Text S1. Statistical estimation of parameters in submodels
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Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 1 Text S1: Statistical estimation of parameters in submodels The following section presents additional details on parameterization of the submodels for the individual-based model (IBM). All parameters were were estimated using Markov Chain Monte Carlo simulations in JAGS v. 4.3.0 [1], with four chains, each run for 110,000 iterations with a burn-in of 10000 iterations. To reduce autocorrelation, we retained every 200th iteration from each chain. Convergence was assessed using the Gelman-Rubin diagnostic with a threshold of 1.1 [2], and by visually examining output from the chains. We assigned noninformative priors to most parameters, with parameters of these non-informative priors constrained to biologically reasonable values (Table S1). Parameter estimates are shown in Table S2. In this supplementary text, we provide details on statistical estimation of parameters in submodels; for further details on data collection of seeds, seedlings and fruit production see [3]. For further details on data collection of trees >1 cm DBH in the 50 ha Forest Dynamics Plot, see [4,5]. Fruit production The fruit production model estimates fruit production as a function of tree size using data on fallen fruit (including peels of fruit dispersed by frugivores) measured directly beneath parent trees in 3 m x 3 m fruit count quadrat h (42 total quadrats). Fruit production was modeled as a ππ Poisson random variable with a mean (λβ ) modeled as a sigmoid function of tree size (DBH), ππ ππ ππ with three free parameters (π½0 , π½1 , π½2 ). ππ πππ’ππ‘π β ~Poisson(λβ ) ππ λβ = ππ π½0 + (1) 1 ππ ππ ππ π½1 +exp(π½2 ×DBHβ ) Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 2 ππ In this equation, π½0 represents baseline fruit production for a 20 cm DBH tree (the observed ππ minimum size of reproductive trees), π½1 represents limiting fruit production as DBH gets large, ππ and π½2 is the per unit increase in fruit production as DBH increases. We assumed that total canopy area of any tree of reproductive size was equal to 89 m2 , the mean canopy area of reproductive Miliusa trees. Alternate models including canopy area as a function of DBH did not improve model fit. Thus, total fecundity per tree g is equal to: πππ’ππ‘π β ππππ’ππππ‘π¦π = ( ) × 89 m2 Unit area (2) Seedling establishment We modeled germination as a binomial random variable using data on 6500 seeds experimentally added to 1 m x 1 m quadrat i in plot j, with the number of experimentally added ππ₯π seeds equal to Nπ,π . As with other submodels for tree growth and survival, we assumed that πππ germination probability, θπ,π , was a function of conspecific seedling density and tree neighborhoods. We accounted for spatial non-independence in our data by including a normally πππ distributed plot-level random effect, ππ . The model is formally described as follows for seed addition quadrat i (281 total quadrats) in plot j (93 plots): πππ ππ₯π Experimental new recruits ~Binomial(θπ,π , Nπ,π ) πππ πππ + π½ πππ × Con. seedlingsπ + DBH. neighbori πππ πΌ πππ × ∑ +ππ πππ .πππ Distance. neighbori logit(θπ ) = π0 (3) Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. πππ In this equation, π0 3 is baseline germination rate, π½ πππ is the linear decrease in seed survival as seedling conspecific density increases, πΌ πππ represents the magnitude of NDD from tree neighborhoods, and πππ . πππ represents the distance decay of NDD. Seed dispersal We estimated seed dispersal parameters by combining data on fruit production by sampled reproductive adults, germination probability, and the spatial distribution of newly germinated seedlings relative to adults. This approach enables us to directly estimate seed dispersal kernels from the distributions of natural seedlings. New recruit abundance in quadrat i (1723 quadrats) within plot j (219 plots) was modeled as a Binomial-distributed random variable πππ with a a probability of success equal to the probability of germination (θπ,π ) and a number of πππ πππ‘ trials equal to natural seed arrival (Nπ,π ). θπ,π is known from the seed addition experiment πππ‘ π ππππ data, and Nπ,π is estimated as a Poisson-distributed random variable with mean equal to λπ,π : πππ πππ‘ Natural new recruitsπ,π ~Binomial(θπ,π , Nπ,π ) (4) πππ‘ π ππππ Nπ,π ~ Poisson(λπ,π ) (5) π ππππ Expected seed arrival (λπ,π ) is the sum of seed rain from adult trees within a 20 m radius around each new recruit quadrat (πΏππππ π πππ πππππ,π ) and seed rain from sources outside of the 20 m radius, (π΅ππ‘βπ.π ): π ππππ λπ,π = (πΏππππ π πππ πππππ,π + exp[π΅ππ‘β]) (6) πΏππππ π πππ πππππ,π = ∑ππ=1 ππππ’ππππ‘π¦π × πππ ππππ πππ (7) Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 4 The Bath term is necessary in the model to represent seedling quadrats where seedlings were counted but no trees were present in the local neighborhood. The Local seed rain term includes the model for fecundity per tree g (ππππ’ππππ‘π¦π,π,π ; Eq. 3), summed over all trees in a 20 m radius around quadrat k in plot j, and a dispersal submodel (πππ ππππ πππ ). The dispersal submodel represents the probability of seed arrival at a certain distance from tree g and was modeled as a two-dimensional Student’s T distribution with three degrees of freedom, following [6,7]. πππ ππππ πππ 1 = [ 2 2 distanceπ ) (π×exp[π’.πππ ππππ ππ])(1+ exp[π’.πππ ππππ ππ] (8) ] In the equation above, the parameter π’. πππ ππππ ππ represents the shape of the dispersal kernel. Initial height of seedlings We parameterized a submodel for the initial height of seedlings using data on the initial height of 628 seedlings from the seed addition experiment, with height measured three months after seed addition. Initial height of the ith seedling in plot j was modeled as a skew normal distribution: ππ.βπ‘ Initial height π,π ~Skew Normal(μππ.βπ‘ , π ππππ ππ.βπ‘ ) π,π , π βπππ (9) We modeled the expectation, μππβπ‘ π,π , of initial seedling height as a constant term with a normally distributed plot-level random effect, ππππβπ‘ , to represent non-independence of seedlings within the same seed addition plot: ππβπ‘ μππβπ‘ + ππππβπ‘ π,π = π½0 (10) Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 5 Seedling survival and growth To parameterize models for seedling survival and growth, we used annual censuses of 1505 seedlings in 174 plots. Seedling survival was modeled as a Bernoulli distribution and seedling growth was modeled as a skew normal distribution. Both models included a normally distributed random effect (ππ,π ), a linear term for size-dependence, (Sizeπ ), and a term for NDD from conspecific seedling density (π½ π π ). Unlike submodels for germination, seedling growth, tree growth and tree survival, preliminary analysis revealed that the tree neighborhood term did not improve model fit for seedling survival. Consequently, we did not include this term in the submodel for seedling survival. π π Seedling survival π,π ~Bernoulli(θπ,π ) π π π π logit(θπ,π ) = π0π π + π ππ§π π π × Sizeπ + π½ π π × Con. seedlingsπ + ππ,π (11) π π Seedling growthπ ~Skew Normal(μπ , π βπππ π π , π ππππ π π ) π π π π μπ = (π0 + π ππ§π π π × Sizeπ ) × exp(π½ π π × Con. seedlingsπ ) πΌπ π × exp (Size × ∑ π DBH.neighborj Distance.neighborj πππ .π π )+ (12) π π ππ,π In the equation above, size represents the linear effect of size on probability of seedling survival and seedling growth. Variance in the skew normal distribution is a function of two parameters, shape and scale. Skewness (in this case a tendency towards positive growth) is a function of the shape parameter. Parameters for NDD are described in Equation 3. Seedling height-DBH allometry Because seedling size was measured in height (cm) and tree size was measured in DBH (cm), the IBM requires an expression to translate seedling height into diameter as seedlings grow Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 6 into saplings. We parameterized two equations for seedling height-DBH allometry using DBH and height measurement data from 45 seedlings. The first allometric expression quantifies the probability that a seedling of a certain height, Height π , is >1 cm DBH: ππππ π π€ππ‘πβπ ~Bernoulli(θππ π€ππ‘πβ ) (13) logit(θππ π€ππ‘πβ ) = π π π€ππ‘πβ + π π π€ππ‘πβ × Height π The second allometric expression applies only to seedlings that have a Tree switch value of 1, and predicts a DBH measurement for new trees as a normally-distributed random variable: βπ‘ Tree DBHπ ~Normal(μβπ‘ π ,π ) (14) μβπ‘ π =π βπ‘ +π βπ‘ × Height π Tree survival and tree growth We estimated tree survival and growth using 2,049 tagged individuals censused at five year intervals from 1994 to 2004 in a 50 ha Forest Dynamics Plot. Survival was modeled as a Bernoulli distribution and growth was modeled as a skew-normal distribution. The probability of survival (θππ ) and the expectation of growth (μππ ) included terms for both size and In these equations, DBHπ represents the DBH of the ith individual, with DBH. neighborπ and Distance. neighborj representing the size and distance of neighbor tree j in a 25 m radius around the target individual. The first term in the models for tree survival and growth is the Hossfeld IV function, a function of tree DBH [8]. The Hossfeld IV function includes two free parameters, G and P. The second term is the function for NDD from tree neighborhood, including two free Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 7 parameters, πΌ, which determines the strength of NDD, and πππ . ππ , which determines the distance decay of NDD over the tree neighborhood. Tree survival was modeled as a Bernoulli distribution and tree growth was modeled as a skew normal distribution. Tree survival ~Bernoulli(θππ π ) logit(θππ π )= 2×πΊ.π ×DBHπ 2 DBHπ 2 (πΊ.π + ) π.π πΌππ + DBH × ∑ π ππ DBH.neighborπ Distance.neighborj πππ .ππ ππ (15) ππ Tree growthπ ~Skew Normal(μπ , π βππππ , π πππππ ) ππ μπ = πΌππ 2×πΊ.π×DBHπ (πΊ.π+ DBHπ 2 ) π.π ×∑ 2 × exp ( DBH π,π DBH.neighborj Distance.neighborj (16) πππ .ππ ) Similar to population structure in other plants [9], our data revealed changing variance as a function of tree size. Consequently, we modeled shape and scale of the skew normal distribution for adult growth as functions of tree size (DBH): ππ π βππππ = ππ βπππ × exp( DBHπ × ππ βπππ ) ππ π πππππ (17) = √ππ ππππ × DBHπ ππ ππππ References 1. Plummer, M. 2003 JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. URL http://citeseer.ist.psu.edu/plummer03jags.html. 2. Gelman, A. & Rubin, D. B. 1992 Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–472. 3. Caughlin, T. T., Ferguson, J. M., Lichstein, J. W., Bunyavejchewin, S. & Levey, D. J. 2013 The importance of long-distance seed dispersal for the demography and distribution of a canopy tree species. Ecology 95, 952–962. (doi:10.1890/13-0580.1) Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 8 4. Bunyavejchewin, S., LaFrankie, J. V. & Ashton, P. S. 2000 Population Ecology of Trees in Huai Kha Khaeng Wildlife Sanctuary, Thailand: Data from the 50-ha Forest Dynamics Plot. Smithson. Trop. Res. Inst. Balboa Panama 5. Bunyavejchewin, S., Baker, P. J., LaFrankie, J. V. & Ashton, P. S. 2004 Huai Kha Khaeng Forest Dynamics Plot, Thailand. Trop. For. Divers. Dynamism Find. Large-Scale Plot Netw. Univ. Chic. Press Chic., 482–491. 6. Clark, J. S., Silman, M., Kern, R., Macklin, E. & Lambers, J. H. 1999 Seed dispersal near and far: Patterns across temperate and tropical forests. Ecology 80, 1475–1494. 7. Muller-Landau, H. C., Wright, S. J., Calderón, O., Condit, R. & Hubbell, S. P. 2008 Interspecific variation in primary seed dispersal in a tropical forest. J. Ecol. 96, 653–667. 8. Zuidema, P. A., Jongejans, E., Chien, P. D., During, H. J. & Schieving, F. 2010 Integral Projection Models for trees: a new parameterization method and a validation of model output. J. Ecol. 98, 345–355. 9. Easterling, M. R., Ellner, S. P. & Dixon, P. M. 2000 Size-specific sensitivity: applying a new structured population model. Ecology 81, 694–708. Tables Table S1. Priors for parameters in submodels. Table S2. Parameter estimates and 95% credible intervals Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 9 Table S1. Priors for parameters in submodels. For normally and lognormally distributed priors, prior 1 is the mean and prior 2 is the standard deviation, with mean and standard deviation on the log-scale for lognormally-distributed priors. For uniform priors, prior 1 and 2 are the minimum and maximum of the distribution, respectively. Submodel Seed dispersal Seed production Seed production Seed production Seed dispersal Germination Germination Germination Germination Germination Germination Germination Seedling survival Seedling survival Seedling survival Seedling growth Seedling growth Parameter π’. πππ ππππ ππ ππ π½0 ππ π½1 ππ π½2 π΅ππ‘β πππ π0 π½ πππ πΌ πππ πππ . πππ μππ.βπ‘ π ππππ ππ.βπ‘ π βπππ ππ.βπ‘ π0π π π½ π π π ππ§π π π π π π0 πΌ π π Distribution Normal Uniform Uniform Uniform Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Normal Prior 1 0 0 0 -100 0 0 0 0 0 0 0 0 0 0 0 0 0 Prior 2 1.00E-06 100 100 0 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 1.00E-06 Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 10 Table S1 Priors for parameters in submodels, cont. Submodel Seedling growth Seedling growth Seedling growth Seedling heightDBH allometry Seedling heightDBH allometry Seedling heightDBH allometry Seedling heightDBH allometry Seedling heightDBH allometry Seedling heightDBH allometry Seedling heightDBH allometry Tree survival Tree growth Tree survival Tree growth Tree growth Tree growth Tree survival Tree survival Tree growth Tree growth Tree growth Tree growth Parameter πππ . π π π ππ§π π π π½ π π π ππππ π π π βπππ π π πβπ‘ π βπ‘ π βπ‘ π π π€ππ‘πβ π π π€ππ‘πβ ππ πΌ πΌ ππ πππ . ππ πππ . ππ πΊ. π π. π πΊ. π π. π ππ βπππ ππ ππππ ππ βπππ ππ ππππ Distribution Normal Normal Normal Prior 1 0 0 0 Prior 2 1.00E-06 1.00E-06 1.00E-06 Gamma 0.01 1.00E-02 Exponential 0.01 - Normal 0 1.00E-06 Normal 0 1.00E-06 Uniform 0 1.00E+03 Normal 0 1.00E-06 Normal Normal Normal Normal Normal Uniform Normal Normal Normal Normal Exponential Normal Normal 0 0 0 0 0 -50 0 0 0 0 1.00E-02 0 0 1.00E-06 1.00E-02 1.00E-02 1.00E-02 1.00E-02 5.00E+01 1.00E-02 1.00E-02 1.00E-02 1.00E-02 1.00E-02 1.00E-02 Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 11 Table S2. Parameter estimates and 95% credible intervals Submodel Parameter 2.50% 50% 97.50% Seed dispersal Seed production Seed production Seed production Seed dispersal Germination Germination Germination Germination Germination Germination Germination Seedling survival Seedling survival Seedling survival Seedling growth Seedling growth Seedling growth Seedling growth Seedling growth Seedling heightDBH allometry Seedling heightDBH allometry π’. πππ ππππ ππ ππ π½0 ππ π½1 ππ π½2 π΅ππ‘β πππ π0 π½ πππ πΌ πππ πππ . πππ μππ.βπ‘ π ππππ ππ.βπ‘ π βπππ ππ.βπ‘ π0π π π½ π π π ππ§π π π π π π0 πΌ π π πππ . π π π ππ§π π π π½ π π 776.9 1.01 0 -0.13 0.14 -2.86 -1.29 -0.05 0.01 4.71 0.83 -0.1 -0.24 -0.06 0.1 2.81 -15.4 0.69 0.13 -0.22 1915.8 1.42 0 -0.12 0.32 -2.49 -0.8 -0.03 0.21 4.91 0.9 0 0.35 -0.04 0.14 5.46 -3.9 1.3 0.17 -0.15 4590.73 2.93 0 -0.12 0.5 -2.17 -0.39 -0.02 0.56 5.1 0.97 0.1 0.91 -0.02 0.19 8.14 -0.83 1.84 0.21 -0.08 π ππππ π π 20.62 22.17 23.79 π βπππ π π 0.05 0.14 0.25 Caughlin T.T., Ferguson J.M., Lichstein J.W., Zuidema P.A., Bunyavejchewin S., Levey D.J. 12 Table S2. Parameter estimates and 95% credible intervals, cont. Submodel Parameter 2.50% 50% 97.50% Seedling heightDBH allometry Seedling heightDBH allometry πβπ‘ -1.04 -0.85 -0.64 π βπ‘ 0.01 0.01 0.01 Seedling heightDBH allometry π βπ‘ 0.12 0.15 0.19 Seedling heightDBH allometry π π π€ππ‘πβ -78.42 -36.52 -14.61 Seedling heightDBH allometry π π π€ππ‘πβ 6.27 15.86 34.19 Tree survival Tree growth Tree survival Tree growth Tree growth Tree growth Tree survival Tree survival Tree growth Tree growth Tree growth Tree growth πΌ ππ πΌ ππ πππ . ππ πππ . ππ πΊ. π π. π πΊ. π π. π ππ βπππ ππ ππππ ππ βπππ ππ ππππ -1.65 -0.29 0.38 1.18 -1.89 4.85 -6.79 3.17 -23.4 0.07 -19.34 0.86 -0.74 -0.15 0.82 1.56 12.75 4.92 -0.06 3.26 -9.28 0.08 0.14 0.89 -0.1 -0.04 1.21 2.06 24.38 4.98 9.05 3.37 -3.25 0.09 18.71 0.92
