Introduction to the Modeling and Analysis of Complex Systems

106 CHAPTER 6. CONTINUOUS-TIME MODELS I: MODELINGdef initialize():global x, result, t, timestepsx = 0.1result = [x]t = 0.timesteps = [t]def observe():global x, result, t, timestepsresult.append(x)timesteps.append(t)def update():global x, result, t, timestepsx = x + r * x * (1 - x / K) * Dtt = t + Dtinitialize()while t < 50.:update()observe()plot(timesteps, result)show()Note that there really isn’t much difference between this and what we did previously. Thiscode will produce a nice, smooth curve as a result of the numerical integration, shown inFig. 6.1. If you choose even smaller values for ∆t, the curve will get closer to the truesolution of Eq. (6.27).Exercise 6.4 Vary ∆t to have larger values in the previous example and see howthe simulation result is affected by such changes.As the exercise above illustrates, numerical integration of differential equations involvessome technical issues, such as the stability and convergence of solutions and thepossibility of “artifacts” arising from discretization of time. You should always be attentiveto these issues and be careful when implementing simulation codes for continuous-time