S. Twombly, Guiming Wang, and N. Thompson Hobbs. 2007. Composite forces shape population dynamics of copepod crustaceans. Ecology 88:658–670.

Appendix A. Details of field sampling methods and the stage-structured model used in analyses.

Field Sampling: Samples were collected in Laguna la Orsinera, Venezuela, from June through December 1984. The lake is small, permanent, and seasonally flooded by the adjacent Orinoco River from late July until late October. During inundation in 1984, lake depth increased from <2 m to 6 m, lake volume increased nine-fold, water transparency increased, and thermal stratification resulted in sporadic oxygen depletion in the deepest (6 m) water. One of us (ST) sampled zooplankton quantitatively at three stations with a 10-cm diameter flexible tube that sampled the entire water column to within 0.5 m of the bottom. The tube was calibrated so that the volume of water collected in vertical samples of varying depth could be determined. Each sample was filtered through a 37-µm net and preserved. Triplicate samples were collected at each station and combined across stations to create three composite replicate samples. From June through mid-September, samples were collected on Monday, Wednesday, Friday, and Saturday of every week (max sampling interval = 2 days). From mid-September until late December, sampling occurred on Monday, Tuesday, Wednesday, and Thursday of every week. These sampling regimes were designed to test a demographic estimation method (Caswell and Twombly 1989) that was being developed at the time. All samples were collected between 0900 and 1200 hours. We also sampled in the channel connecting the lake to the river. This was done weekly during flooding, until flow was negligible and the channel became obstructed with macrophytes. Over the course of this study, lake volume increased approximately nine-fold. To compensate for these changes, zooplankton densities (number of individual per liter) were multiplied by total lake volume; all results are presented as total population size (number of individuals in the entire lake).

Subsamples of each composite replicate were counted at 25× with a stereomicroscope (Twombly 1994). Triplicate subsamples of each sample were regularly counted to ensure that the coefficient of variation among subsamples was 10–15%. Individual development stages (combined N1-N2, N3, N4, N5, N6, CI, CII, CIII, CIV, CV, adult females, females with eggs, adult males) and loose eggs) were counted for D. negrensis; for O. amazonica, all naupliar stages were combined but copepodite stages, adult females, ovigerous females, and adult males were distinguished. The number of eggs carried by each female was also recorded.

Estimates of m, stage-specific developmental rates: Stage-specific durations were estimated directly. Ovigerous females of D. negrensis were collected from the field and placed individually in lake water in small Petri dishes until eggs hatched. Nauplii were then raised individually in small enclosures (70-mL snap-cap vials equipped with two 53-µm plankton mesh "windows;" Elmore 1982) in lake water. Individuals in each enclosure were observed daily, developmental stages were determined, and lake water (with natural phytoplankton densities) in each enclosure was replaced. The data recorded were days, since hatching, required to reach successive developmental stages. From these data, mean durations of successive stages were determined and stage-specific developmental rates were calculated as the reciprocal of mean stage duration (data are presented in Twombly 1994).

We assumed that mean stage durations and developmental rates determined for D. negrensis applied to O. amazonica. We also assumed that developmental rates were constant over time, primarily because temperature – a main determinant of developmental rates in invertebrates – varied little temporally. Although temporal variation in food availability also affects developmental rates (Elmore 1982), we were confident that developmental rates varied less than the other covariates we considered. We also assumed constant developmental rates for practical reasons. As the number of unknown parameters in our models increased, it approached the number of samples or data points we had available to test our models, and model fitting became less precise. By assuming constant stage-specific developmental rates, we also reduced to total number of models that we needed to test with data.

Experimental measures yielded a mean duration for all six naupliar stages of 7.3 days and a mean duration for all five copepodite stages of 10 days. These durations translated into daily developmental rates for nauplii of 0.136 and for copepodites of 0.2; we used these data in our models.

Model Development: Fig. A1 shows the life-cycle diagram we used to develop a two-sex, stage-structured model that considered four composite stages, i.e., eggs, six combined larval stages (nauplii), five combined juvenile stages (copepodites), and adults. The matrix representation for females was:

(A.1)

where E_t, Nf_t, Cf_t, and Af_t are the abundance of eggs, female nauplii, female copepodites, and female adults at day t, respectively; s_e, s_n, s_c, and s_af are the stage-specific daily survival rates of eggs, nauplii, copepodites, and female adults, respectively; m_e, m_n and m_c are stage-specific daily maturation rates; F is the proportion of female adults laying eggs; and r_a is the number of eggs per reproducing female.

The matrix representation for males is given by:

(A.2)

where Nm_t, Cm_t, and Am_t are the abundance of male nauplii, male copepodites, and male adults at day t, respectively, and s_ma is the daily survival rate of male adults.

These matrices can be rewritten as several simultaneous equations:

For example, the third equation above predicts that the number of female copepodites (Cf)   present at time t is equal to the number of female copepodites at time t – 1 that remained copepodites (1 – m_c) and survived as copepodites (s_c) plus the number of female nauplii present at t-1(Nf_t-1) that developed into copepodites (m_n) and survived (s_n) this transition. Because we had independent estimates of stage-specific developmental rates available, we were able in each of equations (that is, for each stage) to separate estimates of survival from recruitment into a stage.

The total population abundance (N_all) is defined as the sum of the abundance of nauplii,

copepodites, and adults: .

Model Selection: Table A1 shows the top three models (lowest AIC_c, second lowest AIC_c, third lowest AIC_c) for each step in the forward selection process, for each species. The first step represents the null model and contained no environmental covariates. Step 1 shows results from the best fitting single parameter models, or those created by adding one of the 36 possible unknown parameters. Step 2 repeats this analysis, adding one of the remaining 35 parameters to create two parameter models. As explained in the text, this procedure was interated until the AIC_c value was no longer improved (Δ AIC_c > 2.0) by adding remaining parameters. These data reinforce results presented in the manuscript that, for both species, five-parameter models provided the best fit to our field data.

LITERATURE CITED

Caswell, H. and S. Twombly. 1989. Estimation of stage-specific demographic parameters for zooplankton populations: methods based on stage-classified matrix projection models. Pages 93–107 in L. McDonald, B. Manly, J. Lockwood, and J. Logan, editors. Estimation and analysis of insect populations. Lecture Notes in Statistics 55, Springer-Verlag, New York, New York, USA.

Elmore, J. L. 1982. The influence of food concentration and container volume on life history parameters of Diaptomus dorsalis Marsh from subtropical Florida. Hydrobiologia 89:215–223.

Twombly, S. 1994. Comparative demography and population dynamics of two coexisting copepods in a Venezuelan floodplain lake. Limnology and Oceanography 39:234–247.

TABLE A1. Results, as AIC_c values, for top three best supported models in forward selection process. (A) Results for D. negrensis, (B) Results for O. amazonica. The best supported model after including all unknown parameters was achieved for each species in Step 5 and is shown in bold type.

FIG. A1. Life cycle diagram for two sex, stage-specific demographic model for Diaptomus negrensis and Oithona amazonica. Circles represent specific stages and the arrows connecting circles or representing loops within a circle represent stage-specific transition rates. In this figure, me, mn, mc represent stage-specific developmental rates for eggs, nauplii, and copepodites respectively. se, sn, sc, sfa, and sma represent survival rates of eggs, nauplii, copepodites, adult females, and adult males, respectively. The arrow that connects adult females with eggs represents reproductive output as defined by r (number of eggs carried per female) and F (proportion of females ovigerous).