Aquaponics requires a lot more knowledge than sticking a Betta splendens in a bowl.
First off, you need to know fish. Bettas and their cousins the Gouramis are members of the family Anabantidae; these are fish who take in oxygen primarily through an internal organ analogous to the mammalian lung. Their gills don't bring in the bulk of the oxygen they use. They breathe out bubbles of CO2 that simply rise to the surface and dissipate.
Furthermore, both plant roots AND fish need oxygen, so you MUST have an aeration system in any setup designed to host plants. The roots do not take in CO2. That's the job of the leaves.
Fish appropriate to aquaponics are warm-water gill-'breathing' fresh-water fish such as sunfishes (crappie, bass, panfish) or relatives of the cichlid family (tilapia are common ones), which can survive the warm temperatures necessary to plant health.
Fish also need a certain level of sodium in the water- yes, even freshwater fish- for their gills to function correctly. This level is higher than the cutoff where Na ions would lock out nutrients that plants need. Fish FOOD is very salty. Custom fish food designed for aquaponics is a must if this balance is to be maintained.
Fish of different sizes and life stages produce different levels of beneficial and toxic substances. Again, you must have aeration and a way to remove excess ammonia, and a correctly designed aquaponics system has a way to adjust the mix of fresh vs fish waste water delivered to the plants based on its EC reading.
Bettas do not eat plant roots, or any other vegetable material for that matter. They eat the larvae of aquatic insects, primarily mosquitoes, but freeze-dried bloodworms are the correct commercial pet food. I don't see how you are feeding that poor fish with the rocks in there.
Oh well this is sort of an old thread anyway.