There is a relationship between the number of ducks per square foot, the number of hours per day the ducks have access, the type of vegetation, and the results.
For the first two, the larger the number, the less vegetation will remain.
The third, there are some plants that can stand up better than others, but that can be undone by the first two numbers being too high.
"Too high" is going to depend on: the particular ducks and their feeding program, availability of water to play in, the climate, and a few other things I am sure I missed (
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If your total area is too small (again, a somewhat relative number), no amount of moving the ducks will help much. One approach used is to have a sacrifice area - an area where the ducks are kept longest, and that would have something like sand or other substrate that drains well and is easy to clean, but with no vegetation.
Then the ducks can be let out into other areas for short periods of time.
That said, I think the original post was about what to plant. If you want something the ducks will like to eat, then greens are a good choice, beans, garlic and onions are no-no's (from what I have been told). They love berries. They will also enjoy a nice fluffy mulch they can work through to eat insects and such. Grass is yummy when it is tender (before it joints). They like watercress, so if you are handy you could create an aquaponic setup with watercress fertilized by duck-soiled water (check out what Wifezilla has begun).
I plan to give mine access to compost piles and garden areas at certain times of the year, but not unlimited. They ought to do a good job of prepping some beds and weeding others.