Modeling the Relationship between Water Level, Wild Rice Abundance, and Waterfowl Abundance at a Central North American Wetland

Abstract

Recent evidence suggests wild rice (Zizania palustris), an important resource for migrating waterfowl, is declining in parts of central North America, providing motivation to rigorously quantify the relationship between waterfowl and wild rice. A hierarchical mixed-effects model was applied to data on waterfowl abundance for 16 species, wild rice stem density, and two measures of water depth (true water depth at vegetation sampling locations and water surface elevation). Results provide evidence for an effect of true water depth (TWD) on wild rice abundance (posterior mean estimate for TWD coefficient, β_TWD = 0.92, 95% confidence interval = 0.11—1.74), but not for an effect of wild rice stem density or water surface elevation on local waterfowl abundance (posterior mean values for relevant parameters overlapped 0). Refined protocols for sampling design and more consistent sampling frequency to increase data quality should be pursued to overcome issues that may have obfuscated relationships evaluated here.

Appendices

Appendix 1. Details of methods used to impute missing data, and corresponding results.

Data were imputed to account for missing observations between sampling periods in the water level and rice data. The Amelia package was used to impute data, in R version 3.3.1 (R Core Team 2016). This package performs multiple imputation using an ‘expectation-maximization with bootstrapping’ algorithm, creating multiple complete datasets using a series of imputations to compile distributions with which to supply mean and variance estimates for each missing data point (Honaker et al. 2011). In each complete dataset, the observed data are not imputed. The average of 10 imputation chains was used in subsequent modeling efforts.

There were 52.2%, and 39.1% of values missing from the TWD and rice density records, respectively. Large proportions of missing data in TWD and rice density led to wide ranges for imputed data in both records (Appendix Figs. 3 and 4). Nevertheless, the median values of the posterior estimates from the Amelia method for imputing missing data were reasonable, given the available observed data.

Fig. 3

Boxplots of the posterior distributions for each year from the Amelia imputation for true water depth at sampling

Fig. 4

Boxplots of the posterior distributions for each year from the Amelia imputation for wild rice density

Appendix 2. Stan code and the associated data used to develop hierarchical water-rice-waterfowl model.

Appendix 3. Simulated data to evaluate the ability of the model to accurately approximate data from a known generating function.

We simulated data starting with 100 values to approximate true water depth at sampling locations (TWD), pulled from a normal distribution with a mean of 75 + (30 × sin(2 × pi × 0.01 × y_i)), with y_i = year i and a standard deviation of 2; the 75 represents typical water depths for optimal wild rice growth, 70 to 77 cm (Pillsbury and McGuire 2009); this generates cyclical patterns in the data with noise. We then generated 100 values meant to represent wild rice density as a function of the mean stem density in a given area (~60 stems per m²), water depth, and random noise (mean stem density - [β_TWD × TWD] + Norm[mean = 0, sd = 5]). Finally, we generated 100 values to represent waterfowl abundance as a function of wild rice density and random noise: ([β_RD × wild rice density] + Norm[mean = 0, sd = 100]).

Table 2 Fixed-effects parameters explained the majority of the variation in the dataset (95% credible intervals (CI) not overlapping 0—represented by “Lower CI” and “Upper CI”). $\widehat{\mathrm{R}}$ is a test for convergence (generally assumed to be achieved when <1.1)

Fig. 5

Posterior distribution estimate for the application of our model to simulated wild rice density (red line, plus 95% credible interval shaded region). The solid black line represents the simulated (input) density data

Fig. 6

Posterior distribution estimate for the application of our model to simulated annual waterfowl abundance (red line, plus 95% credible interval shaded region). The solid black line represents the simulated (input) abundance data

Appendix 4. Comparison of Rice Lake NWR waterfowl abundance with waterfowl breeding population surveys (BPOP). We only show the 11 species that occurred in both the Rice Lake NWR and BPOP records for the period of time for this study (1990–2012). Species are listed using AOU codes; see Table 1 for common and scientific names.

Appendix 5

Fig. 7

Rice density input data (black dots, with 95% confidence intervals for the imputed data) and posterior time series (gray lines, with 95% credible intervals in gray shaded region) for each species. The models are largely identical for each species (as expected, with consistent input data), with a negligible degree of variation present as a result of the random sampling of initial conditions for each application of the model. Species are listed using AOU codes; see Table 1 for common and scientific names