  Motivated by the need to understand the factors driving gentrification, we
introduce and analyze two simple dynamical systems that model the interplay
between three potential drivers of the phenomenon. The constructed systems are
based on the assumption that three canonical drivers exist: a subpopulation
that increases the desirability of a neighborhood, the desirability of a
neighborhood, and the average price of real estate in a neighborhood. The
second model modifies the first and implements a simple rent control scheme.
For both models, we investigate the linear stability of equilibria and
numerically determine the characteristics of oscillatory solutions as a
function of system parameters. Introducing a rent control scheme stabilizes the
system, in the sense that the parameter regime under which solutions approach
equilibrium is expanded. However, oscillatory time series generated by the rent
control model are generally more disorganized than those generated by the
non-rent control model; in fact, long-term transient chaos was observed under
certain conditions in the rent control case. Our results illustrate that even
simple models of urban gentrification can lead to complex temporal behavior.
