Captain America vs. Legolas

Captain America gets his shield and a long tactical knife, Legolas gets his bow and 10 arrows and his two fighting short swords (long elven knives?).

Fight 1: Takes place in a jungle, though both combatants have had 1 day to familiarize themselves with the surroundings (though not allowed to lay traps of any sort until the actual battle begins). Combatants start in random locations roughly 1 km away from each other and hidden from each other's view. They'll need to hunt down and kill each other.

Fight 2: Is the same scenario as Fight 1 except it takes place in a city setting.

Fight 3: Is in a gladiator arena

Discuss.

Legolas is just as accurate as Hawkeye with the arrows, only quicker. I think Cap can get sniped off guard in the first two scenarios. Cap would win the third.