In large volumes of water you can have self sustaining populations of fish, shrimp and plants. However, smaller water bodies don't work well.
The other issue is to do with diseases and chemicals released by the fish. Plants will remove ammonia and help keep algae under control, and provide shelter, habitat and breeding sites for the fish and shrimp. But plants will not remove hormones/ chemicals released by the fish, and nothing will eat or dilute the disease organisms in the water.
Fish live in a soup of microscopic organisms. Without regular water changes, these organisms build up in number and eventually affect the fish. The pH will also drop over time and this can be harmful to the fish, depending on what species they are.
So you can have a smallish tank with some shrimp, fish and plants in, but you will still have to do regular water changes to dilute disease organisms and chemicals released by the fish.
Unfortunately, having a system like this will limit you to how many fish you can keep, and most people that try this sort of set up, have too many fish and end up having to feed them anyway. You would be better off culturing live foods outdoors in ponds or plastic storage containers, and adding the live foods to an aquarium with fish and shrimp in.
