COMMENTARY Equation-free modeling unravels the behavior of complex ecological systems Donald L. DeAngelisa,b,1 and Simeon Yurekb aSoutheast Ecological Science Center, US Geological Survey, Gainesville, FL 32653; and bBiology Department, University of Miami, Miami, FL 33124

Ye et al. (1) address a critical problem con- models in ecology have not passed the crite- fronting the management of natural ecosys- rion of predictive ability (3, 4). There are tems: How can we make forecasts of possible many reasons for this, one being the highly future changes in populations to help guide nonlinear nature of ecological interactions. management actions? This problem is espe- This has led to arguments over whether the cially acute for marine and anadromous fisher- “right” models are being used, but also to ies, where the large interannual fluctuations of broad opinion that, unlike in physics, there populations, arising from complex nonlinear are no right models to describe the dynamics interactions among species and with varying of ecological systems, and that the best that environmental factors, have defied prediction canbedoneistofindmodelsthatareatleast over even short time scales. The empirical good approximations for the phenomena dynamic modeling (EDM) described in Ye they are trying to describe. It is common et al.’s report, the latest in a series of papers in introductions of ecological modeling to by Sugihara and his colleagues, offers a find descriptions of the “modeling cycle,” in promising quantitative approach to building which a question is formulated, hypotheses models using time series to successfully pro- are made, a model structure is chosen in ject dynamics into the future. the form of variables and equations, the With the term “equation-free” in the arti- equations are parameterized according to best cle title, Ye et al. (1) are suggesting broader information, and the model is analyzed and implications of their approach, considering compared with patterns in nature. The cycle the centrality of equations in modern science. can be repeated again and again to obtain the From the 1700s on, nature has been increas- best fit or validation by data. Although this ingly described by mathematical equations, methodology may work for some cases, it has with differential or difference equations become clear that there are limits on the ac- forming the basic framework for describing curacy of models applied to systems, with dynamics. The use of mathematical equations many variables interacting nonlinearly to cre- for ecological systems came much later, pio- ate complex dynamics (e.g., marine ecosys- neered by Lotka and Volterra, who showed tems). Complete parameterization of models that population cycles might be described in of such systems is nearly impossible. Further- terms of simple coupled nonlinear differential more, the nonlinearities of ecological systems equations. It took decades for Lotka–Volterra- cause models to be so sensitive to structure Fig. 1. Two-species competition model of Sugihara type models to become established, but the and parameters that even the most thoroughly et al. (10) showing two populations going in and out of development of appropriate differential equa- and carefully developed model can hardly be mirage correlations at different periods of the simulation. tions is now routine in modeling ecological expected to be predictive (5–7). dynamics. There is no question that the in- The message of Ye et al.’sreport(1)isthat S jection of mathematical equations, by forcing there are ways to make predictive forecasts to population size i through the relationship R = S r – αS “clarity and precision into conjecture” (2), has that do not rely on specifying equations at all, t t exp( t), so that population size S t + led to increased understanding of population thus avoiding the problems outlined above. can be projected ahead to t+1 at time 1. and community dynamics. As in science One can say generally that “the study of nat- Given a set of time series data from a fishery, in general, in ecology equations are a key ural systems begins and ends with the spec- the modeler attempts to find values of r and α method of communication and of framing ification of observables describing such a from the best fit for the equation. Most mod- hypotheses. These equations serve as compact system, and a characterization of the manner els are more complex, linking a population’s representations of an enormous amount of in which these observations are linked” (8). empirical data and can be analyzed by the In traditional models, equations are formu- Author contributions: D.L.D. and S.Y. wrote the paper. powerful methods of mathematics. lated with certain functional forms character- The authors declare no conflict of interest. However, mathematics has not had the izing linkages of variables. For example, the See companion article on page E1569. “ ” unreasonable effectiveness in ecology that it Ricker difference equation model links the 1To whom correspondence should be addressed. Email: don_ has had in physics. Critics point out that number of recruits to a fish population, Rt, [email protected].

3856–3857 | PNAS | March 31, 2015 | vol. 112 | no. 13 www.pnas.org/cgi/doi/10.1073/pnas.1503154112 Downloaded by guest on September 26, 2021 change through time to its own current size, (or embedded) in phase space. An ecological no equations is better than assuming some COMMENTARY the sizes of other populations, and relevant time series plotted in this way will usually traditional equations. abiotic conditions. The functional forms de- collapse to a geometric shape of dimensions The broader message of the Ye et al. (1) scribing these linkages can in principle be less than the number of relevant variables report is that science may be moving into estimated based on measured correlations be- would suggest. Often, three dimensions are a period where equations do not play the tween variables over time. Although such cor- sufficient to view the trajectories forming an central role in describing dynamic systems that relations can in principle be extracted from “attractor,” a criterion for which is that the they have played in the last 300 years. This is data, meta-analyses on marine fisheries show trajectories do not cross. What is important largely the result of the rapidly increasing that apparent correlations between populations is that this preserves the essential features power of computers. Another sign of this trend and environmental variables have not stood up of the dynamics of the N variables. Sugihara is that, from a very different direction, two re- when additional data were analyzed (9). This is lated classes of equation-free models have an important part of the argument of Ye et al. The broader message of emerged as computational power increased. (1) in favor of equation-free modeling. To the Ye et al. report is The methodology of individual- or agent-based illustrate this point, Sugihara et al. (10) gen- modeling does not specify equations at the level erated time series from a nonlinear model of that science may be of populations and communities, and instead two marine populations fluctuating through moving into a period simulates the individual organisms and their time, showing that what appear to be tightly where equations do not interactions with each other and the environ- positive correlations, over a period, suddenly ment (5). A related approach, cellular automata switch to no correlation or even a negative play the central role in models, is based on local laws determining correlation for another period (Fig. 1). Pa- describing dynamic transitions of spatial cells between different rameterizing equations based on these “mi- states. Although it would not seem intuitively rage correlations” from one part of the time systems that they have that cellular automata models can describe bi- series would produce models that make false played in the last ological systems, Wolfram (13), through his predictions. 300 years. Principle of Computational Equivalence, has The mirage correlations reflect deeper proposed that it can form a universal founda- causes involving nonlinear interactions within and his colleagues (10) made important tion for all phenomena of nature. Both indi- the system. Rather than declaring analysis additions to this embedding theorem of vidual- or agent-based models and cellular impossible, Ye et al. (1) show that there is an Takens; in particular, showing that it is automata models are bottom-up approaches alternative that starts with the time series, possible to create a composite time series for constructing models of phenomena, and without a presupposed set of equations. In from ecologically similar species in the although deeply different conceptually from this equation-free or nonparametric approach, same geographic region to form a longer EDM, are similar in not specifying equations no equation is assumed that needs to be pa- time series, and thus create a better model. for population level variables. rameterized. Instead, the dataset is used to Ye et al. (1) apply this approach to sockeye The EDM approach demonstrates progress build a model independent of equations. The salmon stocks in the Fraser River. Dividing toward solving important but seemingly in- basic idea is related to a question asked a time series of nine salmon stocks into tractable problems of ecology; predicting be- by Schaffer and Kot (11): “Do strange attrac- “library data” used to calibrate, and “pre- havior of complex nonlinear dynamic systems tors govern ecological systems?” The authors dict data” used to test the model, the with limited information. This approach also showed that even time series that appear to be authors show that their results are sub- suggests that, with high-performance com- completely random could be following an un- stantially better for most stocks than the putation, the study of dynamic systems is derlying deterministic dynamic. The time parametric models, Ricker and extended moving away from formulation and param- map of any single population is an emergent Ricker. This approach allows projection eterization of equations and toward letting property of interactions with other popula- into the future. One might quibble that it is data directly determine the model. Because of tions and abiotic conditions. Understanding well known that the Ricker is poor at de- the central role equations hold in science, it the dynamics would require plotting all scribing population dynamics, so it is an also raises questions: How will these changes N-interacting variables in an N-dimensional easy target, but the comparison clearly affect the way scientists communicate, the phase space, something that is practically im- demonstrates that, in this case, assuming way they understand, and the way they think? possible for any real system because of the lack of data on most species in a community. However, if there is a sufficiently long time 1 Ye H, et al. (2015) Equation-free mechanistic ecosystem 7 Wood SN, Thomas MB (1999) Super-sensitivity to structure in – series for one population, a theorem of Takens forecasting using empirical dynamic modeling. Proc Natl Acad Sci biological models. Proc Biol Sci (B) 266(1419):565 570. USA 112:E1569–E1576. 8 Casti JL (1992) Reality Rules: I. Picturing the World in (12) solves this problem. Takens showed that 2 May RM (2004) Uses and abuses of mathematics in biology. Mathematics. The Fundamentals (Wiley-InterScience, New York). an N-dimensional phase portrait having the Science 303(5659):790–793. 9 Myers RA (1998) When do environment-recruitment correlations work? Rev Fish Biol Fish 8(3):285–305. 3 Peters RH (1991) A Critique for Ecology (Cambridge Univ Press, same dynamic properties as the portrait 10 Sugihara G, et al. (2012) Detecting causality in complex Cambridge, UK). N ecosystems. Science 338(6106):496–500. from the -independent variables could 4 Pilkey OH, Pilkey-Jarvis L (2007) Useless Arithmetic: Why 11 Schaffer WM, Kot M (1985) Do strange attractors govern Environmental Scientists Can’t Predict the Future (Columbia Univ be constructed by plotting the population ecological systems? Bioscience 35(6):342–350. size x(t)versusx(t+T), vs. x(t+2T)...vs. Press, New York). 12 Takens F (1981) Detecting strange attractors in turbulence. x(t+(M − 1)T), where T is a time delay. 5 Grimm V, Railsback SF (2005) Individual-Based Modeling and Dynamic Systems and Turbulence, eds Rand DA, Young LS (Springer, T Ecology (Princeton Univ Press, Princeton, NJ). New York), pp 366–381. Using an appropriate value of ,asubset 6 Yodzis P (1988) The indeterminacy of ecological interactions as 13 Wolfram S (2002) A New Kind of Science (Wolfram Media, Inc., of these time-lag variables could be plotted perceived through perturbation experiments. Ecology 69(2):508–515. Champaign, IL).

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