Nonlinearity and chaos in ecological dynamics revisited

Historically, both experimental and theoretical ecologists have sought to emulate the development of early theory in the physical sciences: the ideal that a few simple equations may accurately predict the complex movement of celestial bodies or interactions among molecules in mixing gasses. In the environmental sciences, such simple clockworks have rarely been found, and rather than predictable stable or recurring patterns, erratic patterns abound. The discovery in the late 1970s through mid-1980s of certain ecological models—such as the Ricker or discrete logistic maps—suggesting erratic fluctuations through dynamic chaos caused what cautionaries may characterize as ecology’s period of “rational exuberance” with respect to hoping that a small set of mathematical equations may explain the erratic dynamics of real-world ecological communities. Upon much discussion, the field as a whole grew skeptical of this idea. In PNAS, Benincà et al. (1) present evidence that the erratic fluctuations in an intertidal rock-pool ecosystem are caused by competitive interactions that cause chaotic dynamics. This important study sheds new light on a number of threads of discussion in contemporary ecology related to variability and predictability in ecology.

Dynamical Systems Ecology

To understand the significance of the study, it is instructive to consider it through the lens of (a very abridged) history of “dynamical systems ecology” during the last 30 y. After May’s (2) analysis of chaotic dynamics in simple ecological models in the mid-1970s and parallel analyses of epidemiological models in the early 1980s (3), there followed a period of some optimism that ecological variability may be more explicable than previously expected (4, 5). Some beautiful but very stylized laboratory experiments subsequently showed that a chaotic cause of variability is possible in tightly controlled settings (6). However, the relevance of this concept in the field setting was questioned for a number of reasons. First, the “determinism” in the deterministic chaos appeared to presuppose a negligible role of stochasticity, and all ecologists agree that random forces of nature (lightning strikes, droughts, hurricanes, and so forth), although absent in the laboratory, will repeatedly perturb population growth in the field. Subsequent to this followed an important theoretical discussion of the meaning of “noisy chaos” that resolved how the notion of sensitive dependence on initial conditions is relevant also in nonlinear systems with a significant

Benincà et al. present evidence that the erratic fluctuations in an intertidal rock-pool ecosystem are caused by competitive interactions that cause chaotic dynamics.

degree of stochasticity (7, 8). The second question related to the relevance of a theory postulating “low dimensional” dynamics for ecology. Most ecological communities comprise dozens or hundreds of populations, many interacting weakly and some interacting strongly. Subsequent to this worry followed an important discussion of how dynamics may collapse on a low “ecological dimension” even in complex communities (9). The third question related to the pervasiveness of sufficiently tight and sufficiently nonlinear interactions. The models that predict irregular fluctuations only do so if interactions are tightly linked and interactions are highly nonlinear. For ecology, that means that density-dependent feedbacks have to be very strong and strongly overcompensatory. Because interspecific competitive interactions will tend to be compensatory in nature, the most plausible candidates for chaos outside of the laboratory would be in systems with tightly linked, highly specialized consumer–resource interactions, such as lemmings and weasels (10), or the predator/prey-like interactions among susceptible and infected children caused by immunizing infections such as measles (11).

Measles dynamics results from a nonlinear consumer–resource interaction among susceptible, infected, and removed hosts. Historically, irregular periodicity in incidence could thus be taken as a prime candidate of consumer/resource-induced chaos in ecology (11). The erratic fluctuations following World War II, however, turned out to be not caused by chaos but instead by nonstationarities in human birth rates, leading to disease dynamics tracking an ever-changing attractor (12). In 2001, Hastings (13) reviewed the importance of transients in ecology, highlighting how nonstationarities and transients will generally erode any footprint of chaos. Thus, on balance, when Benincà et al.’s (1) monitoring study was initiated 20 y ago, the likelihood that nonlinear dynamic systems theory may play an important role in explaining the rock-pool’s future erratic ecological fluctuations should have seemed remote. For one, it was not a tightly controlled laboratory system but an open-field system; moreover, it was not a consumer–resource system but a community of interspecific competitors. The only plausible aspects were its relatively low dimensionality of only four functional compartments—barnacles, crusty algae, mussels, and free space—and its pristine environment that appeared to be largely unchanging except for its predictable seasonal cycles.

Seasonality and Chaos

Benincà et al. (1), however, show that the predictable seasonal cycle is in fact a key driver that forces the dynamic toward the erratic. In the absence of seasonality, their rock-pool dynamic is predicted to oscillate toward a stable equilibrium with a period of around 1.5 y. However, the annual period of the seasonal cycle forces chaotic fluctuations onto the inherent cyclical clockwork. The emergence of complex dynamics from seasonal forcing has previously been dissected for models of predator–prey interactions (14) and infectious diseases (11). However, in the case of the former, empirical evidence appears sometimes equivocal (15), and in the case of the latter, the seasonally forced nonlinearities suggestive of chaos appear to lead to such violent fluctuations that frequent extinctions break the nonlinear feedbacks (16). From a dynamical point of view, it almost appears as if there is a trade-off: the magnitude of seasonality that permits sustained chains of transmission may be insufficient to drive epidemics into the chaotic regime.

In light of previous theory, it is fascinating that the discovery of a likely case of chaotic dynamics in nature is not in a consumer–resource system but rather in a rock–paper–scissors-like competitive system (17). The successional dynamics share some consumer–resource-like properties, though, in the sense that each species cyclically generates space for recruitment—the ultimate resource in the intertidal—for the next species. Barnacles need bare rock but as they grow they inhibit their own recruitment and facilitate the growth of the algae, which in turn allow the settlement of mussels, which upon death and detachment leave barren pools behind ready for another chaotic, successional cycle.

Theoretical ecology has come a long way from the 1970s and 1980s when models were generally touted as useful for highlighting concepts and providing proof-of-concept of novel ideas. In 1985, two of the most important contributors to modern mathematical ecology, Bill Schaffer and Mark Kot, published a seminal paper that used a simple ecological model to suggest a potential role for chaos in infectious disease ecology (18). After years of working on a vast array of biological problems, Schaffer and Kot soberly summarized their expectation for a role of “simple equations” in ecology (18): “Put another way, the variables studied in nature are generally embedded in more complex systems. As a practical matter, it is unlikely that population dynamicists will ever be able to write down the complete governing equations for any natural system.” Benincà et al. (1) demonstrate that given high-resolution, long-term ecological monitoring data—at least in one select field system—four simple equations are enough to characterize the erratic wiring diagram of nature and how the changing seasons shift the rock-pool system in and out of the chaotic regime. It is a beautiful illustration of just how far the discipline has moved.