Fishing Tournament

A woman wins a four-day fishing tournament with a total of 36 fish. If she had caught two more fish on the first day, two fewer on the second, twice her catch of the third, and half her catch of the fourth, she would have caught the same number of fish each day but lost the tournament. In winning the tournament, how many fish did she catch each day?

Source: Parade, Ask Marilyn, Nov. 22, 2009