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?
I am going to sketch the steps for you and have you fill in the details.
We are given the system:
$$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$ $$\tag 1 \frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$
We need to find the critical points.
We find that there are four critical points at:
$$\displaystyle (N_1, N_2) = (0, 0), (0, 3), (2, 0), (\frac{4}{5},\frac{3}{5}).$$
In order to classify these, we find the Jacobian of the system in $(1)$, evaluate it at 'each' of those critical points by finding the eigenvalues.
Do you know how to find the Jacobain?
Do you know how to evaluate the eigenvalues at each of these four points?
Do you know how to classify the eigenvalues and the type of point each is? We have two stable and two unstable points (you can see these in the diagram).
Next, we need to draw the phase plane, null clines, ... Here is a guide to aid with that.
Lastly, we can take all of this information, draw a phase plane with the direction fields, classified eigenvalues and then superimpose example solution curves (notice the green versus blue lines). You have all of the answers to the questions above in this single graphic.
You should end up with: