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I have an Lotka Volterra type of system as seen below;

$$\begin{align} x' &= 10 x y -1/2 x - 1/10 x^2 \\ y' &= y - y^2 -x y \end{align}$$

I would like to plot the phase portrait of the system.

Critical points (points where $dx/dt$ and $dy/dt$ are simultaneously zero) are $(0,0) , (0,1), (95/101, 6/101)$. (there is $(-5,0)$ as well but as negative values are not meaningful for this system I ignore this Critical point.)

I see $(0,0)$ and $(0,1)$ are saddle points, and $(95/101, 6/101)$ is a stable spiral.

I think unstable manifold (Eigenvector) of $(0,1)$ is joining to spiral of $(95/101, 6/101)$. And no matter which initial points are chosen (except the ones on the eigenvectors) the system ends in $(95/101, 6/101)$. Is this true? I would appreciate if you could show me what the phase portrait of this system looks like.

Moo
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vyaman
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  • yes, sorry for the typo. I now fixed it – vyaman Oct 04 '18 at 16:47
  • Moo, this is very good to me but I can`t see how the unstable eigenvector of (0,1) joins to spiral or does it really join? I can t see this issue on the plots. And do you think all initial points end at(95/101, 6/101) in this system? – vyaman Oct 04 '18 at 17:19
  • I agree with your response. If you answer it I`ll approve. Thank you – vyaman Oct 06 '18 at 15:54

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