  The emergence of oscillations in models of the El-Ni\~no effect is of utmost
relevance. Here we investigate a coupled nonlinear delay differential system
modeling theEl-Ni\~no/ Southern Oscillation (ENSO) phenomenon, which arises
through the strong coupling of the ocean-atmosphere system. In particular, we
study the temporal patterns of the sea surface temperature anomaly of the two
sub-regions. For identical sub-regions we typically observe a co-existence of
amplitude and oscillator death behavior for low delays, and heterogeneous
oscillations for high delays, when inter-region coupling is weak. For moderate
inter-region coupling strengths one obtains homogeneous oscillations for
sufficiently large delays and amplitude death for small delays. When the
inter-region coupling strength is large, oscillations are suppressed
altogether, implying that strongly coupled sub-regions do not exhibit ENSO-like
oscillations. Further we observe that larger strengths of self-delay coupling
favours oscillations, while oscillations die out when the delayed coupling is
weak. This indicates again that delayed feedback, incorporating oceanic wave
transit effects, is the principal cause of oscillatory behaviour. So the effect
of trapped ocean waves propagating in a basin with closed boundaries is crucial
for the emergence of ENSO. Further, we show how non-uniformity in delays, and
difference in the strengths of the self-delay coupling of the sub-regions,
affect the rise of oscillations. Interestingly we find that larger delays and
self-delay coupling strengths lead to oscillations, while strong inter-region
coupling kills oscillatory behaviour. Thus, we find that coupling sub-regions
has a very significant effect on the emergence of oscillations, and strong
coupling typically suppresses oscillations, while weak coupling of
non-identical sub-regions can induce oscillations, thereby favouring ENSO.
