Because when ever you run a lot grass can go away. It is best to put flowers next to the where the grass was gone I hear that it grows faster. Or put paths down and hide where the grass is gone.
There are some dirt spots which are permanent. Maybe Nintendo thought that gave realism to the game, but actually it looks kind of ugly. If only they appeared only by the cliffs... Then there are dirt spots which your character creates by walking on them (running on them makes them even worse), so I suggest you use paths so as to avoid this degradation.
you have some permanent spots that always will be dirt, but also the grass wears off!!! so if you run a lot around your town it will turn int o mud hell sorry ):
What you can also bear in mind is that some people (like me) love grass deteriortion and spend lots of time running only in certain places to create natural dirt paths. There are some great dirt path towns you can look up and visit (if you have the dream suite), definitely worth a look, prettier than you'd think! But yeah, those permanent patches are so annoying!
Just create ur own path and try to cover it up it will cover up the nastiness and prevent it happening in the future, it's coming from sprinting on the grass which wares it down