Quantitative and analytical insights into evolutionary game dynamics on regular lattices

In this paper, the linear invariance of evolutionary game dynamical systems is exploited to systematically realize the spatial partitioning of payoff matrix parameters on a two-dimensional regular lattice with the Neumann neighborhood and the deterministic imitation rule. It is shown that 264 evolutionary game dynamical systems are identified. Meanwhile, a homeomorphism group is set up to establish 132 topological conjugacy equivalence classes. Particularly, for the snowdrift game model, 28 partition regions of the payoff matrix parameters are determined, and the corresponding bifurcation diagram is illustrated. Additionally, some subsystems of the evolutionary snowdrift game dynamical systems are constructed utilizing the glider structures, and the topological conjugacy relationships between these subsystems and infinite product systems of finite-type subshifts are analyzed. Subsequently, their chaotic dynamics is investigated in detail, including properties such as topological mixing and topological entropy, which reflect the complexity and unpredictability of the underlying evolutionary behaviors. It is undeniable that the proposed analytical method can be used to characterize the evolutionary game dynamics on regular grids with symmetric neighborhood structures and deterministic updates, offering a systematic and coherent theoretical framework for analyzing evolutionary dynamics.