If they kept detailed records we don't have them. But we do know that sometimes the Romans had beasts fight each other. One memorable incident was at the inaugural games for the Flavian Amphitheatre, better known as the Colosseum. The writer Martial was an eyewitness and wrote about a rhino beating several other animals.

While in fear the trainers were goading a rhinoceros, and long was the great beast's wrath gathering strength, all despaired of the conflict of the promised war; yet at length the fury, known ere-while, returned. For a heavy bear he tossed with his double horn, even as a bull hurls dummies heavenward, and with as sure an aim as that wherewith the stout right hand of Carpophorus, as yet young, levels the Noric hunting-spear. That beast, agile with pliant neck, stood up against (?) a pair of steers, to him yielded the fierce buffalo and bison; a lion in flight from him ran headlong upon the spears.

Martial, Liber de Spectaculis XXII