Modeling urban land density with Gaussian and inverse S functions by analyzing urban expansion in Zhengzhou City

Urban land density analysis is central to urban expansion research. Different mathematical models have unique strengths and limitations in exploring land density, yet there is little systematic comparison between them. This paper addresses this gap by analyzing land use data from 2000 to 2020 in Zhengzhou City, using two models: the Gaussian function and the inverse S function. It quantifies changes in land density and expansion trends, while also comparing the models' applicability across expansion directions. The findings are as follows: (1) Both models show strong overall fitting abilities. However, the Gaussian model offers a more detailed understanding of urban expansion due to its multi-dimensional parameter settings. (2) In determining urban boundaries and zoning, the Gaussian model is more convenient, reflecting wave-like diffusion patterns that better match actual urban growth trends. (3) In terms of expansion direction, the urban compactness index reveals spatial heterogeneity. The inverse S function performs well, showing a clear compactness trend, while the Gaussian function's fitting degree is weaker, with less distinct compactness patterns. Overall, these two models complement each other in analyzing urban land density, unveiling the form and mechanisms of urban expansion, and providing valuable insights for sustainable urban development.