sumes that depends on the size of the populationn td // tthe corresponding differential equation becomes / the reader is strongly encouraged to repeat the steps in the derivation of the forward euler scheme and establish that we get nc / which computes as easy as for constant rsince nis known when computing nc alternativelyone can use the forward euler formula for the general problem utand use utd / and replace by the simplest choice of is linear functionstarting with some growth value rn and declining until the population has reached its maximumm according to the available resourcesr rn = in the beginningn and we will have exponential growth rtn but as increasesr decreasesand when reaches so there is now more growth and the population remains at td this linear choice of gives rise to model that is called the logistic model the parameter is known as the carrying capacity of the population let us run the logistic model with aid of the ode_fe function in the ode_fe module we choose : montht monthsr : and the complete programcalled logistic pyis basically call to ode_fefrom ode_fe import ode_fe import matplotlib pyplot as plt for dtt in zip(( )( ))ut ode_fe( =lambda ut *( / )*uu_ = dt=dtt=tplt figure(make separate figures for each pass in the loop plt plot(tu' -'plt xlabel(' ')plt ylabel(' ( )'plt savefig('tmp_% pngdt)plt savefig('tmp_% pdfdtfigure shows the resulting curve we see that the population stabilizes around individuals corresponding exponential growth would reach rt :  ; individualsit is always interesting to see what happens with large values we may set and now the solutionseen in fig oscillates and is hence qualitatively wrongbecause one can prove that the exact solution of the differential equation is monotone (howeverthere is corresponding difference