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