Continuum theory revisited: what shape are species responses along ecological gradients?

The shape of species’ responses along ecological gradients has important implications for both continuum theory and community analysis. Most current theories and analytical models in community ecology assume that responses are unimodal and symmetric. However, interactions between species and extreme environmental stress may cause skewed or non-unimodal responses. To date, statistical tools for evaluating response shapes have been either inappropriate, inefficient or biased. Using a data set on vascular plant distributions along an elevation gradient, we show that Huisman–Olff–Fresco (HOF) models are an effective method for this purpose, allowing models of various forms (skewed, symmetric, plateau, monotonic) to be tested for adequacy. HOF modeling was compared with alternative methods for response fitting, including Gaussian responses as Generalized linear model (GLMs), Generalized Additive Models (GAM) and Beta Functions with fixed or estimated endpoints. In our data set, skewed and plateau responses are less common than symmetric ones. Less than half of the species have skewed or plateau responses that can not be adequately modeled by Gaussian models. We show that Beta function models with fixed endpoints are biased, confounding skewness and the location of the mode and should not be used to test response shapes. Beta models with estimated endpoints are fairly consistent with other models. GAMʹs cannot provide clear tests of skewness or kurtosis of response curves, though we show that GAMʹs, in general, confirmed the shapes chosen by HOF modeling. We provide free software for fitting HOF models and encourage further applications to community data collected along different types of ecological gradients.