I would assume that plantation owners would have a tighter grip on the slaves, due to a limited amount of them. Also, wouldn't they become much more expensive too?
Oops, I noticed my mistake in the title. Disregard "lead" in the title
It didn't really. Congress and individual states passed laws either banning or limiting the international slave trade prior to 1808. The slave trade act of 1794 being one of many.
Slaves in the US were more expensive than elsewhere in the Americas due to the distance the slave ships had to travel from West Africa. So a system was created that was obviously not humane but your chance of surviving and having offspring was much better in the US than anywhere else in the New World. There also wasn't the economic demand for slaves like there was in Brazil and the Caribbean and the huge sugar plantations. Until the invention of the cotton gin, that is. As cotton began to take over the economy of the south, in the 1820's, the demand for slaves increased. And because the system was already set up to be self sustaining the number of slaves increased after the international slaved trade ban.
The US system was also very efficient at maintaining control of slaves. This was done by the prohibition of African languages and religions. This was not the case in the Caribbean or Brazil to nearly the same degree. The evidence of this is still seen today in religions like Vodou in Haiti, Candomble in Brazil, or Santaria in Cuba. The only thing comparable in the US is Louisiana Voodoo but it was a remnant of the French system because New Orleans was a part of France until 1805. In many cases these religions and native languages allowed for the preparation of slave revolts, or in the case of Brazil, Quilombos. They caused a constant threat to the economic system of slavery that the US just didn't face. Brazil continued to import hundreds of thousands of slaves illegally after the 1808 international ban because they just couldn't sustain their system without new imports.