy weight method, obtain the economic agglomeration index of the YRD region from 2000 to 2019. Combined with the above research, visualize and analyze three periods of economic agglomeration index in 2000, 2011, and 2019 on the trend surface (Figure 4). According to the trend surface changes, the north-south direction (Y-axis) from 2000 to 2019 showed an inverted U-shaped trend of “high in the middle and low in the north and south,” and the north-south gap has fallen. The east-west direction (X-axis) shows an evident trend of “east is high, and west is low,” extending from west to east as a parabola, but the gap between east and west also narrowed. Further, according to the contour visualization diagram, the economic agglomeration index of the YRD has been continuously improved during the study period. The gap between regional agglomeration and development has gradually narrowed from the “single center” of Shanghai in 2000 to the “multi-center” of Shanghai-Nanjing-Hangzhou-Hefei in 2011, and then to the “continuous economic high-value agglomeration area” with the provincial capital city as the center in 2019. In general, the economic agglomeration in the YRD shows a spatial distribution trend of “high in the middle and low in the north and south, high in the east and low in the west” and gradually forms a distribution pattern of “continuous economic high-value agglomeration area” with the core area as the center. The degree of spatial clustering of economic agglomeration gradually decreases and tends to spread, reflecting the efficacy of the promotion of the integrated development strategy of the YRD.

Using SPSS to explore the correlation between the population-land and economic agglomeration indices by measuring its cross-sectional data for three periods 2000, 2011, and 2019 (Table 4). There is a significant correlation between population-land factors and economic agglomeration, and the Pearson correlation coefficient gradually increases with time. This indicates that the correlation between population, land, and economic agglomeration is constantly increasing, and the correlation between land and economic agglomeration is significantly higher than that of population. Specifically, the Pearson correlation coefficient of each region has increased to different degrees. The correlation coefficients of population and land in the Shanghai metropolitan area increased from 0.518 and 0.555 in 2000 to 0.725 and 0.809 in 2019, respectively. The correlation coefficients of population and land in the core area increased from 0.501 and 0.576 to 0.642 and 0.788, respectively. The correlation coefficients of the overall population and land in the YRD increased from 0.353 and 0.544 to 0.575 and 0.751, respectively. However, except for the correlation coefficient between population and economic agglomeration in the central area increased from 0.522 to 0.606, the correlation between land and economic agglomeration decreased slightly, from 0.781 to 0.753, indicating that the impact of land on economic agglomeration in the central area has been reduced. However, it still plays a significant role compared with the population. In general, with the expansion of the research scope and the change of time, land has a significant promoting effect on economic agglomeration. At the end of the study, forming the overall influence pattern of Shanghai metropolitan area > core area > central area > YRD.

To further measure the difference in the economic agglomeration index from 2000-2011-2019, if the difference is positive, it is a positive effect, indicating that the economic agglomeration index of the city has increased. Otherwise, it is a negative effect. Therefore, combined with the classification of allometric growth types and the evolution of economic agglomeration, dividing the YRD region into four types of cities: “positive-positive,” “positive-negative,” “negative-positive,” and “negative-negative.” In quadrant I, the positive-positive type is the economic agglomeration city with land expansion. In quadrant II, the positive-negative type is the economic diffusion city with land expansion, the negative-positive quadrant III is the economic agglomeration city with population expansion, and in quadrant IV, the negative-negative type refers to the cities with population expansion and economic diffusion. The specific classification basis and categories are as follows (Table 5).

According to the measurement results of the economic agglomeration effect of allometric growth (Figure 5), the cities and directions indicated by the arrows are the evolvement changes of 2012–2019 compared with 2000–2011. From 2000 to 2011, there were 14 positive-positive cities, accounting for 34.1% of all cities, mainly concentrated in the core area of Shanghai, southern Jiangsu, and eastern Anhui. Nine positive-negative cities, accounting for 21.9% of all cities, are concentrated in Zhejiang and Anhui’s northern areas. There were 14 negative-positive cities, mainly concentrated in northern Jiangsu and central Anhui. Only four negative-negative cities, accounting for 9.7% of all cities, are mainly distributed in southern Anhui.

In comparison, most cities experienced leapfrog movements in different quadrants from 2012 to 2019. The number of positive-positive cities increased to 27, mainly in Shanghai, Zhejiang, and eastern Anhui provinces. The number of positive-negative cities increased to 10, mainly distributed in Jiangsu Province. The number of negative-positive cities decreased to 2, namely, Anqing and Lu’an in Anhui Province. The negative-negative cities were also reduced to Xuzhou and Yancheng in Jiangsu province. During the study period, the proportion of cities with the positive effect of allometric growth on economic agglomeration increased to 70.7%, while the proportion of cities with the negative effect decreased to 29.3%. The positive allometric effect of land expansion is dominant, indicating that land expansion significantly promotes economic agglomeration in the YRD.

The land expansion development mode mainly drove the economic agglomeration effect in the YRD from 2000 to 2019. The percentage of cities with positive economic agglomeration effects of allometric growth is 70.7%, which has a significant promoting effect. The allometric growth of the population-land has an essential impact on reconstructing the economic agglomeration pattern in the YRD region. During the study period, the vast majority of cities have leapfrog movements in different quadrants, and only 22% of cities are stable in the stage of positive allometric positive effect. With the expansion of the research scope and scale, the effect of land on economic agglomeration in the YRD showed a pattern of Shanghai metropolitan area > core area > central area > YRD as a whole.