Introduction to the Modeling and Analysis of Complex Systems

116 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSISya/b0 c/dxFigure 7.3: Drawing a phase space (2): Adding directions to the nullclines.Now the phase space is divided into four regions. It is guaranteed that the trajectoriesin each of those regions flow only in one of the following four directional categories:• dxdt• dxdt• dxdt• dxdt> 0,dydt< 0,dydt> 0,dydt< 0,dydt> 0 (to “Northeast”)> 0 (to “Northwest”)< 0 (to “Southeast”)< 0 (to “Southwest”)Inside any region, a trajectory never switches between these four categories, because if itdid, such a switching point would have to have already appeared as part of the nullclines.Therefore, sampling just one point from each region is sufficient to know which directionthe trajectories are flowing in. For example, you can pick (2c/d, 2a/b) as a sample point in