Mynah birds have taken over my garden, displacing the natural avian fauna. I would like to get rid of them, but my being in Australia means that I can't buy a gun.
Any alternative suggestions on how to control them?
An air rifle would do it, if that's permissible where you live (it is in the UK for use with feral pigeons and rats) and I believe its not illegal to shoot myna birds in Australia. However, if you live in a city or a built up area, it may not be possible to use an air rifle because of danger to other residents and pets.
It might be worth consulting with your local authority to see if they have any plan of action in place; from what I've read, culling is under consideration in some areas, but not decided upon. The alternative methods seem to be trapping and killing, or poisoning, and I'm not sure how safe the latter would be in areas where there are plenty of creatures you don't want to harm. This link here seems quite useful though http://indianmynah.blogspot.co.uk