/***********************************************
************************************************
************************************************
Euler_logistic.mac

This software is a part of:
Kai Velten: Mathematical Modeling and Simulation, Wiley-VCH, 2009.

It can be used under the conditions of the GNU General Public License (GPLv3),
see http://www.gnu.org/licenses/gpl.html.
************************************************
************************************************
************************************************/

/* Initializations */

r : 0.07;
K : 1000; /*can store 1/K instead.*/
final_time: 1000;
h:1;
n:final_time/h;
y[0]:100;

for i:1 thru n step 1 do
   y[i]:(1+(h*r*(1 - y[i-1]/K)))*y[i-1];

EulerApprox1:create_list([i*h,y[i]],i,0,n);

h:5;
n:final_time/h;
y[0]:100;
for i:1 thru n step 1 do
   y[i]: (1+(h*r*(1 - y[i-1]/K)))*y[i-1];

EulerApprox2:create_list([i*h,y[i]],i,0,n);

/* Plot the result: */

plot2d([[discrete,EulerApprox1],[discrete, EulerApprox2], K*100/(100 +(K-100)*exp(-r*t))]
,[t,0,100]
,[style, points,  points, lines], [color, magenta,black, blue, green],
 [point_type, circle, asterisk] 
,[xlabel,"Time"]
,[ylabel,"Population Growth"],[legend, "h= 1 years", "h = 5 years" ,"Actual Population Growth"]
,[title, "Logistic Model for the population"]
,[gnuplot_preamble,"set key bottom"]);