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.