In Lord of the Flies, how does the rise and fall of the conch correspond with the rise and fall of the boys on the island?

Piggy and Ralph discover the shell but it is Piggy who first notes how it could be used. Piggy is the intellectual one on the island. Fittingly, he suggests how to use the conch to summon, organize, and civilize the group. When Ralph finally gets a sound out of the shell, it is striking: 

A deep, harsh note boomed under the palms, spread through the intricacies of the forest and echoed back from the pink granite of the mountain. Clouds of birds rose from the treetops, and something squealed and ran in the undergrowth. 

The sound is powerful. Gradually, all of the boys flock to the origin of the sound. The entire structure of the first chapter is built around the effectiveness of the shell. It has allowed Ralph and Piggy to organize everyone. The conch is the symbol and also the tool that initially keeps the boys in a relatively democratic and civilized order. It is democratic because everyone has the opportunity to speak if he holds the conch. 

In Chapter 5, the boys are discussing the beast and they start fighting over the conch. It is now less of an organizing tool. Recognizing this and the inattention to the fire, a frustrated Ralph says to Piggy, "If I blow the conch and they don’t come back; then we’ve had it. We shan’t keep the fire going. We’ll be like animals. We’ll never be rescued." In other words, if the conch stops being useful, Ralph feels that the boys will be doomed. 

Another pivotal moment regarding the conch occurs at the end of Chapter 9. Jack has moved to other side of the island. When Ralph tries to assert his authority as chief, he uses the conch. But Jack has ceased respecting Ralph and anyone who holds the conch. Jack adds, "And the conch doesn’t count at this end of the island." More and more boys shift to Jack's tribe and the conch therefore has less power. Shortly after this, Jack riles up the boys and they inadvertently kill Simon in their frenzy. The conch is useless and savage behavior ensues. 

At the end of Chapter 11, Piggy and the conch are destroyed together. The symbol of order and the intellectual boy of reason perish together. It is a fitting tragedy. Any sense of order is completely gone at this point. In the final chapter, when Ralph is being hunted, the narrator notes "There was no Piggy to talk sense. There was no solemn assembly for debate nor dignity of the conch."