Plant a bunch of flowers on the dirt spots. Supposedly, flowers help grow back grass more quickly. You thought grass dying was bad on New Leaf... You should have seen City Folk when they first introduced it!
http://animalcrossing.wikia.com/wiki/Grass_Deterioration
City Folk :
This feature was originally intended to create paths in frequently-traversed areas, which would add to the town’s own unique character. However, in many instances the feature has run amok, almost completely obliterating all the grass in an entire town, especially when players are chasing after insects such as Banded Dragonflies or Giant Petaltails.
New Leaf:
In Animal Crossing: New Leaf, it is reported that grass deterioration not only happens more slowly, but the damaged grass regrows faster. Also, when a Public Works Project has been completed and built, the grass under it will deteriorate instantly.
Fertilizer will not help restore grass. However, flowers can aid the process of grass regrowth. Rain and snow will also help grass slowly regrow.
Placing patterns on the ground such as paths protects the grass underneath from deterioration. It will not regrow any already-damaged grass, however.
