townhouserOnt - I have had the same query and have found a great plant...Garden Chives or Allium schoenoprasum. Not only does it come quickly in the spring, bloom in just a week from now, or so, and not get ratty or invasive, has lovely pink-rose-colored2" balls of flowers (of course in the Allium Family), and it repels insects that may want a bite of your roses. They are a pretty perennial plant, and you can eat them to your heart's content (no spraying needed), and you're doing your part in the IPM management of your garden! Roses can also get bare around ground level during the season, and I have found they mask this area and look great all year. Hope this helps you in making your choice. Best, PP

r u sure its not a persulane? They r sold at nurseries,& called a weed.I call them zombie weeds,cause they just wont die!!!! If you dig them up,roots and all & leave ANY part(leaf,stem,root) on the ground,it will re-root itself & if u dont seal them in a bag get them far,far away asap,they will project their demon seed all over.They r the devil

Thank you both so much, sorry I was only able to post the "link" -- still learning how to copy photos into the message. You are right also (as I'm sure you realize) as I asked a ladyfriend who has 4-5 acres and has planted mostly natives, etc. for years, also tags and releases Monarchs every year. I realized after she told me that and I read your replies that I had planted Bloodroot 4 years ago and it came up (love the white flower) first year and not after that. BUT, it was 5 feet away from that spot. My friend said that they can move but also that ants can carry the seed and perhaps that's how it "moved". Thanks again for your help.

It is sweet allysum, Lobularia maritima, This is an annual so generally not considered to be a groundcover. However it reseeds rather freely so once planted and left to flower, it will typically come back with such regularity that it almost appears to be a perennial groundcover.