When you talk about the turbine being able to self-sustain, perhaps an analogy will help. Think about a single-cylinder piston engine.
The intake and compression strokes provide a charge of fuel-air mixture into the cylinder, compressed, and ready for ignition. The spark plug ignites the mixture, and the power stroke turns the crankshaft, and the exhaust stroke empties the burned charge.
But what drives the machine into the next cycle? What provides the energy for the next compression stroke?
It's the stored energy - sometimes an external flywheel (or a mower blade, etc.) - that pushes the crank around through the first two strokes of the next cycle.
Similarly in a turbine engine, there is stored energy is the compressed air, and the compressor designer's job is to prevent a stall (surge) - i.e. that air is not back-flowing out the inlet. He makes sure each airfoil sees its incoming flow at a safe angle of attack, and when this rule is satisfied, the cycle happily continues.