How do I deal with a muddy mess under a tree caused by dogs?

Lay down a 3-inch layer of bark mulch under the tree. Decorate the area with container gardens. As the dogs displace the bark mulch, you will have to replace it. Nonetheless, in the long run, I think you will find this less work than trying to grow plants in the ground in this area. When you select plants to grow in pots for this project, make sure that they are not plants poisonous to dogs.