Not always, it would be too inefficient for species, when loser dies. After he lost battle, his testosterone greatly drops, to make him submissive and not agressive anymore, so he can leave female to winner. He just goes live by his own till next season, when he will compete for females once more.
You can clearly see this behavior in humans too. When Chad bullies and incel, his testosterone increases, because he's a winner, while low T incels have his T reduced even more, to avoid any competition and be fully submissive. It's like vicious circle, we was doomed from the start. Since the beginning, we had worst T compared to Chad, yet we have to compete with him, and after we lose, get our T reduced even more, while his increases. Nature pretended us to be genetic losers.