Abstract
Self-organization of spatial patterns is investigated for a scalar field of a system of locations on a hexagonal lattice. Group-theoretic bifurcation analysis is conducted to exhaustively try and find possible bifurcating patterns. All these patterns are proved to be asymptotically unstable for general spatial economic models in new economic geography. Microeconomic interactions among the locations are expressed by a spatial economy model and all bifurcating patterns are demonstrated to be unstable by numerical bifurcation analysis.
Highlights
The mechanism of self-organization of hexagonal distributions of cities was envisaged in central place theory of economic geography (Fig. 1 of [Christaller, 1933]), and has come untangled by group-theoretic bifurcation theory [Ikeda & Murota, 2014]
This paper would contribute to the study of a two-dimensional spatial agglomeration through the introduction of a methodology in group-theoretic bifurcation theory that has been developed mainly on the group D6 + ̇ T2 (T2 expresses the two-torus of translation symmetries)
We introduce a spatial economy model on a hexagonal lattice comprising a system of uniformly distributed n × n places, and prescribe groups expressing the symmetry of this model
Summary
The mechanism of self-organization of hexagonal distributions of cities was envisaged in central place theory of economic geography (Fig. 1 of [Christaller, 1933]), and has come untangled by group-theoretic bifurcation theory [Ikeda & Murota, 2014]. This paper tries to exhaustively find bifurcating hexagonal and nonhexagonal patterns of spatial economy models on a hexagonal lattice. Krugman [1996] regarded the racetrack economy, which comprises a system of identical cities spread uniformly around the circumference of a circle, as one-dimensional and inferred its extendibility to a two-dimensional economy to engender hexagonal distributions In this economy, bifurcation produced a chain of spatially repeated core-periphery patterns a la Christaller and Losch, which denotes a spatial alternation of a core place with a large population and a peripheral place with a small population.. Hexagonal patterns of spatial economy models on a hexagonal lattice were shown to exist by theoretical and numerical studies of the bifurcation problem equivariant to D6 + ̇ (Zn × Zn) [Ikeda et al, 2012b; Ikeda et al, 2014; Ikeda & Murota, 2014].
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