First, it's a myth that only Asian elephants can be worked with. It's such a strange myth since trained African elephants have been around for hundreds of years. Jumbo, the most famous elephant ever, was a trained African bull. Heck, Hannibal: Rome's Worst Nightmare staring Alexander Siddig has trained African elephants in it!

Elephants were introduced to Mediterranean warfare by the Persians/Alexander, who obviously had contact with India.

Initially all these war elephants were imported from India, however once the situation in the east solidified and the Selucids had firm control over Persia, they controlled the supply and if you wanted war elephants and were not friendly with them, you had to get them from Africa.

The Carthaginians first encountered war elephants when they fought Pyrrhus in Sicily, who had loaned some elephants from Ptolemy II, not sure if they Asians or not, the Ptolemies definitely captured and trained African elephants later.

The Carthaginians adopted this new weapon and captured their own elephants.

Breeding elephants in captivity is a tricky business, that we're only just figuring out. There are references to a place where Rome breed elephants just outside the city, but that's about it.

Once the supply of North African elephants died out and hostile Persian and later Islamic powers took control of Iran, the supply to Europe dried up.

After about 100 AD elephants were basically gone from Europe. The ones that do show up were gifts of individual animals, like Harun al-Rashid's gift of Abul-Abbas to Charlemagne.

War Elephants by John M. Kistler has a lot more information on the use of elephants by Mediterranean peoples.

The elephants used by Hannibal were native to North Africa, a now-extinct species.