Natural ponds have earth substrates which do a fine job of nitrite - nitrate conversion via nitrifying bacteria but nitrates are consumed by pond flora. To have a successful aquaponic pond it seems possible but one would have to overstock heavily to provide ample nitrates to supply growbeds with enough excess nutrients. It could be done but I'm not the expert to say just how much loading or grow bed area required to make it successful.
In a conventional aquaponic design all the variables are constant and fairly predictable. The Backyard Aquaponics link is probably the best source of AP knowledge available
