The "vacuum effect" is B.S. spread by animal huggers (hoarders really) to convince people not to bother controlling pests. Yeah, the population will replenish itself but it takes time. When you don't see immediate results, it's because the population is larger than what it appears. Once you exterminate most of them you won't see much activity. Some animals are territorial but a good food source is too good to pass up and that area will be shared. Some feral cat colonies are almost a thousand in number so the theory that the resident animals will keep newcomers out is flawed.
I once had a neighbor who fed them which led to a booming rodent population. I started trapping them but she would release them so I started shooting them with an air rifle. I never saw more than 3-4 at a time so after I shot 3 I thought my work was done but there were several groups of squirrels visiting the feeding station throughout the day. I kept shooting them and after a few weeks, squirrel sightings were far and few beyond. Once in awhile I'd see a newcomer and promptly shoot him but the population stayed low after the initial decimation.
The neighbor wondered what happened and must have thought it was the food because she set out a huge variety of nuts that went untouched. She even constructed a contraption to keep the birds out, perhaps thinking they were somehow responsible for the squirrels' absence. When autumn came and the leaves fell, all the squirrel nests became visible but it was different that year. All the nests were unoccupied and therefore unmaintained so all around the neighborhood you could see the nests falling apart and getting smaller as the wind and rain whittled away at them.
A friend had a similar experience with squirrels raiding his garden. He started shooting them and after the initial depopulation, only a little maintenance shooting was necessary to make them very scarce. All his grapefruits, avocados, and tomatoes were in prime condition but were left completely untouched.