separation anxiety

from The American Heritage® Dictionary of the English Language, 5th Edition.

• noun A child's apprehension associated with separation from a parent or other caregiver.
• noun Anxious behavior exhibited by a domestic animal in the absence of its owner or custodian.

from Wiktionary, Creative Commons Attribution/Share-Alike License.

• noun A psychological condition in some children, characterized by apprehension when separated from a parent
• noun Similar anxious behaviour of a domestic animal in the absence of its owner