Theoretical ecology uses mathematical tools to place the nature of interactions into a formal, analytical framework. A goal is to understand features governing population dynamics and species diversity. Special emphasis has been placed on ecological competition and predator-prey dynamics. These studies often focus on the presence or absence of equilibrium points in model solutions, which indicate the possibility of species coexisting with each other over the long term. Moreover, a fundamental difference between stable equilibrium points and unstable equilibrium points is recognized. In the former, dynamics tend to restore the system to its equilibrium point after a small perturbation, whereas in the latter, like a ball rolling off the peak of a hill, small displacements from equilibrium can result in large changes and even be catastrophic to coexistence.
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