Removing cats from their territory does not solve the cat over population problem. In nature when resources are left unguarded, new animals will take up residents to take advantage of those resources.
Farmers are not interested in taking in city cats because they attract coyotes and pose a danger to their livestock.
When ever there is a largely populated area we can expect to see rodents. Removing cats from those areas removes the preferred natures pest control.
Rats and Mice attract snakes when there are not cats maintaining the echo system.
Cats from up to 3 miles away will travel to answer the call of the scent that the last cats left as a calling card for mating. When they get there they will see mice, rats and the food the humans leave out as resources and take up residents. this is whats called the vacuum effect. When an animal leaves a territory unguarded, and new animal will move in to take advantage of the resources.
Trapping, spay/neutering the cats and putting them back in their territory is the BEST way to stop the over population problem. When the cats are fixed they change their focus from making babies to guarding their territory. This also prevents new cats from moving into the area.