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I want to know about apex predators. Why are they so important for nature? When no one hunt them, how is their population controlled by nature?
Answer 1:

Apex predators are important because they operate as a negative feedback on the prey populations, providing ecological stability. Their population is largely controlled by abundance of food, and apex predators unable to find enough food will die. In general, apex predators exist at low populations and are therefore vulnerable to hunting, both of the predators themselves and of overhunting their prey. There are additional more complicated dynamics too, including a fairly recent "Mesopredator Release Hypothesis." This is the idea that if you remove the apex predators from the food chain, predators just below apex (mesopredators) will quickly expand their populations, leading to overhunting and possible extinction of their prey. Essentially, apex predators promote stability in ecosystems by keeping their prey populations balanced correctly.

Scientists often model these effects on populations with predator-prey dynamics mathematical models, such as the Lotka-Volterra (wolf and rabbit) model.
LotkaVolterra is a good summary of it, and while it can be advanced, it gives some insight about how these populations work.

Answer 2:

As with almost everything in ecology, this is complicated.

Apex predators are the animals (usually animals, but sometimes fungi) that are at the "top" of the food web, i.e. they eat other things, and nothing eats them. Depending on the structure of the ecosystem and the factors forcing it, they may or may not control the populations of every other species in the food web, all of the way down to the plants and algae at the bottom.

Generally, which happens depends on the biology and climate driving the ecosystem from the bottom. If factors such as water, light, inorganic nutrients, or disturbances (e.g. fire, storms) are common and/or unpredictable, then the ecosystem will be structured from the bottom-up, with the plants/algae determining populations of everything else. However, in a more stable environment, often the apex predators will reduce the populations of what they eat, which will in turn increase the populations of the species on the next level down, and so on.

Because of the second law of thermodynamics, energy is lost through each level of the food web. This means that there is very little energy left over for the apex predators, which has the consequence that the apex predators often have smaller populations, and are more likely to be endangered.

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