Introduction to the Modeling and Analysis of Complex Systems

7.2. PHASE SPACE VISUALIZATION 115The first equation givesx = 0, or y = a b . (7.18)These are two straight lines, which constitute one set of nullclines for dx/dt = 0 (i.e., youcould call each line a single nullcline). In the meantime, the second one givesy = 0, or x = c d . (7.19)Again, these two lines constitute another set of nullclines for dy/dt = 0. These results canbe visualized manually as shown in Fig. 7.2. Equilibrium points exist where the two setsof nullclines intersect.yNullclines fordx/dt = 0a/bEquilibriumpointsNullclines fordy/dt = 00 c/dxFigure 7.2: Drawing a phase space (1): Adding nullclines.Everywhere on the first set of nullclines, dx/dt is zero, i.e., there is no “horizontal”movement in the system’s state. This means that all local trajectories on and near thosenullclines must be flowing vertically. Similarly, everywhere on the second set of nullclines,dy/dt is zero, therefore there is no “vertical” movement and all the local trajectories flowhorizontally. These facts can be indicated in the phase space by adding tiny line segmentsonto each nullcline (Fig. 7.3).