Desert people in fierce competition (3,4)
I believe the answer is:
rat race
This clue is a double definition.
'desert people in' is the first definition.
(I've seen this before)
'fierce competition' is the second definition.
(Other definitions for rat race that I've seen before include "The competitive society" , "Fiercely competitive struggle in business" , "unhealthy competition" , "competitive lifestyle" , "constant effort to get ahead" .)