Topological phases in population dynamics with rock-paper-scissors interactions

Topological phases have arisen great interests of physicists. Though most works focus on quantum systems, topological phases can also be found in nonquantum systems. In this work, we study an antisymmetric Lotka-Volterra dynamics defined on a chain of two-site cells with open boundary conditions. We find two edge-localization states, left edge-localization state, and right edge-localization state. In an edge-localization state, there exists a boundary region in which mass distribution displays an exponential decay with the distance away from the boundary. The two edge-localization states are connected by a sharp transition. To comprehend the edge-localization states, we transform the population dynamics into a non-Hermitian quantum system. Based on the generalized topological band theory of the non-Hermitian system with periodic boundary conditions, we use winding number to distinguish the left and the right edge-localization states, and the transition between these two states is identified to be a topological one.