Rabbits harming lawn. Cats the solution?

There are lots of rabbits in my neighborhood. Some of them like to lounge on my lawn which is fine except the loungers usually destroy one-foot- long swaths of zoysia grass after they've been laying on the same spot for a few days. Then they'll go on to destroy a new spot.
There are a couple cats that roam around the neighborhood. Would it be a good idea to feed the cats with tuna (or some other food) whenever I see them, so maybe the cats will scare the rabbits away? Will this work, or will the rabbits just return when the cats leave after finishing the tuna?
Will the cats actually keep a helpful eye on your lawn after they learn that they can get a meal there when they see you? And will this be enough to keep rabbits away?
I guess another question is will cats cause their own problems to the lawn?
If it makes a difference, I think the rabbits might actually be hares, and I live just outside St. Louis, Missouri. Thanks.
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