I am working with the system $$x'=7x-x^2-xy \quad\quad y'=-5y+xy$$ and determining its phase portrait. I have found the x-nullclines to be $x=0$ and $y=7-x$ and the y-nullclines to be $y=0$ and $x=5$.
Having plotted these, I have 5 intersections of these 4 lines: $(0,0),(0,7),(5,0),(7,0),(5,2)$. However, if I plot this system using pplane, it does not appear as if $(0,7)$ or $(5,0)$ are equilibrium points. Why is this? Surely all intersections with the $0$ axis count?
I have looked at other examples and similarly, not all their intersections are equilibrium points either. How can you know which are?
$\endgroup$1 Answer
$\begingroup$An intersection of an $x$-nullcline with a $y$-nullcline will be an equilibrium point. $(0,7)$ is the intersection of two $x$-nullclines, but is not on any $y$-nullcline, so it is not an equilibrium. $(5,0)$ is the intersection of two $y$-nullclines, but is not on any $x$-nullcline, so it is not an equilibrium.
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