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

 Дата канвертавання 22.04.2016 Памер 81.3 Kb.
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

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 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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