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Chesson's Coexistence Theory Reply Ecology, 101(11), 2020, e03140 © 2020 The Authors. Ecology published by Wiley Periodicals LLC on behalf of Ecological Society of America This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. Chesson’s coexistence theory: reply standardized environmental and interaction parameters Ej and Cj via 1,2,4 3 GYoRGY¨ BARABaS´ AND RAFAEL D’ANDREA E ¼ ð ∗Þ C ¼ ð ∗ Þ ð Þ j rj E j,C j , j rj E j ,C j , 1 Citation: Barabas,´ G., and R. D’Andrea. 2020. Chesson’s coex- ∗ ∗ “ ” istence theory: reply. Ecology 101(11):e03140. 10.1002/ecy.3140 where E j and C j are equilibrium values such that ð ∗ ∗Þ¼ C rj E j ,C j 0 (Chesson defines j with an extra negative sign, conforming to the situation where C measures a ’ j Chesson s (2019) comment on our review of modern competitive reduction in growth rates). In our review, we ´ coexistence theory (Barabas et al. 2018) focuses on per- did mention these definitions, though only as a refer- ceived mistakes and provides a re-derivation of key ence. Instead, we defined the standardized parameters as results from his own perspective. The criticisms concern quadratic approximations to these formulas (Barabas´ three main issues: the definition of standardized environ- et al. 2018:279): mental and interaction parameters; the definition of comparison quotients and the meaning of their non- E ¼ α À ∗ þ 1αðÞ2 ð À ∗Þ2 uniqueness; and the status of the theory when there are j j E j E j j E j E j , 2 at least as many limiting factors as species in the system. 1 ðÞ C ¼ β C À C∗ þ β 2 ðC À C∗Þ2 ð2Þ He makes many important observations, and the provi- j j j j 2 j j j sion of his own derivation of the theory gives a unique perspective on what elements of the theory are deemed where the αs and βs are Taylor coefficients. Chesson important and why. Additionally, Chesson (2019) men- (2019) mentions two problems with our definitions. tions in passing several elements of our review that he First, they are less accurate than those of Eq. 1. Second, believes are not handled well. However, we argue that they do not yield critical understanding of the underly- the differences between Chesson’s (2019) account of the ing theory. theory and ours are overstated, turning out to be differ- Eq. 1 is indeed more accurate than Eq. 2, as the ences of emphasis and interpretation, rather than funda- latter is a quadratic approximation to the former. mental mistakes. We therefore take this opportunity to Fig. 1 of Chesson (2019) presents a model example, clarify our point of view, and to synthesize Chesson’s showing that, for large environmental variation, the perspective with ours for an improved outlook on the difference in accuracy is especially pronounced in theory. favor of Eq. 1. While this is true, one must keep in mind that the theory as a whole is based on a small fluctuation approximation. This assumption pervades DEFINITIONS OF THE STANDARDIZED ENVIRONMENTAL AND the development of the theory; e.g., the quadratic INTERACTION PARAMETERS approximation of the standardized interaction parame- After choosing environmental and interaction param- ters Cj in the limiting factors rests on this (Eq. 11 in eters Ej and Cj (both may depend on time), the theory Chesson [2019]; Eq. 9 in Barabas´ et al. [2018]), as starts by writing the per capita growth rates rj as their does the approximation EjCj≈covðEjCjÞ, which = function: rj gj(Ej,Cj). Based on this parameterization neglects the EjCj term. For small environmental vari- of the growth rates, Chesson (1994, 2019) defines the ability, the difference between Eqs. 1 and 2 diminishes. From the point of view of deriving the equations of Manuscript received 21 August 2019; revised 10 June 2020; the theory or applying them to particular ecological accepted 29 June 2020. Corresponding Editor: Sebastian J. models, and making sure that they are accurate to Schreiber. 1Division of Theoretical Biology, Department of IFM, quadratic order, it therefore does not matter which set Linkoping¨ University, Linkoping,¨ Sweden of definitions one uses, and the difference in accuracy 2MTA-ELTE Theoretical Biology and Evolutionary Ecology between Eqs. 1 and 2 is beside the point. To go Research Group, Budapest, Hungary 3 beyond small fluctuation approximations, one way to Department of Plant Biology, University of Illinois at go is to estimate model quantities via simulations in Urbana-Champaign, 505 South Goodwin Avenue, Urbana, Illinois 61801 USA the first place (Ellner et al. 2016, 2019, Chesson 4 E-mail: [email protected] 2019). Article e03140; page 1 Article e03140; page 2 REPLY Ecology, Vol. 101, No. 11 Instead, the goal of our alternate approach in Barabas´ notation or in terminology. We of course did not use the et al. (2018) was to offer a new angle to view the struc- terms “comparison quotient” and “scaling factor” as ture of the theory. This takes us to the other criticism of suggested by Chesson (2018, 2019), as our review was Eq. 2 by Chesson (2019), that it does not yield the same published prior to these works. That said, we have noth- insight into the theory as Eq. 1. Clearly, different ways ing against adopting Chesson’s suggested terminology of arriving at a result emphasize different aspects of the as described above. During review, it was suggested that same underlying concepts. Ours emphasizes how a the di should then also receive their own name. We sug- straightforward quadratic expansion recovers all salient gest calling the di the “(linear) cancellation coefficients,” quantities of the theory. In the end, we would argue that to emphasize their role in eliminating the linear terms in having both developments is better than having just one the limiting factors from the invader-resident compari- or the other. Out of a pool of possible parameterizations, son. it is best to use the one that yields the most insight into Apart from terminology, another possible reason for the problem at hand. Standard parameterizations facili- the impression that we have conflated comparison quo- tate the comparison of different models and results. That tients and scaling factors is the fact that the scaling fac- said, we see no reason to rely on a single one exclusively. tors happen to be the same as the “canonical” choice of invader comparison quotients in the presence of a single limiting factor, because then di = 1/ϕi for any invader THE COMPARISON QUOTIENTS species i (Barabas´ et al. 2018: Eq. 22). That scaling fac- In relation to the comparison quotients qis that are tors and comparison quotients are related in this way is used in invader-resident comparisons (denoted ds/di in itself a noteworthy result. It is an open question whether Barabas´ et al. 2018:283), Chesson (2019) objects that we this should be treated as an opportunity to unify the have confused them with the conceptually different scal- concepts of scaling factors and comparison quotients, or ing factors (which are used, among other things, to if it is better for future developments of the theory to decompose invasion rates into stabilizing and equalizing still treat them as separate entities, despite the fact that contributions), and takes issue with our claim that using they are related. partial derivatives does not properly define the qis, whose Chesson (2019) also points out that we unduly criti- non-uniqueness we seemingly deemed a flaw of the cized the definition of comparison quotients in terms of theory. the partial derivative ∂Ci=∂Cs evaluated at Cs ¼ 0 (Ches- We have not in fact confused comparison quotients son 1994, Chesson 2019: Eq. 4). This was not our inten- with scaling factors, but this is obfuscated by conflicting tion, though we do see that it was possible to read our terminology in the literature. Historically, the qis had not text that way (last paragraph of Partitioning the sum of been assigned a particular name until Ellner et al. (2016) invader and resident growth rates). We regret if we were called them scaling factors. It was only in Chesson interpreted by readers as claiming that this definition is (2019) that the term “comparison quotient” was offered, faulty, and so we would like to emphasize here that this to differentiate them from another set of quantities is not the case. As pointed out by Chesson (2019), the called “scaling factors,” which were independently intro- derivative works just as well as our method, as long as duced by Chesson and Huntly (1997). Their purpose the invader’s Ci is expressible as a function of resident was to cancel linear terms in the limiting factors (Ches- Cs values, which is another way of saying that the num- son and Huntly 1997: Appendix C), and they therefore ber of independent limiting factors should be lower than closely correspond to our dj in Barabas´ et al. (2018); in the number of species competing for them. The deriva- fact, as we pointed out in our review, our inspiration for tive will not be unique unless the number of limiting fac- these factors was Chesson and Huntly (1997) in the first tors is one less than the number of species (in which place. Without being explicitly named, scaling factors case, the di are unique up to a common multiplicative underwent further evolution (e.g., Chesson 2003, 2008, constant that cancels from qis = ds/di).
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