## 3.1 The changing trend of spatial population distribution in urban agglomerations

We took any central city *α* and any peripheral city *β* to analyze the spatial distribution characteristics of urban agglomerations under different development levels. The meaning of each letter is as follows: E is the net utility value, C is the cost, \({C}_{1}\) is the cost of living, \({C}_{2}\)is the cost of housing, s is the initial state, r is the critical state, and l is the final state.

In the initial state of city agglomeration development, the per capita income level of central cities is higher than that of peripheral cities 21, and the housing costs and living costs of central cities are the same as those of peripheral towns 35–37.

\({E}_{\alpha s}\) =\({W}_{\alpha s}\)-\({C}_{1\alpha s}{-C}_{2\alpha s}\) (1)

\({E}_{\beta s}\) =\({W}_{\beta s}\)-\({C}_{1\beta s}{-C}_{2\beta s}\) (2)

Under the initial state of city agglomeration development, \({W}_{\alpha s}\)>\({W}_{\beta s}\),\({C}_{1\alpha s}\)=\({C}_{1\beta s}\)༌\({C}_{2\alpha s}\)=\({C}_{2\beta s}\),, o\({E}_{\alpha s}\)>\({E}_{\beta s}\). The n, t utility value of central cities is greater than that of peripheral cities. From condition 1, the population gathers in the central town 38,39.

In mature urban agglomerations, the per capita income of central cities is higher than that of peripheral cities. The cost of housing in central cities is much greater than the cost of living in surrounding cities, and the cost of living in central cities is the same as that of the surrounding towns 35,37.

\({E}_{\alpha l}\) =\({W}_{\alpha l}\)-\({C}_{1\alpha l}{-C}_{2\alpha l}\) (3)

\({E}_{\beta l}\) =\({W}_{\beta l}\)-\({C}_{1\beta l}{-C}_{2\beta l}\) (4)

In the final state,\({W}_{\alpha l}\)>\({W}_{\beta l}\),\({C}_{1\alpha l}={C}_{1\beta l}\)༌\({C}_{2\alpha l}\)>>\({C}_{2\beta l}\), s, \({E}_{\alpha l}\)<\({E}_{\beta l}\), the c, nclusion is that in the later period of the city agglomeration, the net utility value of peripheral cities is greater than that of central cities. Thus, according to Condition 1, the population began to gather in peripheral cities33,40. The above reasoning process leads to the first conclusion:

Conclusion 1: In the early stage of urban agglomeration development, the population tends to gather in central cities. In the mature period of urban agglomeration, the population is concentrated in peripheral cities.

## 3.2 The changing trend of urban agglomeration population gravity center

From 3.1, cities within urban agglomeration will experience varying population migration due to different development levels. We choose the population center of gravity indicator to visually express the population flow changes. The population center of gravity refers to the point at which the moment of population distribution on the spatial plane reaches equilibrium, usually used to measure the equilibrium of population distribution in the region41. By studying the movement trajectory of the population center of gravity, we explain the characteristics and causes of the spatial change of population distribution, and provide a decision-making basis for formulating population development policies and social and economic development plans. We took the permanent population data of municipal districts as weights and weighted the spatial units to get the population center of gravity. The calculation method is as follows:

X=\(\frac{\sum _{i=1}^{n}{M}_{i}{X}_{i}}{\sum _{i=1}^{n}{M}_{i}}\), Y=\(\frac{\sum _{i=1}^{n}{M}_{i}{Y}_{i}}{\sum _{i=1}^{n}{M}_{i}}\) (5)

Where, \({X}_{i}\)and \({Y}_{i}\) represent the longitude and latitude coordinates of the ith spatial unit, respectively, and \({M}_{i}\) represents the population size of a city in the spatial unit.

It is assumed that there is a central city and a peripheral city in the urban agglomeration, where the location of the central city is (\({X}_{1}\), \({Y}_{1}\)) and the population size is \({M}_{1 }\).The location of the peripheral cities is (\({X}_{2}\), \({Y}_{2}\)), and the population size is \({M}_{2}\).The population center of gravity position of this urban agglomeration is (X, Y).

1. To judge the change in the population center of gravity in the early urban agglomeration, we assume that the population of the central city \({M}_{2}\) remains unchanged, find the derivation of \({M}_{1}\).\({\text{M}}_{1}>0,\) \({\text{M}}_{2}\)>0.

1. Longitude.

\(\frac{{d}_{x}}{{d}_{M1}}\) = \(\frac{{{X}_{1}({\text{M}}_{1}+{\text{M}}_{2})-(M}_{1}{X}_{1}+{M}_{2}{X}_{2})}{{({\text{M}}_{1}+{\text{M}}_{2})}^{2}}\)

=\(\frac{{M}_{2}({\text{X}}_{1}-{\text{X}}_{2})}{{({\text{M}}_{1}+{\text{M}}_{2})}^{2}}\)

If \({\text{X}}_{1}\)>\({\text{X}}_{2}\) (the central city is to the east of the peripheral city), \(\frac{{\text{d}}_{\text{x}}}{{\text{d}}_{\text{M}2}}\)>0. The longitude of the population gravity center increases (moving eastward), that is, toward the central cities. If \({\text{X}}_{1}\)<\({\text{X}}_{2}\) (the central city is to the west of the peripheral city), \(\frac{{\text{d}}_{\text{x}}}{{\text{d}}_{\text{M}2}}\)<0. The longitude of the population gravity center decrease (moving westward), that is, toward the central cities.

In conclusion, when the peripheral city population remains unchanged, the population size of the central city increases (when the population size of the central city increases more than that of the peripheral city increases), the population gravity center's longitude moves toward the central city population.

(2)latitude

\(\frac{{d}_{y}}{{d}_{M1}}\) = \(\frac{{{Y}_{1}({\text{M}}_{1}+{\text{M}}_{2})-(M}_{1}{Y}_{1}+{M}_{2}{Y}_{2})}{{({\text{M}}_{1}+{\text{M}}_{2})}^{2}}\)

=\(\frac{{M}_{2}({\text{Y}}_{1}-{\text{Y}}_{2})}{{({\text{M}}_{1}+{\text{M}}_{2})}^{2}}\)

If \({\text{Y}}_{1}\)>\({\text{Y}}_{2}\) (the central city is to the north of the peripheral city), \(\frac{{\text{d}}_{\text{Y}}}{{\text{d}}_{\text{M}2}}\)>0. The latitude of the population gravity center increases (moving north), that is, toward the central cities. If \({\text{Y}}_{1}\)<\({\text{Y}}_{2}\) (the central city is to the south of the outer city), \(\frac{{\text{d}}_{\text{Y}}}{{\text{d}}_{\text{M}2}}\)<0. The latitude of the population gravity center decrease (moving south), that is, toward the central cities.

In conclusion, when the population of peripheral cities remains unchanged, but the population size of central cities increases (or when the population size of central cities increases more than that of peripheral cities), the population gravity center moves to the central cities in latitude.

2. To judge the change of population center of gravity in the mature urban agglomeration, we assume that the population of the central city \({M}_{1}\) remains unchanged, find the derivation of \({M}_{2}\).\({M}_{1}>0,\) \({M}_{2}\)>0.

(1) Longitude.

If \({\text{X}}_{1}\)>\({\text{X}}_{2}\) (the central city is to the east of the outer city), \(\frac{{\text{d}}_{\text{x}}}{{\text{d}}_{\text{M}2}}\)<0. The longitude of the population gravity center decrease (moving westward), that is, toward the peripheral cities. If \({\text{X}}_{1}\)<\({\text{X}}_{2}\) (the central city is to the west of the outer city), \(\frac{{\text{d}}_{\text{x}}}{{\text{d}}_{\text{M}2}}\)>0. The longitude of the population center of gravity increase (moving eastward), that is, toward the peripheral cities.

To sum up, when the population of the central city remains unchanged, and the population size of the peripheral city increases (or when the increase in the population size of the surrounding cities is more significant than the increase in the population size of the central cities), the longitude of the population gravity center moves to the peripheral cities.

(2) latitude

If \({Y}_{1}\)>\({Y}_{2}\) (the central city is to the north of the outer city), \(\frac{{\text{d}}_{\text{Y}}}{{\text{d}}_{\text{M}2}}\)<0. The center of gravity of the population decreases latitude (moving southward), moving toward the outer cities. If \({Y}_{1}\)<\({Y}_{2}\) (the central city is to the south of the outer city), \(\frac{{\text{d}}_{\text{Y}}}{{\text{d}}_{\text{M}2}}\)>0, the center of gravity of the population increases latitude (moving northward), moving toward the outer cities.

In conclusion, when the population size of the central city remains unchanged, and the population size of the peripheral city increases (the increase of the population size of the peripheral city is greater than that of the central city), the latitude of the population gravity center moves to the peripheral cities. The above reasoning process leads to the second conclusion:

Conclusion 2: The latitude and longitude changes of the population gravity center reveal the characteristics of the spatial distribution of the population in the urban agglomeration. In the initial stage of urban agglomerations, the population size of the central city increased more than that of the surrounding cities, and the population gravity center shifted to the central city. In the mature period of urban agglomeration development, the population growth of surrounding cities is greater than that of central cities, and the population gravity center shifts to surrounding cities.

## 3.3 Changing in equilibrium degree of spatial population distribution in urban agglomerations

Based on four hypotheses and existing research, this paper demonstrated the equilibrium changes of the spatial population distribution of urban agglomerations at different development levels. In the initial state of urban agglomeration development, the productivity level is low, the economy is underdeveloped, the urbanization level is relatively low, and the city's population is generally in a low-level equilibrium state (XiaoJian Li, (2018). Economic Geography. (3rd Ed). Chapter 8). Some cities develop rapidly through external stimulation or long-term accumulation 42. Various favorable factors have driven the population to shift to advantageous cities, and central cities have gradually formed 43,44. The formation of central cities in urban agglomerations broke the balance of the original urban agglomeration population spatial structure. Specifically, the population gap between central and other cities continues to widen. The concentration of population in central cities has caused congestion in central cities, further increasing land costs. A series of urban problems have emerged, ultimately reducing the net utility value of central cities 6. The net utility value of the central city is gradually more minor than the net utility value of the peripheral cities. According to Condition 1, the population begins to migrate to the surrounding towns, the population gap between the central city and the peripheral cities is gradually reduced, and the spatial population distribution of the urban agglomeration tends to be balanced.

Population spatial balance in urban agglomerations is a complex process of urban self-regulation caused by people's pursuit of utility maximization20. The direction of population migration is related to the relative gap between cities at different development levels in urban agglomerations. The population flows in cities with high development levels and immense net utility value, ultimately affecting the spatial equilibrium of people in urban areas agglomerations.8,26. The above reasoning process leads to the third conclusion:

Conclusion 3:In the early stage of urban agglomeration development, the degree of spatial population equilibrium gradually decreases with the development of urban agglomeration. When urban agglomerations reach the mature stage, the degree of spatial equilibrium of the population increases gradually with the development of urban agglomerations.