  This study presents a computational simulation exploring the complex
interactions between population density and economic factors over a 100-year
period. Inspired by the Keller-Segel model, traditionally applied in biological
contexts, my model adapts this framework to analyze urban and economic
dynamics. The simulation employs two coupled partial differential equations to
represent the evolution of population density and money concentration in a
hypothetical region. Population density is initially uniform, while money
concentration begins with a random distribution. The model integrates diffusion
processes for both population and money, coupled with a chemotactic response of
the population towards areas of higher economic activity. Over the course of
the simulation, we observe the emergence of distinct spatial patterns:
population clusters forming around economic hubs and the development of wealth
concentration in certain areas. These patterns highlight the mutual
reinforcement between population density and economic factors. The study
provides insights into the dynamics of urban growth, economic disparities, and
resource distribution, offering a simplified yet powerful lens through which to
view complex socio-economic systems. My findings have implications for urban
planning and policy-making, especially in understanding the long-term evolution
of cities and economic centers.
