Introduction
Species’ performance depends on multiple limiting factors that determine
population size at a certain location and time. However, not all factors
may operate simultaneously as it is sometimes assumed. For instance, a
plant species could be limited by low temperatures in winter or in the
coldest regions of its distributional range, while competition for light
or the scarcity of a specific nutrient could set a limit to its
abundance in regions characterized by milder climates. This idea was
proposed about two centuries ago by Carl Sprengel (1828), who claimed
that, even if most essential nutrients are abundant in the soil, the
scarcity of a single critical nutrient will limit plant growth. This
observation inspired Liebig’s Law of the Minimum (LoM hereafter;
Liebig, 1840), which states that one single factor sets limits to the
maximum performance a species can reach at any given point in space and
time.
The LoM has been recognized to be important for explaining ecological
patterns (e.g. Didham, 2006; Hiddink & Kaiser, 2005; Huston, 2002), yet
it has been underexplored. If as stated by the LoM, a unique
environmental factor limits the abundance of a species at each location,
then the relationship between the species’ abundance and a single
environmental factor should take a polygonal shape characterized by
points scattered on the Y axis from zero to an upper boundary,
resembling an envelope (Fig 1A). These polygon-shaped patterns could
take a triangle (e.g. Carroll et al., 2011), a bell (e.g. de Boer et
al., 2013) or other polygonal shape (Anderson, 2008). The upper limit of
the polygonal point cloud would be defined by the locations where the
abundance is limited by the environmental factor under study, while
points below the upper limit corresponds to locations where other
factors act as limiting for abundance. This polygon-shaped pattern
contrasts with line-shaped patterns generally expected by ecologists,
which would imply that abundance at each location is determined by the
interaction of different factors (Fig 1B).
Although the LoM was formulated
about two centuries ago, its application to interpreting polygon-shaped
patterns and the analytical solution to the estimation of the upper
boundary are relatively recent. In 1996, Thomson et al. linked
polygon-shaped plots to the LoM and highlighted the problem of analyzing
them through classical correlational approaches. Shortly after, Scharf
et al. (1998) and Cade et al. (1999) recommended quantile regression
models (QR hereafter) as the most suitable statistical tool to estimate
the upper limits of polygon-shaped plots. Unlike standard regression
models that are based on central tendency, QR can estimate the response
near the upper boundary of the abundance-environment point cloud, where
the measured environmental gradient is the limiting one (Cade & Noon,
2003). In the last two decades QR have been increasingly used to assess
the limiting effect of abiotic and biotic factors on the performance of
different organisms (e.g. Bissinger et al., 2008; Brennan et al., 2015;
Fornaroli et al., 2015; Jarema et al., 2009; McClain & Rex, 2001; Vaz
et al., 2007). However, despite the rising awareness of the relevance of
limiting relationships in ecology, we lack a quantitative perspective on
their prevalence in nature and the assimilation of this body of theory
by ecologists.
We aim to assess the generality of polygon-shaped patterns in
abundance-environment relationships and to estimate the frequency to
which these patterns are interpreted and analyzed in the context of
ecological limitation. We hypothesize that polygon-shaped patterns are
more common than it is generally recognized, and that the information
contained in their upper limit is systematically neglected in ecological
studies (Thomson et al., 1996). We also argue that, even when the
structure of the data is identified as polygon-shaped, the theory about
ecological limitation is rarely discussed, and the connection with the
LoM seldom established. First, we assessed the universality of
polygon-shaped patterns in abundance-environment plots found in the
literature. We also analyzed how frequently polygon-shaped patterns
found in the literature are conceptually linked to the LoM and modeled
through QR. Second, we assessed the prevalence of polygon-shaped
patterns in abundance-environment relationships across 186 tree and 114
bird species in North America. We used QR to estimate the number of
species for which population size in the region studied is limited by
water balance and energy, illustrating its application in the context of
ecological limitation. Finally,
we present a proof-of-concept of a procedure named filling indexthat aims to differentiate polygon-shaped point clouds from line-shaped
ones.
We demonstrate that polygon-shaped patterns are pervasive in
abundance-environment relationships, though they are rarely interpreted
from a LoM perspective and analyzed with proper statistical tools. We
hope that this work might lay the ground to help ecologists identify
polygon-shaped patterns in their data and boost the incorporation of the
theory of limiting factors in the area.