Ecological Archives E096-070-A2

Richard P. Shefferson and Deborah A. Roach. 2012. The triple helix of Plantago lanceolata: Genetics and the environment interact to determine population dynamics. Ecology 93:793–802.

Appendix B. Details of vital rate estimation, matrix construction, and LTRE analysis.

The main field experiment in this study was designed to assess patterns in adult mortality over time. Thus, we conducted several further studies to parameterize fertility, field germination, and early survival. To estimate the number of seeds produced by individual plants, all inflorescences were collected following seed maturation but prior to seed dispersal, throughout the reproductive season each year. The average mass of these inflorescences was measured by weighing one average-sized, undamaged inflorescence, collected mid-reproductive season for each individual. Inflorescence mass varied across both year and cohort (Proc Mixed, SAS (2003), P < 0.0001, N = 16903, Block as random, log transformation to improve normality). Using a smaller subset of inflorescences, from each cohort and year, the total number of seeds within a weighed inflorescence was counted and a linear regression (Proc Reg, SAS (2003)) was performed to determine the relationship between inflorescence mass and seed number (R² ranged from 0.85–0.96 over the seven years). The total seed output per individual each year was then estimated as: inflorescence number × mean inflorescence weight per cohort per year × slope of seed number regression.

Field germination and early survival were measured in a separate experiment to estimate fertility using the seed production estimates. In Oct. 2002, we planted approximately 60 seeds per cross from 40 dam-sire crosses directly into the field, yielding 2147 planted seeds. Each individual seed was planted in a small hole adjacent to a toothpick and censused weekly for one year. The germination rate was 69%, and the survival of germinated seedlings after one year was low (14.1%). We estimated the fertility for individuals as estimated seed output × field germination rate (69%). Size-class fertility was the mean fertility across all individuals within each size class for either each combination of year, cohort, and block (four cohort analysis, see below), or each combination of year and sire (cohort-specific analyses, see below). Growth from the seedling stage to the adult stages was estimated as the probability of seedling survival (14.1%) multiplied by the probability of surviving adults being in each of the adult classes per year.

We constructed 8 × 8 population projection models to test the influence of cohort, temporal environmental, and spatial environmental variation in the all-cohort analysis, and genetic vs. temporal variation in the single-cohort analyses. Given the eight life cycle classes we used, including a seedling stage and seven adult size classes, each matrix had the following form:


Fi represents the fertility of adult size class i (i.e., the average per-capita contribution of class i in year t to the seedling stage in next year t + 1), Pji represents the probability of progression (i.e., growth) from stage i to larger stage j, Si represents the probability of stasis in adult size class i (seedlings cannot remain seedlings for more than one year), and Rji represents the probability of retrogression (i.e., shrinkage) from adult size class i to smaller class j. All probabilities of progression, stasis, and retrogression incorporated the probability of surviving from year t to year t + 1. Thus, column sums of non-fertility transitions should total the annual survival of the particular life stage associated with that column. Although seed dormancy is possible in Plantago (van Groenendael and Slim 1988), we observed none in the field germination experiment and so it was not included in our model. We then created a reference matrix in which matrix elements were estimated by an element-by-element average from all our matrices.

In life table response experiment (LTRE) analysis, the impact of a particular factor or treatment on population dynamics can be explored in detail by comparing the population projection matrix or matrices resulting from this factor with equivalent reference, control matrices. Each matrix element’s contribution to changes in λ according to tested factors is given as:

Δλ = λm - λr ≈ Σji(aji(m) - aji(r))(δλ / δaji) | (A(m) + A(r)) / 2(B.2)

where λ is the deterministic population growth rate, aji is the matrix element in the jth row and ith column, A(m) refers to a specific treatment matrix, A(r) refers to the reference matrix (calculated as the element-by-element average of all utilized matrices), and Δλ is the difference between the projected population growth rate of the population under treatment (λm) and the projected population growth rate under control conditions (λr). Here, (δλ/δaji)|(A(m) + A(r)) / 2 refers to the sensitivity of λ to changes in matrix element aji, but taken using a matrix in which each element is the arithmetic mean of corresponding matrix elements in A(m) and A(r). The impacts of particular factors on Δλ can be assessed by summing the appropriate matrix elements in the resulting LTRE matrices, which are of the same dimension as the initial projection matrices. LTRE matrix elements are essentially independent contributions of shifts in projection matrix elements between the treatment and reference matrices to Δλ, and are obtained by multiplying the difference in projection matrix elements by the sensitivity of λr to changes in the parameter in question. The sum of all LTRE contributions corresponding to a projection matrix model set is equal to the difference between the projected deterministic population growth rate (λ) in the reference matrix (A(r)) and the treatment matrix (A(m)) (i.e., Δλ = λm – λr). The impact of a particular factor on λ via changes to a particular kind of demographic transition can be assessed by summing LTRE contributions corresponding to that kind of transition. For example, in the current study, the impact of the year 2003 on Δλ via progression from the seedling stage was estimated as the sum of LTRE contributions of all progression-related transitions from the seedling stage in the average 2003 matrix (i.e., summing the LTRE scores associated with P1S, P2S, P3S, P4S, P5S, P6S, and P7S, in matrix B.1).


SAS Institute Inc. 2003. The SAS System 9.1 for Windows. SAS Institute, Inc., Cary, North Carolina, USA.

van Groenendael, J. M. and P. Slim. 1988. The contrasting dynamics of two populations of Plantago lanceolata classified by age and size. Journal of Ecology 76:585–599.

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