This story is science fiction, so a lot of exposition will be needed to explain all the strange happenings to follow. In the case of The Island of Dr. Moreau, it's almost the entire first-half of the novel. Seriously, there's a lot of exposition to be had. It includes Prendick's misadventures at sea, his time aboard the Ipecacuanha, coming to the island, and then realizing things aren't all tropical vacay in Moreau's abode.
We can tell this is the exposition because Prendick sets up all the information we'll need to understand the novel: setting, characters, relationships, and, best of all, the mystery of the island.
Oh, a word of warning. Prendick will keep explaining stuff throughout the novel; this is a natural byproduct of good science fiction, and Prendick's a bit yappy anyhow. However, the plot's exposition ends when setting up the situation gives way to letting the conflict kick into high gear. For our money, that's the end of chapter nine—just after Prendick's run in with Leopard Man.
At the beginning of chapter ten, Prendick discovers Moreau experimenting on the puma, and the action begins its steady rise toward awesome. The rising action includes Prendick's escape to the Beast Folk's village, his confrontation with Moreau, Moreau's explanation of his experiments, and the hunting of the Leopard Man.
We can tell this is rising action because the events get more exciting, the stakes get higher, and the conflict between Prendick's civilized mindset and Moreau's brave new world takes center stage. In short, it's the part of the novel where the pages practically turn themselves.
But wait a second. If this is the rising action, then why does it include scenes like Moreau explaining his experiments or Prendick discussing the Beast Folk? Shouldn't that be considered exposition, too? Well, these scenes are exposition in the sense that they expound information. True. But they are not exposition in terms of plot structure because the information they give us doesn't set up the mystery. Instead, they complicate the mystery, making it more exciting and intense.
The climax is the turning point, the moment where everything changes in the story. Chapter seventeen and eighteen are just that. In these chapters, the puma escapes, Moreau confronts it, and the two end up killing each other. It may seem like more action—and it is—but this action has some earth-shattering consequences for our island inhabitants.
We can tell this is the climax because everything beforehand led up to these moments. For example, it's the reason Wells mentioned that puma over and over and over and over again (this is what English teachers worldwide have dubbed foreshadowing). Also, everything that happens afterward results directly from these events. Speaking of which, if you would please direct your attention to the section below….
With Moreau dead, Prendick decides to bail on the whole island scene. This starts the falling action because it's the final, suspenseful push of the novel where we aren't sure of the protagonist's ultimate fate. If this were a Shakespearean tragedy, everyone would start dying at this part of the story.
And wouldn't you know it, everyone starts dying. Montgomery dies. M'ling kicks the bucket. The Sayer of the Law goes the way of the dodo. The Fat Lady sings for Prendick's companion, Dog Man. Hyena-Swine is put down. The rest of the Beast Folk either die or revert to their animal states. With all this death, one lingering question still invests us in the novel: will Prendick escape the island or not?
Resolution comes when that last question is answered. Basically, it's answered in the final chapter, and the answer is yes. Prendick escapes and travels back to England where he lives the rest of his life in peace, though he is still haunted by the memories of Moreau's island. With Prendick in England and everyone else dead, all of the plot lines revolve themselves nice and neatly.