Let's say the tree lobster on Lord Howe is replaced by rats and cats. Then Lord Howe lost one specie and gained 2, for a net gain of one specie. So local biodiversity increased. The planet lost one specie, so global biodiversity decreased.

I'm sure Lord Howe lost much more than just one species of tree lobster.

You're right. go down to page 26 of this pdf: http://www.environment.nsw.gov.au/resources/threatenedspecie...

Makes for scary reading!

That assumes nothing else in the ecosystem was dependent on the lost species.

You think the rats won't continue to displace other species? And what happens to the rats when there's nothing left to displace? And then what happens to the cats?