Newtonian Dynamical Approach to Fish Schooling

Schools are groups of fish engaging in cohesive movements with parallel orientation. Schooling fish perform a well-organized collective motion by some kind of social interaction even in the absence of leaders or external stimuli. In this paper, a Newtonian model is developed for schooling behavior to study the self-organizing (i.e., decentralized control) mechanism of a school in a setting of collective motion of autonomous individuals, that is, within a Langrangian (i.e., individual-based) framework ofN-fish motion dynamics. The school is modeled as an interacting particle system with behavioral and environmental stochasticity, such as gas molecules following the rules of Newtonʹs mechanics. Two equations for the school mean and the variance of swimming velocity are derived from the Langrangian model which has a great degree of freedom (N>> 1). These two reduced equations of motion of the system illustrate a feed-forward control structure underlying the self-organization of the school. On the basis of these equations, the transient process in which the system approaches the stationary polarized schooling is investigated. Features of the temporal evolution of the system from the motionless state of the non-polarized aggregation to the onset of the polarized school are extracted by calculating the following: transient fluctuation of the individual velocity in a school, transient fluctuation of the centroid velocity of a school, two-time correlation of initial fluctuation of the centroid velocity, and temporal evolution of the mean square velocity of the center of school, which are regulated by individual stochastic movement in a school and nonlinearity of a system. Experimental observations of the transient behavior of schooling are consistent with the theoretical predictions. It is expected that the transient fluctuation of the centroid velocity shows the enhancement at around the onset-time of schooling structure. The fluctuation enhancement is found to be the essential mechanism for the onset of schooling.