Appendix B. Description on how matrix elements were modified to allow for variation among years in the eleven different types of disturbance matrices.
Temporal variable REM matrices were made by use of life-table response experiment analyses with fixed design (LTRE; Caswell 2001). Analyses were made separately for REM1 and the REF population for the 19931994 period, and for REM2 and the REF population for the 19941995 period, respectively. LTRE analyses identified the transitions aij that contributed most strongly (defined as ≥ 2 %) to observed differences in population growth rates between treatment (REM; t) and control (REF; c) as given by:
is the difference in the transition probability aij, between a treatment matrix (REM) and a control matrix (REF), and
is the sensitivity of transition aij in the mean matrix (the matrix ‘halfway’ between At and Ac).
The following transitions (matrix elements aij) gave the highest contributions, all with a proportional contribution 2%, to the variation in in the LTRE analyses [the LTRE analysis accounted for 95.6% (REM1 vs. REF 19931994) and 97.2% (REM2 vs. REF 19941995) of the differences in , respectively]:
Temporal variability was added to each of these transitions in two steps. In the first step, eleven candidate values of aij, aijt*, one for each year t except t = T, the year the experiment in question was performed, were created by adding to aij’ (the probability for the transition from stage j to stage i in a REF matrix a given year) the difference between aijT in the REM matrix and aijT in the REF matrix for year T (addition of deviations provided more realistic transition probabilities than multiplication; K. Rydgren, unpublished results). For all other transitions, aijt* was set equal to aij’. In few exceptional cases single candidate aijt* values calculated this way were unrealistically high, as reflected in a sum Z of aijt* values for survival into S2-S8 that exceeded 1.000. Accordingly, all column vectors in all matrices in which Z exceeded 1.000 were subjected to a second step in which the aijt* were curtailed by multiplication with 1/Z. The matrices so constructed mainly kept the structure of the REF matrices for the different years (indicating that realistic temporal variation had been added), while at the same time provided variation in the transition probabilities that contributed the most to variation in due to disturbance by removal of 50% of the shoots (correlations between REF and REM1 matrices: Kendall’s = 0.870; between REF and REM2 matrices; = 0.788; n = 12, P < 0.001).
A LTRE approach to creation of temporal variation in CLP matrices was not applicable because the Clipped treatment constrained the life cycle of H. splendens and produced reducible matrices (Fig. 1; also see Rydgren et al. 2001). Instead we created two series of matrices from the REF matrices by (1) setting equal to zero all elements that were zero in CLP1 or CLP2, respectively, and (2) tentatively adding to the remaining elements the difference between these transition probabilities in CLP1 and in the 19931994 REF matrices, and in CLP2 and the 19941995 REF matrices, respectively. In the few cases in which the modified aij values were unrealistic, i.e., when the sum Z of survival transition probabilities in a given matrix column were > 1.000, all tentatively modified aij values in a column were curtailed by multiplication with 1/Z.
Temporal variation in the seven other types of matrices, i.e., H2, H3, H4, CLP1/REM1, H2/REM1, H3/REM1, H3/REM2 follows from CLP1, CLP2, REM1, REM2, or REF matrices, since the matrix elements comes from these matrices as detailed in Appendix A.
Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. Second edition. Sinuauer, Sunderland, Massachusetts, USA.
Rydgren, K., H. de Kroon, R. H. Økland, and J. van Groenendael. 2001. Effects of fine-scale disturbances on the demography and population dynamics of the clonal moss Hylocomium splendens. Journal of Ecology 89:395405.