Flocking and invariance of velocity angles

Motsch and Tadmor considered an extended Cucker-Smale model to investigate the flocking behavior of self-organized systems of interacting species. In this extended model, a cone of the vision was introduced so that outside the cone the influence of one agent on the other is lost and hence the corresponding influence function takes the value zero. This creates a problem to apply the Motsch-Tadmor and Cucker-Smale method to prove the flocking property of the system. Here, we examine the variation of the velocity angles between two arbitrary agents, and obtain a monotonicity property for the maximum cone of velocity angles. This monotonicity permits us to utilize existing arguments to show the flocking property of the system under consideration, when the initial velocity angles satisfy some minor technical constraints.