Modeling collective animal behavior: Analysis of the phase diagram from geometric aspect
Many animal groups, for example fish school, bird flock and ant trail, exhibit eye-catching collective motion patterns. In these groups individuals can move cohesively in spite of limitation of their perceptions. These group motions have been studied using mathematical model and computer simulation. However in those models, researchers emphasized similarity in appearance of real movements and gave complex moving processes to the individual units. Accordingly it causes difficulty in understanding of the models. Determining how the moving processes of individuals work the emergence patterns in detail is therefore important to construct advanced models and to help application for experimental data analysis. Here, we constructed a simple model for collective animal movement based on the ideas of moving processes in previous works authored by C. Reynolds (1987) and I. D. Couzin (2002). We attempted to make a valid phase diagram to represent the collective patterns by the relative strength of the moving processes: Alignment (or Orientation) and Cohesion (or Attraction).
We used a property of the model to divide parameter regions for two distinct patterns, which were Marching (Parallel group) and Circle (Torus), on the phase diagram. We derived the boundary between the patterns, calculating a case of only two individuals by geometric calculation and vector analysis. Preliminary results show that the deterministic phase diagram and the formulas of the boundary help to understand stochastic effects toward the collective patterns according to the number of individuals increasing in animal groups.