  A paradigmatic framework to study the phenomenon of spontaneous collective
synchronization is provided by the Kuramoto model comprising a large collection
of limit-cycle oscillators of distributed frequencies that are globally coupled
through the sine of their phase differences. We study here a variation of the
model by including nearest-neighbor interactions on a one-dimensional lattice.
While the mean-field interaction resulting from the global coupling favors
global synchrony, the nearest-neighbor interaction may have cooperative or
competitive effects depending on the sign and the magnitude of the
nearest-neighbor coupling. For unimodal and symmetric frequency distributions,
we demonstrate that as a result, the model in the stationary state exhibits in
contrast to the usual Kuramoto model both continuous and first-order
transitions between synchronized and incoherent phases, with the transition
lines meeting at a tricritical point. Our results are based on numerical
integration of the dynamics as well as an approximate theory involving
appropriate averaging of fluctuations in the stationary state.
