Of these convenience stores that you mention, how are they set up?  company stores?  franchise stores? In a small town, I think a store that an individual owns - as opposed to a company store - would have a better chance of staying around.  If a store is company-owned, if they don't meet the  company sales/ income target, it is more likely for headquarters to close it down.  If you have a individual-owned stores, they might be more likely to ride out the "rough patches".  The big question would be:  how to they find and buy inventory to sell?  Can they find a coop-type purchasing with other similar-minded stores?  

I run across a couple of similar stores in my travels.

Having said that, it seems like Dollar General stores are popping up everywhere around here. It seems like they have a goal to have at least one DG store in every township.