Download Appendix 2S: Bayesian modeling Effects of density and relatedness

Appendix 2S: Bayesian modeling
Effects of density and relatedness on the dispersal kernel
To measure the effects of density and relatedness and the interaction between them on dispersal
distance, we analyzed dispersal kernels by generating decumulative density distributions according to
distance Figures S1a, b. Such functions (Vinatier et al. 2011) represent for each kernel the distance
travelled by all individuals, with maximal probability at the first patch all individuals moved at least
zero distance and a zero probability at distances not reached by any individual during a trial. A
negative power function was fit to the data in its simplest binomial linearized form:
Logitpi=lnµ0-α.lnDi+ε
with pi denoting the proportion of individuals from the populations reaching a certain distance i, µ0 the
intercept, α the slope of distance dependent decay in dispersal and the error term ε. In order to test the
impact of the other parameters, additional terms were added for main effects related to density Dens,
relatedness Rel, and inbreeding Inb. All interactions were modeled and interdependency of individual
dispersal distance probabilities was controlled for by adding replicate ηline as an estimated variance
component. Because distances moved showed largest variance at day seven, with an overall increase
in distance with day in all models data not shown we only present kernel parameters for day seven.
Besides analyses of the overall kernel attributes, we inferred the distance travelled by the furthest 5%
of individuals D95 (Figure S1).
In order to assess proper parameter estimates from kernel parameters and the derived kernel
attributes we applied Bayesian estimation using a Monte-Carlo Markov chain MCMC procedure in
WinBugs v. 1.4. (Spiegelhalter et al. 2003). Because we had no a priori information, flat priors for
regression coefficients were drawn from a normal distribution with a mean of 0 and a standard
deviation SD of 106. Priors for variance components were drawn from a positively constrained
uniform distribution with a mean of 1 and SD 5 and three chains were modeled for each model. To
assure accurate MCMC simulations from the prior distributions, an initial “burn in” of 10 000
iterations was performed and discarded from analysis. This was followed by 20 000 iterations for both
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analyses. After visual inspections for possible autocorrelation and assessing chain convergence
Brooks-Gelman-Rubin diagnostics (Brooks and Gelman 1998), the mean and SD of each posterior
parameter estimate regression coefficients and variance estimates was calculated, as well the 2.5th and
97.5th percentiles of the samples. These were used to describe the 95% Bayesian credibility interval of
the posterior distributions of model parameters and derived kernel attributes.
Model 1: four levels of density Dens and two levels of relatedness Rel
Logitpi=lnµ0-α1lnDi+ δ0.Densi+δ1.Densi.lnDi +β0.Reli+ β1.Reli.lnDi+γ0.Reli.Densi+γ1.Reli. Densi..lnDi
+ηi+ε
Table S1: Effects of density and relatedness on dispersal distance. Parameter estimates from the
negative exponential model based on cumulative density distributions according to distance.
Subscripts indicate intercept 0 or slope 1. Boldness indicates significance.
symbol
Parameter
µ0
α1
Overall
δ0
δ1
β0
β1
γ0
γ1
Density
Relatedness
Density *
Relatedness
mean
sd
MC error
2.50%
median
97.50%
0.07492
0.341
0.02007
-0.565
0.1001
0.7031
0.6502
0.1049
0.00616
0.4465
0.6548
0.8613
184
122.3
7.197
56.89
143.2
519.2
3.916
0.2144
0.01259
3.502
3.919
4.359
-0.6449
0.6707
0.0397
-1.818
-0.7982
0.9216
0.7249
0.1874
0.01103
0.4104
0.7199
1.106
1.733
1.168
0.06915
-0.9722
1.914
3.841
-1.859
0.3858
0.02271
-2.649
-1.846
-1.212
First, density Dens, δ1 significantly affects dispersal distance: the higher the density, the further the
mites disperse Table 1. Second, treatment Rel; β1 affects the dispersal distance: the higher the
relatedness, the further the distance travelled by the mites. There is an interaction between Dens and
Rel γ1. The effect of density is stronger when mites are more genetically related.
Increased relatedness increases dispersal distance
Following the same procedure as in the previous experiment, we analyzed the effects of genetic
relatedness on the dispersal kernels using the following model:
Model 2: five levels of relatedness
Logitpi=lnµ0-α.lnDi+β0.Reli+ β1.Reli.lnDi+ηi+ ε
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Treatment or Rel; β1 affects the dispersal distance: the higher the relatedness, the further the distance
travelled by the mites Table S2. The increase in dispersal distance is significant overall across
treatments, while pairwise comparisons between each treatment gamma 1 are not significant.
Table S2: Effects of relatedness on dispersal distance. Parameter estimates from the negative
exponential model based on cumulative density distributions according to distance. Boldness indicates
significance. Subscripts indicate intercept 0 or slope 1.
symbol
µ0
α.
β0
Parameter
Overall
Relatedness
β1
mean
sd
MC error
2.50%
median
97.50%
120.6
30.17
2.807
75.96
115.5
195.1
-3.309
0.06671
0.005528
-3.182
-3.311
-3.428
-0.3992
0.3786
0.03753
-1.23
-0.3779
0.2746
0.887
0.1006
0.008875
0.6918
0.8899
1.072
Relatedness, but not inbreeding, increases dispersal distance
Following the same procedure as in the previous experiments, we analyzed the effects of genetic
relatedness and inbreeding on the dispersal kernels using the following model:
Model 3: two levels of relatedness, two levels of inbreeding
Logitpi=lnµ0-α.lnDi+β0.Reli+ β1.Reli.lnDi+λ0.Inbi+λ1.Inbi.lnDi+γ0.Reli.Inbi+γ1.Reli.Inbi..lnDi +ηi+ε
The level of relatedness Rel, β1 affects the dispersal distance: the higher the relatedness, the further the
distance travelled by the mites Table 3. The level of inbreeding Inb, λ1 has no effect on the dispersal
distance of the mites.
Table S3: Effects of relatedness and inbreeding on dispersal distance. Parameter estimates from
the negative exponential model based on cumulative density distributions according to distance.
Boldness indicates significance. Subscripts indicate intercept 0 or slope 1.
symbol
µ0
α.
β0
β1
λ0
λ1
γ0
γ1
Parameter
mean
sd
MC error
2.50%
median
97.50%
143.4
76.64
5.299
53.11
121.1
343.4
3.194
0.1005
0.006324
2.998
3.193
3.407
-0.1734
0.2381
0.01473
-0.6756
-0.1724
0.2787
0.8294
0.1236
0.007862
0.5932
0.828
1.09
0.5016
0.2527
0.01568
-0.003361
0.5071
0.9882
-0.1983
0.1406
0.008819
-0.4703
-0.2035
0.08985
-0.4148
0.3312
0.0205
-1.027
-0.4201
0.29
0.07507
0.1726
0.01086
-0.2955
0.07767
0.3981
Overall
Relatedness
Inbreeding
Relatedness*
Inbreeding
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Figure S1: Parameter estimates of the effect of density on the distance moved by the furthest moving
5% of individuals (D95) on the synthetic (SP) and base (BP) populations. Error bars represent
confidence intervals as estimated by the model.
Figure S2: Decumulative distribution of all trials from the density and relatedness experiment on the
7th (last) day of the experiment. Each line represents one trial. Light to dark grey represents increasing
density. Dotted lines represent the synthetic populations while solid lines represent the base
population.
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Figure S3: Decumulative distribution of all trials in response to increasing relatedness at the 7th (last)
day of the experiments. Light to dark grey represents increasing relatedness.
References
1.
Brooks S.P. & Gelman A. (1998). General Methods for Monitoring Convergence of Iterative
Simulations. Journal of Computational and Graphical Statistics, 7, 434-455.
2.
Spiegelhalter D., Thopas A., Best N. & Lunn D. (2003). Winbugs. In. MRC Biostatistics Unit
Cambridge.
3.
Vinatier F., Lescourret F., Duyck P.F., Martin O., Senoussi R. & Tixier P. (2011). Should I Stay or
Should I Go? A Habitat-Dependent Dispersal Kernel Improves Prediction of Movement. Plos
One, 6.
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