Instability of Homogeneous State in Dynamical Models of Self-propelled Particles

We investigate the linear stability of homogeneous state of self-propelled particles based on a microscopic dynamical model where particles obey Newton's equations of motion. The model is constructed to include most of dynamical models for self-propelled particles. It is shown that the homogeneous state is unstable for values of parameters which we expect in natural schools. The result shows that a large school, the inside of which can be considered as the homogeneous state, is unstable, and other interactions or suppositions are necessary to maintain such a large school.