When the hare population increases, what happens to the lynx population? Why?

The lynx population would probably increase as well. That is until the hares destroy the presence of producers and all of the populations start going down.

The lynx population will increase. Then the hare population will drop because they're being eaten by all the lynxes and so the lynx population will drop too because they won't have much food. These things can be modeled with differential equations.