
Transcritical bifurcation

In bifurcation theory, a field within mathematics, a transcritical bifurcation is a particular kind of local bifurcation, meaning that it is characterized by an equilibrium having an eigenvalue whose real part passes through zero.
A transcritical bifurcation is one in which a fixed point exists for all values of a parameter and is never destroyed. However, such a fixed point interchanges its stability with another fixed point as the parameter is varied. In other words, both before and after the bifurcation, there is one unstable and one stable fixed point. However, their stability is exchanged when they collide. So the unstable fixed point becomes stable and vice versa.
The normal form of a transcritical bifurcation is
This equation is similar to the logistic equation, but in this case we allow formula_2 and formula_3 to be positive or negative (while in the logistic equation formula_3 and formula_2 must be non-negative).
The two fixed points are at formula_6 and formula_7. When the parameter formula_2 is negative, the fixed point at formula_6 is stable and the fixed point formula_7 is unstable. But for formula_11, the point at formula_6 is unstable and the point at formula_7 is stable. So the bifurcation occurs at formula_14.
A typical example (in real life) could be the consumer-producer problem where the consumption is proportional to the (quantity of) resource.
For example:
where