Discussion
The LoM as a conceptual framework to analyze abundance patterns in ecology and biogeography is promising, but largely underutilized. There is a striking contrast between the high prevalence of empirical polygon-shaped patterns in abundance-environment relationships (i.e., found in 76% of reviewed studies and an average of 86% of analyzed species) and the frequency at which these relationships are interpreted from a LoM perspective (0% of reviewed studies). Such disconnect raises several questions: what challenges a more widespread use of the LoM? What are the alternatives and challenges to analyzing and interpreting polygonal abundance-environment relationships? What can be learnt by analyzing abundance-environment relationships from a LoM perspective? Our results illustrate with two environmental variables how the LOM can provide a quick way to identify the limiting effect of environmental factors for any taxa group, which has key implications for ecological inference.
For decades, ecologists have focused on abundance-environment relationships from various perspectives: population ecology (e.g. Rosenzweig & Winakur, 1969), habitat selection (e.g. Holt, 1987; Pigeon et al., 2016), or biogeographic (e.g. Dallas et al., 2017; Sagarin & Gaines, 2002). Through a literature review, we found that most abundance-environment plots are characterized by polygon-shaped patterns (76% out of 21 studies). However, the vast majority of these relationships were not approached statistically as limiting relationships (only one study used QR). Moreover, none of these studies discussed the limiting ecological relationships implicit in their polygon-shaped patterns considering the LoM. The implications are clear. Ecologists have very often ignored limiting responses in the distribution of abundance (Konrad et al. , 2008; Greenberget al. , 2015; Stralberg et al. , 2018) by implicitly assuming the simultaneous influence of different environmental factors in each given site rather than acknowledging that the maximum reachable abundance would be locally limited by only one of these factors.
We identify three major challenges that may hinder a more widespread application of the LoM perspective to understand ecological patterns. First, we uncover researchers’ biases towards using standard regression models that estimate measures of central tendency responses (Lancaster & Belyea, 2006). This tendency could be due to researchers not being concerned with inspecting the shape of bivariate relationships between abundance and environmental factors, which would also explain why 23% of 31 reviewed studies did not show a point cloud in their plots. Alternatively, researchers may have explored abundance-environment plots, found polygonal point clouds, and assumed that fitting the central tendency amidst the point cloud would characterize accurately the relationship and that deviations from the mean response are the consequence of sampling error (Lancaster & Belyea, 2006). This assumption overlooks the ecological significance of the upper limit in the cloud of points (Thomson et al. , 1996), which is explicitly accounted for by QR.
To overcome this bias, we encourage ecologists to thoroughly inspect their abundance-environment plots (Thomson et al. , 1996) with the novel filling index procedure provided here (the access code and a database example in Appendix S12). Although this method is presented here as a proof-of-concept and may require further evaluation, it has proven to be useful in detecting polygon-shaped point clouds in the simulated data. To our knowledge, this is the first procedure that identifies polygonal patterns with upper boundaries (but see an alternative approach not focused on the upper limit in Milne et al., 2006). When ecologists come across polygon-shaped patterns, they should consider the role of limiting factors together with the theory and the techniques associated to it, since a misinterpretation of polygon-shaped data (i.e. estimating central tendency rather the maximum limit of abundance) could lead to incorrect inferences about abundance-environment relationships. We expect that this tool, together with the empirical evidence unveiled here, will manifest the pervasiveness of limiting relationships in ecology and promote the usage of the LoM framework. Progressively adopting in a data exploration stage, the practice of assessing the shape of point clouds may help ecologists correct the bias found in the literature towards expecting line-shaped patterns and a using regression techniques based on mean estimates.
A second barrier as to why the LoM perspective is not adopted is the general dismissal of the theory behind limiting relationships in ecology and biogeography. Polygonal relationships and QR are not linked to the LoM even in research acknowledging the limiting nature of different predictors on abundance. We recognize that several alternative methods can accommodate non-stationarities in the importance of different factors along gradients, in the magnitude and direction of effects along gradients or across geographical spaces (i.e. Geographically Weighted Regression – GWR; Fotheringham et al., 2002; Generalized Additive Models for Location, Scale and Shape – GAMLSS; Rigby & Stasinopoulos, 2005; Rollinson et al., 2021). These particularities may be better fit to respond to a different set of questions. For instance, GWR is applied to assess the spatial heterogeneity in the relationship between species richness and climate variability by identifying regions in which some environmental factors are more relevant than others (e.g. Hortal et al., 2011). However, QR is the only statistical method that accounts for non-stationarity along a bivariate relationship, which is the central point of the LoM. Thus, if one aims to model the effect of an environmental gradient on a particular part of the distribution of the response variable, in our case the upper quantiles or the maximum abundance of a species, QR seems best equipped to provide accurate answers (Kneib, 2013).
Finally, a more widespread application of the LoM is hampered by data availability. The lack of large enough abundance samples hinders capturing meaningful clouds of points and estimating their upper limit (Cade et al. , 1999). For example, only 9 out of 24 reviewed studies containing at least one plot, encompass sufficiently large sample sizes (> 100 observations) to estimate upper limiting responses using QR with a reasonable Type-I Error (τ ≥ 0.90; Cade et al., 2005). Moreover, we found that identifying polygon-shaped patterns were positively related with sample size in trees and birds in the USA. Our empirical examples encompass some of the best sampled organisms and regions worldwide, but the exploration of polygon-shaped patterns in other regions and taxa would require good quality abundance data sampled across large spatial (or temporal) scales (Howard et al. , 2014). These data requirements may be better fulfilled by institutional programs dedicated to the systematic sampling of a group of organisms such as FIA, whose data demonstrated to be better fitted for QR models than eBird data (i.e., evidence ratios, were on average higher for trees than for birds; See Appendix S4 and Appendix S5). Systematic sampling at large spatial scales may sometimes require international coordination, but abundance data may not be costlier to collect than presence data in terms of time and number of collectors (Gibbons et al. , 2007). We expect that the increased availability of large-scale data on species abundances will foster the application of LoM-based approaches in the near future.
Here we found polygon-shaped patterns in most American birds and trees (Appendix S8 and S10) showing the constraints imposed by GDD and water balance in their maximum abundances (see Fig 5-6; Cade & Noon, 2003). Contrary to what ecologists may intuitively think, the LoM does not dismiss that abundance patterns depend on a range of environmental conditions and resources. In fact, it considers these multiple limiting factors, that is, the variation below the upper boundary is explained by situations when factors other than the gradient under study limit abundance. But the LoM emphasizes the maximum abundance reachable in each point of the gradient under study which is limited by it (Hiddink & Kaiser, 2005).
The potential applications of the LoM approach are manifold because knowing the maximum number of individuals of a species that a given locality or region can support is fundamental to making informed decisions on wildlife and forest management. First, it is key to predict ecological shifts driven by environmental disturbance (e.g. algal blooms; Carvalho et al., 2013), potential abundance shifts caused by climate change (e.g. Villén‐Peréz et al., 2020), or it could complement forecasts of species invasions induced by any type of environmental change (e.g. Bezeng et al., 2017). Moreover, reintroduction, restoration, and rewilding programs may also benefit from accurate estimates of the maximum number of individuals potentially supported by a system (Johnston et al. , 2015). The LoM approach could also be applied to identify limiting factors in populations and/or determine whether a given habitat could support a viable population (See et al. , 2021). Finally, the LoM approach can be used to estimate the carrying capacity of managed systems such as forest plantations (e.g. Farias et al., 2021) or fishery stocks (e.g. See et al., 2021; Sweka & Mackey, 2010). Overall, the LoM approach can help both to design mitigation actions against global change impacts on biodiversity and to optimize production systems.
We advocate using QR and the LoM to model maximum potential abundances and interpret polygon-shaped patterns (see also Villén‐Peréz et al., 2020), acknowledging that its potential to ecology goes far beyond. For instance, it would be interesting to formally examine the role of the LoM and niche theory. Does the maximum potential abundance that species can reach at each given temperature value reflect species’ fundamental niche respect to temperature (Villén‐Peréz et al. , 2020)? If this is validated, it would set new standards in ecology, which currently assume that distribution data only inform the realized niche of the species and the fundamental niche can only be assessed experimentally (Kearney & Porter, 2009; Soberón, 2010). Distribution data may reveal more information than previously acknowledged, and the LoM approach could enable the quantification of the fundamental niche for a massive number of species and complement the experimental approaches.
Understanding abundance-environment relationships is an integral part of ecology, and it has the potential to aid at planning species conservation and habitat management (Wilson et al. , 2005), or for predicting responses to global change (Ehrlén & Morris, 2015). However, this is only possible when ecological data are correctly interpreted (Thomson et al. , 1996). Here we show that species’ abundance patterns along environmental gradients often adopt polygonal shapes, with an upper limit indicating consistent limiting responses. These patterns conform to the LoM and can be analysed using QR. Despite the many decades of studying abundance-environment relationships, there is still room for a paradigm shift in how we analyze, interpret, and infer such relationships. The LoM as a conceptual framework and QR as an analytical tool offer a promising research avenue in need for further exploration.