Where the respiratory center is located?

The respiratory center is an area located in the pons and
the medulla oblongata, the lowest portion of the brain
stem.

There are actually four separate areas that together
comprise the respiratory center. In the medulla there is the ventral respiratory group,
which controls the force of exhalation. The dorsal group controls timing of inspiration.
In the pons the apneustic center controls the length if inhalations and helps coordinate
the turnover from inhaling to exhaling. It is assisted in this by the pneumotaxic
center.

Interestingly, the medulla oblongata does not sense
oxygen levels in the blood as you might expect, but instead it contains chemoreceptors
which monitor the levels of carbon dioxide  and the blood pH. When carbon dioxide
dissolves in the blood it becomes carbonic acid, which lowers the blood
pH.