Pollen dispersal data used to fit curves and PDFs were drawn from the same data in Butcher et al. (2020a), which examined the relative degree of pollen dispersal among two green roof and two ground-level sites in the New York (NY, United States) metropolitan area. Accordingly, study sites, study species, and greenhouse and field methods are as in Butcher et al. (2020a). In summary, pollen recipient and donor plants were placed in arrays at four sites: (1) Rose Hill Campus (Rose Hill) in Bronx, NY, United States, (2) the Javits Convention Center (Javits) in New York, NY, United States, (3) the Louis Calder Center (Calder) in Armonk, NY, United States, and (4) the Queens Zoo (Queens Zoo) in Corona, NY, United States. Rose Hill and Javits were primarily green roof sites, while Calder and Queens Zoo were ground-level sites.

More specifically, at Rose Hill, most of the plants were located on two separate portions of the roof of a campus building, as well as the roof of Rose Hill’s parking garage, and a small number of plants were located at a ground-level site at the New York Botanical Garden (Table 1 and Figure 1). At Javits, plants were located on two separate portions of the Javits Center’s Sedum-dominated green roof (Table 1). At Calder, the majority of the plants were located at two suburban lawns, which were regularly mowed, and a small number of plants were located at an unmowed meadow (Table 1). And, at the Queens Zoo, most of the plants were located at two urban lawns, which were regularly mowed, and a small number of plants were located at an unmowed meadow (Table 1).

The wind-dispersed Amaranthus tuberculatus and the insect-dispersed Solanum lycopersicum were selected because species of both genera grow well in urban environments (Del Tredici, 2010; Aloisio et al., 2016). Further, A. tuberculatus is dioecious, which allowed us to assess pollen dispersal (i.e., degree of dispersal and dispersal distance) to female plants (pollen receptors) from male plants (pollen donors; Hager et al., 1997). Solanum lycopersicum is most often pollinated by bees in the genera Anthophora and Bombus, although some pollination may also occur via bees in the genera Lasioglossum, Apis, and Halictus (Greenleaf and Kremen, 2006). And, although S. lycopersicum is typically self-compatible and most often self-fertilizes because of its flower morphology, the NC4 breeding line (Panthee and Gardner, 2013), which was used in our study, is heterozygous for the ms-10 male-sterility gene, such that it produces both male-sterile and male-fertile offspring. Accordingly, we used the NC4 line to measure pollen dispersal to male-sterile plants (pollen receptors) from male-fertile plants (pollen donors).

Greenhouse and field methods were as described in Butcher et al. (2020a). In summary, seeds from an Iowa population of A. tuberculatus (Hinz and Owen, 1997) and the NC4 line of S. lycopersicum (Panthee and Gardner, 2013) were germinated and grown in a polycarbonate greenhouse. Once mature and separated by sex or sterility, these plants served as the parent plants in pollen recipient and donor arrays at the four sites. For both species, and at each site, the pollen receptor plants were placed at increasing distances, starting at 1 m, from a group of twelve pollen donors, with the maximum distance varying among sites (Table 1). The experimental array layout for both species at the Rose Hill site are shown in Figure 1 to depict the layout of the experimental array at one site. All sites were organized similarly, taking into account site size and shape.

As with the majority of pollen dispersal studies, we calculated the straight-line Euclidean distance between pollen receptors and the donor group and this distance was used as the effective pollen dispersal distance (Byrne et al., 2007; Ismail et al., 2012; Browne et al., 2018). The number of A. tuberculatus seeds and S. lycopersicum fruits were used as proxies for the number of flowers pollinated (i.e., degree of pollen dispersal).

Curves and PDFs were fit to pollen dispersal data collected for A. tuberculatus at two sites in 2014 and for both A. tuberculatus and S. lycopersicum at four sites in 2015. As methods and results were comparable between the 2 years, and the 2015 study was more inclusive, only results for 2015 are detailed here. A summary of the 2014 results are shown in Supplementary Figures 1, 2.

Paternity was assigned using the rare allele approach, as described in Butcher et al. (2020b). In brief, all A. tuberculatus pollen receptor and donor plants, and a subset of seeds, were genotyped at markers for the external transcribed spacer region (ETS) and putative DEAD box ATP-dependent RNA helicase gene (PutDead). All S. lycopersicum pollen receptor and donor plants, and seeds from a subset of fruits, were genotyped at markers for spotted wilt virus disease resistance (Sw-5) and Fusarium wilt race 3 virus resistance (I-3) genes. We then used simple exclusion methods, similar to the paternity assignment methods presented in Ellstrand (1984), to confirm that pollen dispersed from an experimental donor and not a local pollen donor. We found that two S. lycopersicum fruits were pollinated by a local donor, one from the Queens Zoo site and one from the Rose Hill site. Also, we observed other non-experimental S. lycopersicum near both the Rose Hill and Queens Zoo sites. However, our experimental design (i.e., rare alleles and simple exclusion) does not allow us to definitively locate the donor for these fruits. Therefore, we were not able to determine a pollen dispersal distance, and, thus, these two fruits were removed from the analyses.

Curve fitting was performed using the Curve Fitting Tool in the MATLAB Release 2016b Curve Fitting Toolbox 3.5.4 (The MathWorks, Inc, Natick, MA, United States). We used a non-linear, least-squares approach with a trust-region algorithm to fit exponential decay, inverse power, Weibull, logistic, and Gaussian curves to the relationship between number of seeds (A. tuberculatus) or fruits (S. lycopersicum) produced and distance from the pollen donor group.

The exponential decay curve was included because it has been used extensively in studies that assessed pollen dispersal patterns using curve fitting (e.g., Beckie and Hall, 2008), including studies on A. tuberculatus (Liu et al., 2012; Sarangi et al., 2017). The inverse power curve also has been used to model pollen dispersal in a range of crop and other species (Beckie and Hall, 2008; Buehler et al., 2012; Chang et al., 2018). The Weibull and logistic curves are not widely used in curve fitting studies, but were included to enable comparisons to the results of the PDF analyses, which included both functions. Similarly, although pollen dispersal is expected to decrease more rapidly with increasing distance from the pollen source (i.e., leptokurtic curves) than expected by the Gaussian curve, this curve was included because the Gaussian PDF is often included in PDF fitting studies (Lavigne et al., 1998; Goto et al., 2006), as we did in this study.

For all curves, parameters were not selected a priori, but instead were estimated by the MATLAB Curve Fitting Toolbox when fitting the curves to the data. Curves with any parameters that were not significantly different than zero, based on the 95% confidence interval around the parameter estimate, which was calculated using the t-distribution, or curves that did not converge after 400 fitting iterations were excluded. The remaining were considered candidate curves and we used adjusted R² (Buehler et al., 2012) and root mean squared error (RMSE) to assess fit (Loureiro et al., 2016).

Probability density function fitting was conducted using the Distribution Fitting Tool in the MATLAB Release 2016b Statistics and Machine Learning Toolbox 11.0 (The MathWorks, Inc). For each site, we calculated the mean number of seeds (A. tuberculatus) or fruits (S. lycopersicum) produced by the two pollen receptors at each distance from the pollen donor group. Each seed or fruit was counted as a dispersal event, and a probability histogram was generated by binning these events by distance (i.e., distribution of dispersal distances). We then fit exponential, inverse Gaussian, Gaussian, logistic, and Weibull PDFs to these probability histograms using maximum likelihood.

We chose a wide range of PDFs because PDF fitting has not been used to investigate pollen dispersal dynamics in A. tuberculatus or S. lycopersicum. We included the exponential, logistic, Gaussian, and Weibull PDFs as they have been used in pollen dispersal studies on other species (Lavigne et al., 1998; Austerlitz et al., 2004; Goto et al., 2006; Klein et al., 2006b). We included the inverse Gaussian because this function has been used to model wind-dispersed propagule dispersal patterns (e.g., Quinn et al., 2011; Bullock et al., 2017). We also chose this range of PDFs because the tail-thickness is a distinguishing characteristic of PDF shape, and these PDFs represent a range of tail-shapes, including thin-tailed (Gaussian), exponential-tailed (exponential and inverse Gaussian), and fat-tailed (logistic), as well as one flexible-tailed PDF that can be thin-tailed, exponential, or fat-tailed depending on the shape parameter estimate (Weibull). Specifically, if the Weibull function is a candidate, the PDF is thin-tailed if the shape parameter (b) = 2, exponential if b = 1, or fat-tailed if b < 1.

As with curve fitting, parameters for all PDFs were not selected a priori, but instead were estimated by the MATLAB Curve Fitting Toolbox when fitting the PDFs to the probability histograms. We excluded PDFs with any parameters that were not significantly different from zero, based on the 95% confidence interval around the parameter estimate, which was calculated using the t-distribution. We then used the log-likelihood values estimated during maximum likelihood PDF fitting to calculate Akaike information criterion values (Fenart et al., 2007), corrected for small sample size (AIC_c), and used these values to choose candidate PDFs. Specifically, the PDF with the lowest AIC_c value was considered the top PDF, and PDFs with ΔAIC_c < 4 from the top PDF were considered candidate PDFs.

For the curve fitting analyses, the exponential decay and inverse power curves generally provided the best, or only, fits to pollen dispersal patterns across both green roof and ground-level sites for both A. tuberculatus and S. lycopersicum. After considering parameter significance, the exponential decay and inverse power curves were the only candidate curves for the A. tuberculatus data at the Rose Hill site (exponential – R² = 0.69, RMSE = 883.1; inverse power – R² = 0.51, RMSE = 1009.0) and the Calder site (exponential – R² = 0.74, RMSE = 494.3; inverse power – R² = 0.75, RMSE = 489.7; Figure 2). The exponential decay and inverse power curves were also candidate curves at the Javits site, as well as the Gaussian curve (exponential – R² = 0.78, RMSE = 372.0; inverse power – R² = 0.44, RMSE = 590.3; Gaussian – R² = 0.84, RMSE = 319.3; Figure 2). There were no candidate curves for A. tuberculatus at the Queens Zoo, as data yielded either parameter estimates that were not significantly different from zero or did not converge, or the model itself did not converge.

For S. lycopersicum, the exponential decay and inverse power curves were the only candidate curves at all four sites: Rose Hill (exponential – R² = 0.88, RMSE = 0.86; inverse power – R² = 0.88, RMSE = 0.85), Calder (exponential – R² = 0.92, RMSE = 5.49; inverse power – R² = 0.68, RMSE = 10.75), Javits (exponential – R² = 0.85, RMSE = 0.58; inverse power – R² = 0.81, RMSE = 0.65), and Queens Zoo (exponential – R² = 0.90, RMSE = 0.89; inverse power – R² = 0.84, RMSE = 1.10; Figure 3).

For the PDF fitting analyses, the exponential decay and inverse Guassian curves generally provided the best, or only, fits to pollen dispersal patterns across both green roof and ground-level sites for both A. tuberculatus and S. lycopersicum. Inverse Gaussian was the only candidate PDF for A. tuberculatus at all four sites: Rose Hill (AIC_c = 70753.80, ΔAIC_c = 0.00; mean dispersal distance = 12.5 m), Calder (AIC_c = 14573.35, ΔAIC_c = 0.00; mean dispersal distance = 6.0 m), Javits (AIC_c = 27469.61, ΔAIC_c = 0.00; mean dispersal distance = 14.2 m), and Queens Zoo (AIC_c = 5772.29, ΔAIC_c = 0.00; mean dispersal distance = 7.1 m; Figure 4).

For S. lycopersicum, exponential was the only candidate PDF at the Rose Hill site (AIC_c = 137.22, ΔAIC_c = 0.00; mean dispersal distance = 30.2 m) and the Javits site (AIC_c = 34.45, ΔAIC_c = 0.00; mean dispersal distance = 3.3; Figure 5). And, exponential was a candidate PDF for S. lycopersicum at the Calder site (AIC_c = 1008.68, ΔAIC_c = 0.69; mean dispersal distance = 11.5 m), though Weibull was the top PDF at this site (AIC_c = 1007.99, ΔAIC_c = 0.00, b = 0.90; mean dispersal distance = 11.5 m; Figure 4). Finally, inverse Gaussian was the only candidate PDF for S. lycopersicum at the Queens Zoo site (AIC_c = 63.03, ΔAIC_c = 0.00; mean dispersal distance = 3.4; Figure 5).

As we hypothesized, the exponential decay curve provided a good fit to the relationship between pollen dispersal and distance to the pollen donor group at both the two green roof and the two ground-level sites. More specifically, an exponential decay function accounted for 69% or more of the decrease in the number of A. tuberculatus seeds with distance at three sites, and 85% or more of the decrease in the number of S. lycopersicum fruits at all four sites. The inverse power curve generally also provided a good fit to the relationship, accounting for 41% or more of the variation in A. tuberculatus seed numbers at three sites, and 68% or more of the variation in S. lycopersicum fruit numbers at all sites. The superior fit of the exponential decay curve to the A. tuberculatus data are consistent with previous studies that modeled A. tuberculatus pollen dispersal (Liu et al., 2012; Sarangi et al., 2017). However, both studies chose the exponential decay curve a priori, and did not compare fits of different curves. As a result, our study is the first to suggest that the inverse power curve may also be suitable for modeling A. tuberculatus pollen dispersal patterns. Additionally, although the relationship between pollen dispersal and distance has not been investigated in S. lycopersicum, our results are also consistent with other studies on insect-pollinated species, where the exponential decay (e.g., Van Rossum and Triest, 2010; Van Rossum et al., 2011) and inverse power curves (e.g., Barbour et al., 2005; Buehler et al., 2012) have both been used to model pollen dispersal.

Despite the relatively common use of both exponential decay and inverse power functions to assess pollen dispersal using curve fitting, only two studies have compared them directly. Though Chang et al. (2018) concluded that the exponential decay curve was the best fit for Panicum virgatum pollen dispersal patterns at both of two sites, the inverse power curve showed a comparatively high R² at one of the sites. And, Barbour et al. (2005) concluded that the inverse power curve was the best fit for Eucalyptus nitens pollen dispersal, but the exponential decay curve showed a similarly high R². The results of these studies suggest that both curves may be appropriate for modeling this relationship. Further, the suitability of both curves is consistent with patterns seen in seed dispersal studies, where the exponential decay curve may better fit the relationship between seed dispersal and distance at short distances, while the inverse power curve may better fit this relationship at longer distances (Bullock and Clarke, 2000). Although seed dispersal is often more localized then pollen dispersal, similar fits of these curves to seed dispersal data may also provide support for pollen dispersal studies, as seed and pollen dispersal exhibit analogous patterns (Sant’Anna et al., 2013; Van Tussenbroek et al., 2016).

While both exponential decay and inverse power functions may adequately model pollen dispersal patterns, the choice of these functions a priori may result in over- or under-estimation of long-distance pollen dispersal. We addressed this issue by also using Weibull, logistic and Gaussian curve fitting to analyze our data. In contrast to the exponential decay and inverse power curves, the Weibull and logistic curves did not fit any of the data, and the Gaussian curve only fit the data for one species at one location (A. tuberculatus at Javits). The Gaussian curve likely was a good fit for these data because of the anomalous observation that seed numbers peaked 8 m from the pollen donors, which follows the classic bell-shape of the Gaussian curve (Austin, 1976). Aside from this result, our analyses suggest the Gaussian, Weibull and logistic curves were less appropriate than the exponential decay and inverse power curves for modeling pollen dispersal curves for the conditions in this study.

As we hypothesized, a slightly fat-tailed Weibull PDF (b = 0.90) was the top candidate PDF for the S. lycopersicum pollen dispersal probability histogram at the Calder site. However, the exponential PDF was also a candidate PDF at this site. Further, an exponential-tailed PDF (i.e., inverse Gaussian or exponential) was the only candidate PDF for S. lycopersicum at the other three sites, as well as for A. tuberculatus at all sites. Although other studies have observed that exponential-tailed PDFs fit pollen dispersal probability histograms better than the thin-tailed Gaussian (Austerlitz et al., 2004; Skarpaas and Shea, 2007; de-Lucas et al., 2008; Marchelli et al., 2012), many studies have found that fat-tailed PDFs provide the best fit (e.g., Albaladejo et al., 2012; Colabella et al., 2014), while few studies have observed that exponential-tailed PDFs specifically provide the best fit (e.g., Millerón et al., 2012). This suggests long-distance pollen dispersal is more common than explained by an exponential-tailed PDF, in contrast to our results.

The uniformity of our PDF analyses across sites and species suggests long-distance pollen dispersal was less frequent in the urban sites in our study compared with many studies (Klein et al., 2006a). Urbanization is typified by the impervious matrix (Fernandez-Juricic and Jokimäki, 2001; Ramalho and Hobbs, 2012), which may inhibit pollen dispersal. For example, buildings may interfere with wind patterns that transport pollen (Emberlin and Norris-Hill, 1991). Also, urban cores typically host fewer native plant species than surrounding areas (McKinney, 2008), which may reduce the abundance (Franceschinelli et al., 2017) and species richness of bees (Matteson et al., 2008), the primary pollinators of S. lycopersicum. If so, pollinator richness and abundance, and long-distance dispersal, may be higher at Calder, which may account for why Calder was the only site that exhibited slightly fat-tailed pollen dispersal in S. lycopersicum. Although dispersal was primarily exponential-tailed for both A. tuberculatus and S. lycopersicum in our study, the PDF analyses did not exhibit a truncated tail, and an exponential tail shape implies more long-distance dispersal than a thin-tailed PDF, indicating pollen dispersal at both our green roof and ground-level sites was intermediate between studies showing extensive long-distance dispersal and patterns suggested by thin-tailed PDFs (Klein et al., 2006a).

Pollen and seed dispersal both contribute to gene flow in plants, but pollen is often considered the more important element driving long-distance gene flow (Levin and Kerster, 1974; Ennos, 1994). Consequently, understanding patterns of pollen movement is also key to understanding gene flow in plants. Our study addressed several questions about plant gene flow via pollen in urban environments, but also raises additional questions that warrant future studies to expand on the limited number of studies conducted on this topic.

First, future studies using these modeling techniques should be conducted, including estimating overall dispersal predictions. Our results from the curve-fitting analyses suggest the Gaussian, Weibull and logistic curves were less appropriate than the exponential decay and inverse power curves for modeling pollen dispersal, but because of the widespread use of the Gaussian, Weibull, and logistic curves for PDF fitting, further research is still needed to address whether these curves may provide better fits to pollen dispersal curves in other systems. In this study, we were able to calculate mean dispersal distance and the maximum distance that pollen dispersed to our experimental pollen receptors, which were at known distances from the experimental pollen donor group. However, because long-distance pollen dispersal is critical for gene flow in plant populations, it is also important to estimate how far pollen could potentially disperse. Therefore, theoretical overall long-distance dispersal predictions should be incorporated in future studies.

Second, additional research is needed in larger and diverse habitats, including further studies on urban pollen dispersal. This study was conducted in sites with shorter pollen dispersal distances (i.e., <350 m) compared with some other studies (e.g., >1 km). Thus, future studies on pollen dispersal in both urban areas should examine pollen dispersal at larger spatial scales compared with our study. Further, studies directly comparing pollen dispersal patterns in urban vs. rural areas are needed to confirm whether, and to what extent, long-distance pollen dispersal may be comparatively less in urban areas, as suggested by our results.

Finally, added studies on pollen dispersal in species that are self-compatible and species whose pollen is dispersed by other vectors (i.e., other animal vectors), as well as studies that incorporate climatic variables are needed to further investigate pollen dispersal patterns. In this study, A. tuberculatus and the NC4 Grape line of S. lycopersicum were self-incompatible; however, pollen dispersal patterns of species that are reproductively self-compatible should also be investigated in urban environments. Additionally, although we investigated a wind-dispersed species, as well as an animal-dispersed species, future studies should investigate additional wind and animal-dispersed species, including species that are dispersed by animals other than bees to determine if the pollen dispersal patterns are the result of a difference in dispersal vector. Microclimatic variations between sites may also affect pollen dispersal patterns, so future studies should also measure and account for these variables.