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Stability and slow dynamics of an interior spiky pattern in a one-dimensional spatial Solow model with capital-induced labor migration

Fanze Kong, Jiayi Sun, Shuangquan Xie

TL;DR

This work analyzes a one-dimensional spatial Solow-Swan model with capital-induced labor migration under small capital diffusivity, revealing the formation of interior spike patterns that localize economic activity. Using matched asymptotics and hybrid asymptotic-numerical methods, the authors construct a single interior spike quasi-equilibrium and dissect its stability via large- and small-eigenvalue analyses. They show the spike is stable for small reaction-time constants $\tau$ but undergoes a Hopf bifurcation as $\tau$ increases, producing height-oscillations, and they derive a slow drift dynamics for the spike location through a reduced differential-algebraic system. The results illuminate core–periphery agglomeration mechanisms in spatial economics and lay groundwork for extending to multi-spike patterns and higher dimensions, with potential connections to related reaction-diffusion systems such as Keller–Segel models.

Abstract

One of the most significant findings in the study of spatial Solow-Swan models is the emergence of economic agglomeration, in which economic activities concentrate in specific regions. Such agglomeration provides a fundamental mechanism driving the spatial patterns of urbanization, labor migration, productivity growth, and resource allocation. In this paper, we consider the one-dimensional spatial Solow-Swan model with capital-induced labor migration, which captures the dynamic interaction between labor and capital through migration and accumulation. Focusing on the regime of sufficiently small capital diffusivity, we first construct an interior spike (spiky economic agglomeration) quasi-equilibrium. Next, we perform the linear stability of the corresponding spike equilibrium by using a hybrid asymptotic and numerical method. We show that a single interior spike remains stable for small reaction-time constants but undergoes a Hopf bifurcation when the constant is sufficiently large, leading to oscillations in spike height (economic fluctuation). Finally, we derive a differential-algebraic system to capture the slow drift motion of quasi-equilibrium (core-periphery shift). Numerical simulations are carried out to support our theoretical studies and reveal some intriguing yet unexplained dynamics.

Stability and slow dynamics of an interior spiky pattern in a one-dimensional spatial Solow model with capital-induced labor migration

TL;DR

This work analyzes a one-dimensional spatial Solow-Swan model with capital-induced labor migration under small capital diffusivity, revealing the formation of interior spike patterns that localize economic activity. Using matched asymptotics and hybrid asymptotic-numerical methods, the authors construct a single interior spike quasi-equilibrium and dissect its stability via large- and small-eigenvalue analyses. They show the spike is stable for small reaction-time constants but undergoes a Hopf bifurcation as increases, producing height-oscillations, and they derive a slow drift dynamics for the spike location through a reduced differential-algebraic system. The results illuminate core–periphery agglomeration mechanisms in spatial economics and lay groundwork for extending to multi-spike patterns and higher dimensions, with potential connections to related reaction-diffusion systems such as Keller–Segel models.

Abstract

One of the most significant findings in the study of spatial Solow-Swan models is the emergence of economic agglomeration, in which economic activities concentrate in specific regions. Such agglomeration provides a fundamental mechanism driving the spatial patterns of urbanization, labor migration, productivity growth, and resource allocation. In this paper, we consider the one-dimensional spatial Solow-Swan model with capital-induced labor migration, which captures the dynamic interaction between labor and capital through migration and accumulation. Focusing on the regime of sufficiently small capital diffusivity, we first construct an interior spike (spiky economic agglomeration) quasi-equilibrium. Next, we perform the linear stability of the corresponding spike equilibrium by using a hybrid asymptotic and numerical method. We show that a single interior spike remains stable for small reaction-time constants but undergoes a Hopf bifurcation when the constant is sufficiently large, leading to oscillations in spike height (economic fluctuation). Finally, we derive a differential-algebraic system to capture the slow drift motion of quasi-equilibrium (core-periphery shift). Numerical simulations are carried out to support our theoretical studies and reveal some intriguing yet unexplained dynamics.
Paper Structure (9 sections, 5 theorems, 125 equations, 10 figures)

This paper contains 9 sections, 5 theorems, 125 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The construction of spike solutions
  4. The stability of a interior spike solution
  5. Computation of large eigenvalues
  6. Computation of small eigenvalues
  7. The slow drift dynamics of a single spike
  8. Discussion
  9. Subinner approximation of the NLEP problem

Key Result

Proposition 1

Suppose $ab < \tfrac{\pi^2}{4}$. Then there exists a steady state of SS01 on $[-1,1]$ that is even with respect to the origin and given by where $L_0(x/\varepsilon) := S e^{-S} e^{K_0(x/\varepsilon)}$, $S$ is determined by and $K_0$ is the solution of

Figures (10)

  • Figure 1: Illustration of single spike dynamics in the spatial Solow model \ref{['SS01']}. Parameters: $\varepsilon=1\times 10^{-2}, a=1, b=1.$
  • Figure 2: Numerical results illustrating the divergence of $K_0(0)$ and $\mathcal{I}(S)$ as $S\to 0$.
  • Figure 3: Comparison of $S$ (top panels) and $\xi$ (bottom panels) obtained from different approaches for $\varepsilon = 2\times10^{-2},\; 1\times10^{-2},\; 5\times10^{-3},\; 2.5\times10^{-3},\; 1.25\times10^{-3}$, with parameters $a=1,\; b=0.25$. The $x$-axis is plotted on a logarithmic scale; $S$ is also on a logarithmic scale, while $\xi$ is on a linear scale. Blue curves correspond to direct numerical solutions of the full PDE system \ref{['SS01']} using FlexPDE, red curves are obtained from the numerical solution of \ref{['fullinnermatching']}, and green curves from the solution of \ref{['subalge']}. The agreement between all approaches improves as $\varepsilon$ decreases.
  • Figure 4: Comparison of $S$ and $\xi$ obtained from different approaches for varying values of $\theta$, with parameters $a=1,\; b=1,\; \varepsilon=2.5\times10^{-3}$. All methods show consistent qualitative trends as $\theta$ increases. For fixed $\varepsilon$, the accuracy of the asymptotic approximation decreases with larger $\theta$, in agreement with the scaling \ref{['asyms']}, which indicates that neglected terms are of order $S^2 \sim \varepsilon^2 |\ln \varepsilon|^{\frac{4-2\theta}{1-\theta}}$.
  • Figure 5: The function $f(\lambda)$ for real $\lambda$ at $\tau=0$, with parameters $a=1,\; b=1,\; \varepsilon=1\times 10^{-2}$. In both cases, no positive real roots of $f(\lambda)=0$ are observed.
  • ...and 5 more figures

Theorems & Definitions (5)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1