Survival, reproduction and immigration explain the dynamics of a local Red-backed Shrike population in The Netherlands




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On-line APPENDIX for “Survival, reproduction and immigration explain the dynamics of a local Red-backed Shrike population in The Netherlands”
Details of the elasticity analysis

We modelled the Bargerveen population of the Red-backed Shrike with a two-stage matrix model with the census just after the breeding season (post-breeding). At the census juvenile birds J(t) are up to one year old and adult birds A(t) are older than one year. The population matrix C is based on Fig. 1 and expressed by equation (A.1). We subsequently filled in the parameters: m mean number of fledglings per breeding pair, 0 the survival of juveniles during the first 10 months after fledging and 1 the yearly survival of adult birds resulting from the MARK analysis. The formulas for the matrix entries are the ones defined by Caswell (2001) and we described some of them in words in the main text.



(A.1)

Without change in environmental parameters, the population is assumed to grow at an annual rate equal to the dominant eigenvalue (λd) of the population matrix C. A dominant eigenvalue greater (less) than one, indicates an exponential growing (decreasing) population in absence of limiting factors in the environment. Matrix C includes the probability of survival of each age group from year t to year t+1, and the reproduction of sexually active adults giving rise to new juveniles (the top row). It should be noted that the factor 2 is because of an assumed 1:1 sex ratio in fledglings and the female-based model.

The elasticity of a parameter h is defined as in eqn. A.2.

(A.2)

For the elasticities we first have to derive the characteristic equation (A.3). The dominant eigenvalue is .



(A.3)

The partial derivatives with respect to m, φ0 and φ1 for the dominant eigenvalues are easily derived. Thereafter, these are multiplied with the appropriate factor, giving the expressions (A.4), which are also given in the main text.



, (A.4)





Figure S.1 The population development of the Red-backed Shrike, Lanius collurio, in the Bargerveen area, The Netherlands.


Table S.1 (=Full version of Table 3 from the main text) Model selection for the survival (φ) and resighting probability (p) of Red-backed Shrikes ringed in Bargerveen, The Netherlands, based on a variance inflation factor () equal to 1.20: for each model the quasi likelihood AICc (QAICc), the difference in QAICc between the current model and the best model (ΔQAICc), the relative belief in the model (weight) as expressed by AICc-weight for the ith model , the number of estimable parameters (k), and the deviance (Qdev) are given. The models are sorted by increasing QAICc. Coding; (.): constant value is estimated for this parameter; 2 stage (S2): distinction between “first year” and “older”; 3 stage (S3): distinction between “first year”, ”second year” and “three years and older”; g: females and males separate; t: each year is treated differently; combinations of these codes are possible (see the model names). Below, the most general model (for which the goodness of fit procedure is performed) is given in bold, italic lettertype.

Model

QAICc

ΔQAICc

weight

k

Qdev

φ(S2*g)p(t*g)

964.796

0.000

0.316

18

125.056

φ(S2*g)p(g)

965.732

0.936

0.198

6

150.544

φ(S2)p(t*g)

966.842

2.047

0.114

16

131.232

φ(S2)p(g)

967.889

3.093

0.067

4

156.741

φ(S3*g)p(t*g)

968.204

3.408

0.058

20

124.320

φ(S3)p(t*g)

968.673

3.877

0.045

17

131.000

φ(S2*g)p(S2*g)

969.043

4.247

0.038

8

149.801

φ(S3*g)p(g)

969.328

4.532

0.033

8

150.085

φ(S3)p(g)

969.342

4.547

0.033

5

156.176

φ(S2)p(S2*g)

969.647

4.851

0.028

6

154.459

φ(S3)p(S2*g)

970.251

5.455

0.021

7

153.037

φ(S2*g)p(S3*g)

971.588

6.792

0.011

10

148.276

φ(S2)p(S3*g)

971.767

6.971

0.010

8

152.524

φ(S3*g)p(S2*g)

972.371

7.575

0.007

10

149.059

φ(S2*g)p(1)

972.502

7.706

0.007

5

159.336

φ(S3)p(S3*g)

973.326

8.530

0.004

9

152.050



Model

QAICc

ΔQAICc

weight

k

Qdev

φ(S2*g)p(t)

974.067

9.271

0.003

12

146.670

φ(S2*g)p(S2)

974.139

9.343

0.003

6

158.951

φ(S2*g)p(S3)

975.460

10.665

0.002

7

158.247

φ(S3*g)p(S3*g)

975.470

10.674

0.002

12

148.073

φ(S3*g)p(1)

976.029

11.234

0.001

7

158.816

φ(S3*g)p(S2)

977.395

12.599

0.001

8

158.152

φ(S3*g)p(t)

977.591

12.795

0.001

14

146.095

φ(S3*g)p(S3)

979.054

14.258

0.000

9

157.778

φ(S2)p(1)

983.494

18.698

0.000

3

174.360

φ(S2)p(S2)

985.169

20.374

0.000

4

174.021

φ(S2)p(t)

985.295

20.499

0.000

10

161.982

φ(S3)p(1)

985.298

20.502

0.000

4

174.150

φ(S2)p(S3)

986.452

21.657

0.000

5

173.286

φ(S3)p(S2)

986.651

21.856

0.000

5

173.485

φ(S3)p(t)

987.126

22.331

0.000

11

161.774

φ(S3)p(S3)

988.312

23.516

0.000

6

173.124

φ(1)p(S3*g)

1007.577

42.781

0.000

7

190.363

φ(t)p(S3*g)

1008.719

43.924

0.000

14

177.224

φ(g)p(S3*g)

1009.592

44.796

0.000

8

190.349

φ(1)p(S2*g)

1010.818

46.022

0.000

5

197.652

φ(t)p(S2*g)

1010.927

46.131

0.000

12

183.530

φ(g)p(S2*g)

1012.839

48.044

0.000

6

197.651

φ(t*g)p(S3*g)

1019.561

54.766

0.000

22

171.517

φ(t*g)p(S2*g)

1021.506

56.710

0.000

20

177.622

φ(g)p(S3)

1024.678

59.882

0.000

5

211.512

φ(g)p(S2)

1028.317

63.521

0.000

4

217.169

φ(1)p(S3)

1030.448

65.652

0.000

4

219.300

φ(t)p(S3)

1031.046

66.250

0.000

11

205.694

φ(t*g)p(t*g)

1031.453

66.657

0.000

26

175.043

φ(t)p(g)

1032.712

67.916

0.000

10

209.400

φ(1)p(t*g)

1033.838

69.042

0.000

16

198.228

φ (t)p(S2)

1033.966

69.170

0.000

10

210.654

φ (1)p(S2)

1034.358

69.562

0.000

3

225.224

φ(t)p(t*g)

1034.972

70.176

0.000

22

186.927

φ(g)p(t*g)

1035.525

70.730

0.000

17

197.852

φ(t*g)p(S3)

1035.882

71.086

0.000

19

194.072

φ(1)p(g)

1037.074

72.278

0.000

3

227.941




Model

QAICc

ΔQAICc

weight

k

Qdev

φ(g)p(g)

1037.803

73.008

0.000

4

226.656

φ(t*g)p(S2)

1038.430

73.634

0.000

18

198.690

φ(t*g)p(g)

1043.201

78.406

0.000

18

203.462

φ(g)p(t)

1045.668

80.872

0.000

10

222.355

φ(1)p(t)

1053.006

88.211

0.000

9

231.731

φ(t)p(t)

1054.651

89.856

0.000

15

221.100

φ(g)p(1)

1055.051

90.255

0.000

3

245.917

φ(t*g)p(t)

1057.325

92.529

0.000

23

207.195

φ(t)p(1)

1057.689

92.894

0.000

9

236.414

φ(t*g)p(1)

1061.050

96.254

0.000

17

223.377

φ(1)p(1)

1062.669

97.874

0.000

2

255.547




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