ERS 488/688 LABORATORY 7

MATRIX POPULATION MODELS INTRODUCED

 

1)      Basic Matrix Operations

a)      Use the matrix in A1:D4, sheet 1.

 

b)      Calculate the inverse of the A1:D4 matrix using MINVERSE and show that the result is actually the inverse of the matrix you started with.  To use the MINVERSE function in Excel, first select a range of cells for the inverse to be placed into (left click in the upper left corner and drag the mouse to the lower left corner of the desired area).  Return the mouse to the upper left corner of the selected area and type "= MINVERSE(A1:D4)" then hit F2 and ctrl+shift+enter. 

 

        Matrix inversion replaces the notion of scalar division when we are working with matrices.  For a matrix A, the product AA^-1 = I, which is equivalent to the scalar operation a/a = aa^-1 = 1.

 

 

c)      Use the steps for inverting a matrix from lecture to show that a₁₁ in the inverse is correct.

 

2)   Leslie Matrices

a) Use the martix in A1:D4, sheet 2, which represents an approximate Leslie matrix for black brant.  Multiply this matrix times the population vector in F1:F4 using the MMULT function of Excel.

 

b)  Calculate the total size of the population for each of 50 time steps.  Hint: you don't want the cells for the Leslie matrix to change.  Remember how we keep the cells constant as we drag the mouse from the cell containing the formula.  We do want the cells for the population vector t be continuously updated.  That is, we want to multiply the Leslie matrix times the new population vector at each time step, which models the progression of the population through time.  Remember that the Leslie matrix projects the number of individuals in each age class from one time step to the next.  Dynamics of  a population through time represents the repeated projection of the population vector by the Leslie matrix.

 

d)      Calculate l for each time step.  Remember l = N_t+1/N_t.

 

e)      Calculate the age distribution (the proportion of individuals in each age class) for each time step.

 

2)      Demonstrate that the value of l you calculated above represents the dominant eigenvalue of the Leslie Matrix you are working with.  Hint: you will need the MDETERM function.

 

3)      Would you say that the population you are working with is most sensitive to changes in adult survival, juvenile survival or reproductive rate?