Gene dispersal in Heliconia
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Côrtes et al.
Supporting information:
Data S1. Study site
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Figure S1- Map of the study area with location of continuous forest sites (CF1 and CF2) and
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three 1-ha fragments (F1, F2 and F3). Areas in white are forested areas and in gray are areas
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that were clear-cut (today these areas encompass secondary growth vegetation and abandoned
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pastures). Black lines are access roads.
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Table S1. Pairwise distances in meters among populations and between edges of fragments to
nearest old-growth continuous forest.
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CF1
CF2
F1
F2
F3
Edge to nearest forest
CF1
-
41,400
24,300
39,400
40,200
-
CF2
-
-
17,200
2,100
1,300
-
F1
-
-
-
15,100
15,900
100
F2
-
-
-
-
840
100
F3
-
-
-
-
-
100
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Data S2. Implementation of the Bayesian model
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a)
Implementation:
In the genetic component of the model, the probability of observing the seedling’s
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genotype, given its parental genotypes, P(GkO|GiO,Gi’O) is calculated by producing an (na +1) x
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(na+1) matrix of parentage probabilities for each seedling, with rows representing the identified
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potential mothers within the plot, plus a hypothetical mother from outside the plot, and
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columns representing the identified potential fathers within the plot, plus a hypothetical father
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from outside the plot. The matrix represents the probability that a seedling k is produced by
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mother i’ and father i, relative to all potential parental combinations, given genotyping errors
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and Mendelian inheritance. This step is conducted separately and is independent of ecological
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variables and dispersal parameters.
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Priors were defined based on previous information about pollen and seed vector
movements. A prior of up = 2000 (standard deviation 1500) was used for the pollen dispersal
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kernel, corresponding to a conservative modal pollination distance of 36 m. Although there are
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no data on hummingbird movement in the area, these pollinators can move across the
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heterogeneous landscape at the study site and elsewhere, covering distances of 1000 m in less
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than four hours (Stouffer & Bierregaard 1995; Hadley & Betts 2009). A prior of us = 500
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(standard deviation 1000) was used for the seed dispersal kernel, representing a conservative
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modal dispersal distance of 18 m, which is the average seed dispersal distance found for the
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main frugivores of Heliconia in the study area (Uriarte et al. 2011). Parameters up and us were
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drawn from a truncated normal distribution, with lower and upper bounds of 10 and 15,000 for
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up and 10 and 3,000 for us. The assignment of maternity and paternity is defined from drawing
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the parameters from either the pollination truncated normal distribution (which will define the
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pollination kernel) or the seed dispersal truncated normal distribution (which will define the
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seed dispersal kernel). Therefore, the plant is defined as a mother or father based on the
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distribution the parameter is drawn from. Instead of assigning the nearest parent as the mother
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(on the assumption that seed dispersal is more limited than pollination), the seed dispersal
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distribution has a lower mean and narrower standard deviation, which will most likely result in
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lower distances of seed dispersal compared to pollination, based on prior information we have
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on the system. This method allows room for uncertainty in the assignment of maternity and
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paternity, which makes the model more realistic. We included more details in the supporting
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information. We used different priors to evaluate the effects of changing initial conditions on
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the recovery of the dispersal kernel parameters (Supporting information S2.b).
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The contributions of adult plants from outside the plot can be incorporated in the model
by assigning hypothetical plants at 10 interval distances of 10 m, going outwards from the
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sampled plot. Pollen and seed production outside the plots was calculated as the mean
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production within plots, multiplied by the plant density within each of the rectangular zones
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delimited by these hypothetical plants. For populations in continuous forest sites (CF1 and CF2),
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density of flowering plants outside plots was set as the density of flowering plants within plots,
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under the assumption that density, although variable in space, is homogeneous within a
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continuous forest of similar environmental conditions. By contrast, areas outside fragment
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plots encompass secondary growth vegetation, where density of Heliconia acuminata is
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generally lower than in forested areas (Bruna & Ribeiro 2005). We set the density of flowering
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plants to 0.0008 plants/m2 outside the Dimona plots (F2 and F3) and 0.0012 plants/m2 outside
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F1, based on the results of Bruna & Ribeiro (2005). Genotypes of hypothetical plants, as well as
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of plants with missing genotypes at locus l, were set as the product of the allele frequencies of
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the respective populations (Moran & Clark 2011). Results are sensitive to the density of
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hypothetical parents outside the sampled plots. It is likely that density of flowering H.
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acuminata will increase as secondary forest regenerates in the matrix, which may lead to
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changes in animal space use and movement. Thus we repeated the analysis by setting the
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density of flowering plants outside plots equal to density of plots inside the fragments
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(Supporting information S2.c), to determine whether the change would matter for the
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estimates. As expected, increasing density of flowering plants outside plots consistently
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decreased estimates of pollen and seed dispersal distances in fragments (Supporting
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information S2.c).
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Implementation of Gibbs sampler:
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1.
The iterative chain is initialized by generating for each seedling a random
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pedigree Pk = (mk,fk) following a multinomial distribution of the parentage probability
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matrix. For each step of the Gibbs sampler,
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2.
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The conditional distribution is:
Parameters us and up are sampled from a truncated normal distribution, given Pk.
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𝑝( 𝑢𝑝 , 𝑢𝑠 |𝑃) =
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= ∏𝑘 [(∑
𝑐𝑖′ 𝑠𝑖′𝑖 𝑝(𝑑𝑖′𝑖 |𝑢𝑝 )𝑓𝑖 𝑟𝑖𝑘 𝑝(𝑑𝑖𝑘 |𝑢𝑠 )
)] 𝑝(𝑢𝑝 |𝑚𝑝 , 𝑠𝑝 )𝑝(𝑢𝑠 |𝑚𝑠 , 𝑠𝑠 )
𝑖,𝑖′ 𝑐𝑖′ 𝑠𝑖′𝑖 𝑝(𝑑𝑖′𝑖 |𝑢𝑝 )𝑓𝑖 𝑟𝑖𝑘 𝑝(𝑑𝑖𝑘 |𝑢𝑠 )
(Eqn. S2.1)
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Where i and i’ are the inputted father and mother, mp and ms are the prior mean values of the
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truncated normal distribution, and sp and ss are the a priori standard deviations. If the
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conditional probability, based on the new values (pnew), is greater than the conditional
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probability of the current values (pnow), pnew is accepted. If pnew < pnow then the new values are
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accepted with probability a = pnew / pnow.
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1- Pedigree Pk is sampled, conditioned on the us and up parameters. New mothers and
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fathers are proposed for each seedling by sampling from the parentage probability
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matrix. Now, any parental pair for which parentage probability for seedling k is 0 is
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excluded from the pool of possible combinations. This step avoids sampling parent
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combinations with incompatible genotypes. The conditional probability of the proposed
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pedigree, given proposed parameters is:
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𝑝(𝑃𝑘 = (𝑖, 𝑖 ′ )|𝑢𝑠 , 𝑢𝑝 ) = 𝑝(𝑑𝑖𝑖 ′ , 𝑑𝑖 ′ 𝑘 |𝑢𝑠 , 𝑢𝑝 , 𝑃𝑘 )𝑝(𝐺𝑘𝑂 |𝐺𝑖𝑂 , 𝐺𝑖𝑂′ )
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=∑
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𝑐𝑖′ 𝑠𝑖′𝑖 𝑝(𝑑𝑖′𝑖 |𝑢𝑝 )𝑓𝑖 𝑟𝑖𝑘 𝑝(𝑑𝑖𝑘 |𝑢𝑠 )
𝑖,𝑖′ 𝑐𝑖′ 𝑠𝑖′𝑖 𝑝(𝑑𝑖′𝑖 |𝑢𝑝 )𝑓𝑖 𝑟𝑖𝑘 𝑝(𝑑𝑖𝑘 |𝑢𝑠 )
𝑝(𝐺𝑘𝑂 |𝐺𝑖𝑂 ,𝐺 𝑂′ )
𝑖
∑𝑖,𝑖′ 𝑝(𝐺𝑘𝑂 |𝐺𝑖𝑂 ,𝐺 𝑂′ )
(Eqn. S2.2)
𝑖
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The new pedigree is accepted or rejected using the same approach as described in the
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previous step.
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4.
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iterations. For each chain, we ran 100,000 iterations with a discarded burn in of 50,000
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steps.
Steps 2-3 are repeated until the iterative chain reaches the desired number of
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b)
Effect of varying priors on posteriors
Results may be sensitive to the chosen priors, so to evaluate the effect of different initial
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conditions on the recovery of the parameters, we tested different prior means for the pollen
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(800, 2000, 8000) and seed dispersal (100, 500, 2000) kernels and standard deviations for
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pollen (500, 1500, 4000) and seed dispersal (500, 1000, 1500). Because the routine is time
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expensive, we ran different priors on only one population (F2).
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In general, mean posteriors were more sensitive to standard deviations than to mean
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priors, and variation was higher for pollination parameters, probably because the range of
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values were wider than for seed dispersal. Mean posteriors of pollen dispersal varied from
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2,400 to 14,000, while seed dispersal varied from 1,800 to 2,800. Lower mean posteriors for
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seed dispersal (almost half of the current value) were found at low mean priors (100 and 500)
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and standard deviation (500). Under all other parametric choices, mean posteriors did not
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change substantially (Table S2.1). Mean posteriors of pollen dispersal also increased for low
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standard deviation values, increasing 2.5 times at standard deviations of 500 and only 1.4 times
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at standard deviation of 4000 (Table S2.1). Doubling the standard deviation increased the mean
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posterior of seed dispersal parameter by less than 50% at low priors, but no differences were
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found among the posteriors by increasing standard deviations of high mean priors (Table S2.1).
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For pollen dispersal parameters, tripling the standard deviations doubled the mean posterior on
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average, with decreased difference as the mean prior increased (Table S2.1).
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The results were not highly sensitive to changes in prior means, but were more sensitive
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to standard deviations. Influence of wider standard deviations to parameters and pedigree
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recovery have been associated with low numbers of zero genetic mismatches inside the
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sampled plot, an effect that is pronounced with high rates of genotyping error and small
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numbers of parents within plots (Moran & Clark 2011). Also, higher uncertainty around the
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parameter values were found for populations with lower sample size (e.g. CF2), which was
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further aggravated by the high immigration rate and a concomitantly reduced number of
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parent pairs within sampled plot (Chybicki & Burczyk 2010).
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Table S2.1 - Summary of posterior means (and support intervals) obtained for pollen and seed
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dispersal, based on different priors (mean and standard deviations) for population F2.
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Underlined numbers highlighted in bold face indicate the parameters used in the present study.
Pollen dispersal parameters
Prior mean
Seed dispersal parameters
Posterior mean and
Prior
Credible interval
mean
SD
Posterior mean and
SD
Credible interval
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Gene dispersal in Heliconia
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800
500
2400 [1700-3100]
100
500
1800 [1200-2500]
800
1500
4800 [2100-7600]
100
1000
2700 [2300-3000]
800
4000
11000 [8900-14000]
100
1500
2800 [2400-3000]
2000
500
3100 [2400-3900]
500
500
2100 [1500-5800]
2000
1500
6100 [4300-7600]
500
1000
2700 [2300-3000]
2000
4000
10000 [7300-12000]
500
1500
2800 [2500-3000]
8000
500
8200 [7400-9200]
2000
500
2700 [2200-3000]
8000
1500
2000
1000
2700 [2200-3000]
8000
4000
2000
1500
2700 [2200-3000]
14000 [11000-15000]
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c)
Effect of density of hypothetical parents on posteriors
Fragments may be less isolated if there are large quantities of resources for the vectors
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in the surrounding areas. To test the effect of higher density of plants in the matrix, we
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increased the density of hypothetical parents outside the plots in the gene dispersal model.
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Setting the flowering density equal to flowering density inside forest fragments reduced
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distances of pollination and seed dispersal in all three fragments (Table S2.2). The posterior
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mean of pollen dispersal was about 45% lower in F1 and F2 and 20% lower in F3. The decrease
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in posterior mean of seed dispersal was more subtle in F1 and F2, decreasing by less than 10%
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(Table S2.2). The seed dispersal parameter of population F3, however, was about 55% times
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lower than in the original scenario (Table S2.2).
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Gene dispersal in Heliconia
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Côrtes et al.
Changing the initial conditions in the gene dispersal model altered the pollen and seed
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dispersal distances of H. acuminata. Assuming a greater number of flowering plants outside
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sampled plots in fragmented sites decreased gene dispersal distances in all three fragments.
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This is expected given the potential role that density of flowering plants has on the behavior
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and patterns of space use by foraging birds. That is, we created a scenario of low patch isolation
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so animals have access to a continuum of food resources throughout the environment. Given
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the availability of resources in the neighborhood, animals do not need to move further
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distances to find food, and therefore, do not disperse propagules to greater distances.
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Table S2.2- Average distance values based on the posterior means (and 95% support intervals
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within brackets) of the modal pollen dispersal parameter (up) and seed dispersal parameter (us)
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for the current scenario, in which density of flowering plants outside plots is lower than that
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within fragment, relative to a scenario in which the density of flowering plants outside plots is
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equal to that inside fragments. Values highlighted in bold face indicate non-overlapping support
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intervals between current and new scenarios.
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Fragment
Modal pollination distance
(meters)
Modal seed dispersal distance
(meters)
Current
New
Current
New
F1
58 [47-64]
36 [27-44]
43 [39-45]
39 [27-44]
F2
66 [55-77]
34 [20-46]
40 [30-43]
39 [32-44]
F3
64 [54-71]
50 [41-63]
42 [39-45]
19 [12-28]
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Data S3. Probability of parental identity (PPaI)
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Probability of parental identity was calculated for each Heliconia acuminata population,
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using the r - estimator, as follows:
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𝑋 (𝑋 −1)
𝑘 𝑘
𝑅0 = ∑𝐾
𝑘=1 (2𝑁(2𝑁−1))
(Eqn. S3)
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Where, Xk is the number of seedlings that plant k either fathered or mothered, and N is the
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total number of offspring recruited for which either father or mother are inside the sampled
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plot. The two sets of chromosomes, one coming from each parent, are multiplied by the
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number of offspring. The total number of chromosome sets is used, instead of the number of
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offspring, because we are interested in the contribution of both father and mother to the
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seedling genotype, meaning that one seedling can have up to two inside plants contributing to
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its whole genotype.
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Data S4. Flowering and seedling phenology
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The proportion of flowering plants was very low in all populations but was higher for
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CF1 and F3, with around 15% of all recruited plants flowering at least once during the 11 year-
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study period. In contrast, only about 7% of the plants flowered in CF2, F1 and F2, with some
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years lacking any flowering events (Table 1, Fig. S4). Higher flowering probability indicates that
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more plants have the potential to contribute genes to next generations within the plots,
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Gene dispersal in Heliconia
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whereas populations with lower flowering probability depend on immigration of seeds from
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other locations to secure plant recruitment.
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Figure S4- Heliconia acuminata phenology at the study site: A. Number of flowering plants, and
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B. Number of seedlings recruited along the 11-yr study period. B. depicts the total number of
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seedlings ever recruited, including the ones that were not included in the genetic analysis.
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Data S5. Inbreeding coefficient and fine-scale spatial genetic structure
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Gene dispersal in Heliconia
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-
-
-
-
-
Inbreeding
-
Reproductive plants
-
0.00 0.05 0.10 0.15 0.20
Inbreeding
0.00 0.05 0.10 0.15 0.20
Seedlings
-
-
-
F1
F2
F3
CF1
-
-
0.04
0.02
-
sp
-
-0.02
-
-0.04
-
-
-
-0.02
-
-
CF1
CF2
F1
F2
F3
CF2
-
F1
F2
F3
-
-
-
-
-
0.00
0.02
-
-0.04
sp
-
-0.10
-0.10
CF2
0.04
CF1
0.00
-
-
-
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-
-
-
CF1
CF2
F1
F2
F3
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Fig. S5- Actual inbreeding coefficients and sp-values and 95% confidence intervals computed
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from 2,000 bootstrapped samples of size equal the least dense population (CF2) of seedlings (N
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= 52) and reproductive plants (N = 20) across the five populations of Heliconia acuminata at the
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BDFFP.
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