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

TL;DR

This paper analyzes the stability and dynamics of a localized economic concentration in a one-dimensional spatial Solow model with capital-driven labor migration, revealing conditions for stability, oscillations, and slow shifts.

Contribution

It introduces a new analysis of interior spike stability in a spatial Solow model, combining asymptotic, numerical, and differential-algebraic methods.

Findings

Interior spike remains stable for small reaction times.

Large reaction times induce Hopf bifurcations and oscillations.

Numerical simulations reveal complex slow drift dynamics.

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.