Introduction to the Modeling and Analysis of Complex Systems

118 CHAPTER 7. CONTINUOUS-TIME MODELS II: ANALYSISHaving said that, there is still merit in this analytical work. First, analytical calculationsof nullclines and directions of trajectories provide information about the underlying structureof the phase space, which is sometimes unclear in a numerically visualized phasespace. Second, analytical work may allow you to construct a phase space without specifyingdetailed parameter values (as we did in the example above), whose result is moregeneral with broader applicability to real-world systems than a phase space visualizationwith specific parameter values.Exercise 7.5 Draw an outline of the phase space of the following SIR model (variableR is omitted here) by studying nullclines and estimating the directions of trajectorieswithin each region separated by those nullclines.dS= −aSIdt(7.22)dI= aSI − bIdt(7.23)S ≥ 0, I ≥ 0, a > 0, b > 0 (7.24)Exercise 7.6 Draw an outline of the phase space of the following equation bystudying nullclines and estimating the directions of trajectories within each regionseparated by those nullclines.d 2 xdt 2 − xdx dt + x2 = 0 (7.25)7.3 Variable RescalingVariable rescaling of continuous-time models has one distinct difference from that ofdiscrete-time models. That is, you get one more variable you can rescale: time. Thismay allow you to eliminate one more parameter from your model compared to discretetimecases.Here is an example: the logistic growth model. Remember that its discrete-time version(x t = x t−1 + rx t−1 1 − x )t−1(7.26)K