> with(plots):

> with(DEtools):

> exponential:=diff(y(t),t)=r*y(t);

> logistic:=diff(y(t),t)=r*y(t)*(1-y(t)/M);

> harvesting:=diff(y(t),t)=r*y(t)*(1-y(t)/M)-H;

> expsol:=dsolve({exponential,y(0)=y0},y(t));

> logisticsol:=dsolve({logistic,y(0)=y0},y(t));

> harvestingsol:=dsolve({harvesting,y(0)=y0},y(t));

> expplot:=subs(r=0.2,y0=1400,rhs(expsol));

> logisticplot:=subs(r=0.2,y0=1400,M=100000,rhs(logisticsol));

Do the following commands for t range max of 10, 20, 100, sequentially.

> plot(expplot,t=0..10);

> plot(logisticplot,t=0..10);

> plot({expplot,logisticplot},t=0..20);

> exponential;

> logistic;

> harvesting;

> numexponential:=subs(r=0.2,exponential);

> numlogistic:=subs(r=0.2,M=100000,logistic);

> numharvesting:=subs(r=0.2,M=100000,H=200,harvesting);

Note the y axis in the following 3 plots. All of them go from t=0..20 but after 20 years the population is very different.

> DEplot(numexponential,y(t),t=0..20,[[y(0)=1400]],'arrows'='NONE');

> DEplot(numlogistic,y(t),t=0..20,[[y(0)=1400]],'arrows'='NONE');

> DEplot(numharvesting,y(t),t=0..20,[[y(0)=1400]],'arrows'='NONE',stepsize=0.01);

Now compare the logistic and harvesting models for long times.

> DEplot(numlogistic,y(t),t=0..50,[[y(0)=1400]],'arrows'='NONE',stepsize=0.01);

> DEplot(numharvesting,y(t),t=0..50,[[y(0)=1400]],'arrows'='NONE');

Let's examine the equilibria dy/dt = 0 of the logistic and harvesting models.

> logistic;

> harvesting;

> rhs(logistic);

> solve(%,y(t));

> rhs(harvesting);

> subs(r=0.2,M=100000,H=200,%);

> solve(%,y(t));

>

>

>