  Fish schools and bird flocks exhibit complex collective dynamics whose
self-organization principles are largely unknown. The influence of
hydrodynamics on such collectives has been relatively unexplored theoretically,
in part due to the difficulty in modeling the temporally long-lived
hydrodynamic interactions between many dynamic bodies. We address this through
a novel discrete-time dynamical system (iterated map) that describes the
hydrodynamic interactions between flapping swimmers arranged in one- and
two-dimensional lattice formations. Our 1D results exhibit good agreement with
previously published experimental data, in particular predicting the
bistability of schooling states and new instabilities that can be probed in
experimental settings. For 2D lattices, we determine the formations for which
swimmers optimally benefit from hydrodynamic interactions. We thus obtain the
following hierarchy: while a side-by-side single-row "phalanx" formation offers
a small improvement over a solitary swimmer, 1D in-line and 2D rectangular
lattice formations exhibit substantial improvements, with the 2D diamond
lattice offering the largest hydrodynamic benefit. Generally, our
self-consistent modeling framework may be broadly applicable to active systems
in which the collective dynamics is primarily driven by a fluid-mediated
memory.
