The Republic did not fall in any way whatsoever - the idea that it did is a popular misconception caused by historical categorisation and emphasis on the antics of certain Caesars. The fallacy that it fell is easily overturned when one takes the trouble to realise that nothing about the Republic actually stopped or was dismantled - all it was was a significant political change, with powerful individuals acting as sponsored advisors, or in some cases, self important tyrants. There was no job description for the role of Caesar, no constitutional means of providing for a succession, and the Caesars themselves were not absolute rulers - even Caligula is known to have asked the Senate for permission to stage games. The poers of a particular Caesar were provided by the Senate, not the job, although whether popularity or lots of soldiers were the cause of this provision is another matter. Remember that not until the reign of Commodus diod the Romans accept that a Caesar had been "born to the purple", or in other words, succeded his father in monarchial succession.