Linking the ball‐and‐cup analogy and ordination trajectories to describe ecosystem stability, resistance, and resilience

The ball‐and‐cup diagram for conceptualizing ecosystem stability, resistance, and resilience is often presented as a ball rolling around within and between two or more cups. This analogy has a long history in ecology and has been used to illustrate ecosystem changes over time where the magnitude of changes required to push the ball from one cup to another represents a regime shift to an alternative state. Another approach for visualizing ecosystem stability, resistance, and resilience involves ordinations of repeated measures of community data or environmental variables and tracking trajectories over time in ordination space. Interestingly, the two approaches have not been linked in a meaningful way. Here, we provide a conceptual link between trajectories of ecological change in ordination space to the ball‐and‐cup analogy and show how distance‐based measures calculated from ordination scores can be used to quantitatively classify and evaluate the relative stability and resilience of ecological systems. The ball‐and‐cup diagram is commonly used by scientists and engineers to illustrate complex topics across a variety of scientific disciplines (e.g., ecology, evolution, engineering; Fig. 1). Among the first to use the ball‐and‐cup analogy was Sewall Wright in a presentation at the sixth International Congress of Genetics in 1932 on shifting balance theory (Wright 1931, Ruse 1996, Kaplan 2008). In his presentation, Wright used the ball‐and‐cup metaphor to describe an evolutionary fitness landscape, where the cup‐shaped landscape is comprised of adaptive peaks that represent high‐fitness collections of genotypes and valleys representing genotypes of relatively low fitness (Kaplan 2008). A ball mapped onto the fitness landscape and its trajectory across the landscape over time provided a measure of change in adaptive capacity of a population. Willems (1970) was among the first to interpret the ball‐and‐cup diagram in an ecological context. In his perspective, the ball represented the current ecosystem state (see Table 1 for definitions of italicized words) and the cup represented a basin of attraction (Peterson et al. 1998, Walker et al. 2004) based on the normal range of variation for that ecosystem. One‐or‐more neighboring cups represent alternative stable states in the stability landscape (Walker et al. 2004, Scheffer 2009) for that ecosystem. Willems (1970) used this analogy to describe ecosystem stability, and this diagram has since been used to describe ecosystem responses to disturbance (Hurd and Wolf 1974, Godron and Forman 1983, Hamilton and