# An economic cross-diffusion mutualistic model for cities emergence

## TL;DR

This paper models city emergence through an economic cross-diffusion system with mutualistic interactions, analyzing stability and pattern formation, supported by numerical simulations.

## Contribution

It introduces a novel cross-diffusion model for urban development based on profit-driven mutualism, with stability analysis and numerical validation.

## Key findings

- Conditions for pattern formation identified
- Cities modeled as high and low concentration regions
- Numerical simulations confirm theoretical predictions

## 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.