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In this study we were able to obtain estimates of parameters that had been missing for the southwest subpopulation, including survival probabilities of younger stages of manatees, recovery rates of manatee carcasses, and abundance in years before and between abundance surveys.
Survival probabilities of younger animals are key parameters in population viability analyses of Florida manatees11,23,25. But these probabilities have long been extrapolated from one study of manatees in a small management unit on Florida’s east coast26. The average probabilities of juvenile survival estimated here are lower than those obtained from that extrapolation (Fig. 8). Independent estimates of the younger manatee survival probabilities for the southwest management unit will soon be available from genetic mark–recapture–recovery modeling, but such data are not forthcoming for the other three Florida manatee management units, making the approach used here for estimating these probabilities more readily applicable.
In addition, our model provided estimates of the effects of red tide and cold events on the population. The red tide event of 2013, during which 353 carcasses were recovered in the southwest (of which at least 268 were killed by red tide), contributed to an estimated net drop in the population of 331 (217–459) manatees (Fig. 5) for an annual population growth rate of 0.89 (0.85–0.93; Fig. 4). Our results support the finding that such red tide events (classified as intense) affect calves particularly (Supplementary Fig. S13, online)11. In contrast, the cold event of 2010, which led to 247 recovered carcasses in the southwest region, did not appear to lead to a net drop in population, according to our model. This may be in part because our prior estimate of adult survival that year was relatively high (Fig. 7), and the model assumes (and estimates) a fixed ratio between age-class survival rates across years (Supplementary Table S1, online). These new estimates can be helpful in communicating the impact these disturbance events had on the population. Unusual mortality events that lead to high carcass counts often attract a lot of attention from the press and the public. The IPM provides a way to put such mortality events in perspective and to answer questions such as “What was the impact of a particular mortality event on the population?” In addition, the average population growth rate (1.02, 1.01–1.03) estimated from our data supports the hypothesis that the manatee population was increasing from 1997 to 2016 (Figs. 3 and 4). This is the first rigorous estimate of historical (realized) population growth rate for this population. This information is complementary to and consistent with the projected population growth rate obtained from the CBM projections11.
Our model also provided more precise estimates of many parameters estimated earlier, such as adult survival and abundance for years in which abundance surveys were carried out (Figs. 3 and 7). In some cases, our approach may reduce bias, although it is also possible for IPMs to introduce or increase bias27. Possible biases in some input estimates to our model, such as abundance28,29 and end-of-time-series survival probabilities30, have been noted28,29,30. In some cases, the median estimates obtained from the IPM were substantially different from the original estimates (compare prior abundance survey and posterior estimates in Figs. 3 and 7). The IPM might correct for biases in abundance and end-of-time-series survival estimates, although this idea needs to be further evaluated. Because it includes a recovery model for carcass data, the IPM does not hindcast impossible numbers of deaths, unlike the simulation-based hindcast model (Supplementary Fig. S3, online). The IPM results suggest that these results from the simulation-based hindcast model were off both because the 2011 abundance estimate input was too low and because the survival estimate inputs for juveniles (s1–s4) were too high. By integrating multiple sources of information, we are synthesizing the best available information but also hedging our bets by not relying on just one source of data in estimating critical demographic parameters.
Many of our posterior estimates are consistent with other published results for Florida manatees. Our estimates of realized population growth rates (Fig. 4) are similar to the projected population growth estimates from the CBM and consistent with general trends of growth in synoptic and carcass counts. Our estimates of age structure (Fig. 6), although variable over time, are consistent with the asymptotic stable age structure that projecting from a simple matrix model would provide. Our estimates of the mortality effects of the 2013 red tide (Supplementary Fig. S13, online) are similar to those from the CBM. The pattern of our estimated recovery probabilities by coarse stage (Supplementary Fig. S5, online) is consistent with an earlier estimate of age-specific recovery rates relative to (unknown) adult recovery probability31, although our estimates of subadult and adult recovery probabilities are closer to 1 than we expected. The high estimates of recovery probability may be due to the IPM attempting to harmonize partially incompatible model components (Supplementary Fig. S3, online). When model components generate incompatible results, either due to model misspecification or not referencing exactly the same populations, an IPM must reconcile those results. This reconciliation can generate bias in some estimates, although the generally higher precision of IPM estimates may still mean higher accuracy. Ground truthing or other research may be needed to determine whether FWC is actually recovering such a high proportion of manatee carcasses.
The results of this study are relevant to the management of Florida manatee populations. The manatee recovery plan used by the USFWS under the Endangered Species Act relies on several metrics that can be obtained from the IPM, such as realized population growth rates and population size. The IPM provides one of the most rigorous assessments to date for these quantities and may be used by natural resource managers in assessing the status of the manatee population. It can also be used to update key model parameters of the CBM, which at present is the primary population assessment tool for managers.
Another important regulatory framework relevant to marine mammal conservation in the United States is the Marine Mammal Protection Act. Here again, an IPM can help in addressing some of the act’s requirements. Indeed, the act specifies a formula for computing potential biological removal (PBR; the maximum number of animals that can be removed from a stock while allowing it to reach or remain at its optimum sustainable population)32,33,34,35

$$begin{aligned} PBR & = N_{min} frac{{R_{max} }}{2}F_{r} \ N_{min} & = frac{{hat{N}}}{{exp left( {0.842sqrt {log left( {1 + {text{CV}} left( {hat{N}} right)^{2} } right)} } right)}} \ end{aligned}$$
(1)

where Nmin is the minimum population abundance estimate (20th percentile of abundance estimate distribution), Rmax is the theoretical maximum rate of increase for the stock, Fr is a recovery factor (generally 0.5 for threatened species, but see Moore et al.35), and (hat{N}) is the point estimate of population abundance. Based on our estimate from the last year of the analysis (2016), Nmin for the southwest population of Florida manatees is about 2780. This is lower than Nmin would be based on the abundance survey (prior) estimate from the same year (about 3140); (CVleft(hat{N}right)) from the IPM posterior was lower than from the prior (Supplementary Fig. S2, online) but (hat{N}) was as well (Fig. 3). Estimation of Rmax requires extrapolating growth rates to conditions of low population density and absence of anthropogenic mortality; our IPM is not designed for that purpose, but future extensions could be developed to address this need. A merging of our IPM, or other matrix model approach, with an allometric approach to estimating Rmax would allow a more accurate estimate of this parameter36. Both matrix model (individual population) and allometric (cross population) approaches to estimating Rmax are strongly affected by biases caused by using empirical estimates of adult survival instead of what adult survival would be under ideal conditions; however, these biases run in opposite directions, so an integration of these approaches greatly reduces any bias in Rmax36.
Another benefit of the IPM is its usefulness for planning monitoring activities, including how to allocate resources to various aspects of the monitoring program, such as aerial surveys, photo-identification, genetic sampling, and carcass recovery. Various sampling scenarios (e.g., 40% of carcasses recovered; 200 genetic samples per year; one aerial survey every 5 years) can be combined with simulated data generated under those scenarios to see how the accuracy of model parameter estimates differs among scenarios. Trade-offs between parameter accuracy/precision and budget allocation can then be examined to improve monitoring efficiency. Optimizing the sampling with an IPM also makes sense in the context of targeted monitoring for adaptive management37. In such applications, the IPM can be used to estimate state variables (e.g., abundance) that keep track of system changes, allow managers to implement state-dependent decisions, and update beliefs about which model is the best approximation of reality (through Bayes theorem)37,38. A now classic example of an implementation of this adaptive management process is for the sustainable harvesting of waterfowl in North America37, where the optimal state-dependent harvest policies are driven, at least partially, by waterfowl abundance. IPMs are now being used to increase precision of abundance and other state variables in adaptive management of waterfowl39,40.
A monitoring component that could be streamlined is the carcass-recovery and necropsy program. The present protocol is that almost all carcasses reported must be recovered and necropsied, which, along with the growth in the manatee population, is making this program increasingly labor-intensive and expensive. The IPM gives us the first true estimates of carcass recovery probabilities for Florida manatees. These estimates are now being used by FWC in evaluating and improving the efficiency of these programs.
Monitoring populations of marine mammals involves special challenges, such as the difficulty, cost, and risk to researchers involved in counting the population, often through aerial surveys. Several other studies that involved the development of IPM for marine mammals16,41,42,43 had at least one thing in common with ours: population surveys were not conducted every year, which differs from most IPMs used for terrestrial birds and mammals. Our approach, like those applied to other marine mammals, could be valuable for filling in abundance estimates for other sirenians and small cetaceans, where estimating survival and reproductive probabilities from mark–recapture data is often easier than obtaining abundance estimates. As explained earlier, the IPM can then be used to determine the optimal frequency of surveys and optimal spatial sampling effort (e.g., how much area to survey and how many survey visits at each location to estimate detection)28.
Studies of other marine mammals16,41,42,43 collected explicit data on age or stage structure, while for manatees, reliable data were not available for these parameters. We were able to estimate age class structure for the years 2002–2016 using neither stage structure data nor particularly informed priors (Fig. 6). This is likely because of the weak ergodic theorem of demography, which shows that the initial stage structure becomes less relevant with more years of known (or, in our case, estimated) survival and reproductive probabilities3,44. Our approach may be useful for other marine species without reliable stage structure information. Modeling stage structure and transient dynamics can be important to improving understanding of the dynamics of wild populations and can have important management implications. For instance, Johnson et al.45 found that the initial stage structure could have substantial policy consequences for the management of an invasive species.
Our IPM and the associated input models are based on a series of assumptions (Supplementary Table S1, online). One of the assumptions of the IPM is the independence of the data sources for the input analyses. This assumption is violated in our case; the adult survival analysis shares carcass data with the recovery analysis and mark–recapture data with the reproductive analysis. Two simulation studies17,46 found that violating this assumption had little effect, but as their analyses were not identical to ours, this assumption violation still might diminish the accuracy of our estimates. Simulations by Rieke et al.47 show that assumption violations in one of the model components can dramatically reduce the accuracy of estimates of latent parameters. Therefore, in our case, the estimates of juvenile survival, recovery probabilities, and abundance in years without abundance surveys should all be interpreted cautiously.
There are several possible extensions of this model, for example for use in the other three Florida manatee management units (Fig. 1). Because we are uncertain about winter within-coast manatee distribution29, two coast-wide IPMs that each jointly model the two management units on that coast might be most appropriate. With an initial abundance distribution and yearly vital rate estimates for each management unit (possibly including movement rates between regions, if they become available), subsequent coast-wide abundance estimates could be shared between them. This would allow relaxation of the assumption that the proportion of the winter population in each of the two management units remains fixed over time.
Possible extensions could demonstrate whether and to what extent the IPM decreases bias in input estimates, through simulating estimates with known biases and carcass data, running the IPM with the simulated data, and repeating this process many times. One could similarly test the model’s robustness to different assumption violations.
Preliminary analyses suggest that our use of earlier analyses as priors in the integrated model does not bias results but that it might reduce precision. Therefore, it may be useful to estimate more parameters from data directly within a future version of this IPM. In addition, incorporating additional data sources (such as genetic mark–recapture and age estimates using tympanoperiotic ear bones) could improve parameter estimation. Since each parameter can have only one prior, this too requires performing more of the data analysis within the IPM.
Despite these limitations, we believe that this manatee IPM is the most rigorous means of retrospective assessment of the population dynamics of the Florida manatee. Because the model is modular (e.g., abundance module, survival module), as each module is improved, the model as a whole is improved. This offers a compelling framework within which to synthesize and update information about population dynamics. We have shown here that an IPM can be used: (1) to infer historical trends in abundance, improving our understanding of population dynamics and therefore our ability to forecast; (2) to model the transient dynamics of stage distribution, which can be important to some populations; (3) to assess the conservation status of wild populations and to communicate that information to stakeholders (e.g., we can now quantify the impact of the 2013 red tide event on the manatee population); and (4) to improve allocation of effort in complex monitoring programs.
Our modeling frameworks are relevant to population status assessment protocols for management and conservation, such as recovery plans under the Endangered Species Act and potential biological removal under the Marine Mammal Protection Act. Other marine mammal conservation programs, such as that of the Hawaiian monk seal, also have complex monitoring components48. We hope that our ideas can inform other programs that focus on the conservation of marine mammals. More

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Neighbor GWAS
Basic model
We analyzed neighbor effects in GWAS as an inverse problem of the two-dimensional Ising model, named “neighbor GWAS” hereafter (Fig. 1). We considered a situation where a plant accession has one of two alleles at each locus, and a number of accessions occupied a finite set of field sites, in a two-dimensional lattice. The allelic status at each locus was represented by x, and so the allelic status at each locus of the ith focal plant and the jth neighboring plants was designated as xi(j)∈{−1, +1}. Based on a two-dimensional Ising model, we defined a phenotype value for the ith focal individual plant yias:

$$y_i = beta _1x_i + beta _2mathop {sum }limits_{ < i,j > } x_ix_j$$
(1)

where β1 and β2 denoted self-genotype and neighbor effects, respectively. If two neighboring plants shared the same allele at a given locus, the product xixj turned into (−1) × (−1) = +1 or (+1) × (+1) = +1. If two neighbors had different alleles, the product xixj became (−1) × (+1) = −1 or (+1) × (−1) = −1. Accordingly, the effects of neighbor genotypic identity on a particular phenotype depended on the coefficient β2 and the number of the two alleles in a neighborhood. If the numbers of identical and different alleles were the same near a focal plant, these neighbors offset the sum of the products between the focal plant i and all j neighbors (mathop {sum }nolimits_{ < i,j > } x_ix_j) and exerted no effects on a phenotype. When we summed up the phenotype values for the total number of plants n and replaced it as E = −β2, H = −β1 and εI = Σyi, Eq. 1 could be transformed into ({it{epsilon }}_I = – {E}mathop {sum }nolimits_{ < i,j > } x_ix_j – {H}mathop {sum }x_i), which defined the interaction energy of a two-dimensional ferromagnetic Ising model (McCoy and Maillard 2012). The neighbor effect β2 and self-genotype effect β1 were interpreted as the energy coefficient E and external magnetic effects H, respectively. An individual plant represented a spin and the two allelic states of each locus corresponded to a north or south dipole. The positive or negative value of Σxixj indicated a ferromagnetism or paramagnetism, respectively. In this study, we did not consider the effects of allele dominance because this model was applied to inbred plants. However, heterozygotes could be processed if the neighbor covariate xixj was weighted by an estimated degree of dominance in the self-genotypic effects on a phenotype.
Association tests
For association mapping, we needed to determine β1 and β2 from the observed phenotypes and considered a confounding sample structure as advocated by previous GWAS (e.g., Kang et al. 2008; Korte and Farlow 2013). Extending the basic model (Eq. (1)), we described a linear mixed model at an individual level as:

$$y_i = beta _0 + beta _1x_i + frac{{beta _2}}{L}mathop {sum }limits_{ < i,j > }^L x_ix_j^{(s)} + u_i + e_i$$
(2)

where β0 indicated the intercept; the term β1xi represented fixed self-genotype effects as tested in standard GWAS; and β2 was the coefficient of fixed neighbor effects. The neighbor covariate (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}) indicated a sum of products for all combinations between the ith focal plant and the jth neighbor at the sth spatial scale from the focal plant i, and was scaled by the number of neighboring plants, L. The number of neighboring plants L was dependent on the spatial scale s to be referred. Variance components due to the sample structure of self and neighbor effects were modeled by a random effect (u_i in {boldsymbol{u}}) and ({boldsymbol{u}}sim {mathrm{Norm}}(textbf{0},sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2)). The residual was expressed as (e_i in {boldsymbol{e}}) and ({boldsymbol{e}}sim {mathrm{Norm}}(textbf{0},sigma _e^2{boldsymbol{I}})), where I represented an identity matrix.
Variation partitioning
To estimate the proportion of phenotypic variation explained (PVE) by the self and neighbor effects, we utilized variance component parameters in linear mixed models. The n × n variance-covariance matrices represented the similarity in self-genotypes (i.e., kinship) and neighbor covariates among n individual plants as ({boldsymbol{K}}_1 = frac{1}{{q – 1}}{boldsymbol{X}}_1^{mathrm{T}}{boldsymbol{X}}_1) and ({boldsymbol{K}}_2 = frac{1}{{q – 1}}{boldsymbol{X}}_2^{mathrm{T}}{boldsymbol{X}}_2), where the elements of n plants × q markers matrix X1 and X2 consisted of explanatory variables for the self and neighbor effects as X1 = (xi) and ({boldsymbol{X}}_2 = (frac{{mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}}}{L})). As we defined (x_{i(j)} in){+1, −1}, the elements of the kinship matrix K1 were scaled to represent the proportion of marker loci shared among n × n plants such that ({boldsymbol{K}}_1 = left( {frac{{k_{ij}, + ,1}}{2}} right));(sigma _1^2)and (sigma _2^2) indicated variance component parameters for the self and neighbor effects.
The individual-level formula Eq. (2) could also be converted into a conventional matrix form as:

$${boldsymbol{y}} = {boldsymbol{X}}{boldsymbol{beta }} + {boldsymbol{Zu}} + {boldsymbol{e}}$$
(3)

where y was an n × 1 vector of the phenotypes; X was a matrix of fixed effects, including a unit vector, self-genotype xi, neighbor covariate (({mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}})/{L}), and other confounding covariates for n plants; β was a vector that represents the coefficients of the fixed effects; Z was a design matrix allocating individuals to a genotype, and became an identity matrix if all plants were different accessions; u was the random effect with Var(u) =(sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2); and e was residual as Var(e) = (sigma _e^2{boldsymbol{I}}).
Because our objective was to test for neighbor effects, we needed to avoid the detection of false positive neighbor effects. The self-genotype value xi and neighbor genotypic identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}) would become correlated explanatory variables in a single regression model (sensu colinear) due to the minor allele frequency (MAF) and the spatial scale of s. When MAF is low, neighbors (x_j^{(s)}) are unlikely to vary in space and most plants will have similar values for neighbor identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}). Furthermore, if the neighbor effects range was broad enough to encompass an entire field (i.e., s→∞), the neighbor covariate and self-genotype xiwould become colinear according to the equation: (left(mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}right)/L = x_ileft( {mathop {sum }nolimits_{j = 1}^L x_j^{left( s right)}} right)/L = x_ibar x_j), where (bar x_j) indicates a population-mean of neighbor genotypes and corresponds to a population-mean of self-genotype values (bar x_i), if s→∞. The standard GWAS is a subset of the neighbor GWAS and these two models become equivalent at s = 0 and (sigma _2^2) = 0. When testing the self-genotype effect β1, we recommend that the neighbor effects and its variance component (sigma _2^2) should be excluded; otherwise, the standard GWAS fails to correct a sample structure because of the additional variance component at (sigma _2^2, ne ,0). To obtain a conservative conclusion, the significance of β2 and (sigma _2^2) should be compared using the standard GWAS model based on self-effects alone.
Given the potential collinearity between the self and neighbor effects, we defined different metrics for the proportion of phenotypic variation explained (PVE) based on self or neighbor effects. Using a single-random effect model, we calculated PVE for either the self or neighbor effects as follows:
‘single’ PVEself = (sigma _1^2/(sigma _1^2 + sigma _e^2))when s and (sigma _2^2) were set at 0, or
‘single’ PVEnei = (sigma _2^2/(sigma _2^2 + sigma _e^2))when (sigma _1^2) was set at 0.
Using a two-random effect model, we could focus on one variable while considering relationships between two variables (sensu partial out) for either of the two variance components. We defined such a partial PVE as:
‘partial’ PVEself = (sigma _1^2/(sigma _1^2 + sigma _2^2 + sigma _e^2)) and
‘partial’ PVEnei = (sigma _2^2/(sigma _1^2 + sigma _2^2 + sigma _e^2)).
As the partial PVEself was equivalent to the single PVEself when s was set at 0, the net contribution of neighbor effects at s ≠ 0 was given as
‘net’ PVEnei = (partial PVEself + partial PVEnei) − single PVEself,
which indicated the proportion of phenotypic variation that could be explained by neighbor effects, but not by the self-genotype effects.
Simulation
To examine the model performance, we applied the neighbor GWAS to simulated phenotypes. Phenotypes were simulated using a subset of the actual A. thaliana genotypes. To evaluate the performance of the simple linear model, we assumed a complex ecological form of neighbor effects with multiple variance components controlled. The model performance was evaluated in terms of the causal variant detection and accuracy of estimates. All analyses were performed using R version 3.6.0 (R Core Team 2019).
Genotype data
To consider a realistic genetic structure in the simulation, we used part of the A. thaliana RegMap panel (Horton et al. 2012). The genotype data for 1307 accessions were downloaded from the Joy Bergelson laboratory website (http://bergelson.uchicago.edu/?page_id=790 accessed on February 9, 2017). We extracted data for chromosomes 1 and 2 with MAF at >0.1, yielding a matrix of 1307 plants with 65,226 single nucleotide polymorphisms (SNPs). Pairwise linkage disequilibrium (LD) among the loci was r2 = 0.003 [0.00–0.06: median with upper and lower 95 percentiles]. Before generating a phenotype, genotype values at each locus were standardized to a mean of zero and a variance of 1. Subsequently, we randomly selected 1,296 accessions (= 36 × 36 accessions) without any replacements for each iteration and placed them in a 36 × 72 checkered space, following the Arabidopsis experimental settings (see Fig. S1).
Phenotype simulation
To address ecological issues specific to plant neighborhood effects, we considered two extensions, namely asymmetric neighbor effects and spatial scales. Previous studies have shown that plant–plant interactions between accessions are sometimes asymmetric under herbivory (e.g., Bergvall et al. 2006; Verschut et al. 2016; Sato and Kudoh 2017) and height competition (Weiner 1990); where one focal genotype is influenced by neighboring genotypes, while another receives no neighbor effects. Such asymmetric neighbor effects can be tested by statistical interaction terms in a linear model (Bergvall et al. 2006; Sato and Kudoh 2017). Several studies have also shown that the strength of neighbor effects depends on spatial scales (Hambäck et al. 2014), and that the scale of neighbors to be analyzed relies on the dispersal ability of the causative organisms (see Hambäck et al. 2009; Sato and Kudoh 2015; Verschut et al. 2016; Ida et al. 2018 for insect and mammal herbivores; Rieux et al. 2014 for pathogen dispersal) or the size of the competing plants (Weiner 1990). We assumed the distance decay at the sth sites from a focal individual i with the decay coefficient α as (wleft( {s,alpha } right) = {mathrm{e}}^{ – alpha (s – 1)}), since such an exponential distance decay has been widely adopted in empirical studies (Devaux et al. 2007; Carrasco et al. 2010; Rieux et al. 2014; Ida et al. 2018). Therefore, we assumed a more complex model for simulated phenotypes than the model for neighbor GWAS as follows:

$$y_i = beta _0 + beta _1x_i + frac{{beta _2}}{L}mathop {sum }limits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)} + beta _{12}frac{{x_i}}{L}mathop {sum }limits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)} + u_i + e_i$$
(4)

where β12 was the coefficient for asymmetry in neighbor effects. By incorporating an asymmetry coefficient, the model (Eq. (4)) can deal with cases where neighbor effects are one-sided or occur irrespective of a focal genotype (Fig. 2). Total variance components resulting from three background effects (i.e., the self, neighbor, and self-by-neighbor effects) were defined as (u_i in {boldsymbol{u}}) and ({boldsymbol{u}}sim {mathrm{Norm}}(textbf{0},sigma _1^2{boldsymbol{K}}_1 + sigma _2^2{boldsymbol{K}}_2 + sigma _{12}^2{boldsymbol{K}}_{12})). The three variance component parameters (sigma _1^2), (sigma _2^2), and (sigma _{12}^2), determined the relative importance of the self-genotype, neighbor, and asymmetric neighbor effects in ui. Given the elements of n plants × q marker explanatory matrix with ({boldsymbol{X}}_{12} = (frac{{x_i}}{L}mathop {sum }nolimits_{ < i,j > }^L w(s,alpha )x_ix_j^{(s)})), the similarity in asymmetric neighbor effects was calculated as ({boldsymbol{K}}_{12} = frac{1}{q-1}{boldsymbol{X}}_{12}^{mathrm{T}}{boldsymbol{X}}_{12}). To control phenotypic variations, we further partitioned the proportion of phenotypic variation into those explained by the major-effect genes and variance components PVEβ + PVEu, major-effect genes alone PVEβ, and residual error PVEe, where PVEβ + PVEu + PVEe = 1. The optimize function in R was used to adjust the simulated phenotypes to the given amounts of PVE.
Fig. 2: Numerical examples of the symmetric and asymmetric neighbor effects.

The intercept, distance decay, random effects, and residual errors are neglected, to simplify this scheme. a Symmetric neighbor effects represent how neighbor genotype similarity (or dissimilarity) affects the trait value of a focal individual yi regardless of its own genotype. b Asymmetric neighbor effects can represent a case in which one genotype experiences neighbor effects while the other does not (b) and a case in which the direction of the neighbor effects depends on the genotypes of a focal individual (c). The case b was considered in our simulation as it has been empirically reported (e.g., Bergvall et al. 2006; Verschut et al. 2016; Sato & Kudoh 2017).

Full size image

Parameter setting
Ten phenotypes were simulated with varying combination of the following parameters, including the distance decay coefficient α, the proportion of phenotypic variation explained by the major-effect genes PVEβ, the proportion of phenotypic variation explained by major-effect genes and variance components PVEβ + PVEu, and the relative contributions of self, symmetric neighbor, and asymmetric neighbor effects, i.e., PVEself:PVEnei:PVEs×n. We ran the simulation with different combinations, including α = 0.01, 1.0, or 3.0; PVEself:PVEnei:PVEs×n = 8:1:1, 5:4:1, or 1:8:1; and PVEβ and PVEβ + PVEu = 0.1 and 0.4, 0.3 and 0.4, 0.3 and 0.8, or 0.6 and 0.8. The maximum reference scale was fixed at s = 3. The line of simulations was repeated for 10, 50, or 300 causal SNPs to examine cases of oligogenic and polygenic control of a trait. The non-zero coefficients (i.e., signals) for the causal SNPs were randomly sampled from −1 or 1 digit and then assigned, as some causal SNPs were responsible for both the self and neighbor effects. Of the total number of causal SNPs, 15% had self, neighbor, and asymmetric neighbor effects (i.e.,β1 ≠ 0 and β2 ≠ 0 and β12 ≠ 0); another 15% had both the self and neighbor effects, but no asymmetry in the neighbor effects (β1 ≠ 0 and β2 ≠ 0 and β12 ≠ 0); another 35% had self-genotypic effects only (β1 ≠ 0); and the remaining 35% had neighbor effects alone (β2 ≠ 0). Given its biological significance, we assumed that some loci having neighbor signals possessed asymmetric interactions between the neighbors (β2 ≠ 0 and β12 ≠ 0), while the others had symmetric interactions (β2 ≠ 0 and β12 ≠ 0). Therefore, the number of causal SNPs in β12 was smaller than that in the main neighbor effects β2. According to this assumption, the variance component (sigma _{12}^2) was also assumed to be smaller than (sigma _2^2). To examine extreme conditions and strong asymmetry in neighbor effects, we additionally analyzed the cases with PVEself:PVEnei:PVEs×n = 1:0:0, 0:1:0, or 1:1:8.
Summary statistics
The simulated phenotypes were fitted by Eq. (2) to test the significance of coefficients β1 and β2, and to estimate single or partial PVEself and PVEnei. To deal with potential collinearity between xi and neighbor genotypic identity (mathop {sum }nolimits_{ < i,j > }^L x_ix_j^{(s)}), we performed likelihood ratio tests between the self-genotype effect model and the model with both self and neighbor effects, which resulted in conservative tests of significance for β2 and (sigma _2^2). The simulated phenotype values were standardized to have a mean of zero and a variance of 1, where true β was expected to match the estimated coefficients (hat beta) when multiplied by the standard deviation of non-standardized phenotype values. The likelihood ratio was calculated as the difference in deviance, i.e., −2 × log-likelihood, which is asymptotically χ2 distributed with one degree of freedom. The variance components, (sigma _1^2) and (sigma _2^2), were estimated using a linear mixed model without any fixed effects. To solve the mixed model with the two random effects, we used the average information restricted maximum likelihood (AI-REML) algorithm implemented in the lmm.aireml function in the gaston package of R (Perdry and Dandine-Roulland 2018). Subsequently, we replaced the two variance parameters (sigma _1^2) and (sigma _2^2) in Eq. (2) with their estimates (hat sigma _1^2) and (hat sigma _2^2) from the AI-REML, and performed association tests by solving a linear mixed model with a fast approximation, using eigenvalue decomposition (implemented in the lmm.diago function: Perdry and Dandine-Roulland 2018). The model likelihood was computed using the lmm.diago.profile.likelihood function. We evaluated the self and neighbor effects for association mapping based on the forward selection of the two fixed effects, β1 and β2, as described below:
1.
Computed the null likelihood with (sigma _1^2, ne ,0) and (sigma _2^2 = 0) in Eq. (2).

2.
Tested the self-effect, β1, by comparing with the null likelihood.

3.
Computed the self-likelihood with (hat sigma _1^2), (hat sigma _2^2), and β1 using Eq. (2).

4.
Tested the neighbor effects, β2, by comparing with the self-likelihood.

We also calculated PVE using the mixed model (Eq. (3)) without β1 and β2 as follows:
1.
Calculated single PVEself or single PVEnei by setting either (sigma _1^2) or (sigma _2^2) at 0.

2.
Tested the single PVEself or single PVEnei using the likelihood ratio between the null and one-random effect model.

3.
Calculated the partial PVEself and partial PVEnei by estimating (sigma _1^2) and (sigma _2^2) simultaneously.

4.
Tested the partial PVEself and partial PVEnei using the likelihood ratio between the two- and one-random effect model.

We inspected the model performance based on causal variant detection, PVE estimates, and effect size estimates. The true or false positive rates between the causal and non-causal SNPs were evaluated using ROC curves and area under the ROC curves (AUC) (Gage et al. 2018). An AUC of 0.5 would indicate that the GWAS has no power to detect true signals, while an AUC of 1.0 would indicate that all the top signals predicted by the GWAS agree with the true signals. In addition, the sensitivity to distinguish self or neighbor signals (i.e., either β1 ≠ 0 or β2 ≠ 0) was evaluated using the true positive rate of the ROC curves (i.e., y-axis of the ROC curve) at a stringent specificity level, where the false positive rate (x-axis of the ROC curve) = 0.05. The roc function in the pROC package (Robin et al. 2011) was used to calculate the ROC and AUC from −log10(p value). Factors affecting the AUC or sensitivity were tested by analysis-of-variance (ANOVA) for the self or neighbor effects (AUCself or AUCnei; self or neighbor sensitivity). The AUC and PVE were calculated from s = 1 (the first nearest neighbors) to s = 3 (up to the third nearest neighbors) cases. The AUC was also calculated using standard linear models without any random effects, to examine whether the linear mixed models were superior to the linear models. We also tested the neighbor GWAS model incorporating the neighbor phenotype (y_j^{(s)}) instead of (x_j^{(s)}). The accuracy of the total PVE estimates was defined as PVE accuracy = (estimated total PVE − true total PVE) / true total PVE. The accuracy of the effect size estimates was evaluated using mean absolute errors (MAE) between the true and estimated β1 or β2 for the self and neighbor effects (MAEself and MAEnei). Factors affecting the accuracy of PVE and effect size estimates were also tested using ANOVA. Misclassifications between self and neighbor signals were further evaluated by comparing p value scores between zero and non-zero coefficients. If −log10(p value) scores of zero β are the same or larger than non-zero β, it infers a risk of misspecification of the true signals.
Arabidopsis herbivory data
We applied the neighbor GWAS to field data of Arabidopsis herbivory. The procedure for this field experiment followed that of our previous experiment (Sato et al. 2019). We selected 199 worldwide accessions (Table S1) from 2029 accessions sequenced by the RegMap (Horton et al. 2012) and 1001 Genomes project (Alonso-Blanco et al. 2016). Of the 199 accessions, most were overlapped with a previous GWAS of biotic interactions (Horton et al. 2014) and half were included by a GWAS of glucosinolates (Chan et al. 2010). Eight replicates of each of the 199 accessions were first prepared in a laboratory and then transferred to the outdoor garden at the Center for Ecological Research, Kyoto University, Japan (Otsu, Japan: 35°06′N, 134°56′E, alt. ca. 200 m: Fig. S1a). Seeds were sown on Jiffy-seven pots (33-mm diameter) and stratified at a temperature of 4 °C for a week. Seedlings were cultivated for 1.5 months under a short-day condition (8 h light: 16 h dark, 20 °C). Plants were then separately potted in plastic pots (6 cm in diameter) filled with mixed soil of agricultural compost (Metro-mix 350, SunGro Co., USA) and perlite at a 3:1 ratio. Potted plants were set in plastic trays (10 × 40 cells) in a checkered pattern (Fig. S1b). In the field setting, a set of 199 accessions and an additional Col-0 accession were randomly assigned to each block without replacement (Fig. S1c). Eight replicates of these blocks were set >2 m apart from each other (Fig. S1c). Potted plants were exposed to the field environment for 3 weeks in June 2017. At the end of the experiment, the percentage of foliage eaten was scored as: 0 for no visible damage, 1 for ≤10%, 2 for >10% and ≤25%, 3 for >25% and ≤50%, 4 for >50% and ≤75%, and 5 for >75%. All plants were scored by a single person to avoid observer bias. The most predominant herbivore in this field trial was the diamond back moth (Plutella xylostella), followed by the small white butterfly (Pieris rapae). We also recorded the initial plant size and the presence of inflorescence to incorporate them as covariates. Initial plant size was evaluated by the length of the largest rosette leaf (mm) at the beginning of the field experiment and the presence of inflorescence was recorded 2 weeks after transplanting.
We estimated the variance components and performed the association tests for the leaf damage score with the neighbor covariate at s = 1 and 2. These two scales corresponded to L = 4 (the nearest four neighbors) and L = 12 (up to the second nearest neighbors), respectively, in the Arabidopsis dataset. The variation partitioning and association tests were performed using the gaston package, as mentioned above. To determine the significance of the variance component parameters, we compared the likelihood between mixed models with one or two random effects. For the genotype data, we used an imputed SNP matrix of the 2029 accessions studied by the RegMap (Horton et al. 2012) and 1001 Genomes project (Alonso-Blanco et al. 2016). Missing genotypes were imputed using BEAGLE (Browning and Browning 2009), as described by Togninalli et al. (2018) and updated on the AraGWAS Catalog (https://aragwas.1001genomes.org). Of the 10,709,466 SNPs from the full imputed matrix, we used 1,242,128 SNPs with MAF at >0.05 and LD of adjacent SNPs at r2  More

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