Center for Complex and Nonlinear Science

We study how an ODE description of schools of fish by Birnir (2007) changes in the presence of a random acceleration. The model can be reduced to one ODE for the direction of the velocity of a generic fish and another one for its speed. These equations contain the mean speed v and a Kuramoto order parameter for the phases of the fish velocities, r. We show that their stationary solutions consist of an incoherent unstable solution with r=v=0 and a globally stable solution with r=1 and a constant v > 0. In the latter solution, all fishes move uniformly in the same direction with v and the direction of motion determined by the initial configuration of the school.In the second part, the directional headings of the particles are perturbed, in two distinct ways, and the speeds accelerated in order to obtain two distinct classes of non-stationary, complex solutions. We show that the system has similar behavior as the unperturbed one, and derive the resulting constant value of the average speed, verified numerically. Finally, we show that the system exhibits a similar bifurcation to that in Cirok and Vishek et al. 1995, between phases of synchronization and disorder. In one case, the variance of the angular noise, which is Brownian, is varied, and in the other case, varying the turning rate causes a similar phase transition.