I have to say I've not looked into Aquaponics too deeply, mainly because I can't see it working.
On the other hand I have four Hydroponics systems operating here, with the first one running continuously now for 7 years, so I do know a bit about hydroponics.
It does sound a perfect system, with plants cleaning the water for the benefit of the fish and the fish "fertilising" the water for the benefit of the plants. But I'm pretty sure it's not that simple.
The only system I can see working is where the foul water is "watered" onto plants growing in soil, but not growing plants in inert media with the water recycled.
For instance vegetable plants need a wide range of trace elements and brassicas (for instance) need high levels of nitrates, which would not be produced in high enough concentrations by a small quantity of fish. Which means either the plants would be stunted or you would have to add extra, but ONLY THE EXACT AMOUNT the plants would absorb at that time, or the fish would suffer from high nitrate levels and the fish tank would become a pea soup of algae.
When I grow cauliflowers in my hydro I have to add a largish quantity of magnesium (Epsom Salts) once a week to prevent leaf tip burn which I'm sure would be detrimental to fish. The Epsom Salts I add to the tank is enough for 7 days, if there were fish to consider I would have to add it daily in exact quantities, the same as the nitrates.
So I'm sure if totally necessary it could be done, but it would be far too complicated and time consuming for me, and as I'm not that keen on fresh water fish anyway, I'll just grow my fruit and veg on their own.

OK, I stand corrected.
I will now have to spend some time Googling to see what I'm missing (and misunderstanding)
It's a good job we have 2 thoroughly wet days forecast.

Seva Tokarev wrote:
Still, what about trace elements? Are they supposed to come from fish feed? Does it impose additional requirement on the latter?