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Analysis of population projection matrices

Population projection matrices are powerful tools for the analysis of population dynamics and life-history strategies. But procedures for analysis of population projection matrices are not routinely available in standard software. PopTools tries to plug this gap. The background to these methods is discussed in the book by Hal Caswell. The example shown here is from Page 76. Example 4.9.

The life-history diagram looks like this:

As a first step, it is always useful to draw the life-cycle for a matrix model, because you can use it to understand the assumptions of the model. Moreover, if you follow Caswell's directions, the diagram allows you to construct the correct projection matrix. 
In this example there are three stages (labelled 1, 2 and 3). The red arcs represent reproduction (roughly speaking, the number of females per female that survive to stage 1), and the blue lines indicate survival rates during a single time step. To construct the appropriate projection matrix, look at the source and destination of each arrow. An arrow leading from stage X to stage Y, should be represented in the projection matrix by an entry in column X and row Y of the projection matrix.. The colour coding on the rates shows where each should appear in the matrix.

The population projection matrix (A) therefore looks like this:


You will get something like the following:

Each of the coloured blocks produced by PopTools represents an array formula that can be shifted (in its entirety only - you can't split an array formula) around the spreadsheet. The pink block lists all the eigenvalues of the projection matrix. The dominant eigenvalue should always be the first entry in the top left of the pink block. The right (yellow) and left (blue) eigenvectors of the dominant eigenvalue give the stable age structure and reproductive values. They are "normalised" so that each sums to one (100%). The pale blue cells lists the rate of increase and the generation times that are discussed in Caswell's book.

You can also enter these array formulas by hand anywhere on your spreadsheet

Use Ctrl-Shift-Enter to enter these formulas - as described in the PopTools help file

Technical topic: Use the array formula {=EIGS(A)}, where A is the address of the projection matrix to return a full eigenanalysis

The values in the coloured area are the output of a single array formula. Each coloured row represents an eigenvalue (real and imaginary parts are in the last two columns) and an associated eigenvector (in the first three columns). For example, an eigenvector associated with the dominant eigenvalue (1.21) is w1 = [1, 0.82, 0.97].

You can calculate the sensitivity of the dominant eigenvalue (ie, the finite rate of increase) to any of the matrix elements by entering the array formula "=SENSITIVITY(A)", where A is the range containing the matrix

Elasticities are easily calculated using Excel formulas - once you have the sensitivities - or you can use the array formula "=ELASTICITY(A)"

For a quick graphical display of the elasticities, use POPTOOLS/COLOUR SCALE to get this output

Richer colour indicates greater elasticity. (You will need a high video resolution for this routine to work properly)

Final note: If you want the sensitivity to a compound parameter, you will have to use the numerical sensitivity routines. See the Sensitivity/Elasticity demo sheet