This symbiosis goes back to the original novelization of the stage play, called "Peter and Wendy", published in 1911.
In there it explains that Neverland is basically an attempt at a map of a child's mind, and each one is different.
I don't know whether you have seen a map of a person's mind. Doctors sometimes draw maps of other parts of you, and your own map can become intensely interesting, but catch them trying to draw a map of a child's mind, which is not only confused, but keeps going round all the time. There are zigzag lines on it, just like your temperature on a card, and these are probably roads on the island, for the Neverland is always more or less an island...
Further corroborating that each one has their own island:
Of course, the Neverlands vary a good deal. John's for instance, had a lagoon with flamingoes flying over it at which John was shooting, while Michael, who was very small, had a flamingo with lagoons flying over it.
Later, they recognize their islands when flying to Neverland with Peter, which annoys him, detailed on pages 64-65. The final suggestion that without their owner, the Neverlands fade away is detailed on page 75, the beginning of the chapter titled "The Island Come True".
Feeling that Peter was on his way back, the Neverland had again woke into life. We ought to use the pluperfect and say wakened, but woke is better and was always used by Peter.
In his absence, things are usually quiet on the island. The fairies take an hour longer in the morning, the beasts attend to their young, the redskinds feed heavily for six days and nights, and when pirates and lost boys meet they merely bite their thumbs at each other. But with the coming of Peter, who hates lethargy, they are all under way again: if you put your ear to the ground now, you would hear the whole island seething with life.
While this doesn't explicitly say that Neverland dies without Peter Pan, there is a clear progression of each individual island being the creation of a small child, and that without the child, the island goes dormant or disappears (For children that grow up and forget about Neverland). It also shows that Peter is the driving force for his Neverland.