An economic cross-diffusion mutualistic model for cities emergence

Abstract

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.

Economic Structures and Dynamics: A Morphogenetic View

“…Turing bifurcation is conspicuous by its relative absence in the context of models dealing with economic dynamics. Exceptions to this include Fujita et al [1999], Akamatsu et al [2012], de Córdoba andGaliano [2020] and Velupillai [2005, ch. 7].…”

Solution of the system of nonlinear PDEs characterizing CES property under quasi-homogeneity conditions

“…Notice that there are two important production models exhibiting the CES property widely utilized in economics (see for instance the recent works [3][4][5][6][7]). The first one is the Cobb-Douglas (CD) production function introduced in [8] for two inputs (labor and capital).…”