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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 ∗ þ 1αðÞ2 ð ∗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 C∗ þ β 2 ðC 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).

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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). Otherwise, the Yuan and Chesson 2015) and became a tool for nondi- invader Ci can be expressed as a function of the resident mensionalizing invasion growth rates and decomposing Cs in multiple ways; Chesson (2019) explains very clearly them into stabilizing and equalizing contributions. why this is a natural consequence of the theory. In the Finally, Chesson (2018) gave scaling factors their official end, both our method and that of Chesson (1994, 2019) name, notation (as bi), and a general definition. yield the same results, and both can be non-unique for In Barabas´ et al. (2018), we consistently denote what the exact same reason. Chesson would call the comparison quotients qis by ds/di, We opted for our approach because it reduces the and call the individual di scaling factors (in line with problem of finding the comparison quotients to the solu- Chesson and Huntly 1997). In turn, the bi of Chesson tion of a system of linear algebraic equations. This felt (2018), which he calls scaling factors, we denote by ϕi more direct than computing ∂Ci=∂Cs, which requires the and do not give them any particular name. In no place implicit function theorem to even interpret correctly, do we refer to one as if it was the other, either in and potentially obscure methods such as generalized November 2020 REPLY Article e03140; page 3 inverses to actually evaluate. As an added benefit, the having to cancel it out, so the number of limiting factors fact that a set of linear equations may have a non-unique to be eliminated is reduced from S to S − 1. This is now solution is well known, and we hope this helps shed new less than the number of species, so either the method of light on the origins of the non-uniqueness of the com- Chesson (2019: Eq. 14) or our method of the cancella- parison quotients (this is why, despite the criticisms, tion coefficients dj (Barabas´ et al. 2018, Eq. 21) may be Chesson (2019) acknowledges that our method has applied to cancel the remaining S − 1 factors. All in all, merit). We reiterate that the purpose of our approach is here indeed comparison quotients can be obtained even not to dismiss that of Chesson (1994, 2019). Both are though there are as many limiting factors as species. On equally adequate and lead to the same result, and it is up the downside, this method requires a separation of time to the user to decide which formulation they prefer. scales which may not always hold. Relaxing this require- ment to obtain comparison quotients when L < S is still avenue for future research (Barabas´ et al. 2018: THE NUMBER OF LIMITING FACTORS REACHING OR 295–296). EXCEEDING THE NUMBER OF SPECIES Our claim that Chesson’s framework loses its advan- Chesson (2019) argues that we underestimate the util- tages compared to other methods when there are as ity of the theory when there are at least as many limiting many or more limiting factors than species was based on factors L as species S. This is in response to our claim the fact that the theory is rooted in invasion analysis. that the theory “does not offer any advantages, and may What distinguishes it from “standard” textbook invasion even work worse than other methods, if there are as analysis is the meaningful partitioning of invasion ´ many or more limiting factors as species” (Barabas et al. growth rates into the Δρi, ΔNi, ΔIi, and Δκi coexistence- 2018:300). To be fair, we did follow that sentence up by affecting terms via the comparison quotients (Barabas´ saying that this limitation, along with others, may not be et al. 2018: Eqs. 20 and 47). However, these quotients do fundamental to the theory and could be amended by not exist (or are all zero, if one uses our di) when L ≥ S, future work. In light of this, it is especially helpful that unless we are in the realm of the example discussed Chesson (2019) cites example applications with as many above. Without the comparison quotients, the method or more limiting factors than species, with a worked-out effectively reduces to standard invasion analysis, and example in an Appendix. can do only as much. That said, there is more to the the- The example of Chesson’s Appendix has two species ory than the comparison quotients, and other aspects in a MacArthur-style consumer–resource model may still be useful for analysis, e.g., the quadratic expan- (MacArthur 1970, Chesson 1990). In this case, due to sion of growth rates. Furthermore, even when L ≥ S, the assumption of a timescale separation between one can choose a subset of up to S − 1 limiting factors resources and consumers (the former are much faster to eliminate explicitly, leaving the others as contributions ´ than the latter), the limiting factors become the species’ to the Δρi term (Barabas et al. 2018:283). Though this population densities themselves, and the competitive leaves the Δρi term unevaluated, one still ends up with a ¼ ∑S σ factors are weighted sums of them: C j k¼1 jkEkNk, meaningful partitioning of the invasion growth rates. where S is the number of species, σjk is the interaction Therefore, while the theory may not offer benefits over coefficient between species k and j based on the degree others in performing calculations, it still has strong of overlap in their resource use, Ek is the kth environ- heuristic power and can facilitate improved understand- mental parameter, and Nk is the density of species k. ing even when L ≥ S. We take it that this is what Ches- Then, one can indeed obtain the comparison quotients son meant by pointing out that the theory is not using ∂Ci=∂Cs evaluated at Cs ¼ 0 (Chesson 2019: primarily methodological, but a theory of coexistence in Eq. 14). The reason this works is that the invader spe- ecological communities: an important and valid point. i ¼ cies is absent from the community: Ni 0, which also means that the corresponding limiting factor is absent. FURTHER POINTS In the two-species case for instance, letting i be the inva- i ¼ σ der and s the resident, we have Ci isEsNs and Chesson (2019) lists a number of additional issues that i ¼ σ “ ” Ci ssEsNs, where Ns is the monoculture equilibrium were not handled well by our review. As these were not density of the resident. We can therefore write expounded on, we do not respond to any specific criti- i ¼ σ ¼ σ σ =σ ¼ iσ =σ Ci isEsNs ssEsNs is ss Cs is ss, which is cisms here. Instead, we simply provide our own perspec- i now explicitly a function of Cs . Taking the derivative tive on some of them, those about which we feel we have with respect to this variable yields the comparison quo- something relevant to say. ¼ ∂ i=∂ i ¼ σ =σ tients: qis Ci Cs is ss. More generally, in any S-species community with lim- Species average fitness iting factors proportional to the population densities, putting species i in the invader state will also set the ith In our review, we suggested replacing the term “aver- factor to zero. This factor therefore disappears without age fitness difference” with “competitive advantage.” We Article e03140; page 4 REPLY Ecology, Vol. 101, No. 11 wish to emphasize that this was not because the original unlikely (Song et al. 2019): unless one chooses very care- terminology cannot be justified (Chesson 2018). Rather, fully which parameters to vary and exactly how, both the problem is that subdisciplines such as adaptive quantities will simultaneously change in response to the dynamics (Geritz et al. 1998, Meszena´ 2005, Metz 2012) change in parameters. That said, this is a rapidly devel- operate with a slightly different fitness concept. In adap- oping area, and while perfect independence may be the tive dynamics, “fitness” corresponds to the realized per exception, the same need not be true of quasi-indepen- capita growth rate ri of a rare invader in the environment dence, whereby some mechanism mainly affects either A defined by the resident species (Metz and Geritz 2016: or the fi, leaving the other relatively intact. For instance, Invasion fitness and fitness proxies: a short review). This analysis of a competition-predation tradeoff revealed differs from the “species average fitness” of Chesson that it mostly affected only the fi (Stump and Chesson (2018), which determines the identity of the competi- 2017). Such quasi-independence is an intriguing possibil- tively superior species in the absence of any coexistence- ity and an avenue for future work. affecting mechanisms; and also from the “average fitness difference” (adjusted for the presence of such mecha- The significance of the ratio of intra- to interspecific nisms) that was denoted by ξ in Chesson (2018) and by i competition fi in Barabas´ et al. (2018: Eq. 52). As we have stated in Barabas´ et al. (2018:286–287), we believe coexistence For two-species Lotka-Volterra competition, intra- theory and adaptive dynamics combined could yield a specific competition must exceed interspecific competi- method for studying evolution in variable environments. tion for coexistence. This statement has strong But then one may wish to avoid a potential terminologi- heuristic power: coexistence requires that a species lim- cal clash, where “fitness” means multiple, subtly different its itself more than its competitor. Generalizations of things. While the qualifier “species average” in front of this principle to multiple species and other models all “fitness” ought to clarify the distinction from the general emphasize that stable and robust coexistence requires fitness concept of adaptive dynamics, the term is often that species’ growth rates are, to an extent, regulated not used this way. As we have pointed out in Barabas´ by different factors (Levin 1970, Chesson 2000, et al. (2018:286), various aliases are in use, such as “rela- Meszena´ et al. 2006, Pasztor´ et al. 2016), leading to tive fitness difference,”“fitness difference,” or just “fit- the same effect. ness.” While we do not wish to insist on our own Its heuristic power notwithstanding, a naive applica- terminology, we hope this clarifies why we felt that there tion of the “intra > inter” principle can lead to incorrect is a risk of a terminological clash, and opted for “com- conclusions. We have shown an example in Barabas´ et al. petitive advantage” instead. (2018: Appendix S8) where increasing the ratio of intra- to interspecific competition coefficients does not neces- sarily move the system in the direction of stability. This is Community average stabilization and the size of the due to indirect effects that are absent in a two-species set- coexistence region ting (Barabas´ et al. 2016). For instance, reducing As long as the community average stabilization A and intraspecific competition may allow two species to coexist the average fitness differences fi are independently adjus- that together are able to prevent a third from establishing, table, A will measure the size of the parameter region even though all three possible species pairs would form allowing for coexistence (Yuan and Chesson 2015). We stable communities. In some situations it is even possible have argued in Barabas´ et al. (2018:298) that, in general, to increase the intraspecific competition strength of a sin- A and fi are not independent, thus limiting the use of A gle species and get a non-monotonic effect on community to measure the coexistence region. Since the publication stability, with a stable configuration achieved only for of our review, there have been several new developments. intermediate strengths (Barabas´ et al. 2017). Chesson (2018) proves a theorem stating that in models Despite these examples (which Peter Chesson with sufficiently many parameters, one can always find acknowledges; see Chesson 2018: Eq. 37 and surround- some combination of them such that A and fi are inde- ing text), one may justifiably argue that the heuristic is pendently adjustable, at least for A sufficiently small. nevertheless important, as it usefully guides our thinking While this theorem shows that one can vary A and fi about what kinds of effects can promote coexistence. independently in a broad class of models, it is silent on Naive applications notwithstanding, the general princi- how likely this is to be the case in nature. Should field ple that species must be somewhat independently regu- ecologists expect species differences in traits such as lated for stable coexistence still stands. We therefore beak shape or flower color to contribute only to A, only stress that our purpose is not to dispel the principle, but to fi, or both, confounding whether those differences to warn against adopting an unquestioning acceptance promote coexistence by increasing stabilization or hinder of it, and using it where it does not apply. The examples it by increasing competitive advantages? We argue else- in Barabas´ et al. (2016, 2018) are designed to bring where that the independence of A and fi is in fact highly attention to the non-intuitive role of indirect effects and November 2020 REPLY Article e03140; page 5 show how naive applications can run into trouble, rather Barabas,´ G., M. J. Michalska-Smith, and S. Allesina. 2017. Self- than to question the heuristic itself. regulation and the stability of large ecological networks. Nat- ure Ecology & Evolution 1:1870–1875. Chesson, P. 1990. MacArthur’s consumer-resource model. The- Niche overlap and the ratio of species average fitnesses oretical Population Biology 37:26–38. Chesson, P. 1994. Multispecies competition in variable environ- While this was not explicitly mentioned by Chesson ments. Theoretical Population Biology 45:227–276. (2019), we bring it up here nevertheless, as it is a topic Chesson, P. 2000. Mechanisms of maintenance of species we have indeed not handled well. Unlike with the other diversity. Annual Review of Ecology and Systematics – issues where we see differences in emphasis but no out- 31:343 366. right error, this topic was treated far too superficially in Chesson, P. 2003. Quantifying and testing coexistence mecha- ´ nisms arising from recruitment fluctuations. Theoretical Pop- Barabas et al. (2018:287), without explanation but with ulation Biology 64:345–357. notation that could easily cause confusion. To remediate, Chesson, P. 2008. Quantifying and testing species coexistence we have included an Appendix S1 where we explain the mechanisms. Pages 119–164 in F. Valladares, Camacho, A., origin of the often-used formulas Elosegui, A., Gracia, C., Estrada, M., Senar, J. C., and Gili, J. M., editors. Unity in diversity: Reflections on ecology after rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi κ the legacy of Ramon Margalef. Fundacion´ BBVA, Bilbao, ρ ¼ ajkakj j ¼ bj akjakk ð Þ , κ 3 Spain. ajjakk k bk ajjajk Chesson, P. L. 2018. Updates on mechanisms of maintenance of species diversity. Journal of Ecology 106:1773–1794. ’ ρ κ Chesson, P. L. 2019. Chesson s coexistence theory: Comment. in detail, along with the meaning of and the -ratio, Ecology, e02851. http://dx.doi.org/10.1002/ecy.2851 and their connection to the community average stabiliza- Chesson, P., and N. Huntly. 1997. The roles of harsh and fluctu- tion A and average fitness differences fi. ating conditions in the dynamics of ecological communities. American Naturalist 150:519–553. Ellner, S. P., R. E. Snyder, and P. B. Adler. 2016. How to quan- CONCLUSIONS tify the temporal storage effect using simulations instead of – We have argued that the deviations of Barabas´ et al. math. Ecology Letters 19:1333 1342. ’ Ellner, S. P., R. E. Snyder, P. B. Adler, and G. Hooker. 2019. An (2018) from Chesson s formalism (Chesson 1994, 2018, expanded modern coexistence theory for empirical applica- 2019) are not errors but differences in terminology, empha- tions. Ecology Letters 22:3–18. sis, and perspective, which we believe can help clarify the Geritz, S. A. H., E.´ Kisdi, G. Meszena,´ and J. A. J. Metz. 1998. theory’s strengths and weaknesses. Regardless of which Evolutionary singular strategies and the adaptive growth and description one prefers, the theory has the potential to branching of evolutionary trees. Evolutionary Ecology – continue developing and contributing to our understand- 12:35 57. Kremer, C. T., and C. A. Klausmeier. 2013. Coexistence in a ing of coexistence. Alongside many other possibilities, one variable environment: eco-evolutionary perspectives. Journal untapped direction is its application to problems of adap- of Theoretical Biology 339:14–25. tive dynamics in variable environments. While the adaptive Levin, S. A. 1970. Community equilibria and stability, and an dynamics of two species in cyclic environments has been extension of the competitive exclusion principle. American explored before (Kremer and Klausmeier 2013), coexis- Naturalist 104:413–423. tence theory could help generalize these results to station- MacArthur, R. H. 1970. Species packing and competitive equi- libria for many species. Theoretical Population Biology ary nonperiodic fluctuations and an arbitrary number of 1:1–11. species. Furthermore, the theory will continue to remain Meszena,´ G. 2005. Adaptive dynamics: the continuity argu- useful in guiding our thinking about coexistence in spa- ment. Journal of Evolutionary Biology 18:1182–1185. tiotemporally variable environments. Meszena,´ G., M. Gyllenberg, L. Pasztor,´ and J. A. J. Metz. 2006. Competitive exclusion and limiting similarity: a unified ACKNOWLEDGMENTS theory. Theoretical Population Biology 69:68–87. Metz, J. A. J. 2012. Adaptive dynamics. Page 7–177 in A. Hast- We thank Peter Chesson for the ongoing discussion, and ings and Gross, L. J., editors. Encyclopaedia of theoretical Sebastian Schreiber, Stephen Ellner, and Robert Holt for their ecology. California University Press, Berkeley, California, input. G. Barabas´ acknowledges funding by the Swedish USA. Research Council (grant VR 2017-05245). Metz, J. A. J., and S. A. H. Geritz. 2016. Frequency dependence 3.0: an attempt at codifying the evolutionary ecology perspec- tive. Journal of Mathematical Biology 72:1011–1037. LITERATURE CITED Pasztor,´ L., Z. Botta-Dukat,´ G. Magyar, T. Czar´ an,´ and G. Barabas,´ G., R. D’Andrea, and S. M. Stump. 2018. Chesson’s Meszena.´ 2016. Theory-based ecology: a Darwinian coexistence theory. Ecological Monographs 88:277–303. approach. Oxford University Press, Oxford, UK. Barabas,´ G., M. J. Michalska-Smith, and S. Allesina. 2016. The Song, C., G. Barabas,´ and S. Saavedra. 2019. On the conse- effect of intra- and interspecific competition on coexistence in quences of the interdependence of stabilizing and equalizing multispecies communities. American Naturalist 188:E1–E12. mechanisms. American Naturalist 194:627–639. Article e03140; page 6 REPLY Ecology, Vol. 101, No. 11

Stump, S. M., and P. Chesson. 2017. How optimally foraging Yuan, C., and P. Chesson. 2015. The relative importance of rela- predators promote prey coexistence in a variable environ- tive nonlinearity and the storage effect in the lottery model. ment. Theoretical Population Biology 114:40–58. Theoretical Population Biology 105:39–52.

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