How to get rid of stonecrop sedum in lawn
I think this is a related plant called mossy stonecrop (Sedum acre), which is much more common in Michigan. Mossy stonecrop is a perennial weed that was originally introduced as an ornamental, but has escaped cultivation in some areas. It reproduces by seeds, stolons (stems that grow along the ground), and stem fragments.
Control will be the same ultimately regardless of which stonecrop it is. For small patches it can be removed manually by digging the plants out and disposing of all material (above and below-ground). For larger infestations, you may want to explore herbicide options. Standard turfgrass herbicides that target emerged broadleaf weeds (e.g. dicamba, 2,4-D, triclopyr, etc.) may have some effect, but may only suppress mossy stonecrop and not achieve complete control. Another option is to apply glyphosate (e.g. Roundup Weed and Grass Killer, though there are many others).
When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid green plant material, exposed roots, and injured bark of desired plants. You will likely kill some of the lawn in this case (near the treated area). Third, glyphosate is relatively safe in the environment when used as labeled. It adsorbs strongly to soil in most cases (i.e. clay and organic matter), allowing even sensitive crops to be planted shortly after application; meaning no carry over issues are expected. Fourth, glyphosate alone can take up to 14 days to show full activity under ideal growing conditions. Retreatment of the area may be needed depending on the degree of infestation. Glyphosate is most effective for perennial control in the fall but can be applied anytime the plants are actively growing (temperatures consistently above 50F). Finally, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).
You'll want to replant grass once you are sure the mossy stonecrop is controlled. Once you replant the grass, keep an eye out for any individual plants that may come up from seed in the following seasons as there is likely a "seed bank" present that may come up if the opportunity arises.
Once you are rid of the weed there are some steps you can take to help give your grass a competitive edge. Consider fertilizing particularly in the late summer, around Labor day, mow at a height of 3" or greater, and water when the weather turns dry in the heat of summer. You can read more information about summer lawn care in the free MSUE bulletin E-3180 (https://www.canr.msu.edu/resources/michigan_fresh_summer_lawn_care)
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