Appendix A. Methodology and results for state–space modeling of duck time series data.
We used the state-space modeling methods of Dennis et al. (2006) and Knape (2008) to compare parameter estimates of population dynamics between time periods and within duck species, while accounting for sampling error. Our state space model of duck dynamics is:
X_{t} = X_{t-1}+ r_{max}-γX_{t-1}+η_{t}, η_{t}~ N(0,σ^{2}),
Y_{t}= X_{t}+ ε_{t}, ε_{t}~ N(0, τ^{2}),
where X_{t} is the true population abundance, Y_{t} is the observed population abundance, τ_{t} is variation in true abundance due to process error (with variance s^{2}), and ε_{t} is variation in observed abundance due to sampling error (with variance t^{2}).
Briefly, we estimated γ, r_{max}, σ and τ by using the PROC MIXED procedure in SAS (version 9.1.3, SAS Inc., Cary, NC) to conduct a repeated measures analysis of variance with ln-transformed observed density as the response variable, an AR(1) covariance structure (i.e., possible density-dependence) on the repeated measures, and a random effect of time (Dennis et al. 2006). We also estimated γ, r_{max}, and σ, while specifying τ from observed standard errors, using a Kalman filter in the SSPIR package of R (version 2.8.1, R Foundation for Statistical Computing; see Knape 2008). In this latter method, a different, observed τ was included for each time-step. For both methods, maximum likelihood was used to fit the model to the data. These methods fit a Gompertz model for density-dependence, rather than the Ricker model used in our main analyses. Thus, the objective of the present exercise was not direct comparison of individual estimates of population processes between methods. Instead, we focus on comparisons of qualitative results describing changes in population dynamics between time periods.
Results generated using the above methods were similar to those generated in our main analyses (Tables 1, 2, 3, and 4 supporting Appendix A). Fundamental changes in patterns of density-dependence between the first and second time segments were similar to those observed in our main analyses using the Dennis et al. (2006) method. Estimates indicate that all species except gadwall experienced at least qualitative reductions in r_{max} (mean Δr_{max}: -1.434 ± 0.411; W+ = 54, W- = 1, P = 0.004; Table A2). Similarly, γ also declined with time (mean Δγ: -0.212 ± 0.065; W+ = 54, W- = 1, P = 0.004; Table A2). Neither process error nor sampling error changed significantly with time (mean Δσ2: -0.0048 ± 0.0058; W+ = 33, W- = 22, P = 0.63; mean Δt2: -0.0057 ± 0.0035; W+ = 41, W- = 14, P = 0.193; Table A2). Similarly, sampling error was usually substantially lower than process variance and the ratio of process error to sampling error did not change significantly over time (W+ = 33, W- = 22, P = 0.63; Table A2).
Fundamental changes in patterns of density-dependence between first and second time segments were also similar to those observed in our main analyses, using the Knape (2008) method. Most species (except gadwall, American wigeon, and northern pintail) experienced qualitative reductions in r_{max} (mean Δr_{max}: -1.65 ± 0.568; W+ = 50, W- = 5, P= 0.020; Table A4). Similarly, γ also declined with time (mean Δγ: -0.224 ± 0.085; W+ = 49, W- = 6, P = 0.027; Table A4). Process error did not change significantly with time (mean Δσ2: -0.0055 ± 0.0035; W+ = 43, W- = 12, P = 0.13; Table A4), but sampling error did (mean Δt2: -0.0032 ± 0.0008; W+ = 54, W- = 1, P = 0.004; Table A4). However, sampling error was usually substantially lower than process variance and the ratio of process error to sampling error did not change significantly over time (W+ = 15, W- = 40, P = 0.23; Table A4).
TABLE A1. Parameter estimates for population growth models fit to time-series data for 10 duck species (1955–2005) using the Dennis et al. (2006) method. Parameter r_{max} is the intrinsic growth rate (at N_{0}), γ is an index of density dependence, σ^{2} is the estimates of process error, and τ^{2} is the estimate of sampling error.
Species |
r_{max} |
γ |
τ^{2} |
σ^{2} |
σ^{2} / τ^{2} |
Mallard |
1.756 |
0.219 |
0.0005 |
0.0156 |
30.52 |
Gadwall |
0.237 |
0.037 |
0.0070 |
0.0199 |
2.85 |
American wigeon |
2.617 |
0.370 |
0.0019 |
0.0201 |
10.86 |
Green–winged teal |
0.708 |
0.105 |
0.0311 |
0.0076 |
0.24 |
Blue–winged teal |
2.037 |
0.270 |
0.0074 |
0.0186 |
2.51 |
Northern shoveler |
0.519 |
0.075 |
0.0136 |
0.0126 |
0.93 |
Northern pintail |
0.029 |
0.039 |
0.0165 |
0.0209 |
1.26 |
Redhead |
1.430 |
0.248 |
0.0078 |
0.0191 |
2.46 |
Canvasback |
2.886 |
0.508 |
0.0078 |
0.0220 |
2.84 |
Lesser scaup |
0.378 |
0.050 |
0.0085 |
0.0040 |
0.46 |
Total ducks |
1.994 |
0.214 |
0.0017 |
0.0093 |
5.53 |
TABLE A2. Parameter estimates for population growth models fit to time–series data for 10 duck species (1955–2005) using the Knape (2008) method. Parameter r_{max} is the intrinsic growth rate (at N_{0}), γ is an index of density dependence, σ^{2} is the estimate of process error, and τ^{2} is the mean in the actual observation standard error.
Species |
r_{max} |
γ |
τ^{2} |
σ^{2} |
σ^{2} / τ^{2} |
Mallard |
1.636 |
0.184 |
0.0020 |
0.0126 |
6.30 |
Gadwall |
0.265 |
0.037 |
0.0073 |
0.0191 |
2.63 |
American wigeon |
2.392 |
0.305 |
0.0046 |
0.0153 |
3.31 |
Green–winged teal |
1.853 |
0.247 |
0.0089 |
0.0212 |
2.37 |
Blue–winged teal |
2.640 |
0.314 |
0.0046 |
0.0229 |
5.01 |
Northern shoveler |
1.265 |
0.165 |
0.0054 |
0.0280 |
5.15 |
Northern pintail |
0.770 |
0.093 |
0.0041 |
0.0417 |
10.07 |
Redhead |
1.258 |
0.196 |
0.0128 |
0.0139 |
1.09 |
Canvasback |
2.524 |
0.400 |
0.0145 |
0.0146 |
1.01 |
Lesser scaup |
0.720 |
0.085 |
0.0047 |
0.0069 |
1.47 |
TABLE A3. Parameter estimates for population growth models fit to time–series data for 10 duck species (1955–2005) using the Dennis et al. (2006) method. Models were fit to 1955–1980 (period 1) and 1981–2005 (period 2) times series separately. Parameter r_{max} is the intrinsic growth rate (at N_{0}), γ is an index of density dependence, σ^{2} is the estimate of process error, and τ^{2} is the estimate of sampling error.
Species |
Time Period |
r_{max} |
γ |
τ^{2} |
σ^{2} |
σ^{2} / τ^{2} |
Mallard |
1 |
2.747 |
0.340 |
0.0008 |
0.0180 |
22.58 |
Mallard |
2 |
1.495 |
0.188 |
<0.0001 |
0.0128 |
>100 |
Gadwall |
1 |
0.421 |
0.067 |
0.0118 |
0.0218 |
1.84 |
Gadwall |
2 |
0.621 |
0.091 |
0.0014 |
0.0182 |
13.17 |
American wigeon |
1 |
3.721 |
0.521 |
<0.0001 |
0.0230 |
>100 |
American wigeon |
2 |
2.851 |
0.406 |
<0.0001 |
0.0213 |
>100 |
Green–winged teal |
1 |
5.693 |
0.563 |
<0.0001 |
0.0650 |
>100 |
Green–winged teal |
2 |
1.665 |
0.243 |
0.0027 |
0.0150 |
5.61 |
Blue–winged teal |
1 |
4.244 |
0.563 |
<0.0001 |
0.0216 |
>100 |
Blue–winged teal |
2 |
1.583 |
0.209 |
0.0103 |
0.0196 |
1.91 |
Northern shoveler |
1 |
1.603 |
0.238 |
0.0185 |
0.0073 |
0.40 |
Northern shoveler |
2 |
0.772 |
0.110 |
0.0098 |
0.0169 |
1.73 |
Northern pintail |
1 |
1.771 |
0.228 |
0.0141 |
0.0323 |
2.28 |
Northern pintail |
2 |
1.294 |
0.181 |
0.0154 |
0.0143 |
0.93 |
Redhead |
1 |
1.882 |
0.329 |
0.0172 |
0.0143 |
0.83 |
Redhead |
2 |
1.370 |
0.235 |
<0.0001 |
0.0216 |
>100 |
Canvasback |
1 |
4.417 |
0.779 |
0.0289 |
0.0048 |
0.17 |
Canvasback |
2 |
1.660 |
0.291 |
<0.0001 |
0.0199 |
>100 |
Lesser scaup |
1 |
1.511 |
0.194 |
0.0107 |
0.0041 |
0.38 |
Lesser scaup |
2 |
0.350 |
0.046 |
0.0059 |
0.0049 |
0.83 |
Total ducks |
1 |
2.622 |
0.280 |
0.0023 |
0.0094 |
4.11 |
Total ducks |
2 |
1.884 |
0.203 |
0.0015 |
0.0086 |
5.79 |
TABLE A4. Parameter estimates for population growth models fit to time–series data for 10 duck species (1955–2005) using the Knape (2008) method. Models were fit to 1955–1980 (period 1) and 1981–2005 (period 2) times series separately. Parameter r_{max} is the intrinsic growth rate (at N_{0}), γ is an index of density dependence, σ^{2} is the estimates of process error, and τ^{2} is the mean in the actual observation standard error.
Species |
Time Period |
r_{max} |
γ |
τ^{2} |
σ^{2} |
σ^{2 }/ τ^{2} |
Mallard |
1 |
2.620 |
0.292 |
0.0024 |
0.0137 |
5.57 |
Mallard |
2 |
1.366 |
0.155 |
0.0015 |
0.0103 |
6.85 |
Gadwall |
1 |
0.655 |
0.094 |
0.0102 |
0.0251 |
2.47 |
Gadwall |
2 |
0.638 |
0.084 |
0.0048 |
0.0158 |
3.31 |
American wigeon |
1 |
2.888 |
0.363 |
0.0058 |
0.0153 |
2.64 |
American wigeon |
2 |
3.144 |
0.404 |
0.0035 |
0.0138 |
3.93 |
Green–winged teal |
1 |
4.942 |
0.668 |
0.0135 |
0.0370 |
2.74 |
Green–winged teal |
2 |
1.527 |
0.200 |
0.0052 |
0.0127 |
2.46 |
Blue–winged teal |
1 |
4.008 |
0.478 |
0.0057 |
0.0152 |
2.67 |
Blue–winged teal |
2 |
2.152 |
0.257 |
0.0035 |
0.0280 |
7.95 |
Northern shoveler |
1 |
4.238 |
0.566 |
0.0076 |
0.0236 |
3.12 |
Northern shoveler |
2 |
1.359 |
0.173 |
0.0036 |
0.0285 |
7.94 |
Northern pintail |
1 |
2.870 |
0.334 |
0.0040 |
0.0467 |
11.80 |
Northern pintail |
2 |
3.702 |
0.470 |
0.0043 |
0.0274 |
6.33 |
Redhead |
1 |
1.958 |
0.307 |
0.0148 |
0.0146 |
0.98 |
Redhead |
2 |
1.125 |
0.174 |
0.0109 |
0.0140 |
1.28 |
Canvasback |
1 |
6.236 |
0.989 |
0.0160 |
0.0221 |
1.38 |
Canvasback |
2 |
1.345 |
0.213 |
0.0130 |
0.0089 |
0.69 |
Lesser scaup |
1 |
3.167 |
0.366 |
0.0056 |
0.0086 |
1.52 |
Lesser scaup |
2 |
0.724 |
0.086 |
0.0038 |
0.0073 |
1.93 |
LITERATURE CITED
Dennis, B., Ponciano, J. M., Lele, S. R., Taper, M. L., and Staples, D. F. 2006. Estimating density dependence, process noise, and observation error. Ecological Monographs 76:323–341.
Knape, J. 2008. Estimability of density–dependence in models of time series data. Ecology 89:2994–3000.