Rats reproduce faster than... well faster than rats.
Noah saved one pair of rats to what purpose I have yet to understand. Ever since that day humans have struggled to stay even with Herr Rat. Some times i think that the rats are gaining on us.
Perhaps training your dog to increase the accuracy of his "biological byproduct applications" would produce better results. Get him to aim squarely between the rats' beady little eyes. I know that this won't work but for those of us who have dealt with rats before, you gotta love the imagery.