The only thing I know that might get rid of it naturally would be a flock of chickens:D They can tear up quite a bit of ground in short order. hmmm....a good argument for a handful of hens.
I have quite a bit of it myself. It's slowing creeping along:p
It's in my lawn, I just mow it - looks green in the summer - LOL I don't free range, but try very hard to not use any chemicals unless absolutely necessary.
I do have success using a mixture of dish soap, salt and vinegar to...
Do your ducks ever eat it? As far as I always knew, it's edible. So I did look that up. If you want to rid yourself of it and your ducks are actively foraging, the safest way would be to pull it up. Chemicals would kill it I'm sure, but that is not practical with foraging animals...