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Math 314 Computer Project  \hfill Due: Friday, April 19

Subject: Management of Domestic Sheep Populations \hfill Points: 25

\vspace{8pt}
{\it Description of the problem:}
You are working for the New Zealand Department of Agriculture on a 
project for sheep farmers.  The species of sheep that these farmers
raise have a lifespan of 12 years.  Of course, some live longer but
they are sufficiently few in number that they may be ignored in your
population study.  Accordingly, you divide sheep into 12 age classes,
namely those in the first year of life, etc.  You have conducted an
extensive survey of the demographics of this species of sheep and
obtained the following information about the demographic parameters
$a_i$ and $b_i$, where $a_i$ is the reproductive rate for sheep in the
$i$th age class and $b_i$ is the survival rate for sheep in that age
class (i.e., the fraction of sheep in that age class that survive to
the $i+1$th class.)

\vspace{.2in}
\begin{tabular}{||c||c|c|c|c|c|c|c|c|c|c|c|c||} \hline
$i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline
$a_i$ & .000 & .023 & .145 & .236 & .242 & .273 & .271 & .251 & .234 & 
.229 & .216 & .210 \\ \hline
$b_i$ & .845 & .975 & .965 & .950 & .926 & .895 & .850 & .786 & .691 &
.561 & .370  & - \\ \hline 
\end{tabular}

\vspace{.2in}
The problem is as follows:  in order to maintain a constant population 
of sheep, farmers will harvest a certain number of sheep each year.
Harvesting need not mean slaughter; it can be accomplished by selling
animals to other farmers, for example.  It simply means removing sheep
from the population.  Denote the fraction of sheep which are removed 
from the $i$th age group at the end of each growth period (a year in our 
case) by $h_i$.  If these numbers are constant from year to year, they 
contsitute a {\it harvesting policy.}  If, moreover, the yield of each 
harvest, i.e., total number of animals harvested each year, is a 
constant and the age distribution of the remaining populace is 
essentially constant after each harvest, then the harvesting policy is
called {\it sustainable.} If all the $h_i$'s are the same, say $h$,
then the harvesting policy is called {\it uniform}.  An advantage of
uniform policies is that they are simple to implement: one selects the
sheep to be harvested at random.

Your problem is as follows: find a uniform sustainable harvesting
policy to recommend to farmers, and find the resulting distribution of
sheep that they can expect with this policy.  Farmers who raise sheep
for sale to markets are also interested in a sustainable policy that
gives a maximum yield.  If you can find such a policy that has a larger
annual yield than the uniform policy, then recommend it.  On the other
hand, farmers who raise sheep for their wool may prefer to minimize the
annual yield.  If you can find a sustainable policy whose yield is
smaller than that of the uniform policy, make a recommendation
accordingly.  In each case find the expected distribution of your
harvesting policies.  Organize your results for a report to be read by
your supervisor and an informed public.

\vspace{12pt}
{\it Procedure:} Express this problem as a discrete linear dynamical 
system $x^{k+1} = L x^k$, where $L$ is a Leslie matrix of the form

$$L = \left[ \begin{array}{cccccc} a_1 & a_2 & a_3 & \cdot & a_{n-1} & a_n \\ 
b_1 & 0 & 0 & \cdots & 0 & 0 \\
0 & b_2 & 0 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & b_{n-1} & 0 
\end{array}\right]. $$

The facts you need to know (and may assume as standard facts about
Leslie matrices) are as follows: Such a matrix will have exactly one
positive eigenvalue which turns out to be a simple eigenvalue (not
repeated).  Moreover, if at least two adjacent entries of the first row
are positive, this eigenvalue will be a {\it dominant} eigenvalue,
i.e., it is strictly larger than any other eigenvalue in absolute
value.  In particular, if the positive eigenvalue is 1, then starting
from any nonzero initial state with non-negative entries, successive
states converge to an eigenvector belonging the the eigenvalue 1 which
has all non-negative entries.  Scale this vector by dividing it by the
sum of its components and one obtains an eigenvector which is a
probability distribution vector, i.e., its entries are non-negative and
sum to 1.  The entries of this vector give the long term distribution of 
the population in the various age classes.

In regards to harvesting, let $H$ be a diagonal matrix with the harvest 
fractions $h_i$ down the diagonal.  (Here $0 \le h_i \le 1$.)  Then the 
population that results from this harvesting at the end of each period 
is given by $x^{k+1} = L x^k - H L x^k = (I - H)Lx^k$.  But the matrix 
$(I - H)L$ is itself a Leslie matrix, so the theory applies it as well.  
There are other theoretical tools, but all you need to do is to find a 
matrix $H = hI$ so that 1 is the positive eigenvalue of $(I - H)L$.  You 
can do this by trial and error, a method which is applicable to any 
harvesting policy, uniform or not.  However, in the case of uniform 
policies it's simpler to note that $(I - H)L = (1-h)L$, where $h$ is the 
diagonal entry of $H$.

{\it Implementation Notes:} The Leslie matrix of this problem is stored 
for you in the notebook /Users/Students1/Math314HFiles/stuff.  Make a 
copy of this file in your own home directory and double click it to run 
it.  You can do all your work in your copy of this notebook.  For the
uniform policies, $H$ is easy to build: use ``{\tt h*IdentityMatrix[12]}'';
for a nonuniform policy, first build the diagonal by a ``{\tt h = \{0.0,
0.1, 0.2, 0.2, 0.3, 0,3, 0.5, 0.1, 0.2, 0.3, 0.4, 0.2\}}'' command, then
use ``{\tt H = DiagonalMatrix[h]}'' to build the harvest matrix.

{\it About reports:}  Use the template for a report which I have 
provided in the directory /Users/Students1/M314HFiles called 
Report\_Template.  Make a copy of it in your own directory, rename it
and work on it.  Expand or contract the table of contents as you like. 
Here are a few simple points to bear in mind when writing your report:
1. Keep it simple.  2. Remember your audience.  3.  Check your spelling 
and grammar.

{\it References:} 1.  Howard Anton, {\it Elementary Linear Algebra},
 
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