BEIJING — Much more is at stake in the American decision to challenge China by sending a destroyer near islands it built in the South China Sea than a handful of rocks, even if they sit on major shipping lines and deposits of natural resources.

China, analysts say, is seeking to establish a sphere of influence in these waters — and edge out the United States.

What that means — whether it represents a crisis, or a natural and inevitable shift given China’s economic strength — depends on whom you ask. But there is little doubt that China is thinking big about how these islands could limit America’s military options, about how control over these waters could give it leverage over key trade routes and about how making the United States look hapless could strengthen its diplomatic clout in the region.

“They have a game plan; it is very clear what it is,” said Christopher K. Johnson, senior adviser on China at the Center for Strategic and International Studies in Washington at a recent seminar. “Sometimes, I think it is easy to get lost in the weeds on what has been built on which island.”