MATH 1170
MATHEMATICS FOR LIFE SCIENTISTS
Computer Assignment X
Due November 4, 2003
1. We will use Maple to study the eects of harvesting cost on the sh harvesting optimization problem. Input the updating function
> g := F -> r*F*(1-F)-h*F;
where represents the harvesting eort and is the maximum population growth rate.
The sh population is measured in millions. Use solve to nd the equilibria. Dene a
function of called Fstar which has as its value the positive equilibrium. The equilibrium
prot as a function of can be given as
h
r
h
h
> Pstar := h -> h*p*Fstar(h)-c*h;
The rst term is the total harvest times the price per sh . Assume = 1, meaning
here 1 million dollars for 1 million sh. The second term is the cost of harvesting. with
equal to the cost per unit eort of harvesting (again measured in millions of dollars).
(a) Set = 2 5, and = 0. Find the value of that maximizes Pstar by using the
derivative. First, nd the derivative of Pstar with the command
p
p
c
r
:
c
h
> dPstar := diff(Pstar(h),h);
Then use the solve command to nd where this is equal to 0,
> hstar := solve(dPstar=0,h);
Find Fstar(hstar) and Pstar(hstar). Set equal to hstar (this is necessary to
get the cobweb to work), and cobweb the function for 10 steps starting from 0 01
with the command
h
g
:
> cobweb(g,10,0.01,0..1);
Write on your diagram the number of sh at equilibrium, the number of sh harvested, and the prot.
(b) Set c:=0.2; and h := 'h'; to reinitialize. Repeat a. Why are there more sh and
lower prots? Make sure to write the number of sh harvested at equilibrium, the
revenue from those sh, and the cost of harvesting those sh. For an extra challenge,
how many years would it be (starting from 0.01 sh) before a prot was made?
(c) Repeat b with c:=0.7;. Does your answer make sense? What is the optimal harvesting eort? Why?