Our paver patio is surrounded by a raised planting bed, as seen in this photo, which is a bit dated. I've sinced tried to plant shrubs around the patio:

I have 2 dogs that INSIST on ONLY running back and forth throughout the mulch, it's like they're allergic to grass. I've tried planting shrubs all the way around to limit where they can run, but they kill anything I plant within a season. They destroy the edge between the mulch and grass, turn the mulch into mud, and then track mud all over the patio. By May the entire patio is a disaster, and we don't even want to spend time out there.

Any ideas for eliminating this mulch bed? Plant grass all the way to the patio? Seating wall? River rock? Help!