We can use this example to how to explore the nature of equilibrium points and trajectories on phase space diagrams. The equilibrium points are the solutions of
a - bXl - cxZ = 0 (2.15) and
xl 0 (2.16) which intersects with those of
- e + fXl = 0 (2.17) and
Xz = 0 (2.18) We concentrate on the interesting equilibrium point where both solutions take a non-zero value and so ignore the axes (2.16) and (2.18). The lines (2.15) and (2.17) are shown plotted on Figure 2.16. The lines divide the positive quadrant of the
state space diagram into four quadrants and the next step is to add arrows on each quadrant which show the direction of change of the variables. (This can be carried out in practice simply by calculating Xl and xz at a number of sample points: XZ/Xl then gives the gradient of the trajectory at each such point.) This is done in the figure, using information from Equation (2.13) and (2.14). These indicate the nature of a typical trajectory which is shown on part (b) of the figure: it is like that in Figure 2.1, case (b), and 'spirals in' to the equilibrium point, which is thus seen to be stable.