In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis as a parameter crosses a threshold value.
Domestication is a multi-generational relationship between humans and other organisms, where humans take control over their reproduction and care to have a steady supply of the organisms' resources. It can be argued that domestication is a form of mutualism, where both humans and the organisms are benefited. The domestication of plants and animals by humans was a major cultural innovation ranked in importance with the conquest of fire, the manufacturing of tools, and the development of verbal language.