What prompts David, Petra, and Rosalind to flee in The Chrysalids?

David, Rosalind, and Petra had to run away from Waknuk society because their telepathic abilities had been discovered.  If they didn't flee, their lives would be in danger. 

By chapter eleven, the group of telepaths is fairly competent at hiding their abilities.  Unfortunately, they are not flawless at controlling their abilities.  In chapter eleven, Petra is spooked by a creature, while she is playing in the woods.  She lets out a big telepathic scream, and several of the telepaths run to Petra's aid.  Another member of the Waknuk society happens across the group, and he is mystified at how so many of the kids managed to converge on the same spot.

I explained that my sister's pony had been attacked, and that we had answered her calls for help. He wasn't willing to take that at its face value. He looked at me steadily, and then turned to regard Sally and Katherine.

"Maybe. But what brought you two here in such a hurry?" he asked them.

"Naturally we came when we heard the child calling," Sally told him.

"I was right behind you, and I heard no calling," he said.   

This incident forces the telepathic group to be even more cautious than normal, but definite suspicion surrounds the group.  

In chapter twelve, Sally and Katherine are caught.  The two are tortured, and they are forced to reveal who several other telepaths are.  

"Sally, dear, of course we're not blaming you — either of you. We understand. But we must know what you've told them. How much do they know?"

"About thought-shapes — and David and Rosalind. They were nearly sure about them, but they wanted it confirmed."

"Petra, too?"

"Yes ... Oh, oh, oh...!" 

Rosalind, David, and Petra are all forced to flee.  Their goal is to make it to the place called "Sealand."  

What challenges do David and Rosalind encounter in The Chrysalids?

David and Rosalind are both "Deviants." They have a genetic mutation that gives them telepathic abilities. The entire Waknuk society is scared of any kind of change to the gene pool that falls outside of what they consider normal. Consequently, the entire society is watchful for any kind of genetic mutation, and the society deals quick and brutal punishments to anybody displaying genetic abnormalities.  David and Rosalind face this problem. If they are discovered, they risk being banished to the Fringes, being sterilized, or even being killed.

If the worst comes to the worst, and you can't save Petra, it would be kinder to kill her than let her go to sterilization and banishment to the Fringes—a lot more merciful for a child. You understand? Do the rest of you agree?

Many of the problems that David and Rosalind face are a result of having the telepathic gene mutation. They are forced to keep their secret, and they must behave in such a way as to not make people suspicious of their gift. They are eventually discovered, and they are forced to run away to the Fringes. The Waknuk people then try to hunt them down, and the people in the Fringes are not welcoming either. Throughout the novel, David and Rosalind are either in hiding or on the run to protect their lives.