monsters have a stamina bar like the player. they eat or try to when it reaches zero. pin attacks also regenerate stamina. some attacks they do takes away bits of this bar. weapons like Hammer and Hunting Horn that cause impact damage also cause exhaust damage. There're other attacks too like SnS and Lance shield bash. staggering them hits their stamina bar too
you know how enemies have a tolerance to sleep and other status effects where future applications get harder to stick? exhaust is the same way. let's say you're fighting a monster with 500 stamina, exhaust tolerance of 300, and an increase of 75. after ten minutes, it uses a lot of stamina draining attacks. it's now down to 400. you hit it a bunch of times with your HH, and you dealt enough exhaust damage to reach the 300 tolerance. 400-300 = 100, and it will stagger. it does more attacks after and the bar reaches zero. it's now tired - drooling, some attacks fail, sits there doing nothing, and its tolerance raises up to 375. rage mode ignores fatigue and monsters are back to their original stamina values while keeping its current tolerance after a few minutes of being tired.
drawing a blank on what monsters tire easily.
