Explain why no two organisms can occupy the same niche at the same
time.

The competitive exclusion principle, also called Gause's Principle, states that when two species compete for exactly the same resources (thus, they occupy the same niche), one is likely to be more successful. ... This principle means: no two species can sustain coexistence if they occupy the same niche.

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if the hares very faster than they are now, and thus the lynx's hunting success rate drops because it will be harder to catch them, the rate that will change will be the pray-rate. the pray-rate will change because more hares will be able to survive, thus more individuals for reproduction, so the end result will be increasing population of hares.

the predator-rate will remain pretty much the same because there will be more than enough pray, but the hunting success rate is stagnant.