Invariant Patterns on Square and Hexagonal Lattices

This chapter presents a study of invariant patterns in replicator dynamics on square and hexagonal lattices with periodic boundary conditions. Invariant patterns are solutions to the governing equation for any value of the bifurcation parameter. We propose a theoretical procedure that can exhaustively identify invariant patterns, including uniform, monocentric (full agglomeration), and polycentric distributions. We introduce many theoretically possible invariant patterns and identify those that are stable in an economic geography model. Solution curves of invariant patterns are connected by those of non-invariant patterns to form a complex mesh-like structure, akin to warp and weft threads. Chapter 2 provides the foundation for the economic geography model. Chapters 4 and 5 are prerequisites for the direct bifurcation of the lattices.