The irony of absurdism
This novel contains absurdist elements, meaning that much of the plot is intentionally wrong, off-putting, and insincere. The novel celebrates bestiality, for instance. This absurdism is inherently ironic. In fact, to be so unpredictable and random is to stretch irony to its limits. It's like a long unbelievable joke.
The irony of Adam's rejection
Madelene rejects Adam, whose namesake is the legendary Biblical character Adam who was the first man to ever live on the earth. In this case, it's appropriate to analyze the religious significance of his name, because both religion and names are issues raised in other ways (Erasmus is a theologian; the captain constantly changes his own name). This all means that when Madelene rejects "Adam," she is ironically rejecting the claims of human culture and society. She is saying "no" to being a human and saying "yes" to being an animal.
The irony of Erasmus
Erasmus is a character invented by irony. He is unusually tranquil in his cage, meditating and waiting for his escape. When he gets out, he easily outsmarts the humans. The humans confront him and learn that astonishingly, he can speak English, and he's a brilliant moral and religious philosopher. These ironies are crazy, but they're not exactly random—Erasmus's unlikely qualities make the reader consider their own sense of superiority above nature and the animals.
The ironic marriage of Madelene
Madelene's story is ironic to her, but perhaps to experienced readers, it seems typical. The irony is that Madelene gets sweet-talked into settling for a pleasant life as a house wife. She expects to be fulfilled and thrilled by her marital bliss with Adam, but actually, she instantly begins to feel dread and panic, and she drinks to calm her nerves. She becomes a serious alcoholic; not exactly the marital bliss she had hoped for.
The irony of the anti-paradise
This novel features an Edenic paradise for much of the second half of the story when Madelene and the ape are making a life for themselves. This is supposed to be the other half of Madelene's dilemma in Adam's house. When Adam was her spouse, she felt claustrophobic, controlled, and hopeless, but in nature, she can return to her primal instincts and be fully herself. Ironically, that's the exact opposite of what Adam thinks heaven should be like.