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  "body": "\n## Summary\n\nA transcritical bifurcation is a change in a family of dynamical systems at which two branches of [[equilibrium points|0N8NGKDZDZ]] cross and exchange their stability.\nBoth equilibria exist on both sides of the critical parameter.\nIt is typical when one equilibrium is forced to exist for all parameter values, such as the zero population in a model of population growth.\nSee the [[definition|1K8Y6BP097]].\n\n## Explanations\n\n### L1 human\n\nThe model is $x' = \\mu x - x^2$, with the equilibria $x = 0$ and $x = \\mu$.\nFor $\\mu \u003c 0$ the equilibrium $0$ is stable and $\\mu$ is unstable.\nFor $\\mu \u003e 0$ the roles are swapped: $0$ is unstable and $\\mu$ is stable.\nAt $\\mu = 0$ the two equilibria meet and pass through each other, trading their stability.\nIn a population model, this is the threshold at which a species can start to grow from small numbers.\n",
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