According to the 1875 Dictionary of Greek and Roman Antiquities by John Murray:
We do not know on what occasion a venatio was first exhibited at Rome; but the first mention we find of any thing of the kind is in the year B.C. 251, when L. Metellus exhibited in the Circus 142 elephants, which he had brought from Sicily after his victory over the Carthaginians, and which were killed in the Circus according to Verres, though other writers do not speak of their slaughter (Plin. H.N. viii.6). But this can scarcely be regarded as an instance of a venatio, since the elephants are said to have been only killed because the Romans did not know what to do with them, and not for the amusement of the people. There was, however, a venatio in the later sense of the word in B.C. 186, in the games celebrated by M. Fulvius in fulfilment of the vow which he had made in the Aetolian war; in these games lions and panthers were exhibited (Liv. xxxix.22). It is mentioned as a proof of the growing magnificence of the age that in the Ludi Circenses, exhibited by the curule aediles P. Cornelius Scipio Nasica and P. Lentulus B.C. 168, there were 63 African panthers and 40 bears and elephants (Liv. xliv.18). From about this time combats with wild beasts probably formed a regular part of the Ludi Circenses, and many of the curule aediles made great efforts to obtain rare and curious animals, and put in requisition the services of their friends (compare Caelius's letter to Cicero, ad Fam. viii.8). Elephants are said to have first fought in the Circus in the curule aedileship of Claudius Pulcher, B.C. 99, and twenty years afterwards, in the curule aedileship of the two Luculli, they fought against bulls (Plin. H.N. viii.7). A hundred lions were exhibited by Sulla in his praetorship, which were destroyed by javelin-men sent by king Bocchus for the purpose. This was the first time that lions were allowed to be loose in the Circus; they were previously always tied up (Senec. de Brev. Vit. 18). The games, however, in the curule aedileship of Scaurus B.C. 58 surpassed anything the Romans had ever seen; among other novelties he first exhibited an hippopotamos and five crocodiles in a temporary canal or trench (euripus, Plin. H.N. viii.40). At the venatio given by Pompey in his second consulship B.C. 55, upon the dedication of the temple of Venus Victrix, and at which Cicero was present (Cic. ad Fam. vii.1), there was an immense number of animals slaughtered, among which we find mention of 600 lions, and 18 or 20 elephants: the latter fought with Gaetulians, who hurled darts against them, and they attempted to break through the railings (clathri) by which they were separated from the spectators (Senec. l.c.; Plin. viii. 7.20). To guard against this danger Julius Caesar surrounded the arena of the amphitheatre with trenches (euripi).
In the games exhibited by J. Caesar in his third consulship, B.C. 45, the venatio lasted for five days and was conducted with extraordinary splendour. Camelopards or giraffes were then for the first time seen in Italy (Dion Cass. xliii.23; Suet. Jul. 39; Plin. H.N. viii.7; Appian, B. C. ii.102; Vell. Pat. ii.56
). Julius Caesar also introduced bull-fights, in which Thessalian horsemen pursued the bulls round the circus, and when the latter were tired out, seized them by the horns and killed them. This seems to have been a favourite spectacle; it was repeated by Claudius and Nero (Plin. H.N. viii.70; Suet. Claud. 21 ; Dion Cass. lxi.9). In the games celebrated by Augustus, B.C. 29, the hippopotamos and the rhinoceros were first exhibited, according to Dion Cassius (li.22), but the hippopotamos is spoken of by Pliny, as mentioned above, in the games given by Scaurus. Augustus also exhibited a snake 50 cubits in length (Suet. Aug. 43), and thirty-six crocodiles, which are seldom mentioned in spectacles of later times (Dion Cass. lv.10).
I also searched the web hoping to find a copy of Dr. Chris Epplett’s dissertation “The 'Creation' of the Roman Beast-Hunts” but I was unsuccessful. Here is an abstract:
The Roman beast-hunts (venationes), despite being popular arena spectacles, have generally received far less scholarly attention than the gladiatorial games with which they were associated. While numerous attempts have been made by modern scholars to explain the origin and purpose behind gladiatorial combat in the Roman empire, this and other aspects of the venationes have been relatively ignored by academics (Jennison's Animals for Show and Pleasure in Ancient Rome, written over 60 years ago, is a notable exception).
Antecedents for beast-hunts have been identified in Greek and Italic contexts, but the increasing violence of these events in the Late Republic can plausibly be linked to contemporary developments in gladiatorial combats. In my paper I investigate closely the process by which animal processions became animal slaughters. I do so first of all by noting that the violence of gladiatorial spectacles in Rome, which first became especially pronounced during the Second Punic War, was soon after followed by increasingly bloody animal spectacles in the course of the second century B.C. While this development may well have been linked to the increased supply of exotic animals which Roman conquests in this period made available, I will argue that a more important cause was competition between magistrates staging gladiator and animal spectacles; had the latter events remained mere processions of animals, they would have swiftly lost their appeal for Roman audiences otherwise becoming accustomed to public displays of bloodshed in the arena.
I checked the price of a copy of Animals for Show and Pleasure in Ancient Rome by George Jennison (1937) but it appears to be rather spendy to purchase it ($87 - $91). However, I found a copy in our research library.
Here is another website with a number of links to articles about Roman bloodsport: