MATH 304 -- Ordinary Differential Equations

September 15-17, 2004

Logistic Growth with Periodic Harvesting

We will study solutions of the 1st order ODE

     x' =  

Note:  This is a case where no technique from calculus

applies to give analytic solutions (i.e. solutions defined

by formulas).  Hence, we must either use approximate

numerical methods to graph approximate solutions

(as here), or apply qualitative techniques, or both.

Here are some plots of solutions for the parameter values

a = 2, h = .3:

>	a:=2: h:=.3:

>	with(DEtools):

>	HarvestEq:=diff(x(t),t) = a*x(t)*(1-x(t))-h*(1+sin(2*Pi*t));

>	DEplot(HarvestEq,x(t),t=0..6, [[x(0)=.6],[x(0)=.7],[x(0)=.8],[x(0)=.9],[x(0)=1],[x(0)=.232]],stepsize=.01,linecolor=black);

>	DEplot(HarvestEq,x(t),t=0..5, [[x(0)=.6],[x(0)=.7],[x(0)=.8],[x(0)=.9],[x(0)=1],[x(0)=.231]],stepsize=.01,linecolor=black);

What do these pictures show us?  Some conjectures we might make based on this evidence:

  1)  There is apparently a periodic solution  (period 1 in t )  that oscillates between  

       roughly  x = . 76 and  x = .85.   That is, some initial condition in this range

       gives this periodic solution.

  2)  This solution seems to ``attract''   nearby solutions in the sense that the

       difference goes to zero as   t   increases.

  3)  There is apparently another periodic solution oscillating between

       roughly   x = . 13  and   x = .23.  

  4)  This second periodic solution appears to ``repel''   nearby solutions in the

       sense that if the initial condition is not right to land exactly on that solution,

       then the solution either grows and tends toward the first periodic solution,

       or else ``crashes'' and goes to  -   as   t   increases.