Introduction to the Modeling and Analysis of Complex Systems

7.2. PHASE SPACE VISUALIZATION 117the upper right region. You plug this coordinate into the model equations to obtaindxdt ∣ = a 2c(x,y)=( 2cd , 2a b ) d − b2c 2ad b = −2ac < 0, (7.20)d dydt ∣ = −c 2a(x,y)=( 2cd , 2a b ) b + d2c 2ad b = 2ac > 0. (7.21)bTherefore, you can tell that the trajectories are flowing to “Northwest” in that region. If yourepeat the same testing for the three other regions, you obtain an outline of the phasespace of the model shown in Fig. 7.4, which shows a cyclic behavior caused by theinteraction between prey (x) and predator (y) populations.ya/b0 c/dxFigure 7.4: Drawing a phase space (3): Adding directions of trajectories in each region.This kind of manual reconstruction of phase space structure can’t tell you the exactshape of a particular trajectory, which are typically obtained through numerical simulation.For example, in the phase space manually drawn above, all we know is that the system’sbehavior is probably rotating around the equilibrium point at (x, y) = (c/d, a/b), but wecan’t tell if the trajectories are closed orbits, spiral into the equilibrium point, or spiral awayfrom the equilibrium point, until we numerically simulate the system’s behavior.