Appendix C. Additional methods of life table response analysis and results for site summary matrices.
LIFE TABLE RESPONSE ANALYSIS
Life table response analyses were conducted to determine the contribution each vital rate had to differences in λ across temporal and spatial scales. In contrast to elasticity, which measures the potential effect of a change in a given transition on λ, a life table response analysis can be used to decompose the population effect of a treatment (i.e., time) into the effect on the underlying transitions (Caswell 1996, 2001).
In the first analysis, changes in λ from the 2000–2003 to the 2003–2006 Conch Reef summary matrices, A^{(0003)} and A^{(0306)} were examined. From the mean matrix of A^{(0003)} and A^{(0306)}, a matrix of sensitivities was calculated. Then, with A^{(0003)} used as a reference matrix, a matrix of contributions was calculated by multiplying each element of the matrix of sensitivities by the respective element of A^{(0306)}  A^{(0003)} (Caswell 1996). In the second analysis, differences in λ between sites were examined. Analyses were completed as described above, with the exception that a mean matrix of the transition matrices from all sites was constructed and used as a reference matrix in which comparisons with each site was made (Caswell 2001).
RESULTS FOR SITE SUMMARY MATRICES
Table C1 presents the contributions of each transition to differences in λ between the site summary matrices. Transitions of size classes IV and V contributed the most to observed differences in λ between sites. Significantly greater λ for A^{(CR30)} compared to A^{(CR15)} was largely due to a large positive contribution from stasis of size class IV individuals at the 30 m Conch Reef site (Table C1). This is in agreement with the high elasticity value for this transition (Appendix J) and the higher frequency of this transition for A^{(CR30)} compared to A^{(CR15)}.
TABLE C1. Contributions of transitions to differences in λ between sites. Contributions were calculated by comparing each site summary matrix to a reference matrix, which is a mean of all site summary matrices. Negative values indicate transitions that contributed to accelerated population decline and positive values indicate transitions that contributed to slowed population decline compared to the reference matrix. Contributions > ±0.01 are in bold.
Location and size class  Size class 

15 m Conch 
Base 
I 
II 
III 
IV 
V 
Base 
0.0008 
0 
0.0006 
0.0018 
0.0082 
0.0118 
I 
0.0017 
0.0012 
0.0001 
0 
0 
0 
II 
0.0033 
0.0019 
0.0003 
0.0030 
0 
0 
III 
0.0044 
0.0001 
0.0003 
0.0098 
0.0099 
0.0091 
IV 
0.0024 
0 
0 
0.0034 
0.0134 
0.0368 
V 
0.0012 
0 
0 
0.0004 
0.0064 
0.0077 
20 m Conch 






Base 
0.0043 
0 
0.0005 
0.0006 
0.0002 
0.0101 
I 
0.0008 
0.0002 
0.0007 
0 
0 
0 
II 
0.0044 
0 
0.0005 
0.0040 
0 
0 
III 
0.0013 
0 
0.0015 
0.0042 
0.0016 
0.0065 
IV 
0.0015 
0 
0 
0.0009 
0.0227 
0.0150 
V 
0.0004 
0 
0 
0.0004 
0.0068 
0.0370 
30 m Conch 






Base 
0.0029 
0 
0.0003 
0.0010 
0.0049 
0.0081 
I 
0.0014 
0.0001 
0.0005 
0 
0 
0 
II 
0.0062 
0.0005 
0.0007 
0.0008 
0 
0 
III 
0.0029 
0 
0.0012 
0.0049 
0.0207 
0.0033 
IV 
0.0022 
0 
0 
0.0035 
0.0843 
0.0336 
V 
0.0004 
0 
0 
0.0006 
0.0383 
0.0305 
15 m Pickles 






Base 
0.0001 
0 
0.0002 
0.0012 
0.0041 
0.0071 
I 
0.0002 
0.0002 
0.0007 
0 
0 
0 
II 
0.0004 
0 
0.0002 
0.0028 
0 
0 
III 
0.0022 
0.0001 
0.0003 
0.0001 
0.0037 
0.0180 
IV 
0.0010 
0 
0 
0.0032 
0.0143 
0.0100 
V 
0.0002 
0 
0 
0.0016 
0.0045 
0.0241 
Reference matrix 






Base 
0.6568 
0 
0.0268 
0.0215 
0.0219 
0.0147 
I 
0.0379 
0.4618 
0.0369 
0 
0 
0 
II 
0.0535 
0.2816 
0.5244 
0.0304 
0 
0 
III 
0.0349 
0.0035 
0.3326 
0.5800 
0.0340 
0.0071 
IV 
0.0270 
0 
0 
0.3160 
0.7007 
0.0420 
V 
0.0069 
0 
0 
0.0081 
0.1846 
0.8153 
LITERATURE CITED
Caswell, H. 1996. Analysis of life table response experiments. II. Alternative parameterizations for size and stagestructured models. Ecological Modelling 88:73–82.
Caswell, H. 2001. Matrix population models. Sinauer, Sunderland, Massachusetts, USA.