Page - 222 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

222 8 SolvingOrdinaryDifferentialEquations Thecorrespondingdifferentialequationbecomes N′ = r(N)N . Thereaderisstronglyencouragedtorepeatthestepsin thederivationoftheForward Euler schemeandestablish thatwe get Nn+1 =Nn+Δtr(Nn)Nn, whichcomputesaseasy as fora constant r, since r(Nn) is knownwhencomputing Nn+1.Alternatively,onecanuse theForwardEulerformulafor thegeneralproblem u′ =f(u,t)andusef(u,t)= r(u)uand replaceubyN. Thesimplestchoiceofr(N) isa linearfunction,startingwithsomegrowthvalue r¯ and declininguntil the populationhas reached its maximum,M, according to the available resources: r(N)= r¯(1−N/M). In the beginning, N M and we will have exponential growth er¯t, but as N increases, r(N) decreases, and whenN reachesM, r(N)= 0 so there is no more growth and the population remains atN(t)=M. This linear choice of r(N)gives rise to a model that is called the logistic model. The parameterM is known as the carryingcapacityof thepopulation. Letusrunthelogisticmodelwithaidoftheode_FEfunction.WechooseN(0)= 100, Δt = 0.5 month, T = 60 months, r = 0.1, and M = 500. The complete program,calledlogistic.py, is basicallya call toode_FE: from ode_FE import ode_FE import matplotlib.pyplot as plt for dt, T in zip((0.5, 20), (60, 100)): u, t = ode_FE(f=lambda u, t: 0.1*(1 - u/500.)*u, \ U_0=100, dt=dt, T=T) plt.figure() # Make separate figures for each pass in the loop plt.plot(t, u, ’b-’) plt.xlabel(’t’); plt.ylabel(’N(t)’) plt.savefig(’tmp_{:g}.png’.format(dt)) plt.savefig(’tmp_{:g}.pdf’.format(dt)) Figure8.9showstheresultingcurve.Weseethat thepopulationstabilizesaround M = 500 individuals. A corresponding exponential growth would reachN0ert = 100e0.1·60 ≈40,300individuals! What happens if we use “large” Δt values here? We may set Δt = 20 and T = 100. Now the solution, seen in Fig. 8.10, oscillates and is hence qualitatively wrong,because one can prove that the exact solution of the differential equation is monotone.