The evolution of urban areas plays a major role in crafting public policy and spurring investment. Gentrification, involving the “transformation of a working-class or vacant area of the central city into middle-class residential or commercial use”, is inherently intertwined with neighborhood ascent and descent. We propose an agent-based model to study the existence and dynamics of gentrification, centered on a two-dimensional lattice with wealth and an inherent amenities interacting via particular feedback processes. Various parameter regimes beget distinct sets of dynamics, including that of wealth hotspots – as consistent with empirical observation. We then derive a continuum model of partial differential equations (PDE) from the agent-based model and analyze the instability regime of the continuum system; the resultant regime agrees with observations of simulation of the discrete model.