So, when the pump is off, no water flows through the stream?
It could be the plumbing as others have stated, but isn't it also possible the leak is in the stream itself? This can be the case more often.
Many times the stones in the stream can get kicked around or misplaced by wildlife which can divert the water enough to cause leakage.
Is the stream liner sufficiently shaped to ensure water doesn't flow over the edges? Maybe a slight berm of soil under the liner along the edges is needed.
Or you could have an actual puncture in the stream liner, again caused by wildlife possibly stepping on a sharp stone.