The growth of a population with an age structure can the projected using matrix algebra. Leslie matrices have the information about birth and death rates of different age classes of a population and are a robust way of figuring out the population growth and make projections for different scenarios. A generalization of the Leslie matrix occurs when the population is classified due to development stages instead of age classes (Lefkovitch matrices). In this scenario, an individual may reproduce, die, grow from one stage to another, or stay in the same stage. In this generalization, the basic vital rates are built into the transition matrix elements, which are used to figure out the effect that individuals from having a number of individuals in each class on the number of individuals in each class at the next time step.
The objective of this exercise is understand how can we study structured populations with these matrix models.
Before that, let's make some matrix multiplications in a spreadsheet.
Let's use one of the examples in Chapter 5 of the book Vegetable Ecology (Gurevitch et al. 2009) (see bibliography).
I4
, then click the INSERT / FORMULA / MATRIX (Excel) or INSERT / FUNCTION / MATRIX (Calc) menu and choose the matrix multiplication function, (M.MULT or MATRIX.MULT depending on the program version). Indicate in the function dialog box, what should be multiplied: first the transition matrix and then the population vector. Attention: before any movement (or breathing) do steps 3 and 4 , otherwise you run into the infinite vortex of Excel !! N2
) with the number of individuals at the next time instant (t + 1) for each of the classes (the three rows in column N2
).Don't panic!
That should solve! The three rows of the column referring to the vector “N2” must be filled at the end of this operation.
Note: If you have a MacOS computer, the sequence of keys to be pressed is a bit different. Press the control + U keys and then command + return (or command + enter)
$
), copy and paste in the next column. Repeat this procedure for multiple columns (that is, multiple future times) to the column you want to project the population to; 2) Select any vector (the three cells of the column with the result - make sure the formula is with the symbols $
), then find the +
sign that appears in the lower right corner position the mouse, click and drag horizontally to the column you want to project the population.Do not panic!
Understanding the worksheet
Note that the matrix is based on stages of development rather than age classes, so it is possible for individuals to remain in the same class from one time to another. In these cases, the transition matrix (called the Leftkovitch Matrix) also has probabilities of permanence. Find the probabilities of staying in the matrix.
J
column). In this way, the results of the multiplication formula will be updated, resulting in the values for the next time (ie the new values that will appear in the L
column). Each time you should copy these new values to the RESULTS TABLE, in the corresponding column. Repeat this until time 15 or more.SHORTCUT
L12
using the $
. The formula should look like this: = K12 * (100- $ M $ 2) / 100
. Copy and paste the cells K6: L12
(you can select the two columns at the same time) in the M6
cell. Two new columns of numbers should appear. The second column represents the population values after extraction. Repeat this procedure by always pasting the cells side by side. C18
the formula = L6
, enter and drag to complete all the cells referring to the size classes at time t2. For time t3, in cell D18
type = N6
and copy to other size classes, for time t4, in cell E18
type = P6
and copy to the other size classes, and so on, always remembering to skip a column.Gotelli, N. J. 2007. Ecologia. Cap.3- Crescimento Populacional Estruturado. Pp. 49-82. Ed. Planta.
Gurevitch, J, Scheiner, S.M, Fox, G.A. 2009. Ecologia Vegetal. Cap. 5 - Ed. Artmed, São Paulo.
An Intuitive Guide to Linear Algebra, from Better explained.
Freckleton, R.P., Silva Matos, D.M., Bovi, M.L.A & Watkinson, A.R. 2003. Predicting the impacts of harvesting using structured population models: the importance of density-dependence and timing of harvest for a tropical palm tree. Journal of Applied Ecology, 40: 846-858.
Silva Matos, D.M., Freckleton, R.P. & Watkinson, A.R. 1999. The role of density dependence in the population dynamics of a tropical palm. Ecology, 80: 2635-2650.
Stubben, C., & Milligan, B. (2007). Estimating and analyzing demographic models using the popbio package in R. Journal of Statistical Software, 22(11), 1-23.