Hildegunn Viljugrein, Nils Chr. Stenseth, Graham W. Smith, and Gunnhilder H. Steinbakk. 2005. Density dependence in North American ducks. Ecology 86:245–254.

Appendix A. Tables showing data characteristics and estimated density dependence from data.

Table A1. Direct density dependence, , estimated from a Bayesian state-space model (Eqs. 1 and 2) where numbers of ponds are not included (c_i = 0) for populations of (A) the Mallard and (B) the Canvasback.

Strata		90% HPD		95% CI		95% CI
A) Bayesian state-space model (Mallards)
26	0.00	[-0.05, 0.06]	3.3	[3.0, 3.6]	0.13	[0.08, 0.19]
27	-0.06	[-0.19, 0.06]	2.8	[2.1, 3.5]	0.29	[0.21, 0.39]
28	-0.18	[-0.41, 0.03]	2.0	[1.5, 2.8]	0.31	[0.23, 0.42]
29	-0.16	[-0.36, 0.07]	1.9	[1.5, 2.4]	0.21	[0.13, 0.31]
30	-0.15	[-0.38, 0.05]	2.8	[2.4, 3.3]	0.23	[0.15, 0.33]
31	-0.05	[-0.16, 0.07]	3.1	[2.4, 3.7]	0.20	[0.12, 0.30]
32	-0.14	[-0.31, 0.03]	2.5	[2.1, 3.0]	0.25	[0.18, 0.34]
33	-0.15	[-0.32, 0.03]	2.0	[1.5, 2.5]	0.30	[0.21, 0.41]
34	-0.22	[-0.52, 0.04]	3.0	[2.4, 4.2]	0.40	[0.29, 0.53]
35	-0.40	[-0.68, -0.07]	2.3	[2.0, 3.1]	0.44	[0.32, 0.59]
36	-0.66	[-1.14, -0.19]	0.4	[0.1, 0.7]	0.34	[0.22, 0.51]
37	-0.43	[-0.77, 0.02]	1.7	[1.4, 2.4]	0.27	[0.17, 0.39]
38	-0.41	[-0.78, -0.02]	0.7	[0.2, 1.3]	0.41	[0.27, 0.59]
39	-0.56	[-0.91, -0.22]	1.9	[1.7, 2.3]	0.33	[0.24, 0.44]
40	-0.13	[-0.41, 0.06]	3.2	[2.5, 4.1]	0.30	[0.21, 0.42]
43	0.00	[-0.07, 0.08]	-1.0	[-2.5, 0.4]	0.37	[0.24, 0.54]
44	-0.14	[-0.40, 0.10]	0.5	[-0.8, 1.2]	0.36	[0.25, 0.50]
45	-0.35	[-0.62, -0.06]	1.6	[1.1, 2.1]	0.42	[0.31, 0.55]
46	-0.26	[-0.52, 0.01]	1.5	[0.8, 2.1]	0.43	[0.32, 0.58]
47	-0.24	[-0.55, 0.08]	-0.5	[-2.5, 0.5]	0.59	[0.35, 0.94]
48	-0.25	[-0.52, 0.06]	1.2	[-0.1, 1.8]	0.45	[0.33, 0.60]
49	-0.37	[-0.67, -0.06]	0.9	[0.2, 1.4]	0.46	[0.34, 0.62]
B) Bayesian state-space model (Canvasbacks)
26	-0.37	[-0.80, 0.05]	0.1	[-0.2, 0.5]	0.20	[0.07, 0.34]
27	-0.95	[-1.45, -0.45]	-1.7	[-2.0, -1.3]	0.64	[0.44, 0.90]
28	-0.81	[-1.61, 0.07]	-2.1	[-2.4, -1.9]	0.14	[0.03, 0.38]
29	-0.44	[-1.00, 0.02]	-4.2	[-10.1, -1.7]	1.66	[1.06, 2.56]
30	-1.32	[-2.08, -0.29]	0.2	[0.1, 0.4]	0.14	[0.03, 0.32]
31	-1.05	[1-87, -0.32]	-0.1	[-0.3, 0.1]	0.22	[0.05, 0.43]
32	-0.24	[-0.46, -0.02]	-0.8	[-1.6, 0.1]	0.52	[0.37, 0.71]
33	-0.73	[-1.30, -0.17]	-2.9	[-3.9, -2.0]	1.28	[0.88, 1.87]
34	-0.49	[-0.87, -0.08]	0.0	[-0.4, 0.6]	0.49	[0.34, 0.68]
35	-0.30	[-1.61, 0.02]	-0.5	[-1.2, 0.8]	0.52	[0.33, 0.77]
36+37†	-1.26	[-1.85, -0.68]	-0.8	[-1.0, -0.6]	0.34	[0.13, 0.54]
38+39†	-0.33	[-0.80, 0.12]	-0.8	[-1.3, -0.5]	0.21	[0.07, 0.38]
40	-0.67	[-1.35, 0.07]	0.9	[0.7, 1.3]	0.19	[0.05, 0.34]
45	0.00	[-0.05, 0.06]	-6.4	[-10.1, -2.9]	0.84	[0.59, 1.16]
46+47†	-0.62	[-0.92, -0.34]	-1.2	[-1.7, -0.8]	0.77	[0.59, 0.99]
48	-0.73	[-1.35, -0.13]	-3.1	[-3.7, -2.5]	0.68	[0.37, 1.07]

   Notes: The estimated parameters of Eqs. 1 are given as posterior means and 90 or 95% credible intervals (in brackets) – the Bayesian alternative to confidence intervals. For  the credible interval is given as the highest posterior density (HPD) interval. If the 90% HPD interval of  do not include zero, the probability of density dependence is found to be at least 0.95. The equilibrium value, , is estimated and is only given a vague prior.

^†Neighboring strata with similar habitat types were combined in order to avoid the problem of too many zero-observations.

Table A2. Density dependence estimated from two different modeling approaches standardizing data by their short-term mean: (i) a second version (V2) of the Bayesian state-space model [m_i = mean({x_i,t}) and c_i = 0] and (ii) a first-order autoregressive model [AR(1)], for populations of (A) the mallard and (B) the canvasback.

	State-space model (V2)		AR(1)-model			Abundance data
Strata		90% CI		SE		mean(Y)	SD(Y)
A) Mallard
26	-0.29	[-0.73, 0.14]	-0.19	0.09	0.17	2.72	0.34	0.14
27	-0.31	[-0.63, -0.01]	-0.27	0.11	0.31	2.10	0.51	0.16
28	-0.41	[-0.72, -0.10]	-0.40	0.13	0.34	1.71	0.49	0.13
29	-0.69	[-1.02, -0.37]	-0.41	0.13	0.30	1.74	0.39	0.24
30	-0.51	[-0.94, -0.09]	-0.45	0.13	0.29	2.48	0.37	0.17
31	-0.96	[-1.29, -0.59]	-0.34	0.12	0.27	2.44	0.42	0.22
32	-0.36	[-0.72, -0.01]	-0.32	0.12	0.29	2.21	0.41	0.14
33	-0.38	[-0.70, -0.08]	-0.32	0.11	0.35	1.68	0.49	0.20
34	-0.55	[-0.95, -0.16]	-0.48	0.13	0.40	2.54	0.52	0.17
35	-0.59	[-0.95, -0.24]	-0.55	0.14	0.47	2.24	0.55	0.20
36	-0.90	[-1.32, -0.50]	-0.82	0.16	0.57	0.42	0.57	0.62
37	-1.05	[-1.33, -0.75]	-0.73	0.15	0.32	1.61	0.36	0.23
38	-0.68	[-1.06, -0.32]	-0.83	0.16	0.51	0.76	0.52	0.32
39	-0.84	[-1.20, -0.46]	-0.68	0.15	0.36	1.87	0.39	0.18
40	-0.53	[-0.94, -0.12]	-0.48	0.13	0.32	2.56	0.42	0.17
43	-0.21	[-0.43, -0.02]	-0.81	0.16	0.80	0.42	2.07	0.30
44	-0.45	[-0.77, -0.16]	-0.42	0.13	0.44	0.88	0.56	0.26
45	-0.59	[-0.88, -0.30]	-0.48	0.13	0.44	1.69	0.52	0.22
46	-0.46	[-0.75, -0.19]	-0.53	0.14	0.51	1.53	0.57	0.21
47	-0.74	[-1.17, -0.34]	-0.73	0.15	0.85	-0.01	0.88	0.60
48	-0.44	[-0.73, -0.18]	-0.47	0.13	0.48	1.43	0.56	0.22
49	-0.56	[-0.88, -0.26]	-0.53	0.14	0.50	1.02	0.57	0.28
B) Canvasback
26	-0.82	[-1.25, -0.42]	-0.66	0.15	0.36	0.10	0.39	0.28
27	-0.97	[-1.34, -0.61]	-0.95	0.16	0.72	-1.33	0.74	0.47
28	-0.99	[-1.41, -0.56]	-1.05	0.16	0.62	-1.72	0.63	0.56
29	-0.92	[-1.36, -0.48]	-1.01	0.16	2.20	-2.87	2.21	0.60
30	-1.07	[-1.51, -0.61]	-0.89	0.16	0.44	0.42	0.45	0.38
31	-1.03	[-1.44, -0.62]	-0.87	0.16	0.44	0.11	0.50	0.41
32	-0.42	[-0.69, -0.16]	-0.42	0.13	0.60	-0.68	0.73	0.32
33	-0.98	[-1.40, -0.55]	-1.00	0.16	1.91	-2.26	1.91	0.59
34	-0.74	[-1.11, -0.39]	-0.66	0.15	0.54	0.07	0.60	0.33
35	-0.65	[-1.02, -0.31]	-0.73	0.15	0.71	-0.56	0.77	0.45
36+37†	-1.11	[-1.52, -0.69]	-0.98	0.16	0.52	-0.56	0.54	0.34
38+39†	-0.83	[-1.25, -0.42]	-0.82	0.16	0.40	-0.63	0.42	0.33
40	-1.11	[-1.50, -0.70]	-0.86	0.16	0.36	0.98	0.38	0.31
45	-0.57	[-0.85, -0.29]	-0.91	0.16	1.69	-1.27	1.71	0.40
46+47†	-0.72	[-1.06, -0.40]	-0.68	0.15	0.78	-1.14	0.83	0.46
48	-0.97	[-1.38, -0.56]	-0.95	0.16	0.89	-2.46	0.89	0.70

   Notes: For the Bayesian model, the estimated density dependence is given as posterior means and 90% credible intervals (in brackets). Estimates of direct density dependence obtained from the Bayesian model and the AR(1)-model are represented by  and , respectively. The standard error of the noise term is represented by  for the AR(1)-model. The mean and standard deviation of duck log_e-abundance estimates (for the time period 1957–1997) is denoted by mean(Y) and sd(Y), respectively. For each stratum the precision of the data is given by an average  (the higher , the lower the precision).

^†Neighboring strata with similar habitat types were combined in order to avoid the problem of too many zero-observations.