Effect of vision angle on the phase transition in flocking behavior of animal groups

The nature of the phase transition in a system of self-propelling particles has been extensively studied during the past few decades. A theoretical model was proposed by [T. Vicsek et al. Phys. Rev. Lett. 75, 1226 (1995)] with a simple rule for updating the direction of motion of each particle. Based on the model of Vicsek et al., in this paper, we consider a group of animals as particles moving freely in a two-dimensional space. Due to the fact that the viewable area of animals depends on the species, we consider the motion of each individual within an angle $\phi =\varphi /2$ $(\varphi$ is called the angle of view) of a circle centered at its position of radius $R$. We obtained a phase diagram in the space $\left(\phi ,{\eta }_c\right)$ with ${\eta }_c$ being the critical noise. We show that the phase transition exists only in the case of a wide view's angle $\phi \ge 0.5\pi$. The flocking of animals is a universal behavior of the species of prey but not the one of the predator. Our simulation results are in good agreement with experimental observation [C. Beccoa et al., Physica A 367, 487 (2006)].