Transcritical bifurcation

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Illustration of the transcritical bifurcation. The stable (red) rest position becomes unstable (blue) and vice versa.

Bifurcation diagram of a transcritical bifurcation. Stable fixed points are shown in red, unstable in blue.

The transcritical bifurcation describes a process in which the stability (“attractive” or “repulsive”) of two rest positions of a system is exchanged. It is therefore a certain type of bifurcation of a nonlinear system .

The normal form of the transcritical bifurcation is:

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu x-x^{2},}$

where is the bifurcation parameter. ${\displaystyle \mu }$

The transcritical bifurcation has the following equilibrium points:

${\displaystyle x_{1}^{*}=0}$

${\displaystyle x_{2}^{*}=\mu }$

If, with the normal form of a (d. E. To interfere with the fixed point) and neglect all terms of order , one obtains ${\displaystyle x=(x_{1/2}^{*}+\delta )}$ ${\displaystyle \delta \ll 1}$ ${\displaystyle \delta ^{2}}$

${\displaystyle {\frac {\mathrm {d} \delta }{\mathrm {d} t}}={\begin{cases}\;\;\mu \delta \;\;\;\mathrm {bei} \;x_{1}^{*}\\-\mu \delta \;\;\;\mathrm {bei} \;x_{2}^{*}\end{cases}}}$

for the temporal development of the disorder . ${\displaystyle \delta }$

So for is a stable fixed point (i.e. the disturbance decreases with time) and an unstable one (the disturbance grows). For it is the other way around. At the critical value of the bifurcation parameter , the (in this case the only) fixed point is indifferently stable . ${\displaystyle \mu <0}$ ${\displaystyle x_{1}^{*}}$ ${\displaystyle x_{2}^{*}}$ ${\displaystyle \mu >0}$ ${\displaystyle \mu =0}$ ${\displaystyle x^{*}=0}$

Discrete system

For a discrete system, the differential equation turns into a difference equation:

${\displaystyle x_{t+1}=x_{t}+\mu \cdot x_{t}-x_{t}^{2}}$

The position of the fixed points remains unchanged compared to the continuous system.

example

In the case of logistical growth , the change in a resource over time is proportional to its current value and the difference between this value and a limit , for example the number of animals in a certain area. Let the constant of proportionality be . If this resource is also consumed proportionally to its current availability with a constant of proportionality , for example through hunting, then the differential equation reads ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle p}$ ${\displaystyle k}$

${\displaystyle {\frac {\mathrm {d} R}{\mathrm {d} t}}=pR(S-R)-kR=(pS-k)R-pR^{2}}$

This can be converted into normal form by the variable transformation and identified . So for is a stable fixed point: If an animal were released into the area, the hunters would shoot it immediately and prevent it from growing. The fixed point , however, is unstable: If the hunters shoot too much game, even for a short time, it cannot recover and dies out if the hunt remains the same (strives against ). The behavior of the fixed points changes: becomes unstable, if the population increases for a short time, not enough game is shot to prevent it from growing onto the fixed point . This is stable, which means that the population only fluctuates when both too much and too little game are shot for a short time . ${\displaystyle x=pR}$ ${\displaystyle \mu =pS-k}$ ${\displaystyle k>pS}$ ${\displaystyle R_{1}=0}$ ${\displaystyle R_{2}=S-k/p}$ ${\displaystyle R_{1}}$ ${\displaystyle k<pS}$ ${\displaystyle R_{1}}$ ${\displaystyle R_{2}}$ ${\displaystyle R_{2}}$