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An economic cross-diffusion mutualistic model for cities emergence

Gonzalo F. de-Córdoba, Gonzalo Galiano

TL;DR

An evolution cross-diffusion problem with mutualistic Lotka-Volterra reaction term is studied to modelize the long-term spatial distribution of labor and capital and finds conditions under which the uniform optimum of profits becomes unstable, leading to pattern formation.

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.

An economic cross-diffusion mutualistic model for cities emergence

TL;DR

An evolution cross-diffusion problem with mutualistic Lotka-Volterra reaction term is studied to modelize the long-term spatial distribution of labor and capital and finds conditions under which the uniform optimum of profits becomes unstable, leading to pattern formation.

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.

Paper Structure

This paper contains 7 sections, 2 theorems, 58 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A historical example
  3. The mathematical model
  4. The reaction term: parameters of competence and mutualism
  5. Example: The CES production function
  6. Turing instability analysis
  7. Numerical simulations

Key Result

Theorem 1

Let $w^*,~r^*$ be positive numbers, and consider the CES production function (def:ces) with parameters satisfying (param). Then, the optimum of the profits function, $\Pi$, is $( L_e^*,K_e^{*})$, with Moreover, $(K_e^*, L_e^*)$ is also the coexistence equilibrium of the Lotka-Volterra system (eq:LVeq1)-(eq:LVeq2) with coefficients given by (def:alphabeta), which satisfy In addition, if $\epsilon

Figures (3)

  • Figure 1: Supercritical case. First row: Experiment 1. New equilibrium reached after the onset of instabilities due to a small perturbation around the uniform equilibrium $(L^*,K^*)\approx(0.287,0.176)$. Second row: Experiment 2. Like previous, with $(L^*,K^*)=(0.6,0.09)$. Mind the different ordinate scales. Third row: Experiment 3. Like previous, with $(L^*,K^*)\approx(2.35,1.31)$.
  • Figure 2: Subcritical case. Experiment 4: Evolution of the instability formation. Only the labor is shown, the capital following a similar trend. For large times, uninhabited regions arise separating densely populated areas.
  • Figure 3: Experiments 1 to 4: Determinant of the eigenvalue matrix $A_k$, see (\ref{['def:Ak']}), as a function of $k$, and for two values of the bifurcation parameter: the critical value $b_c$ and the value used in the experiments, $b=1.01b_c$.

Theorems & Definitions (5)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Remark 3