In Lord of the Flies, how does Jack contribute to the survival of the boys?
I think Jack's key contribution to the survival of the boys is to provide food, though you could maybe also argue that he is the one who best understands the need for some sort of protection. Here's Golding's panoramic shot of Jack as chief:
A fire burned on the rock and fat dripped from the roasting pigmeat into the invisible flames. All the boys of the island, except Piggy, Ralph, Simon, and the two tending the pig, were grouped on the turf. They were laughing, singing, lying, squatting, or standing on the grass, holding food in their hands. But to judge by the greasy faces, the meat eating was almost done; and some held coconut shells in their hands and were drinking from them. Before the party had started a great log had been dragged into the center of the lawn and Jack, painted and garlanded, sat there like an idol. There were piles of meat on green leaves near him, and fruit, and coconut shells full of drink.
The constant diet of fruit gives the littluns diarrhoea, and stomach aches. Jack is the one responsible for giving the meat and protein (which undoubtedly is part of the reason the boys don't get seriously ill on the island!). Jack provides food.
Yet also, Jack's hunting and hunters, his emphasis on a violent "army" under his control, makes the boys feel less vulnerable to an attack from the (anyway imaginary) beast. Jack is much better than Ralph at keeping security and a creating threat of violence.
Jack is a boy of action who wants to do things: "We hunt!" He and his group are baser than Piggy and Ralph; they are not the thinkers of the group because they are concerned only with primal needs, they do not consider maintaining the fire in the abstract hope of being rescued. After he has regressed to the savage level, Jack does not seem to be concerned about being rescued. Instead, his energies are put into asserting his power, becoming cruel and destructive as he crushes Piggy's glasses and shouts at Ralph, "Bullocks to the rules!"
As he becomes more powerful, Jack embraces the "beast" within him, paints his body, and regresses further. He is more concerned about the control he wields by manipulating the boys' fear than he is in providing for the boys' survival as his hunters kill Simon and Piggy. (Of course, he does not want the boys under his control to die as he must use them.
When the officer arrives on the island, Jack is thus depicted:
A little boy who wore the remains of an extraordinary black cap on his red hair and who carried the remains of a pair of spectacles at his waist, started forward, then changed his mind and stood still
For Jack there probably is no rescue; he has degenerated too far.