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# nullcline

Let

$\displaystyle\dot{x}_{1}$ | $\displaystyle=$ | $\displaystyle f_{1}(x_{1},\ldots,x_{n})$ | ||

$\displaystyle\vdots$ | ||||

$\displaystyle\dot{x}_{n}$ | $\displaystyle=$ | $\displaystyle f_{n}(x_{1},\ldots,x_{n})$ |

be a system of first order ordinary differential equation. The $x_{j}$ *nullcline* is the set of points which satisfy $f_{j}(x_{1},\ldots,x_{n})=0$. Note that at an intersection point of all the nullclines implies that

$\displaystyle 0$ | $\displaystyle=$ | $\displaystyle f_{1}(x_{1},\ldots,x_{n})$ | ||

$\displaystyle\vdots$ | ||||

$\displaystyle 0$ | $\displaystyle=$ | $\displaystyle f_{n}(x_{1},\ldots,x_{n}).$ |

Hence the intersection point of all the nullclines is an equilibrium point of the system.

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