We study an evolution cross-diffusion problem with mutualistic Lotka-Volterra
reaction term to modelize the long-term spatial distribution of labor and
capital. The mutualistic behavior is deduced from the gradient flow associated
to profits maximization. We perform a linear and weakly nonlinear stability
analysis and find conditions under which the uniform optimum of profits becomes
unstable, leading to pattern formation. The patterns alternate regions of high
and low concentrations of both labor and capital, which may be interpreted as
cities. Finally, numerical simulations based on the weakly nonlinear analysis,
as well as in a finite element approximation, are provided.Comment: Keywords: Cross-diffusion, mutualism, Turing instability, weakly
nonlinear analysis, labor, capital, cit
