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.