Transport and concentration of wealth: Modeling an amenities-based-theory

We derive a reaction-advection-diffusion equation based framework for analyzing the movement of wealth in urban environments. Gentrification is a core issue that affects many urban areas, and the dynamics of such are not fully understood. To understand the process using a few physically relevant variables, we develop an approach to model the interplay between local amenities and wealth from existing analyses of factors influencing gentrification. We conduct a linear stability analysis on model parameters that results in spatially homogeneous solutions and determine directions for parameter changes to induce instabilities. From these parameters, we determine quantities that lead to the formation of areas of wealth and amenity concentration in the long term on bounded and unbounded domains. We present a global bifurcation result in two dimensions, leading to the existence and stability of non-constant equilibrium solutions, which represents solutions with wealth and amenity hotspots. Finally, we present a theory for discerning the one-dimensional pattern formation in the transition between stability and instability and verify numerically through a weakly nonlinear analysis. The analysis provides a promising framework for further verification using publicly available geospatial data on the relevant variables.