Seite - 107 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

4.1 PopulationGrowth 107 sumes that r dependson thesizeof thepopulation,N: N.tC t/ N.t/D r.N.t//N.t/: Thecorrespondingdifferential equationbecomes N 0 D r.N/N : ThereaderisstronglyencouragedtorepeatthestepsinthederivationoftheForward Euler schemeandestablish thatweget NnC1 DNnC tr.Nn/Nn; whichcomputesaseasyas foraconstantr, sincer.Nn/ isknownwhencomputing NnC1.Alternatively,onecanusetheForwardEulerformulaforthegeneralproblem u0 Df.u;t/andusef.u;t/D r.u/uandreplaceubyN . Thesimplestchoiceofr.N/ isa linearfunction,startingwithsomegrowthvalue Nr anddeclininguntil thepopulationhas reached itsmaximum,M, according to the available resources: r.N/D Nr.1 N=M/: In the beginning,N M and wewill have exponential growth eNrt, but asN increases, r.N/ decreases, andwhenN reachesM, r.N/ D 0 so there is now moregrowthand thepopulation remainsatN.t/DM. This linear choiceofr.N/ gives rise to amodel that is called the logistic model. The parameterM is known as thecarrying capacityof thepopulation. Let us run the logistic model with aid of the ode_FE function in the ode_FE module. We chooseN.0/ D 100, t D 0:5month,T D 60months, r D 0:1, andM D 500. The complete program, calledlogistic.py, is basically a call to ode_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’ % dt); plt.savefig(’tmp_%g.pdf’ % dt) Figure 4.7 shows the resulting curve. We see that the population stabilizes aroundM D 500 individuals. A corresponding exponential growthwould reach N0e rt D100e0:1 60 40;300 individuals! It is always interesting to seewhat happenswith large t values. Wemay set t D 20 and T D 100. Now the solution, seen in Fig. 4.8, oscillates and is hence qualitatively wrong, because one can prove that the exact solution of the differential equation ismonotone. (However, there is a corresponding difference