How to Slow the Problem Down
Right now, solving the problem completely is near impossible due to a multitude of reasons, including but not limited to financial resources and manpower. But, there are a variety of ways to slow down the the impact of domestic cats on the environment.

Solutions for Scientists
The most effective solution for removing feral cats from islands is exterminating them
Sounds cruel, but the cats don't belong on the islands, they are very harmful to the ecosystems
Takes a lot of manpower
Time consuming
Could capture and relocate
Expensive and inefficient
have you ever tried to chase a house cat? Now imagine capturing thousands of FERAL cats on large islands.
Takes a lot of manpower, cats can reproduce very quickly
Time consuming
Could spay and neuter
Again, expensive, and inefficient for the same reasons as before

How Can You Help
Keep outdoor cats indoors
they contribute to the problem as well, if allowed to go outside
Also lowers the chance that your furry friend will contract diseases or parasites, get hit by a car, get lost, or get attacked by other animals or kids
Spay and neuter your pets
Most rescue places spay/neuter the cats before allowing you to adopt them
Actually very humane
Prevents accidental pregnancies
Help out with local rescue groups to capture and re-home strays
Help out with local organizations that capture, spay/neuter, and release feral cats
Educate others on the problems that outdoor cats and feral cats cause
Urge them to take the same steps
Adopt, don't shop
Most pet stores source their animals from animal mills, and they tend to be expensive and sickly
Adopting allows for more feral cats to be taken off of the streets
These won't automatically solve the problem, but it will help mitigate and slow down the destruction
They are cute, cuddly, funny little creatures.... But they also wreak havoc on the ecosystem...