Appendix B. Methods for sensitivity, elasticity, and life table response experiment (LTRE) analyses.
Following Caswell (2001), I calculated the sensitivity matrix for each period as,
(B.1) |
where D_{h}^{T} is the transpose of the product of the periodic matrices used to calculate A_{h}, excluding B_{h}. For example, for h = 1, D_{h} = B_{3} B_{2}. S_{A}_{h} is the sensitivity matrix of A_{h} equaling vw^{T} where v and w are vectors of the reproductive values and stable stage distribution associated with A_{h}, respectively (v and w are scaled so that their scalar product is equals 1). I then used the chain rule on matrix elements to calculate sensitivities of individual vital rates.& Elasticity matrices for each period are calculated as,
(B.2) |
where ◦ indicates element-by-element multiplication.
The LTRE analysis for each shrub microhabitat is,
(B.3) |
where vr_{i} is the ith vital rate and λ^{mean} is calculated using the mean shrub and intershrub vital rates (vr^{mean}). In other words, λ^{shrub} is proportional to λ^{interhsrub} plus the summed contributions of each vital rate difference. Essentially, the contribution of each vital rate to population growth is the product of its treatment-induced difference and mean sensitivity.
LITERATURE CITED
Caswell, H. 2001. Matrix population models: construction, analysis, and interpretation. Second edition. Sinauer Associates, Sunderland, Massachusetts, USA.