---
tag: KFY58YKR3N
kind: concept
name: Transcritical bifurcation
msc: [34C23, 37G10]
sources:
  - url: https://en.wikipedia.org/wiki/Transcritical_bifurcation
same_as:
  wikidata: Q1502581
  wikipedia: Transcritical bifurcation
---

## Summary

A transcritical bifurcation is a change in a family of dynamical systems at which two branches of [[equilibrium points|0N8NGKDZDZ]] cross and exchange their stability.
Both equilibria exist on both sides of the critical parameter.
It is typical when one equilibrium is forced to exist for all parameter values, such as the zero population in a model of population growth.
See the [[definition|1K8Y6BP097]].

## Explanations

### L1 human

The model is $x' = \mu x - x^2$, with the equilibria $x = 0$ and $x = \mu$.
For $\mu < 0$ the equilibrium $0$ is stable and $\mu$ is unstable.
For $\mu > 0$ the roles are swapped: $0$ is unstable and $\mu$ is stable.
At $\mu = 0$ the two equilibria meet and pass through each other, trading their stability.
In a population model, this is the threshold at which a species can start to grow from small numbers.