"灭亡的原因" can be translated into English as "the causes of extinction" or "the reasons for downfall/demise", depending on the context.
If referring to the extinction of a species, "the causes of extinction" is more appropriate.
If referring to the fall of a civilization, empire, or organization, "the reasons for downfall/demise" might be more fitting.
For example:
"Scientists are studying the causes of extinction for endangered species." （科学家们正在研究濒危物种的灭亡原因。）
"Historians analyze the reasons for the downfall of ancient civilizations." （历史学家分析古代文明灭亡的原因。）