Analysis of phase transition points for a two-color agent-based model

The agent-based model treated in the present study describes dynamics of two types of population in a gravity-like potential field. In previous studies, the model was known to exhibit various spatiotemporal patterns on two-dimensioanl lattice systems. However, the patterns were classified depending purely on eye observations, and the underlying dynamics of these patterns were not fully explored. It remained a question to be answered if these eye observation-based classifications could be confirmed by any analytical means. To pursue the question, we first suggest several analytic quantities, such as convergence time steps and reaction speed, to replace the eye observations. As a result, we show that a phase diagram can be reasonably drawn on the contour diagram of the time steps. In addition, we find a power-law scaling in the reaction speed, confirming that a phase transition really is involved there. Next, as a main part of the present study, we apply analytical methods to calculate two important phase transition points from the system. The results from the analytical approach agreed well with the numerically obtained phase transition points from the agent-based model. In general, the paper serves as an example study of estimating global phenomena of complex systems in terms of local parameters of the system.