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.