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We study the properties of an ODE description of schools of fish ( B. Birnir, An ODE model of the motion of pelagic fish, J. Stat. Phys. 128( 1/ 2) ( 2007), 535- 568.) and how they change 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 ODE for its speed. These equations contain the mean speed (v) over bar and a Kuramoto order parameter r for the phases of the fish velocities. In this paper, we give a complete qualitative analysis of the system for large number of particles. We show that the stationary solutions of the ODEs consist of an incoherent unstable solution with r = (v) over bar = 0 and a globally stable solution with r = 1 and a constant (v) over bar > 0. In the latter solution, all the fish move uniformly in the same direction with (v) over bar and the direction of motion determined by the initial configuration of the school. This is called the " migratory solution". In the second part of the paper, 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 perturbed systems have similar behavior as the unperturbed one, and derive the resulting constant value of the average speed, verifying the numerical observations. Finally, we show that the system exhibits a similar bifurcation to that in Vicsek and Czirok ( T. Vicsek, A. Czirok, E. Ben- Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self- driven particles, Phys. Rev. Lett. 75( 6) ( Aug 1995), 1226- 1229.) between phases of synchronization and disorder. Either increasing the variance of the Brownian angular noise, or decreasing the turning rate, or coupling between the particles, cause a similar phase transition.
fish schools; synchronization; order parameters; phase-transition; kuramoto model; pelagic fish; oscillators; simulation; agents