  Collective behavior of pulse-coupled oscillators has been investigated
widely. As an example of pulse-coupled networks, fireflies display many kinds
of flashing patterns. Mirollo and Strogatz (1990) proposed a pulse-coupled
oscillator model to explain the synchronization of South East Asian fireflies
({\itshape Pteroptyx malaccae}). However, transmission delays were not
considered in their model. In fact, the presence of transmission delays can
lead to desychronization. In this paper, pulse-coupled oscillator networks with
delayed excitatory coupling are studied. Our main result is that under
reasonable assumptions, pulse-coupled oscillator networks with delayed
excitatory coupling can not achieve complete synchronization, which can explain
why another species of fireflies ({\itshape Photinus pyralis}) rarely
synchronizes flashing. Finally, two numerical simulations are given. In the
first simulation, we illustrate that even if all the initial phases are very
close to each other, there could still be big variations in the times to
process the pulses in the pipeline. It implies that asymptotical
synchronization typically also cannot be achieved. In the second simulation, we
exhibit a phenomenon of clustering synchronization.
