It dies at the end of three months (5)
I believe the answer is:
ember
I believe this is a double definition.
'it dies' is the first definition.
(the embers are the dying remains of a fire)
'the end of three months' is the second definition.
(September, November and December)
'at' is the link.
(Other definitions for ember that I've seen before include "what causes a warm glow" , "Burning coal" , "Still smouldering fragment of wood or coal" , "Just a glimmer" , "Wood still burning" .)