Harvesting an Age-Distributed Population

Part 1.  Sustainable harvesting

Build the Leslie growth matrix L, as you did in the Leslie Growth Models module.
[Notes:  (1) We first define L to be a matrix of zeros. (2) Next we insert symbolic birth rates a[j] in the first row of L.  (3) Then we insert symbolic survival rates on the first subdiagonal.  (4) The Do command is a way to tell Mathematica to do the same operation over and over and has the structure:  Do[expr, {i, imax}] evaluates expr with the variable i successively taking on the values 1 through imax (in steps of 1). ]

L  =  Table[0,  {12}, {12} ]; 
Clear[a];
Do[Part[L, 1, j] = a[j] ,  {j,  1,  12}];
Clear[b];
Do[Part[L, j + 1, j]  =  b[j],  {j,  1,  11} ];
L  //  MatrixForm

Construct the matrices H, (Subscript[I, 12]), and ((Subscript[I, 12]) - H) L in symbolic form:

Clear[h];
H  =  DiagonalMatrix[ Array[h,  12] ];
I12  =  IdentityMatrix[12];
(I12  -  H)  .  L  //  MatrixForm

Reconstruct the characteristic polynomial p(λ).

p[λ_] =  Det[λ*IdentityMatrix[12]  -  L]

Reconstruct the auxiliary function q(λ).

Clear[c]; 
    Do[c[j] = Product[b[i], {i, 1, j}],  {j,  1,  
        11}];
q[λ_] = 
      a[1]/λ + Sum[a[j]*c[j - 1]/λ^j, {j, 2, 12}]

Assign birth and survival rates for the New Zealand sheep population.

\(\({a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10], a[11], 
          a[12]}\ 
 = \ {0, 0.045, 0.391, 0.472, 0.484, \ 
          0.546, 0.543, 
		\ \ \ \ \ \ \ 0.502, 0.468, 0.459, 0.433, 
          0.421};\)\)
\(\({b[1], b[2], b[3], b[4], b[5], b[6], b[7], b[8], b[9], b[10], 
          b[11]}\ 
 = \ {0.845, 0.975, 0.965, 0.950, \ 
          0.926, \ 
		\ \ \ \ \ \ \ 0.895, 0.850, 0.786, 0.691, 0.561, 
          0.370};\)\)
Do[Part[L, 1, j] = a[j] ,  {j,  1,  12}];
Do[ Part[L, j + 1, j]  =  b[j],  {j,  1,  11} ];
L  //  MatrixForm

Now let Mathematica compute the eigenvalues of L.

Eigenvalues[L]

Recall that we assigned the name λ1 to the uniqe positive eigenvalue of L.Fill in that dominant eigenvalue here:

λ1=

Find a uniform harvesting policy: What fraction of every age group can be harvested to return the population to its starting state?

Part 2.  Harvesting the youngest class

Calculate the net reproduction rate R.  Recall that we earlier defined (Subscript[c, j]) to be (Subscript[b, 1])(Subscript[b, 2])...(Subscript[b, j]) for each j. The following line computes the sum of products of appropriate a's and c's.

R  =  a[1];
Do[R  =  R + a[j]*c[j - 1] ,  {j,  2,  12}]
R

Calculate the fraction of lambs to be harvested in a sustainable harvesting policy.

Calculate the stable population distribution for this policy.

Part 3.  Optimal harvesting

Calculate the age distribution for the optimal harvesting policy.

Part 4.  Summary