I have the two differential equations: Also, the next part says: Sketch the phas
ID: 31973 • Letter: I
Question
I have the two differential equations:
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?
Explanation / Answer
Classification of equilibrium points is done on the basis of the eigenvalues.
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.