What is the largest population that an ecosystem can support over time?

The largest population that an environment can support is called its carrying capacity.

Determining the carrying capacities for most organisms is fairly straightforward. For humans carrying capacity is much more complicated. The definition is expanded to include not degrading our cultural and social environments and not harming the physical environment in ways that would adversely affect future generations.

For populations which grow exponentially, growth starts out slowly, enters a rapid growth phase and then levels off when the carrying capacity for that species has been reached.

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