Spatial Period-Doubling Agglomeration of a Core-Periphery Model with a System of Cities

The orientation and progress of spatial agglomeration for Krugman's core--periphery model are investigated in this paper. Possible agglomeration patterns for a system of cities spread uniformly on a circle are set forth theoretically. For example, a possible and most likely course predicted for eight cities is a gradual and successive one---concentration into four cities and then into two cities en route to a single city. The existence of this course is ensured by numerical simulation for the model. Such gradual and successive agglomeration, which is called spatial-period doubling, presents a sharp contrast with the agglomeration of two cities, for which spontaneous concentration to a single city is observed in models of various kinds. It exercises caution about the adequacy of the two cities as a platform of the spatial agglomerations and demonstrates the need of the study on a system of cities

Spatial Period-Doubling Agglomeration of a Core-Periphery Model with a System of Cities

The orientation and progress of spatial agglomeration for Krugman's core--periphery model are investigated in this paper. Possible agglomeration patterns for a system of cities spread uniformly on a circle are set forth theoretically. For example, a possible and most likely course predicted for eight cities is a gradual and successive one---concentration into four cities and then into two cities en route to a single city. The existence of this course is ensured by numerical simulation for the model. Such gradual and successive agglomeration, which is called spatial-period doubling, presents a sharp contrast with the agglomeration of two cities, for which spontaneous concentration to a single city is observed in models of various kinds. It exercises caution about the adequacy of the two cities as a platform of the spatial agglomerations and demonstrates the need of the study on a system of cities.Agglomeration of population; Bifurcation; Core-periphery model; Group theory; Spatial period doubling

Group-theoretic Study of Economic Agglomerations on a Square Lattice

The present paper aims to elucidate the mechanism of economic agglomerations in two-dimensional economic spaces equipped with square road networks, which prosper worldwide (e.g., Chicago and Kyoto). A series of theoretical approaches provided in the present thesis makes it possible to investigate the spatial patterns of economic agglomerations on such spatial platforms systematically. The present paper focuses on square distributions on the square lattice economy, which has not somewhat been given much attention. We apply group-theoretic predictions to the investigation of bifurcation behavior of economic geography models. The present paper provides a systematic analysis procedure that is applicable to a wide range of economic geography models

Group-theoretic analysis of a scalar field on a square lattice

In this paper, we offer group-theoretic bifurcation theory to elucidate the mechanism of the self-organization of square patterns in economic agglomerations. First, we consider a scalar field on a square lattice that has the symmetry described by the group $\textrm{D}_{4} \ltimes \mathbb{Z}_{n} \times \mathbb{Z}_{n})$ and investigate steady-state bifurcation of the spatially uniform equilibrium to steady planforms periodic on the square lattice. To be specific, we derive the irreducible representations of the group $\textrm{D}_{4} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})$ and show the existence of bifurcating solutions expressing square patterns by two different mathematical ways: (i) using the equivariant branching lemma and (ii) solving the bifurcation equation. Second, we apply such a group-theoretic methodology to a spatial economic model with the replicator dynamics on the square lattice and demonstrate the emergence of the square patterns. We furthermore focus on a special feature of the replicator dynamics: the existence of invariant patterns that retain their spatial distribution when the value of the bifurcation parameter changes. We numerically show the connectivity between the uniform equilibrium and invariant patterns through the bifurcation. The square lattice is one of the promising spatial platforms for spatial economic models in new economic geography. A knowledge elucidated in this paper would contribute to theoretical investigation and practical applications of economic agglomerations

Spatial Discounting, Fourier, and Racetrack Economy: A Recipe for the Analysis of Spatial Agglomeration Models

We provide an analytical approach that facilitates understanding the bifurcation mechanism of a wide class of economic models involving spatial agglomeration of economic activities. The proposed method overcomes the limitations of the Turing (1952) approach that has been used to analyze the emergence of agglomeration in the multi-regional core-periphery (CP) model of Krugman (1993, 1996). In other words, the proposed method allows us to examine whether agglomeration of mobile factors emerges from a uniform distribution and to analytically trace the evolution of spatial agglomeration patterns (i.e., bifurcations from various polycentric patterns as well as a uniform pattern) that these models exhibit when the values of some structural parameters change steadily. Applying the proposed method to the multi-regional CP model, we uncover a number of previously unknown properties of the CP model, and notably, the occurrence of “spatial period doubling bifurcation” in the CP model is proved

Spatial Discounting, Fourier, and Racetrack Economy: A Recipe for the Analysis of Spatial Agglomeration Models

We provide an analytical approach that facilitates understanding the bifurcation mechanism of a wide class of economic models involving spatial agglomeration of economic activities. The proposed method overcomes the limitations of the Turing (1952) approach that has been used to analyze the emergence of agglomeration in the multi-regional core-periphery (CP) model of Krugman (1993, 1996). In other words, the proposed method allows us to examine whether agglomeration of mobile factors emerges from a uniform distribution and to analytically trace the evolution of spatial agglomeration patterns (i.e., bifurcations from various polycentric patterns as well as a uniform pattern) that these models exhibit when the values of some structural parameters change steadily. Applying the proposed method to the multi-regional CP model, we uncover a number of previously unknown properties of the CP model, and notably, the occurrence of “spatial period doubling bifurcation” in the CP model is proved

Bifurcation theory of a racetrack economy in a spatial economy model

Racetrack economy is a conventional spatial platform for economic agglomeration in spatial economy models. Studies of this economy up to now have been conducted mostly on $2^k$ cities, for which agglomerations proceed via so-called spatial period doubling bifurcation cascade. This paper aims at the elucidation of agglomeration mechanisms of the racetrack economy in a general setting of an arbitrary number of cities. First, an attention was paid to the existence of invariant solutions that retain their spatial distributions when the transport cost parameter is changed. A complete list of possible invariant solutions, which are inherent for replicator dynamics and are dependent on the number of cities, is presented. Next, group-theoretic bifurcation theory is used to describe bifurcation from the uniform state, thereby presenting an insightful information on spatial agglomerations. Among a plethora of theoretically possible invariant solutions, those which actually become stable for spatial economy models are obtained numerically. Asymptotic agglomeration behavior when the number of cities become very large is studied

Time evolution of city distributions in Germany

This paper aims to capture characteristic agglomeration patterns in population data in Germany from 1987 to 2011, encompassing pre- and post-unification periods. We utilize a group-theoretic double Fourier spectrum analysis procedure (Ikeda et al., 2018) as a systematic means to capture characteristic agglomeration patterns in population data. Among a plethora of patterns to be self-organized from a uniform state, we focus on a megalopolis pattern, a rhombic pattern, and a core--satellite pattern (a downtown surrounded by hexagonal satellite cities). As the technical contribution of this paper, we newly introduce a principal vector as a superposition of these patterns in order to grasp the multi-scale nature of agglomerations. Benchmark spectra for these patterns are advanced and are found in the population data of Germany in 2011. An incremental population is investigated using this principal vector to successfully detect a shift of predominant population increase/decrease patterns in the pre- and post-unification periods

Spatial Period-Doubling Agglomeration of a Core-Periphery Model with a System of Cities

The orientation and progress of spatial agglomeration for Krugman's core--periphery model are investigated in this paper. Possible agglomeration patterns for a system of cities spread uniformly on a circle are set forth theoretically. For example, a possible and most likely course predicted for eight cities is a gradual and successive one---concentration into four cities and then into two cities en route to a single city. The existence of this course is ensured by numerical simulation for the model. Such gradual and successive agglomeration, which is called spatial-period doubling, presents a sharp contrast with the agglomeration of two cities, for which spontaneous concentration to a single city is observed in models of various kinds. It exercises caution about the adequacy of the two cities as a platform of the spatial agglomerations and demonstrates the need of the study on a system of cities