You Do Need power plug adapter while traveling from Saint Barthélemy to Ecuador.

Here is Why You Need An Adapter

The outlet and plug type used in Ecuador are A and B and in Saint Barthélemy are C and E. This is how they look like.

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E

Ecuador

In Ecuador, outlets of type A and B are in use.

Type A

Type B