I've always thought of zoysia as a sun-loving grass species....as are most turf grasses. I guess you could call it somewhat tolerant of mildly shady sites, but don't expect it to thrive under the canopy of a large shade tree. In the deep South, promises that At. Augustine will do well under the canopy of live oaks trees run rampant. It doesn't thrive, but barely hangs on until it eventually dies. St. Augustine is more shade tolerant than zoysia. You can probably help it to languish longer by careful cultural practices: set the mower higher than usual, water deeply but infrequently, avoid excessive fertilization (or even normal doses) to help shade grown turf grasses do better, longer. If the bare patches coincide with the tree canopy, then you might consider a nice thick layer of mulch, rather than grass. Then you can plant some evergreen ground cover here and there. Many ground covers can really take off under such conditions! Where are you located? We might be able to come up with some good ground cover options for you, but can't do so without know your general location.

Plant hormones called Gibberellins have been successfully used to shorten and even eliminate the dormancy periods of some seeds, however, they also frequently have other effects, such as stimulated growth and alteration of sex expression. The only way to determine if it will work with D. americana is to try it and see what happens. The common method for dealing with this type of dormancy is called cold stratification. Refrigerate the seeds in damp paper towels for the prescribed period, and follow with normal indoor seed starting. Winter sowing will provide essentially the same service, outdoors, but under controlled conditions. See the Winter Sowing Forum for details.

I would recommend avoiding the horseherb as it's almost too hardy and propogates very quickly. As I've written in a previous post, you can subject it to the severest of treatment and it will still come back to haunt you in exponential numbers, propagating and invading every square inch of dirt in your yard and every surrounding neighbors' yard: you can mow it, (better yet) scalp it, starve it, deprive it of water, beat it, wack it to the ground, poison it in 100-degree heat, and even curse at it. There's no killing it! Horseherb firmly roots, which makes it time consuming to weed by hand, but it's the only way to get rid of any strays that end up in your flower beds AND BELEIVE ME, YOUR FLOWER BEDS WILL REQUIRE WEEKLY MAINTANANCE DURING THE GROWING SEASON. What I did in your situation is replace my St. Augustine Floratam grass with St Augustine DelMar, which is very hardy and has superior shade tolerance of any of the St. Augustine and has excellent drought tolerance. See the link below. Here is a link that might be useful: St. Augustine DelMar

I suspect the idea that Tifgrand is shade tolerant is a lot of marketing BS, but you're the first I've read from someone who actually installed it. Weeds would be expected in a normal bermuda lawn in the shade, so that sounds like what you're getting. By fertilize regularly are you saying you use a high N fertilizer every month? That would be normal for bermuda if you want it to look good. How high/low are you mowing? Bermuda will become the most dense (to keep weeds out) when mowed low. It thins out when it gets up above 1.5 inches. Are you watering once a week? When you water more frequently you are encouraging weed seeds to sprout and take root. If you let the soil surface dry out for several days and even get really warm, that usually keeps the weeds out. Of course sometimes Mother Nature will send rain that you can't control. If you're doing all the above, then the problem in the shade almost has to be the grass itself not being all that shade tolerant. There is a zoysia variety called Shadow Turf that sort of looks like hybrid bermuda and will tolerate deep shade. You might look into Diamond zoysia. Supposedly it is shade tolerant and is likely more available in your area. St Augustine does not look anything like bermuda but it would likely grow dense enough to keep weeds out in the shade. Or you might decide to change to a different kind of ground cover altogether.