rapid population increases drive the search for more productivity.
What is the function of "for more productivity"? is it a complement or an adverbial?
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If we rephrase it to say,
A rapid population increase drives the search for more productivity
it becomes self evident: A rapid increase in population [an increase of people] creates a push for more productivity, or, gives people the drive, or motivation, to find ways to become more productive [in tandem with population growth].
Productivity might refer to producing more food, more housing, more living needs, etc.