Matrix algebra can be used to model the state transitions of a set of habitat patches, just like we did for the states of individuals in a population in the models of Leslie and Lefkowitch (do you remember them?). Behind these models are also Markov chains, to which we will return when we study the Hubbell Neutral model. The idea here is exactly the same as in matrix models for populations. We will create a transition matrix representing the likelihood of transition from each state to another over time, which multiplied by a vector containing the number of patches in each state gives us the number of patches in each state in the next time interval. We will build these models in Excel just as we did with populations of palm trees, including also a disturbance term, which represents deforestation.
|status at time t|
|status at t+1||Open||0,10||0,10||0,10||0,01|
|Number of patches at start|
To remember matrix multiplication in Excel we will resume the explanation of the exercise on population dynamics. If you have any doubt, go to the script in Population matrix models - Tutorial in spreadsheets.
In the formula for the multiplication, you can put the \$ symbol in the rows and columns selection for the transition matrix (ex: \$C\$4:\$E\$6). This will “freeze” the selection in the formula, and may help you to make projections for the population size. The result of this multiplication is a vector (N2) with the number of individuals in the next time step (t+1) for each stage. If the formula does not produce a vector, select the cells with the formula and the next lines, related to each class for the time t+1, press F2 (to open the formula) and then Control+Shift+Enter. That should fix it!
Warning: after you do the trick above, whenever you try to change a cell in the matrix, Excel will show an error message. You can't escape this trap with Enter, but you can press Esc to exit.
From this first example, you can create the following situations in different worksheets:
Working on the Sonora desert (Californa, USA), McAuliffe observed the dynamic of the desert for three very slow successional stages. These stages are characterized by the greasewood (Larrea tridentata), the canyon ragweed (Ambrosia ambrosioides) and open spaces.
The transition matrix created after the data collection is the following:
|stages at time t|
Draw a diagram for this model, and create a transition worksheet on Excel.
The same study above also measured the frequency of patches in each state. The collected data are:
| | ^ Observed frequency ^ ^ State ^ Open | 0,99854 | |::: ^ //Ambrosia// | 0,0013 | |::: ^ //Larrea// | 0,00016 |
Compare the observed values with those estimated by the matrix model.