Swarming, schooling, milling: phase diagram of a data-driven fish school model

We determine the basic phase diagram of the fish school model derived from by Gautrais etal (PLoS Comp. Biol. 8, e1002678 (2012)), exploring its space beyond the parameter values determined experimentally on groups barred flagtails (Kuhlia mugil}) swimming in a shallow tank. A modified is studied alongside the original one, in which an additional frontal is introduced in the stimulus/response function to account for the weighting of interactions. Our study, mostly limited to groups of size (in the order of 100 individuals), focused not only on the to schooling induced by increasing the swimming speed, but also on conditions under which a school can exhibit milling dynamics and the behavioral transitions. We show the existence of a transition between milling and schooling, in which the school exhibits and intermittency between schooling and milling for the same of individuals parameters. We also show that milling does not occur arbitrarily large groups, mainly due to a distance dependence interaction the model and information propagation delays in the school, which cause reactions for large groups. We finally discuss the biological of our findings, especially the dependence of behavioural on social interactions, which were reported by Gautrais etal to be in the experimental conditions.