Understanding the Dynamic Mechanism of Urban Land Use and Population Distribution Evolution from a Microscopic Perspective

Abstract

With the advancement of urbanization, the contradiction in the man–land relationship becomes more and more difficult to ignore. Investigation of the change in urban land use, population distribution and its mechanism can provide powerful assistance for urban planning. Since the changes in urban land use and population distribution is a complex process with spatial heterogeneity, the current methods for describing them are still lacking in both interpretability and spatial differences. In this paper, we combine the expansion phenomena of urban land use and population distribution with the heat equation to understand the mechanism. The particle swarm optimization (PSO) algorithm is used to identify the diffusion coefficient to obtain the diffusion law in the city’s development. In this way, the diffusion coefficient identified from the data is directly associated with urban changes. The mechanism of changes in urban land use and population distribution can be explained with the diffusion equation and the diffusion coefficient. Our model is first validated on land use and land cover data, followed by further refinement of the spatial differences in the artificial impervious surface data. The experiment’s results imply that by applying the model to the population data, the model’s generalization ability has been significantly improved.

1. Introduction

As the main space for human life, the city provides vital basic needs for human survival [1]. One of the phenomena is that with the growth of the urban population, the scope of the city also expands. However, the upper limit of urban area development is constrained, even though the urban population is still increasing year by year without a predictable end. This contradiction makes higher demands on the population-bearing capacity of cities [2,3]. Meanwhile, urban sprawl has resulted in some unfortunate expansion effects, such as environmental pollution [4,5], land desertification [6] and deforestation [7]. All of this further exacerbates the contradiction between man and land. Understanding the temporal and spatial evolution of urban land use and population distribution will help with properly planning urban development [8,9], which would help alleviate the contradiction between man and land and realize the city’s sustainable development [10].

Although the changes in urban land use and population distribution are limited by objective factors such as the level of urban development and natural conditions, the trend of diffusion is still a major pattern of change [11,12]. Various models have made efforts to understand and simulate urban sprawl. These models can be divided into black box models, white box models and gray box models.

A typical approach in black box models is machine learning. Machine learning methods generally do not assume the distribution of data, so they are better for handling nonlinear relationships in the data [13]. Different machine learning methods have similarities in acquiring urban evolution rules. Rule mining is usually reduced to a classification problem. Although the classification rules of urban evolution obtained by machine learning can help with prediction, the model itself is a black box model, which makes explaining the change mechanism difficult [14,15].

The methods of the white box model are cellular automata (CA) [16], the agent-based model (ABM) [17,18], fractals, complex networks and so on. In CA, the cell state is changed by local simple rules, thereby producing the macroscopic result of land use change [19,20,21]. Similar to the CA, the ABM continuously makes corrections through the interaction between agents and the whole changes of the overall situation. Through this continuous adjustment, the model finally achieves a simulation of the evolution of urban land. Both the CA and the ABM attempt to understand the development process of urban elements from a microscopic perspective. Fractal theory gives a new way of understanding the evolution of urban development from the perspective of complex systems [22,23,24] and shows the unified spatial distribution characteristics of urban forms at different spatial scales [25,26,27]. Inspired by complex systems, some scholars take the land use types as nodes, and the transformation relationship between types is regarded as the edges. Based on this, they build a complex network to analyze the stability of the evolution of urban land use spatial patterns [28,29]. Although this type of model can observe the evolution of the city as a whole, it is still insufficient for describing the inter-city relationship.

Between the above two types of models is the gray box model, which contains system dynamic (SD) models and complex adaptive systems. The SD model is based on system theory, information theory and cybernetics, and it studies the urban land use system from a macro perspective [30,31,32]. The model can reflect the interaction between the structure, function and various behaviors of the land use system, but it requires expert knowledge and lacks of the depiction of spatial dynamics [33]. In the complex adaptive system model, the transformation rule is determined by the current state and the state of its environment. These transformation rules or behavior rules which are set artificially directly determine the quality of the model’s simulation.

There are also some scholars who combine the advantages of the black box model and the white box model to construct a new model to describe the development and changes in urban land and population. Those models use machine learning methods to determine the conversion rules, which are used in the CA model to avoid the uncertainty caused by artificial settings [20]. For example, the random forest algorithm has a good tolerance for outliers and noise and is not prone to overfitting, so it can be used to improve the simulation performance of the CA model [34]. The artificial neural network (ANN) is one of the most widely used machine learning methods for mining CA model transformation rules. Liu et al. introduced the adaptive inertia coefficient and roulette competition mechanism into the CA model and proposed the FLUS model by combining an ANN and SD [35]. With examples, it was proven that the FLUS model has high accuracy in the prediction of land use change.

Incorporated by specific physical processes, Blumenfeld pointed out that according to the fluctuation of population density in a 100-year cycle, the area of the most remarkable growth will move outward from the center of the city in a specific cycle [36]. Therefore, the wave theory can be used to describe the development of cities. Newling [37] used the wave profile as a theoretical basis to identify the population densities of cities at different stages of development. In recent years, physical models that have received much attention related to cities include percolation models [38,39]. In the percolation model, by setting different thresholds, urban spatial forms are evolved, and then urban spatial texture boundaries and spatial hierarchy relationships can be obtained. In addition, some other physical models, such as the diffusion equation, are also used in many city-related performances. The earliest studies use a diffusion equation to describe the typically complex physical processes related to cities, focusing on the characterization of the diffusion process of pollutants in cities [40]. Jin et al. extended the application of the diffusion equation to urban land and used the heat equation to describe the change in land use or land cover in the city [41].

Given the advantages and disadvantages of the white box and black box models mentioned above, we propose partial differential equations to simulate the changes in urban land use and population distribution in this paper. First, since the diffusion equation is a theoretical equation derived from statistical physics, the partial differential equation can provide land use and population distribution changes with a good explanation. At the same time, the diffusion equation contains the nonlinear terms of the first-order partial derivative in time and the second-order partial derivative in space, which can characterize the nonlinear process in space and time. Our model in each micro unit on the specific parameters differs from the white box model. Therefore, the results of the model have spatial heterogeneity. In the end, the model equation generally has a fixed form to ensure the overall consistency of certain research areas.

The remainder of this paper is organized as follows. Materials and Methods describes the preliminary information and the problem definition. Results presents the study area and data and the experiment’s results, including the land use and land cover data, the artificial impervious data and the population density data. The discussion and conclusions are summarized in Discussion and Conclusions.

2. Materials and Methods

In this section, the definition of micro units will be shown first. After introducing the overall process of the algorithm, we sequentially explain the definitions of the related variables involved, auxiliary algorithm and problem formulation.

For the micro unit, referring to the CA and multi-agent models, each basic unit goes through a nonlinear change process under the influence of multiple factors and finally presents a complex evolutionary shape as a whole. Each cell and agent can be viewed as a microscopic unit. The interaction between these units constitutes the dynamic process of the whole system. At the same time, the spatial inconsistency of this force eventually forms a system with spatial heterogeneity. As for the evolution of urban land use and population distribution, from a microscopic point of view, the heterogeneous micro-dynamic process of spatial distribution makes urban land use and population distribution present intricate characteristics in spatial form. To further illustrate this microscopic process, we take the impervious surface data [42] of Guangming District in Shenzhen as an example, as shown in Figure 1. The data are raster data, and each raster can be regarded as a microscopic unit. With the change in time, the impervious surface of the Guangming District continues to spread outward. This diffusion process, shown in a three-dimensional space, is a distribution of the ups and downs, and every micro unit of change is affected by the surrounding neighborhood.

After the concept of the micro unit was clarified, we adopted different methods to calculate the micro unit concentration value for different data types. Then, the relative position in the diffusion process was determined according to the concentration value. Once the relative positions in the diffusion equation were determined, the PSO algorithm would be used to identify the coefficients of the diffusion equation. Finally, we obtained the diffusion coefficient distribution for each data type. The flow of the whole method is shown in Figure 2.

The following section shows the definition of the concentration and diffusion directions. Then, we are given an introduction to the PSO algorithm. The problem formulation subsection demonstrates the processing flow of the diffusion equation and the final objective function.

2.1. Preliminaries

Diffusion is a phenomenon in which a substance moves along a concentration gradient. The concentration difference moves the element from a high-concentration area to a low-concentration place until evenly distributed. As an example of the urban development process, the built-up areas in cities have been increasing. This phenomenon can be seen as the concentration of the built-up area continuously growing within a certain space. For China, although policies guide the direction and trend of urban development, we can still observe the expansion of artificial impervious surfaces or built-up areas from high-concentration (central city) to low-concentration (suburbs) areas. Thus, there seems to be a diffusion in the process of urban development, which at the same time follows a particular law of diffusion.

2.1.1. The Definition of Concentration

Since the diffusion model delimits the direction of the substance concentration over time, we needed to define the concentration of the substance corresponding to the data. The different clarifications for labeled data and unlabeled data are as follows:

• Labeled data: The labeled data refers to the data in the grid that represent a particular type, and the specific value of the label does not convey any meaning; it only symbolizes a specific type. For example, in the land use and land cover data, there is a correspondence between the numbers and categories, which is shown in

  Table 1. The correspondence between value and label in the land use and land cover data.

  Value	Label
  1	Forest
  2	Grassland
  3	Cultivated land
  4	Built-up

  . The value represents a specific type which can be changed freely, as this adjustment does not affect the characteristics of the data itself. For this kind of data, the key is to ensure the uniqueness of this mapping, since the number itself does not have a numerical meaning. In this case, we take the proportion of the grid of the same type as the central grid in the spatial neighborhood as the concentration value of the grid. For example, for the spatial neighborhood of $3\times 3$, four grids in this spatial neighborhood are the same type as the central grid. Then, the concentration of the center grid is $\frac{4+1}{9}=0.5556$, with four decimal places.
• Unlabeled data: The opposite of labeled data is unlabeled data. The unlabeled data refer to the data in the grid that represent a specific numerical meaning. For example, in population data, the value on each grid represents the region’s total population in the corresponding year. For this kind of unlabeled data, the specific value on the grid can be directly defined as the concentration on the grid. However, the order of magnitude of such data is usually gigantic, and the data’s measurement units are also inconsistent. Therefore, this kind of data needs to be normalized to eliminate this influence.

2.1.2. The Definition of Diffusion Direction

During the diffusion process, the concentration gradient being at different spatial positions causes the substance to change over time. These inconsistent concentrations also objectively define the diffusion direction of the substance and affect the diffusion process objectively. When we use the diffusion equation to identify the law of urban diffusion, it becomes crucial to directly define the direction of material diffusion from the spatial geographic location.

After we obtained the material concentration of each micro unit, the difference in the material concentrations of different diffusion units appeared, so the diffusion direction of the material could show up. Therefore, in the algorithm, we define that within the neighborhood of the central diffusion unit, the diffusion unit with the highest concentration diffuses to the diffusion unit with the lowest concentration. In other words, the position with the highest concentration is the previous position in the diffusion process, and the position with the lowest concentration is the position where the diffusion process will be affected in the future.

2.1.3. The PSO Algorithm

The PSO algorithm is an intelligent calculation method for solving optimization problems. The core idea of the PSO algorithm is to update the particles by tracking the current local and globally optimal solutions of the particles. When the termination condition is reached (after a certain number of iterations or the latter meets an empirical error), the current local optimal solution is regarded as the optimal solution to the problem. Assuming that the size of the particle swarm is N, and the velocity and position of the particles j are represented by the N dimensional vectors $V_j$ and $P_j$, respectively, the iterative equation of the PSO algorithm is

$$\left\{\begin{array}{cc}\hfill V_j(e+1)=& \omega V_j\left(e\right)+c_1{r}_1(p_j\left(e\right)-P_j\left(e\right))\hfill \\ \hfill & +c_2{r}_2(p_g\left(e\right)-P_j\left(e\right))\hfill \\ \hfill \left|V_k(e+1)\right|\le & V_{max},k\in [1,N]\hfill \\ \hfill P_j(e+1)=& V_j(e+1)+P_j\left(e\right).\hfill \end{array}\right.$$

(1)

where $V_j\left(e\right)$ is the velocity of the particle j in generation e, $P_j\left(e\right)$ is the position of the particle j in generation e, $p_j\left(e\right)$ is the individual historical optimal position of the particle j in generation e, $p_g\left(e\right)$ is the historical optimal position of the particle swarm in generation e. w is the weight of the inertia, $c_1$ is the cognitive coefficient, $c_2$ is the social coefficient and $r_1$ and $r_2$ are random numbers in $[0,1]$. The updated rules of $p_j\left(e\right)$ and $p_g\left(e\right)$ are

$$p_j\left(e\right)=\left\{\begin{array}{cc}{p}_i(e-1),\hfill & \mathrm{if}f\left(p_j\left(e\right)\right)>f\left(p_j(e-1)\right),\hfill \\ P_j\left(e\right),\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(2)

and

$$p_g\left(e\right)=\mathrm{argmin}\{f\left(p_j\left(e\right)\right)|j=1,2,\dots ,n\}$$

(3)

Here, $f\left(p_j\left(e\right)\right)$ represents the fitness value of $p_j\left(e\right)$. The smaller the fitness value, the better the corresponding fitness for $p_g\left(e\right)$.

2.2. Problem Formulation

After clarifying the definitions of the concentration and diffusion direction in the model, the following section focuses on the formula processing procedure of the model, the determination of the objective function and the identification process of the coefficients.

For a city, according to the first law of geography [16], the spatial auto-correlation contained in this particular law shows the magnitude of the neighborhood effect. When focusing on the observed changes in the key elements of the city, we divided the data into grids, and each grid could be regarded as a basic diffusion unit. In terms of measuring the diffusion process of the current central grid, since each category in the key elements tends to occupy a larger area over time, this diffusion process is affected by its growing demand and the inhibition of diffusion of the surrounding areas. Based on these considerations, we consider the following diffusion equation:

$$\frac{\partial \Phi (x,t)}{\partial t}=a(x,t)\frac{{\partial }^2\Phi (x,t)}{\partial x^2}+F(x,t),$$

(4)

where x represents the spatial position of the current diffusion unit, t means the time of diffusion. $\Phi (x,t)$ is the state of the diffusion unit at the position x at the time t and $F(x,t)$ plays the part of neighbors’ influence. By combining the determination of species concentration and diffusion direction for each grid, we obtained the correspondence shown in the Figure 3.

2.2.1. The Discretization of the Diffusion Equation

Since the data of the key elements of the city obtained are discrete, so is the computer when they are calculated. Therefore, we needed to discretize the diffusion equation given above (Equation (4)). For the first-order partial derivatives and the second-order partial derivatives in the equation, we used the central difference method for discretization, and then the following relation was obtained:

$$\begin{array}{cc}\hfill \frac{\Phi (x,t+1)-\Phi (x,t)}{\Delta t}=& F(x,t)+\frac{a(x,t)}{{(\Delta x)}^2}[\Phi (x+1,t)\hfill \\ \hfill & -2\Phi (x,t)+\Phi (x-1,t)].\hfill \end{array}$$

(5)

By rearranging the above equation, we find that

$$\begin{array}{cc}\hfill \Phi (x,t+1)=& F(x,t)\Delta t+\frac{a(x,t)\Delta t}{{(\Delta x)}^2}[\Phi (x+1,t)\hfill \\ \hfill & -2\Phi (x,t)+\Phi (x-1,t)]+\Phi (x,t).\hfill \end{array}$$

(6)

Let $D=\frac{a(x,t)\Delta t}{{(\Delta x)}^2}$. Then, we have

$$\begin{array}{cc}\hfill \Phi (x,t+1)=& (1-2D)\Phi (x,t)+D(\Phi (x-1,t)\hfill \\ \hfill & +\Phi (x+1,t))+F(x,t)\Delta t.\hfill \end{array}$$

(7)

In this way, we completed the discretization process of the diffusion equation, making it convenient for computer simulation calculation. At the same time, each item in the equation after discretization can correspond to the actual observed data. For example, when we collected the observational data in 2013 and 2015, combined with Equation (7), we could obtain the following relation and iterative formula:

$$\begin{array}{cc}\hfill \Phi (x,2015)=& [1-4a(x,2013\left)\right]\Phi (x,2013)+2F(x,2013)\hfill \\ \hfill & +2a(x,2013)[\Phi (x-1,2013)+\Phi (x+1,2013\left)\right],\hfill \end{array}$$

(8)

where $\Delta t=2$ and $\Delta x=1$.

2.2.2. Solving for the Diffusion Coefficient

We knew that the diffusion coefficient is closely related to the data changes between different years, in line with the previous content. Since we constructed the expression relationship between the data, our task transformed into finding the diffusion coefficient that satisfied the expression based on the observation data. For the relationship between the two years we derived, we used PSO to identify the unknown diffusion coefficient. Naturally, Equation (4) became our objective function in the PSO algorithm.

More precisely, according to the discretizing objective function in Equation (7), only the coefficient D is unknown. Furthermore, only the variable $a(x,t)$ is unknown because both $\Delta t$ and $\Delta x$ can be obtained from the data. That aside, we can also find the corresponding ${\Phi }_{t+1}^x,{\Phi }_t^x,{\Phi }_t^{x-1},{\Phi }_t^{x+1}$ from the data based on the definition of the diffusion direction. Then, for each iteration e, the fitness function for each particle j in the PSO algorithm was set as follows:

$$f\left(e\right)=\mathrm{argmin}\parallel \Phi (x,t)-{\Phi }^e{(x,t)\parallel }_2^2.$$

(9)

3. Results

This section will be divided into three parts to introduce our experimental results. The first is our study area and data, and then we validate our algorithm on the land use and land cover data. Next, we show the results with the artificial impervious area data. The last part is our experiments on the population density data.

3.1. Area and Data

Shenzhen is a coastal city in southern China adjacent to Hong Kong. It is located south of the Tropic of Cancer between ${113}^{\circ }{43}^{\prime }$ E to ${114}^{\circ }{38}^{\prime }$ E and ${22}^{\circ }{24}^{\prime }$ N to ${22}^{\circ }{52}^{\prime }$ N. The administrative region in Shenzhen city is shown in Figure 4. Since the establishment of Shenzhen in 1979, Shenzhen’s population has grown 40-fold from 310,000 in 1979 to 13.43 million in 2019. The regional GDP jumped from CNY 270 million in 1980 to CNY 2.69 trillion in 2019, with an average annual increase of 20.7%, ranking third in Mainland China and top five in Asian cities. As of 2017, the developed land in Shenzhen has increased from 27 ${\mathrm{km}}^2$ to 946 ${\mathrm{km}}^2$, with an average annual increase of 24 ${\mathrm{km}}^2$ [43]. Shenzhen took 40 years to complete the transformation from a small fishing village to a mega city, making it a very representative case.

Since our model was a microscopic model, the higher the spatiotemporal resolution of the data we used, the better. On the other hand, the Landsat data are multi-spectral, multi-resolution and multi-temporal, so the monitoring of LULC changes and urban expansion is mainly based on Landsat data. Combining our study area and the free availability of data, we adopted the land use and land cover and the artificial impervious area data products of Landsat data, as well as the population density distribution data provided by WorldPOP:

• Land use and land cover data: The land use and land cover data are taken from data that have been publicly shared on the Internet. The time range is from 1988 to 2015, and the time interval ranges from 2 to 6 years. The spatial resolution of the data is 30 m [44]. Land use and land cover includes six categories: forest, water, grassland, cultivated land, built-up area and bare land.
• Artificial impervious area data: The artificial impervious area data can be downloaded directly from the working website of Tsinghua University. The spatial resolution is 30 m, and the mean overall accuracy is higher than 90% [45]. This accuracy for the data in the study of global scales is acceptable. The artificial impervious area data can be downloaded directly from the working website of Tsinghua University.
• Population density data: The population data were taken from data published by WorldPOP. The time interval of the data is one year from 2011 to 2020, and the spatial resolution of the population data is 100 m. The population data generated during the current study are available in the WorldPOP repository [46].

3.2. The Case Study

In this section, we set up three experiments, and the spatial neighborhood used in the experiment was the classical Moore neighborhood. First, the diffusion coefficient of each district in Shenzhen was identified from the built-up area in the land use and land cover data. With this, we used the artificial impervious surface data to refine the diffusion coefficient to each grid cell. Finally, we identified the diffusion coefficient in the unlabeled data and extended the whole model to different data.

3.2.1. The Land Use and Land Cover Data

In this part, for each year’s data, we take the built-up area concentration distribution in each district of Shenzhen as a subset and investigate the diffusion trend of built-up areas in different districts in this subset.

Based on the previous experimental principles and experimental settings, we identified the diffusion coefficient of the built-up area in the land use and land cover data of Shenzhen from 1988 to 2015. Figure 5 shows the change in the diffusion coefficient in the built-up area of Shenzhen. From the results, the changes in the diffusion coefficient in Shenzhen could be roughly divided into three categories: the first category included the Dapeng District, the second category included the three districts of Yantian, Pingshan and Luohu, and the remaining six districts were classified into the third category.

We mapped the spatial location of each district in Shenzhen to obtain Figure 6. From the figure, it can be seen that the diffusion coefficient of the built-up area of Shenzhen decreased from east to west. Among them, the built-up area of Dapeng New District always had the highest diffusion coefficient, which indicates that the built-up area of Dapeng District was in a strong diffusion phenomenon during the study period. This situation is consistent with the development of various districts in Shenzhen. Dapeng District, as a new district, embodies strong vitality and development potential.

In order to clearly show the difference in the diffusion coefficients in those three variations, we separately extracted the changes in the built-up areas of the four districts of Shenzhen. Among them, Pingshan District was the representative of the second category. Since the third category contained more administrative regions, we extracted Baoan District, which covers the largest area, and Futian District, which had the lowest average diffusion coefficient performance, as representatives. In Figure 7, we can see the different diffusion patterns in each district. The higher the diffusion coefficient, the clearer the characteristics of the built-up area’s outward expansion, and the smaller the overall disturbance (i.e., the less cross-diffusion between each other), and Dapeng New District is given as an example. The lower the diffusion coefficient, the less obvious the overall outward expansion forms, such as in Baoan District and Futian District. Between these two types, the overall outward expansion of Pingshan District was very conspicuous. However, it was often accompanied by cross-diffusion in the region, and the uniform outward diffusion direction was not apparent from the overall spatial perspective.

3.2.2. The Artificial Impervious Area Data

We carryied out a spatial refinement of the diffusion coefficient based on the extracted artificial impervious area in Shenzhen. Different numbers represent newly added impervious surface areas in different years in the artificial impervious area data. The correspondence between this number and the year is available on the data download web page. According to the definition of label data concentration in the previous section, we obtained the concentration values for the impervious surface in a different year.

In this part, we further refine the identification of the diffusion coefficient to each spatial grid and explore the variation in the impervious surface concentration in each grid. Then, diffusion coefficient identification down to the grid level is carried out with the diffusion equation we constructed. Moreover, we make a one-step prediction from the diffusion coefficient in Guangming District specifically. Based on the identified diffusion coefficients from 1985 to 2017, the spatial distribution of the artificial impervious area in 2018 is predicted. The prediction results are shown in Figure 8.

The results demonstrate that the expansion of the artificial impervious area was more likely to occur in the region where the concentration changed sharply. Furthermore, when the predicted concentration in 2018 exceeded 0.95 or was lower than −0.75, it could be considered that expansion of the impervious surface occurred with an overall accuracy of 58.53% correspondingly. For the diffusion equation, when the diffusion coefficient is negative, this indicates that the diffusion process in this region is unstable and may be accompanied by sudden changes. Therefore, it is still understandable that the increase in the impervious surface occurs in the region where the diffusion coefficient is negative. At the same time, this result also shows how effective the model is. Because the increase in impervious surfaces resulted from a combination of factors, the historical trend of the new impervious surface represents only a part of the driving force. In another way, the experimental results show that nearly 60% of the new areas were still caused by historical inertia during the expansion of the impervious surface in the following year. This means that the new impervious surface in Guangming District still has an unavoidable time delay under the implementation of the policy.

According to the prediction results for Guangming District, we found that the spatial fluctuation of the corresponding diffusion coefficient was more conspicuous in the area where the impervious surface changed drastically and densely. We then compared the overall artificial impervious diffusion in Shenzhen in 2018 with the predicted impervious surface concentration value in 2018. The results are shown in Figure 9. We also found that even in the whole of Shenzhen, the conclusion that impervious surface expansion is more likely to occur in places with sharp concentration gaps still held.

3.2.3. The Population Density Data

Diffusion coefficient identification was also conducted with the population density data of Shenzhen, and the results are shown in Figure 10a. The change in the population density and diffusion trends of Shenzhen were relatively consistent, but the strength of the diffusion capacity was different. On the whole, the diffusion coefficient of each district in Shenzhen was similar to the slope of the population density change in Shenzhen from 2011 to 2020 (see Figure 10b). In other words, the diffusion coefficient can reflect the extent of population growth to a certain extent: the larger the diffusion coefficient, the more obvious the population growth.

To further explain the diffusion coefficients, we resampled the population density data in conjunction with the zoning of the streets. The 10 districts of Shenzhen were predicted to obtain the diffusion coefficient distribution in each district. As a reference, we found the data of the seventh population census in Shenzhen, which can be downloaded from the public website. Thus far, only Futian, Guangming, Longhua, Luohu, Nanshan and Pingshan Districts have published the census bulletin at the street level in Shenzhen. Therefore, the comparison of our results is focused on these six districts in Shenzhen.

In Figure 11, the blocks circled by the black lines are the blocks with apparent population growth in the districts of Shenzhen, as given in the Shenzhen Seventh Census Bulletin. The pictures from (a) to (e) correspond to Futian District, Guangming District, Longhua District, Nanshan District, and Luohu District. It can be seen from the figure that in the above five districts, the regional diffusion coefficients with population growth were generally higher. This is consistent with our definition for the diffusion coefficient, which further illustrates the effectiveness of the diffusion coefficient defined in this article in actual situations.

4. Discussion

This paper started from the main aspects of urban development and changes such as land use and land cover, the artificial impervious area and population density. According to the changes in the diffusion coefficients of different data, we explored some laws of the development and changes in cities. For the land use and land cover data, it can be seen that the diffusion coefficient of the built-up area of Shenzhen decreased from east to west. Among them, the built-up area of Dapeng New District always had the highest diffusion coefficient, indicating that the built-up area of Shenzhen’s Dapeng New District was in a strong diffusion phenomenon during the study period. This situation is consistent with the development of various districts in Shenzhen. Dapeng New District, as a younger district, embodies strong vitality and development potential.

In the artificial impervious area data, we found that the new impervious surface was usually accompanied by a sharp change in the diffusion coefficient. Without considering other factors, and only based on the historical impervious surface data in Guangming District, it was possible to simulate nearly 60% of the new impervious surface in the next year. This also means that the impact of historical trends on Shenzhen’s impervious surface is inertial. Since the addition of impervious surfaces in Shenzhen is usually determined by policy, this also shows that the implementation of the policy is delayed to a certain extent.

For the population density data, overall, the results showed that the population diffusion coefficients were larger in regions where the population density increases were more pronounced. From the five jurisdictions’ situation, compared with Shenzhen’s central urban area, the diffusion coefficient of the non-central urban area was generally higher than that of the central urban area. This phenomenon means that Shenzhen’s current population growth potential is mainly distributed in non-central urban areas.

All in all, from the experimental results of the three data, the population density, built-up area and artificial impervious area had the characteristics of outward diffusion. Moreover, the growth rates of the built-up areas in different regions were different, and the growth rate of the western region was significantly faster than that of the eastern region. The simulation and prediction of the impervious surface show that there is a delay in the actual implementation of policy planning, which will provide some assistance for subsequent policy designation. Since the planning of land policy is always based on people’s needs, the diffusion of population density can help the overall evaluation of the policy.

5. Conclusions

In this article, we proposed a diffusion equation to simulate the evolution of urban elements from a microscopic perspective. Throughout this work, we first explained how to establish the diffusion model. According to the experiments of the land use and land cover data, we verified that it was feasible to explain the evolution of key elements in cities through diffusion theory. The diffusion coefficients identified by the PSO algorithm can represent the differences in the expansion of built-up areas in different urban areas and reflect the expansion modes in micro units. Next, we expressed the spatial differences more finely in each grid, in which the diffusion coefficient was further refined in the impervious surface data. From the diffusion coefficients of each micro unit, we found that regions where the diffusion coefficient changed drastically were usually accompanied by new amplifications. Finally, the model’s generalization performance was improved by applying it to population density data. The results show that our model worked well not only with the labeled data but also the unlabeled data.

While the results show our model’s effectiveness and good generalization ability, some aspects still need improvement. During the diffusion process caused by the concentration gradient, the whole substance will shift to a certain extent, and this process is called convection. For this change, our model only uses the second-order partial derivative to consider the simple diffusion process and does not describe the convection process that occurs during the diffusion process. Simultaneously, the interaction between multi-source data was not considered. Although the interaction between those data may not be straightforward, the interaction cannot be ignored. The diffusion process of urban elements under these multi-data will be our future research direction.