Ecological Transients and the Ghost of Equilibrium Past

By Matthew R. Francis

The sight and smell of eutrophication—in the form of a layer of stinking green algae on a lake or pond—is likely familiar to many readers. The result is detrimental, even toxic, to other species that rely on the water, ranging from tiny animals to birds and even humans. For example, eutrophication on Lake Erie affects millions (see Figure 1). But the real culprit is actually the substance that feeds the algae: excess phosphorous that is produced by human activities like fertilizer runoff and leaky septic systems.

To manage eutrophication, one must know whether the affected body of water resides in a eutrophic stable state, or if its state is a long transient. The second case mimics stability because it can last a long time but is sustained by another source of phosphorous in the lakebed sediments. According to Tessa Francis, an ecologist at the University of Washington Puget Sound Institute, the wrong management choice has major consequences in terms of costs and trade-offs.

Figure 1. A satellite photo of Lake Erie in North America (taken in October 2011) shows toxic algae in green. The spread pattern is due primarily to phosphorus runoff from agriculture. Public domain image.

Mathematical biologists, field ecologists, and other researchers study the dynamics of ecological long transients to understand how to best distinguish them from stable or oscillatory states. This theme was the topic of a session at the virtual 2021 American Association for the Advancement of Science Annual Meeting this February.

“Transient dynamics are everything that you see when a system is away from equilibrium,” Karen Abbott, a theoretical ecologist at Case Western Reserve University, said. “It’s either in the process of approaching equilibrium, or permanently disturbed in a way that will never allow [the system] to fully approach equilibrium.”

The Ghost in the Saddle

Dynamical systems research has historically focused on the identification of asymptotic behaviors, including stable mathematical attractors, regular oscillations, and other limit cycles. But as Abbott asserts, transients (rather than stability) may be the norm in real ecological systems. “If we’re going to use observations of the natural world to discover the rules through which ecosystems are structured, we need to understand that what we’re looking at is probably not in equilibrium,” she said. Between internal fluctuations and external influences like weather or human intervention, ecological systems balance on metastable mathematical ledges that are far from the clean asymptotic behaviors that dynamicists prefer.

To further complicate the situation, these transients can last a long time and mimic asymptotic conditions. In fact, it is difficult for researchers to precisely define “long.” “In ecological applications, a system usually has a certain characteristic time,” Sergei Petrovskii of the University of Leicester said. This timescale can encompass the life span of an individual or multiple generations of a species in a habitat. “A common-sense approach is to call ‘long’ anything that will substantially exceed this characteristic time by, say, an order of magnitude,” he continued. “As mathematicians, we tend to call a transient ‘long’ when it can be made infinitely long if we play with a certain parameter.”

Petrovskii is concerned with the classification of the mathematical behavior of long transients. “Ghost attractors,” which are asymptotic steady-state solutions under slightly different values of the controlling parameter, are one important class (see Figure 2a). But when the ghost is present, the system is attracted into the long transient and lingers there for an extended period of time before moving on to a real equilibrium state [3].

Figure 2. Visualization of two major mathematical types of long transients. 2a. A “ghost attractor” would be a stable equilibrium point under different conditions, but the system eventually moves away from the ghost because the mathematical shape of the transient contains an escape. 2b. A “crawl-by” is a saddle point in the dynamical space, meaning that it is attractive in one direction and repulsive in the other. Because it is nearly flat, it mimics stable equilibrium. Image courtesy of [3]. Reprinted with permission from AAAS.

A second case occurs when the system contains a saddle point: a state that resembles an attractor in one direction but is not at stable equilibrium. If the slope away from the saddle is shallow, the system lingers near the unstable equilibrium, mimicking stability and producing the long transient behavior known as a crawl-by (see Figure 2b). This behavior occurs in systems with multiple timescales—such as interacting species with different life spans—or large stochastic effects.

Abbott is particularly interested in stochasticity’s role in long transients. In the ubiquitous Lotka-Volterra model for competing species, for instance, the coupled equations are deterministic and cannot account for many important ecological parameters.

“Stochastic disturbances jostle the system to other combinations of population sizes or other states,” Abbott said. “Lots of different species are ectotherms, so they respond directly to variations in temperature. Things like feeding rates and birth rates—very fundamental demographic properties—are going to respond to temperature.”

To demonstrate this phenomenon, Abbott and her colleagues began with the Lotka-Volterra model for the population densities of two species \((u_1, u_2)\) in the form

\[\frac{du_1}{dt}=r_1u_1(1-u_1-\rho\alpha u_2)+\varepsilon_1\]

\[\frac{du_2}{dt}=r_2u_2(1-u_2-\alpha u_1)+\varepsilon_2.\]

Here, \(r_k\) are the natural growth rates, \(\alpha\) and \(\rho\) are interaction parameters, and \(\varepsilon_k\) is Gaussian noise with a mean of 0 and variance of \(\sigma\) [1]. The group considered a case wherein species 1 is native and species 2 is invasive, so that \(r_2>r_1\) and \(\rho>1\) to give the invaders a slight advantage. In the absence of noise, this model has an equilibrium point at \((u_1,u_2)=(1,0)\) that represents a high density of the native species and (local) extinction of the invasive population.

If the relative amount of competition \(\rho\) is fixed but \(\alpha\) varies, the system has two bifurcations in the absence of noise. When \(\alpha<1/\rho\), neither species has a competitive edge and they coexist; when \(1/\rho<\alpha<1\), the system favors the invaders at the expense of the endemic species; and when \(\alpha>1\), either species can “win” the competition, thus producing a bistable state. Including noise in the equations changes the equilibrium state to a saddle point, so that local extinction of the invasive species is a long transient and stochasticity allows the invaders to return. This incidence has a profound effect on potential management decisions, since adding native species alone is not enough; active removal of invasive species or other mitigation options must also occur [2].

“We tend to monitor ecosystems on relatively short timescales and assume that if what we’re observing looks like equilibrium, it therefore is at equilibrium,” Francis said. “If you don’t have long-term monitoring of ecosystems, then you’re unable to see the sort of long-term dynamics that play out. This may affect your ability to adequately manage the system.”

Management and Modeling

Although additive white noise is of course not a good model for realistic ecological systems, it has the virtue of simplicity. After all (to paraphrase the first sentence of Leo Tolstoy’s Anna Karenina), Gaussian noise is all alike, but each non-Gaussian noise type is non-Gaussian in its own way.

“The longer-term goal is to become more sophisticated and begin asking, ‘How do we think these systems are really being perturbed in nature?’” Abbott said. “If a storm comes through and kills a lot of individuals, do we want to represent that as a random spike in the death rate? Or do we want to randomly perturb the whole population density downward? These are two ways to represent the same thing, but they don’t always yield the same outcome mathematically.”

Another challenge stems from the fact that real ecosystems are more complex than even the most sophisticated dynamical models. “Cases are few and far between where you can actually match a model of a long transient to a real-world example,” Francis said. “In some cases, we can use these models to understand the controlling parameters on long transients.”

Instead, much of modeling’s value lies in the qualitative understanding of system behaviors. Without this mathematical insight, it is difficult for researchers to adjust intervention strategies for invasive species or identify why eutrophic lakes resist mitigation.

Current work on long transients mirrors early research on nonlinear dynamics, when scientists devoted much effort to identifying, cataloging, and developing techniques to better comprehend complex systems. Because ecosystems are intrinsically out of equilibrium, we must adjust our thinking about stability and timescales to match the world we live in — and change how we interact with it.

Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.