also rat race, 1934 in reference to aviation training, from rat (n.) + race (n.1). Rat-run is from 1870 in a literal sense.
A rat race is ... a simple game of "follow the leader" in fighter planes. The leader does everything he can think of -- Immelmanns, loops, snap rolls, and turns, always turns, tighter and tighter. ["Popular Science," May 1941]
Meaning "competitive struggle" is from 1945.