Transcritical bifurcation
Encyclopedia
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 logistic equation but in this case we allow
and 
to be positive or negative (while in the logistic equation 
and 
must be non-negative).
The two fixed points are at
and 
. When the parameter 
is negative, the fixed point at 
is stable and the fixed point 
is unstable. But for 
, the point at 
is unstable and the point at 
is stable. So the bifurcation occurs at 
.
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
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
, a field within mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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 logistic equation but in this case we allow
and to be positive or negative (while in the logistic equation and must be non-negative).The two fixed points are at
and . When the parameter is negative, the fixed point at is stable and the fixed point is unstable. But for , the point at is unstable and the point at is stable. So the bifurcation occurs at .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
-
is the logistic equation of resource growth; and -
is the consumption, proportional to the resource 
.