Page - 103 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.1 PopulationGrowth 103 N[0] = N_0 for n in range(N_t+1): N[n+1] = N[n] + r*dt*N[n] import matplotlib.pyplot as plt numerical_sol = ’bo’ if N_t < 70 else ’b-’ plt.plot(t, N, numerical_sol, t, N_0*exp(r*t), ’r-’) plt.legend([’numerical’, ’exact’], loc=’upper left’) plt.xlabel(’t’); plt.ylabel(’N(t)’) filestem = ’growth1_%dsteps’ % N_t plt.savefig(’%s.png’ % filestem); plt.savefig(’%s.pdf’ % filestem) Thecompletecodeaboveresides in thefilegrowth1.py. Let us demonstrate a simulationwherewe startwith 100animals, a net growth rate of 10 percent (0.1) per time unit, which can be onemonth, and t 2 Œ0;20 months. Wemayfirst try t of half amonth (0.5),which impliesNt D 40 (or to be absolutely precise, the last time point to be computed according to our set-up above is tNtC1 D 20:5). Figure 4.4 shows the results. The solid line is the exact solution,while thecirclesare thecomputednumerical solution. Thediscrepancy is clearly visible. What ifwemake t 10 times smaller? The result is displayed in Fig. 4.5,wherewenowuse a solid line also for thenumerical solution (otherwise, 400 circleswould lookverycluttered, so the programhas a test onhow todisplay the numerical solution, either as circles or a solid line). Wecanhardlydistinguish the exact and the numerical solution. The computing time is also a fraction of a second on a laptop, so it appears that the Forward Eulermethod is sufficiently accurate for practical purposes. (This is not always true for large, complicated simulationmodels inengineering, somore sophisticatedmethodsmaybeneeded.) Fig.4.4 Evolutionof apopulationcomputedwith timestep0.5month