Effect of Vision Angle on the Phase Transition in a Flocking Behavior of Animal Groups

The nature of the phase transition in a system of self-propelling particles has been extensively studied during the last few decades. A theoretical model was proposed by T. Vicsek, {\it et. al.} [Phys. Rev. Lett. {\bf 75}, 1226 (1995)] with a simple rule for updating the direction of motion of each particle. Based on the Vicsek's model (VM) [1], in this work, we consider a group of animals as particles moving freely on 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 $\varphi=\Phi/2$ ($\Phi$ is called angle of view) of a circle centered at its position, of radius $R$. We obtained a phase diagram in the space ($\varphi$, $\eta_c$) with $\eta_c$ being the critical noise. We show that, the phase transition exists only in the case of a wide view's angle $\varphi \geq 0.5\pi$. The flocking of animals is an universal behavior of the species of prey, but not the one of the predator. Our simulation results are in good agreement with experimental observation [2].