  Community structure and interaction delays are common features of ensembles
of network coupled oscillators, but their combined effect on the emergence of
synchronization has not been studied in detail. We study the transitions
between macroscopic states in coupled oscillator systems with community
structure and time delays. We show that the combination of these two properties
gives rise to non-monotonic transitions, whereby increasing the global coupling
strength can both inhibit and promote synchronization, yielding both
desynchronization and synchronization transitions. For relatively wide
parameter choices we also observe asymmetric suppression of synchronization,
where communities compete to suppress one another's synchronization properties
until one or more win, totally suppressing the others to effective incoherence.
Using the ansatz of Ott and Antonsen we provide analytical descriptions for
these transitions that confirm numerical simulations.
