  We study the dynamics of the sea surface temperature (SST) anomaly using a
model of the temporal patterns of two sub-regions, mimicking behaviour similar
to El Ni\~no Southern Oscillations (ENSO). Specifically, we present the
existence, stability, and basins of attraction of the solutions arising in the
model system in the space of these parameters: self delay, delay and
inter-region coupling strengths. The emergence or suppression of oscillations
in our models is a dynamical feature of utmost relevance, as it signals the
presence or absence of ENSO-like oscillations. In contrast to the well-known
low order model of ENSO, where the influence of the neighbouring regions on the
region of interest is modelled as external noise, we consider neighbouring
regions as a coupled deterministic dynamical systems. Different parameters
yield a rich variety of dynamical patterns in our model, ranging from steady
states and homogeneous oscillations to irregular oscillations and coexistence
of oscillatory attractors, without explicit inclusion of noise. Interestingly,
if we take the self-delay coupling strengths of the two sub-regions to be such
that the temperature of one region goes to a fixed point regime when uncoupled,
while the other system is in the oscillatory regime, then on coupling both
systems show oscillations. We explicitly obtain the basins of attraction for
the different steady states and oscillatory states in the model. Our results
might be helpful for forecasting of El Ni\~no (or La Ni\~na) progress, as it
indicates the combination of initial SST anomalies in the sub-regions that can
result in a El Ni\~no/La Ni\~na episodes. In particular, the result suggests
using an interval as a criterion to estimate the El-Ni\~no or La-Ni\~na
progress instead of the currently used the single value criterion.
