Discrete and continuous models of the dynamics of pelagic fish: Application to the capelin

In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek and Czirók et al. [A. Czirók, T. Vicsek, Collective behavior of interacting self-propelled particles, Physica A 281 (2000) 17–29; A. Czirók, H. Stanley, T. Vicsek, Spontaneously ordered motion of self-propelled particles, Journal of Physics A: Mathematical General 30 (1997) 1375–1385; T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Novel type of phase transition in a system of self-driven particles, Physical Review Letters 75 (6) (1995) 1226–1229; T. Vicsek, A. Czirók, I. Farkas, D. Helbing, Application of statistical mechanics to collective motion in biology, Physica A 274 (1999) 182–189]. Recently, a system of ODEs was derived from this model [B. Birnir, An ODE model of the motion of pelagic fish, Journal of Statistical Physics 128 (1/2) (2007) 535–568], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.