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I have the two differential equations:

$$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$ $$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$

I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ and $N_2 = 0, \frac{3}{5}$. I then calculated all the Jacobi matrices and worked out the eigenvalues (and eigenvectors). I now have to classify these equilibria. How do I do that? Is there a set of rules I follow to classify them?

Also, the next part says:

Sketch the phase portrait of this system in the biologically sensible region: draw the the null- clines of the system and determine the crude directions of trajectories in parts of the phase plane cut by the null clines, designate the equilibria in the phase plane, and sketch a few typical trajectories.

I can do the null - clines and I think once I have found out the stability of the equilibria, I can determine the directions of trajectories of the bits cut by the null -cline, but how would I sketch the trajectories of the equilibria?

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closed as off topic by MattDMo, terdon, Artem Kaznatcheev, jonsca, Rory M Apr 16 '13 at 0:16

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The fundamental question here is math, not biology, even though it may have been presented as a biological system. You may have better luck on math SE. – dd3 Apr 14 '13 at 16:09
up vote 4 down vote accepted

Classification of equilibrium points is done on the basis of the eigenvalues.

  • If the two eigenvalues have no real parts, it is a hyperbolic fixed point and represents undamped oscillation.
  • If both have a negative real part, it is a stable fixed point. If any of the eigenvalues has an imaginary part then it represents damped oscillations (in that case the equilibrium point is called a focus).
  • If one or both eigenvalue have a positive real part, it is an unstable fixed point.
  • If one eigenvalue has a positive real part and the other a negative real part, it is a saddle point.*

You sketch the trajectories of the equilibria by starting a tiny distance away from the point (where it is not in equilibrium) and trace the quiver plot. Drawing trajectories around unstable points is tricky and you have to plot the function in negative time i.e. instead of plotting f(x1,x2,t), you plot -f(x1,x2,t). This page has representative trajectories for the different types of equilibrium points. This page has a more thorough explanation, generalized to multiple dimensions.

*This classification appears to be slightly variable (more or less categories), on the basis of the two links I found. I think the one I list is the simplest version, but any classification should suffice. They are all based on the same principles. You can refer to some books on non-linear dynamics:

  • Nonlinear Systems by Hassan Khalil
  • Nonlinear Dynamics and Chaos by Steven Strogatz
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@WYSIWYG added the references and some additional information - I would like to second the recommendation for Strogatz's book. – dd3 Apr 15 '13 at 4:45
yeah.. its such an easy and comfortable read.. – WYSIWYG Apr 15 '13 at 4:53

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