Consider phase portrait below. Be prepared to match the blue line in Figure 1 with the graphs
in Figure 2 if the lines are not color-coded. Relax. It’s actually quite easy.
Follow the blue line in in Figure 1 along the direction of the arrows. Notice that the line gets
steadily farther from the vertical axis so, b is always increasing. Notice that the blue line gets farther
from the horizontal axis at first but then gets closer so, m is increasing and then decreasing.
1000
800
m
600
400
200
0
0
200
400
600
800 1000
b
Figure 1: This phase-plane diagram shows nulclines for b (magenta) and m (cyan), equilibria (yellow
dots) and direction arrows. Three trajectories (red, green, blue) are shown, for three different initial
conditions.
1000
1000
800
800
600
b
m
400
b&m
b&m
Now look at the graphs for b and m in Figure 2 and notice that these lines show the same pattern
of increase or decrease in m and p that I just described above. It might help to realize that the
arrows in Figure 1 point in the direction of increasing time, while in Figure 2 time increases along
the horizontal axis. See if the green lines in figures 1 and 3 make sense to you.
The cyan and magenta lines are nulclines for m and b respectively. Recall that equilibria are points
where the nulclines for different variables (here shown with different colors) cross. The intersection
of two nulclines for the same variable (same color) means nothing. Recall that arrows point towards
stable equilibria and away from unstable equilibria. I might ask about that. An equilibrium is
marginally stable if some arrows point towards the equilibrium while other point away.
600
400
200
200
0
0
0
2
4
6
t
Figure 2: The squares and diamonds
in these graphs identify corresponding
points along the blue trjectory in the
phase diagram above.
b
m
0
1
2
t
3
4
Figure 3: The squares and diamonds
in these graphs identify corresponding
points along the green trjectory in the
phase diagram above.