It is the alternator, this supplies 100% of the current needed to run things when the engine is running, the battery is there for the starter and sidelights etc is left on at night.
On a lot more recent ones they are ecu controlled.
It's a vague possibility to be a seriously knackered battery, if so duff it is soaking up so much current it is overloading the alt, maybe charge battery overnight and see what occurs.
tabetha