The Equilibrium Conundrum

Abstract

The idea that natural systems tend to be at equilibrium dates back to the origin of the field of ecology and continues to underlie most ecological theory. However, empirical evidence for equilibrium dynamics in nature and in experiments is surprisingly elusive. Here, we address this conundrum by first exploring the history of equilibrium in ecological theory and the evidence for equilibrium dynamics in natural systems. We then search the literature to quantify how empiricists deal with equilibrium in their research and address barriers to integrating the concept of equilibrium into empirical work by providing step-by-step instructions for determining whether a population is at equilibrium. Next, we lay out three ways that equilibrium is embedded in theory, and for each, outline when meeting the equilibrium assumption in empirical tests is critical for scientific inference, and when it may be possible to relax this assumption. And finally, we present concrete steps that empiricists and theoreticians can each take in order to meet in the middle when it comes to equilibrium. We hope that this paper will stimulate new discussions from researchers from across the theory-empirical divide about this longstanding issue.

Keywords: balance of nature; empirical tests; equilibrium; mathematical models; theory.

FIGURE 1

Graphical representation of concepts related to equilibrium. Panel (A) depicts the abundance of a single species (y‐axis) over time (x‐axis). The initial trajectory (blue line) demonstrates the population at equilibrium (here, carrying capacity). A perturbation then disrupts this equilibrium, causing the population to deviate from its equilibrium state. It then recovers towards its previous equilibrium at a given return rate (red line). The final trajectory (black line) illustrates the concept of stationarity, in which the population fluctuates around a constant mean with constant variance. Panel (B) extends these concepts to two interacting species, illustrating their abundances in a phase space (a graphical representation of the system's possible states, with each axis representing one species' abundance). Focusing on varying perturbation levels at a fixed point reveals different stability states: constancy (no perturbation and no change in the system's state), local stability (robustness to small perturbations) and global stability (robustness to large perturbations). These stability concepts can also be generalised beyond a fixed point.

FIGURE 2

How do empirical studies account for the equilibrium assumption? Flow diagram showing the results of a Web of Science search exploring how empirical tests of ecological theories account for the equilibrium assumption. We developed search strings to capture substantial portions of the literature on five major theories in ecology that invoke an equilibrium assumption. Papers were screened to ensure that they presented new empirical tests of the theory in question, and the top 10 most cited articles within each theory were retained for data extraction. We then identified which articles acknowledged the equilibrium assumption and the steps the authors took (if any) to account for the assumption in their study approach.

FIGURE 3

Visualisation of a method to determine whether populations are at equilibrium, applied here to three replicate populations of flour beetles ( Tribolium castaneum ). Points indicate log‐abundances of flour beetle populations. Dashed lines represent the population dynamics of the fitted model. Solid lines represent the median estimates of equilibrium population size; dark blue and light blue bands represent 66% and 95% credible intervals of the equilibrium state, respectively.

FIGURE 4

How to determine when breaking the equilibrium assumption is a problem. Whether or not meeting the equilibrium assumption in an empirical test is necessary for scientific inference depends on the research goals and the nature of the equilibrium assumption being made, and empiricists can use this flow diagram to help determine whether they need to ensure equilibrium conditions in their study system, when they should exercise caution in making inferences, and when to pursue alternative non‐equilibrium approaches.

FIGURE 5

What can empiricists working in non‐equilibrium systems do? To illustrate some options that empiricists have when faced with a system that is unlikely to reach equilibrium and when scientific inference is likely to be compromised by not meeting this assumption (Figure 4), we use an example of specialist yeast that inhabit floral nectar and are dispersed by pollinators between plants. While we use this study system as an example, the options outlined below apply to a broad range of systems, in particular those characterised by regular disturbance, an ephemeral habitat, or seasonality. The first step is to (A) determine whether the system reaches an equilibrium, for example by using the method illustrated in Box 2. In some cases it may be obvious that equilibrium is not reached, as in this example where the yeast population within a host flower stops growing (apparent K), but this state is only temporary until the host flower begins to senesce. This dynamic is likely to occur in many systems—for example whenever the host dies, the resource is non‐renewing, or waste products accumulate. In these cases, (B) the first option is to reconsider if the research question could be better addressed in another system that does reliably reach a stable equilibrium. Alternatively, one could (C) consider whether an equilibrium is reached at some larger spatial or temporal scale (here patch occupancy at a metapopulation scale), for example using the method we outline in the Supporting Information. Or, (D) if the system allows, it might be possible to experimentally create equilibrium conditions, here shown by regularly replenishing resources and using artificial flowers that do not senesce. And finally, (E) one may choose to pivot the response variable of interest towards something compatible with non‐equilibrium dynamics, such as time to extinction.