A saiyan is a species that is biologically similiar to humans, except in one particular way. Every time they come back from the brink of death, their strength and speed increases. They also become more resistant to the thing that almost killed them. They love to fight, and their entire culture revolves around battle and war. At some point, they reach a level of power that allows them to transform into a more powerful version of themselves.
I need to create a "realistic" version of this race, and they must meet these set of criteria:
Their strength, speed, and agility increase after surviving a fatal attack, and become more resistant to damage.
After a lifetime of fighting, they can learn to overclock their bodies like a computer for a limited amount of time. They become stronger, faster, and tougher for that time period, but burn out quickly. It also takes them a long time to recover.
How can I biologically make this species to meet these requirements?