Sounds like a bad alternator. Is it a 2.6? The fuel pump relay energizes the fuel pump from battery on start-up then the relay flips over to power the pump from the alternator. The reasoning is if you're in a crash and break a fuel line you don't want to keep pumping gas. Could be a bad relay. Switch it out with a matching one to test that theory. Changed your fuel filter?

You might load test the battery too. I know when my alternator went, the battery quit about two weeks later. If the battery is on the low side of good I'd replace both the alternator and battery.