It appears that the mentality is to allow them to live so they reproduce and over-populate & wipe out the food sources faster so their population crashes due to starvation & disease?

I kill every one I see, and when one takes it's place I'll kill it as well. There has to be a limited number reached somewhere, but I assure you there's no limit to how many I'm willing to kill.

I do like the parvo-puppy idea....