A 2-D Diffusion-Based Population Dynamics Model Using Bicubic B-Spline Interpolation

Abstract

Understanding how urban populations change over time and across space is important for studying city growth. Traditional models often focus only on population growth over time and ignore spatial effects, which limits their accuracy. In this study, a 2-D diffusion-based model is developed to describe population changes in both time and space using an analogy with the 2-D heat equation. The model is solved numerically using bicubic B-spline interpolation for space and the Crank–Nicolson scheme for time. This approach provides smooth spatial results, stable calculations and accurate time integration. The model is applied to Surat city, India, divided into 25 subregions, using census-based initial conditions and regression-based boundary conditions. Predicted populations for 2011 closely match actual census data, and error measures such as Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) show good predictive performance. The results show that the model can capture patterns of population movement and growth, making it useful for urban planning. Future work may include socio-economic factors and variable spatial coefficients to improve local accuracy.