Ok, first of all, I am no expert, but I do not think you have anything to worry about if your duck drank the "water" or ate the leaves if at least six months has passed since treatment. I use imidacloprid on my young non bearing trees, and in less than six months, the citrus leaf miners start damaging the leaves again. Also many plants, even some vegetables and teas contain small amounts of nicotine, so I would think their systems would be adapted to it somewhat. After heavy rains, I would think it would be diluted to a safe level. Maybe someone more knowledgeable than I can chime in.
TFN