Is animal over-population a serious problem?

Hello! I have been a vegetarian for about 4 years now and I LOVE it! I am actually thinking about becoming a vegan and have been for a while. I have written multiple papers on why vegetarianism (when right for the person) is the overall better choice. However, there is one argument that meat-eaters make that I can never counter: what about overpopulation? If no one kills and eats animals, wouldn't they become a problem in our society? I would love to hear some information or ideas on how vegetarianism does not lead to overpopulation of animals or how overpopulation would not be a problem. Thanks!