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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 1 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 2 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 3 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. 4 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. 5