When Rarity Has Costs: Coexistence Under Positive Frequency- Dependence and Environmental Stochasticity

When rarity has costs: coexistence under positive frequency- dependence and environmental stochasticity

1,4 2,3 1 SEBASTIAN J. S CHREIBER, MASATO YAMAMICHI, AND SHARON Y. S TRAUSS 1Department of Evolution and Ecology and Center for Population Biology, University of California, Davis, California 95616 USA 2Hakubi Center for Advanced Research, Kyoto University, Sakyo, Kyoto 606-8501 Japan 3Center for Ecological Research, Kyoto University, Otsu, Shiga 520-2113 Japan

Citation: Schreiber, S. J., M. Yamamichi, and S. Y. Strauss. 2019. When rarity has costs: coexistence under positive frequency-dependence and environmental stochasticity. Ecology 100(7):e02664. 10.1002/ecy.2664

Abstract. Stable coexistence relies on negative frequency-dependence, in which rarer species invading a patch benefit from a lack of conspecific competition experienced by residents. In nature, however, rarity can have costs, resulting in positive frequency-dependence (PFD) particularly when species are rare. Many processes can cause positive frequency-dependence, including a lack of mates, mutualist interactions, and reproductive interference from heterospecifics. When species become rare in the community, positive frequency-dependence creates vulnerability to extinction, if frequencies drop below certain thresholds. For example, environmental fluctuations can drive species to low frequencies where they are then vulnerable to PFD. Here, we analyze deterministic and stochastic mathematical models of two species interacting through both PFD and resource competition in a Chessonian framework. Reproductive success of individuals in these models is reduced by a product of two terms: the reduction in fecundity due to PFD, and the reduction in fecundity due to competition. Consistent with classical coexistence theory, the effect of competition on individual reproductive success exhibits negative frequency-dependence when individuals experience greater intraspecific competition than interspecific competition, i.e., niche overlap is less than one. In the absence of environmental fluctuations, our analysis reveals that (1) a synergistic effect of PFD and niche overlap that hastens exclusion, (2) trade-offs between susceptibility to PFD and maximal fecundity can mediate coexistence, and (3) coexistence, when it occurs, requires that neither species is initially rare. Analysis of the stochastic model highlights that environmental fluctuations, unless perfectly correlated, coupled with PFD ultimately drive one species extinct. Over any given time frame, this extinction risk decreases with the correlation of the demographic responses of the two species to the environmental fluctuations, and increases with the temporal autocorrelation of these fluctuations. For species with overlapping generations, these trends in extinction risk persist despite the strength of the storage effect decreasing with correlated demographic responses and increasing with temporal autocorrelations. These results highlight how the presence of PFD may alter the outcomes predicted by modern coexistence mechanisms. Key words: Allee effects; alternative stable states; coexistence theory; competition; competitive exclusion; environmental stochasticity; invasion success; niche overlap; positive frequency-dependence; reproductive interference; storage effect.

INTRODUCTION Understanding mechanisms of multispecies coexis- functions of competing species exhibit negative fre- tence is one of the central topics in community ecology. quency-dependence (NFD) in which the rare species Stabilizing forces of niche differentiation (intraspecific gains a growth rate advantage. These properties have suppression being stronger than interspecific suppres- been considered primarily in light of resource competi- sion) and fitness differences among species are thought tion between species, with fitness functions that give fit- to lie at the heart of stable coexistence by species (Ches- ness advantages to the rarer species through competitive son, 2000). When stabilizing forces are sufficiently strong release. The rarer species escapes intense conspecific relative to fitness differences, the per-capita growth rate competition, while the more common species strongly suppresses itself. Rare species may, however, experience costs that out- Manuscript received 30 June 2018; revised 29 November weigh the fitness gains of competitive release through a 2018; accepted 3 January 2019; final version received 25 January 2019. Corresponding Editor: Cheryl J. Briggs. variety of mechanisms involving positive density- or fre- 4 E-mail: [email protected] quency-dependence. Costs due to positive density- or

Article e02664; page 1 Article e02664; page 2 SEBASTIAN J. SCHREIBER ET AL. Ecology, Vol. 100, No. 7 frequency-dependence (PDD or PFD, respectively) experiences years where environmental conditions are include a range of Allee effects (reviewed in Taylor and more favorable to it than the other species, (2) the more Hastings 2005, Berec et al. 2007) including reproduction common species are more limited by competition in their (e.g., a lack of mates when rare [Courchamp et al., 1999, favorable years than the rare competitors, and (3) species Schreiber, 2003, Zhou and Zhang, 2006]), survival (e.g., a exhibit buffered growth through unfavorable years. loss of required mutualists [Nunez~ et al., 2009, Chung and Using data-driven models, empirical support for the Rudgers, 2016, Lankau and Keymer, 2016] or a reduction storage effect exists in communities of zooplankton in predator saturation [Schreiber, 2003, Gascoigne and (Caceres, 1997), prairie grasses (Adler et al., 2006), Lipcius, 2004]), and reduced vigor owing to inbreeding desert annual plants (Angert et al., 2009), tropical trees and genetic drift (Fischer et al., 2000, Willi et al., 2005). (Usinowicz et al., 2012), phytoplankton (Ellner et al., Thus, rarity may have costs that outweigh benefits. 2016), sagebrush (Chu and Adler, 2015, Ellner et al., Positive frequency-dependence also arises when a spe- 2016), and nectar yeasts (Letten et al., 2018). Despite cies experiences stronger negative interactions with the empirical support for the storage effect and PFD, heterospecifics than with conspecifics. Mechanisms the simultaneous effects of PFD and environmental underlying PFD include niche construction via allelo- stochasticity on species coexistence are not understood. chemicals (Reinhart et al., 2003), changes in disturbance In particular, it is possible that PFD may disrupt coexis- regime (Crandall and Knight, 2015), or nutrient cycling tence mechanisms, like the storage effect, which rely on (Peay, 2016). Rarity of conspecifics also increases the like- species having the advantage when rare. lihood and costs of reproductive interference, in which Here, we use models to explore how positive frequency- mating attempts by one species have fitness costs on dependence (PFD), environmental stochasticity, and reproductively isolated co-occurring heterospecifics asymmetry in PFD interact to influence the coexistence (Groning€ and Hochkirch, 2008, Burdfield-Steel and Shu- of species. Previous theoretical studies have considered ker, 2011, Tastard et al., 2014, Kyogoku, 2015). Negative species coexistence with resource competition and PFD effects of heterospecific mating pressure can reduce use of by numerical simulations (Waser, 1978, Ribeiro and Spiel- otherwise suitable habitat (Takakura and Fujii, 2015), man, 1986, Feng et al., 1997, Molofsky et al., 2001, may be mediated by gamete interactions that reduce fer- Ruokolainen and Hanski, 2016) as well as with graphical tility (Hettyey et al., 2014), or even cause death. Each of approaches (Levin and Anderson, 1970, Kuno, 1992, these effects can impact population demography (Ting Yoshimura and Clark, 1994, Kishi and Nakazawa, 2013, and Cutter, 2018). The fitness costs due to reproductive Kyogoku and Sota, 2017). Although these studies interference are also often asymmetric between interact- revealed alternative stable states arising due to PFD, they ing species (see examples of magnitude, asymmetry, and are not well integrated into the framework of modern costs in Appendix S1: Table S1). PFD may, in general, be coexistence theory (Chesson, 2000). To facilitate this inte- asymmetric, for example if shared mutualists provide rel- gration, we formulated a new discrete-time model atively more benefit to one species than another (Sakata, accounting for the interactive effects of competition and 1999). In sum, rarity is a double-edged sword, potentially PFD on individual fitness. This model builds on a model providing benefits through negative frequency-depen- that has been used extensively to empirically test and fur- dence and competitive release, but also causing costs ther develop coexistence theory (Adler et al., 2007, Levine through diverse PFD mechanisms in both plants and ani- and HilleRisLambers, 2009, Godoy et al., 2014, 2017, mals when organisms are rare. Hart et al., 2016). We present an analysis of the determin- Environmental stochasticity may magnify the impor- istic and stochastic versions of the model to address the tance of PDD and PFD. If environmental fluctuations following questions: How strong does niche differentia- reduce population sizes of a species to the point where tion have to be in the face of PFD to generate negative PDD or PFD kicks in, then such stochasticity may result frequency-dependence and allow for coexistence? How do in the loss of that species from the system. This phe- asymmetries in PFD and fecundity differences influence nomenon has been demonstrated in single species mod- whether coexistence occurs, and can asymmetries in PFD els with an Allee effect (Dennis, 2002, Liebhold and result in non-additive effects of niche differences and Bascompte, 2003, Roth and Schreiber, 2014, Schreiber, PFD on coexistence? How robust is species coexistence to 2016). For example, using models coupled with historical environmental fluctuations? How does this robustness data, Liebhold and Bascompte (2003) found that envi- depend on the degree of correlation between the species ronmental stochasticity could cause extinction of local demographic responses to these fluctuations and tempo- gypsy moth populations (Lymantria dispar) in North ral autocorrelations in these fluctuations? What role does America, even when their densities were well above the the storage effect play in maintaining coexistence in the Allee threshold: the density at which the per-capita face of PFD and environmental stochasticity? growth rate, on average, equals zero. In sharp contrast, environmental fluctuations can, via the storage effect, MODEL AND METHODS mediate coexistence between competing species (Ches- son and Warner, 1981, Chesson, 1994). The storage To integrate the dynamics of competition and PFD, effect stabilizes coexistence, (1) when each species we build on the Leslie-Gower model of competing July 2019 POSITIVE FREQUENCY DEPENDENCE Article e02664; page 3

species (Leslie and Gower, 1958), which has been used where Ni,t denotes the density of species i in year t. extensively for describing the dynamics of competing To account for environmental stochasticity, we replace annual plants and insects (Leslie and Gower, 1958, Ches- the per-capita maximal fecundities ki with random terms son, 1994, Adler et al., 2007, Godoy and Levine, 2014, ki,t that are log-normally distributed with log means li, r2 Godoy et al., 2014). The dynamics of these models are log variances i , log cross-correlation r, and log tempo- fully characterized and serve as discrete-time analogs of ral autocorrelation s. The correlation r determines to the classical, continuous-time Lotka-Volterra competi- what extent the fecundities of the two species respond in tion models (Cushing et al., 2004). Unlike earlier models a similar manner to the environmental fluctuations. For accounting for PFD (Kuno, 1992, Yoshimura and Clark, r = 1, the species respond identically to the fluctuations. 1994, Kishi and Nakazawa, 2013), this model choice For r = 0, their responses are uncorrelated, while for allows us to directly account for the interactive effects of r = 1, good years for one species are bad years for the PFD and competition on the ecological dynamics. other species. The temporal autocorrelation s determines whether favorable years for one species tend to be fol- lowed by favorable years for that species (i.e., s 1) or The model are uncorrelated to the environmental conditions in the The model has two competing species with densities next year (i.e., s 0). As in the face of environmental N1 and N2. The maximal number of offspring produced stochasticity, the storage effect can promote coexistence by an individual of species i is maximal per-capita fecun- of competing species [Chesson and Warner, 1981, Ches- dity ki. Intra- and interspecific competition reduce this son, 1994), we also modify the model to account for fecundity by a linear function of the species densities. overlapping generations that ensure population buffer- That is, let aii and aij be the strengths of intra- and inter- ing, a necessary component of the storage effect. Specifi- specific competition for species i, respectively. Then the cally, we add terms, +siNi,t, to the right hand sides of expected number of offspring produced by an individual Eq. 2 where si is the survival probability of an individual of species i experiencing no PFD is of species i.

ki Methods with j 6¼ i: 1 þ a N þ a N ii i ij j Our results for the deterministic model focus on highly fecund species (i.e., k ≫ 1) for which the analysis is sub- PFD independently reduces the fitness of an individ- i stantially simpler yet still captures the full dynamical ual by a frequency-dependent factor complexity of the model. Specifically, in Appendix S2, we show that the highly fecund model reduces to a one- N i (1) dimensional system. For this reduced model, we find N þ b N i i j implicit expressions for the equilibria, identify their stability, and classify the dynamics into two types: contin- where b determines the negative impact of species j on i gent exclusion in which there is one unstable coexistence i species . One mechanistic interpretation of expression 1 equilibrium and two stable single species equilibria, and can be given in terms of reproductive interference for contingent coexistence in which there is one stable coex- species whose fecundity is more resource limited than istence equilibrium, two unstable coexistence equilibria, limited by processes involved in mating, as is found in and two stable single species equilibria. To extend our many plants (Ghyselen et al. 2016) and animals (Gittle- analysis to the full model, we use the theory of mono- man and Thompson 1988). For such species, expression tone maps (Smith, 1998, Hirsch and Smith, 2005) and 1 can be interpreted as the probability of a successful show, as in the high-fecundity case, that the dynamics conspecific mating. always converge to a one-dimensional system whose Multiplying these components of the per-capita growth dynamics are governed by a finite number of equilibria. rate together yields the following deterministic model: We use this analysis and numerical computations to explore the structure of these equilibria and to determine zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{PFD reduction maximumz}|{ fecundity how niche overlap, fecundity differences, and PFD inter- ¼ k N1;t act to determine whether the models exhibit contingent N1;tþ1 N1;t 1 þ N1;t b1N2;t coexistence or exclusion. zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{competitive reduction For the stochastic model, Appendix S3 uses results of Roth and Schreiber (2014) to show that stochastic fluc- 1 ¼: ð ; Þ N1;tf1 N1;t N2;t (2) tuations ultimately result in species loss whenever the 1 þ a11N1;t þ a12N2;t species responses to the environmental fluctuations are N2;t N2;tþ1 ¼ N2;t k2 not perfectly correlated. When species loss occurs, it N ; þ b N ; 2 t 2 1 t occurs asymptotically in time at a super-exponential 1 ¼: ð ; Þ rate i.e., Ni,t C exp (-exp(rt)) for some r, C > 0 þ a þ a N2;tf2 N1;t N2;t 1 22N2;t 21N1;t and for large t. To estimate extinction risk, we used a Article e02664; page 4 SEBASTIAN J. SCHREIBER ET AL. Ecology, Vol. 100, No. 7 quasi-extinction threshold of 0.001 i.e., when a species’ when this contingent coexistence occurs, we focus on the density falls below 0.001, the species is declared extinct. case of highly fecund species (i.e., ki ≫ 1) for which This extinction threshold is approximately 100,000 times competition is more likely to be severe, and present anal- smaller than the typical species density at equilibrium ysis of the general case in Appendix S2. In this case, for the corresponding deterministic model. Our results competitive outcomes depend on the relative per-capita were not qualitatively sensitive to the value of this quasi- growth rate (R1: = f1/f2) of species 1 as a function of its extinction threshold. We numerically explore how the frequency x :¼ N1=ðN1 þ N2Þ. The relative per-capita probability of species loss over finite time intervals growth rate of species 1 is a product of three terms (see depends on PFD, the standard deviations ri of the envi- Eq. S3 in Appendix S2): ronmental fluctuations, and the interspecific correlation r and temporal autocorrelation s in these fluctuations. relative maximumz}|{ fecundities To understand the role of the storage effect, we allowed k ð þ Þ ð Þ¼ 1 x 1 x b2x for overlapping generations (i.e., positive survivorship R1 x k ð Þ ð þ ð ÞÞ 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}1 x x b1 1 x terms si > 0) and positive temporal autocorrelations relative strength of PFD (s > 0) in the log per-capita maximal fecundities ki,t.The overlapping generations ensure that the populations are relativezfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{ strength of competition a ð Þþa buffered and the temporal autcorrelations ensure there is 22 1 x 21x : a positive correlation between the log per-capita fecundi- a11x þ a12ð1 xÞ ties and the strength of competition. These features, which are required for a storage effect (Ellner et al., 2016), are The second term increases with the frequency of spe- verified in Appendix S3. For this model, we computed the cies 1 whenever b > 0 (dotted curves in Fig. 1A, C; competitive component of the invasion growth rate of i Appendix S2). each species (Chesson, 1994; Schreiber et al., 2011, Schrei- Frequency-dependence in the third term, the relative ber, 2012). Specifically, for species 2, we simulated the strength of competition, can be positive or negative. The dynamics of species 1, in the absence of species 2, for sign of this frequency-dependence depends on the niche T = 10,000 time steps, to estimate its stationary distribu- overlap of these two species (Chesson, 2013, Godoy and tion Nb . Then we estimated the invasion growth rate of 1 Levine, 2014): species 2 without the PFD term as rffiffiffiffiffiffiffiffiffiffiffiffiffiffi a a q ¼ 12 21: k XT k a a E 2;t 1 2;t : 11 22 log b log (3) 1 þ a N T 1 þ a21N1;t 21 1 t¼1 If there is partial niche overlap (q < 1), the relative per-capita growth rates without PFD exhibits negative The larger the value of Eq. 3, the more quickly the frequency-dependence (dashed curves in Fig. 1A, C; species 2 would increase in the absence of positive fre- Appendix S2). Intuitively, as a species becomes more frequency-dependence. Analytical details are provided in quent in the community, it experiences more intraspeci- Appendix S3. R code for simulating both the determinis- fic competition than interspecific competition; as tic and stochastic models are available online (Schreiber intraspecific competition is stronger than interspecific and Yamamichi, 2019). competition, the per-capita growth rate without PFD decreases. When there is perfect niche overlap (q = 1), the relative per-capita growth rate without PFD is fre- RESULTS quency-independent. In this case, the relative per-capita growth rate only exhibits PFD and coexistence is not Frequency-dependence, coexistence, and exclusion possible. Consequently, from now on, we assume q < 1. Our analysis begins with the deterministic model. To Provided there is sufficiently low niche overlap, the rel- ensure each species i can persist in isolation, we assume ative per-capita growth rate R1 of species 1 exhibits neg- that the maximal fecundity ki is greater than one for ative frequency-dependence at intermediate species each species. Under this assumption, species i in isola- frequencies (Fig. 1A, B). When this occurs, there are tion converges to the positive equilibrium, the carrying two critical frequencies, xlow\xhigh of species 1 such that ki 1 capacity Ki :¼ . When there is PFD (b > 0), the low (1) the per-capita growth rate of species 1 is greater than aii j density per-capita growth rate of species j 6¼ i is zero at the per-capita growth rate of species 2 when its fre- this equilibrium as individuals fail to reproduce (e.g., quency is slightly above xlow, and (2) the per-capita failure to reproduce with conspecifics due to their low growth rate of species 2 is greater than the per-capita frequency). Consequently, species j is excluded whenever growth rate of species 1 when species 1’s frequency is it reaches such low frequencies, and the equilibria (N1, slightly below xhigh. When species 1’s frequency lies N2) = (K1, 0) and (0, K2) are locally stable. between xlow and xhigh, negative frequency-dependent Despite these stable, single-species equilibria, coexis- feedbacks dominate and the species approach a unique tence may occur at another stable equilibrium. To see stable coexistence equilibrium. In contrast, when species July 2019 POSITIVE FREQUENCY DEPENDENCE Article e02664; page 5

5.00 Species 1 A B Species 2 100

1.00

0.20 Relative per capita growth w/o PFD Species 2 density Relative PFD 20

0.05 Relative per capita growth 0 0.0 0.2 0.4 0.6 0.8 1.0 020 60 100 Species 1 frequency Species 1 density

5.00 Species 1 C D Species 2 100

1.00

0.20 Species 2 density 20

0.05 0 0.0 0.2 0.4 0.6 0.8 1.0 020 60 100 Species 1 frequency Species 1 density

FIG. 1. Frequency-dependent feedbacks and the dynamics of (A, B) contingent coexistence and (C, D) contingent exclusion. In A and C, relative strength of positive frequency-dependence (PFD, black dotted line), relative strength of competition (black dashed line), and relative per-capita capita growth rate R1(x) (blue line) for species 1 are shown as a function of the frequency x of species 1. In B and D, colored curves correspond to the zero-growth nullclines, and trajectories for different initial conditions are gray lines. In all panels, stable equilibria/frequencies are filled circles and unstable equilibria/frequencies are unfilled circles. In A and B, low niche overlap results in negative frequency-dependence at intermediate species frequencies and coexistence. In C and D, large niche overlap result in PFD in the relative per-capita growth rate of species 1 at all of species 1’s frequencies. Parameter values are b1 = b2 = 0.25 in A and B, and b1 = 0.75 and b2 = 0.25 in C and D. Other parameter values are ki = 100, aii = 1, and aij = 0.2.

1’s frequency falls below xlow or exceeds xhigh, PFD feed- k k q\ 1 and q\ 2 : backs dominate and either species 1 gets excluded by k2 k1 species 2 or excludes species 2, respectively. When niche overlap is too great, PFD dominates at all In this case, coexistence is not contingent upon ini- species frequencies and coexistence is not possible tial conditions. This coexistence condition is sharp: if (Fig. 1C, D). Consequently, there is a critical frequency it is satisfied, the species coexist, else they do not xbistable of species 1 below which species 1 is excluded (Fig. 2A). and above which species 2 is excluded. When species experience both positive frequency- dependence as well as negative frequency-dependence Niche overlap, fecundity differences, and contingent due to interspecific competition, coexistence requires coexistence that the additive effects of niche overlap and the strength of PFD are less than the ratio of maximal To better understand when coexistence or exclusion fecundities: occurs, we focus on the case where the species are demo- graphically similar with respect to competition (a11 = a22 k1 k2 and a12 = a21) but potentially differ in their maximal q þ \ q þ \ : b1 k and b2 k (4) fecundities (ki) or their susceptibility to PFD (bi). The 2 1 general case is presented in Appendix S2. If there is no PFD (b1 = b2 = 0), coexistence occurs if the niche overlap If the conditions in expression 4 are not satisfied, neg- is less than the the ratio of maximal fecundities ki/kj: ative frequency-dependent feedbacks are too weak to Article e02664; page 6 SEBASTIAN J. SCHREIBER ET AL. Ecology, Vol. 100, No. 7

No PFD Equal maximum fecundity 3.0 1.0 2 λ A B 1 λ 0.8 2.5 ρ

0.6

2.0 Coexistence Exclusion ive n 0.4 n−addit io No ibut tr con 1.5 Niche overlap 0.2

Maximum fecundity ratio ratio Maximum fecundity 1.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 ρ Strength of niche overlap Strength of PFD b1 = b2 10% niche overlap, symmetric PFD 10% niche overlap, asymmetric PFD 3.0 3.0 2 2 λ C λ D 1 1 λ λ 2.5 2.5

2.0 2.0

1.5 1.5 Maximum fecundity ratio ratio Maximum fecundity

1.0 ratio Maximum fecunidty 1.0 0.0 0.2 0.4 0.6 0.8 1.0 123456 b = b Strength of PFD 1 2 Relative PFD b1 b2

0.0 0.2 0.4 0.6 0.8 1.0 Frequency of species 1 at coexistence equilibrium

FIG. 2. Effects of PFD, niche overlap, and fecundity differences on species coexistence. Colored region corresponds to contingent coexistence, while the white region corresponds to exclusionary dynamics. Colored shading indicates frequency of species 1 at the stable coexistence equilibrium. In A, B, and C, species are equally sensitive to PFD (b1 = b2) or niche overlap (a12 = a21). In A, there is no PFD (bi = 0), while both PFD and niche overlap occur in B (aij > 0 and bi > 0). In C and D, species differ in their maximal fecundities and experience either symmetric (in C) or asymmetric (in D) PFD. Dashed lines show the coexistence boundary from Eq. 4 which only accounts for the additive contributions of PFD and niche overlap. The solid black lines are determined by the analytic criterion presented in Appendix S2. In C and D, there is symmetric 10% niche overlap (aij = 0.1). Other parameter values are k2 = 100 and aii = 1.

promote coexistence. Satisfying expression 4, however, interactive effect equals 3qb and coexistence occurs if need not ensure coexistence due to nonlinear, interac- (Appendix S2) tive effects between PFD and niche overlap (the dis- q þ b þ 3qb\1: (5) tance between the dashed lines and the coexistence regions in Fig. 2B–D). Eq. S5 in Appendix S2 pro- As niche overlap and PFD contribute equally to this vides an explicit condition for this nonlinear interac- nonlinear interactive effect, coexistence is least likely tive. In general, this expression is difficult to interpret when the strength of PFD and niche overlap are equally biologically. However, for species that exhibit no dif- strong (Fig. 2B). ferences in maximal fecundities (k1 = k2) and are Differences in the maximal fecundities or asymmetries equally susceptible to PFD (b1 = b2 = b), this in the strength of PFD lead to larger, nonlinear effects July 2019 POSITIVE FREQUENCY DEPENDENCE Article e02664; page 7 on coexistence (the greater distance between the dashed environmental fluctuations ultimately will drive one of line and the coexistence region in Fig. 2C, D than B). the species extinct, whether or not deterministic coexis- When PFD is symmetric, larger differences in the maxi- tence is possible (Appendix S3, Fig. 4). Intuitively, envi- mal fecundities (e.g., larger values of k1/k2) always inhi- ronmental fluctuations can push one of the species to a bit coexistence (Fig. 2C). Numerical simulations suggest sufficiently low frequency that the deterministic effects that interactive effects of PFD and niche overlap con- of PFD rapidly drive the species to extinction. Larger tinue to be symmetric in this case. When differences in environmental fluctuations increase the likelihood of the maximal fecundities are too large to permit coexis- these events and, thereby, increase the probability of spe- tence, the species with the fecundity disadvantage can be cies loss. As negative correlations increase fluctuations excluded despite being at an initially higher frequency. in frequencies, they also increase the likelihood that one When there are sufficiently strong asymmetries in species falls below its critical frequency and rapidly goes PFD (b1/b2 > 3.5 in Fig. 2D), coexistence occurs at extinct. Consequently, the probability of extinction intermediate differences in the maximal fecundities. If decreases with positive correlations. In fact, when species the fecundity advantage of the species 1 is not suffi- responses to the environmental fluctuations are identical r2 ¼ r2 = ciently high, coexistence is not possible and this species (i.e., 1 2 and r 1), environmental stochasticity can be excluded even when it is initially at the higher fre- does not drive any species extinct provided they are ini- quency (Fig. 2D). Alternatively, if the fecundity advan- tially near a stable, coexistence equilibrium of the deter- tage of species 1 is too large, coexistence is not possible ministic model (Appendix S3). In contrast to the effects and species 2 has a lower threshold frequency below of positive cross-correlations, positive temporal autocor- which it is excluded (Fig. 2D). relations increase quasi-extinction risk (Appendix S3: Fig. S1A–C). Intuitively, positive temporal autocorrela- Stochastic environments tion leads to longer runs of unfavorable conditions to one species thereby making this species more vulnerable When the maximal fecundities k fluctuate stochasti- i,t to PFD. cally, the fluctuations in the frequency dynamics are To understand whether the storage effect could alter determined by the fluctuations in the ratio of these the predictions about the effects of cross-correlations on fecundities k /k . As these fecundities k are log-nor- 1,t 2,t i,t extinction risk, we simulated the model with overlapping mally distributed with log-mean l , log-variance r2 and i i generations (i.e., s > 0) and varying levels of temporal correlation r, their ratio k /k is log-normally dis- i 1,t 2,t autocorrelation s. To strengthen the potential for a stor- tributed with age effect (Chesson, 1994), species had no difference in their mean maximal fecundities, exhibited complete niche ¼ l l ¼ r2 r r þr2: log-mean 1 2 and log-variance 1 2r 1 2 2 overlap (i.e., a11 = a22 = a12 = a21), experienced weak (6) positive frequency-dependence (i.e., b1 = b2 = 0.01), and had high survival (i.e., si = 0.5). The competitive compo- Eq. 6 implies that positively correlated responses nent of the invasion growth rate (see definition in Meth- > (r 0) of the two species to the environmental fluctua- ods) when rare decreased with positive cross-correlations tions decrease the log-variance in the frequency dynam- and increased with positive temporal autocorrelations ics. Intuitively, environmental fluctuations cause the (Fig. 5A). Therefore, the storage effect is greatest with fecundities of each species to change by the same factor negative cross-correlations and positive temporal auto- and, thereby, reduces the effect of these fluctuations on correlations. Despite this trend, the probability of extinc- the ratio of maximal fecundities (Fig. 3B). Indeed, when tion increased with negative cross-correlations and the responses to environmental fluctuations are of the positive temporal autocorrelations in the maximal fecun- r2 ¼ r2 same magnitude and perfectly correlated ( 1 2 and dity values (Fig. 5B). Namely, the impact of these corre- = r 1), there are no fluctuations in these fecundity ratios lations on extinction risk by generating low frequencies and species may coexist indefinitely. of one species outweighed their impact of increasing the In contrast, when species exhibit opposing responses long-term per-capita growth rates when rare. < to environmental fluctuations (r 0), environmental The effects of asymmetries in maximal fecundities and fluctuations that drive one species to higher densities PFD on extinction risk largely follow patterns suggested simultaneously drive the other species to low densities. by the deterministic model: when a species is at low fre- This behavior results in larger fluctuations in the species quency at the stable, coexistence equilibrium, extinction frequencies (Fig. 3A). In the extreme case where the risk is greater (compare Fig. 2D to 6). In particular, for responses to the environmental fluctuations are of the a given level of asymmetry in PFD, persistence of both same magnitude and are perfectly negatively correlated species is most likely at an intermediate fecundity advan- r2 ¼ r2 = ( 1 2 and r 1), the fluctuations in the log ratio of tage of the species more susceptible to PFD. For smaller fecundities are twice as large as those for uncorrelated differences in maximal fecundity, the species with the r2 r2 fluctuations (i.e., 4 1 vs. 2 1). fecundity advantage is more likely to go extinct. For lar- When species exhibit some differentiated responses ger fecundity differences, the species less susceptible to < r2 6¼ r2 to environmental fluctuations (r 1or 1 2), PFD is more likely to go extinct. As larger fecundity Article e02664; page 8 SEBASTIAN J. SCHREIBER ET AL. Ecology, Vol. 100, No. 7

r = −1 r = 0.95 150 Species 1 nullclines Species 1 nullclines A Species 2 nullclines B Species 2 nullclines Community trajectory Community trajectory

100 ● ● ● ●

● ● 50 Species 2 density

● ● 0 ● ● 050100150050100150 Species 1 density Species 1 density

FIG. 3. Fluctuations, correlations, and coexistence. In A and B, 100 yr simulations (gray lines) of the stochastic model with fluctuating maximal fecundities are plotted in the phase plane. The nullclines for the mean field model are shown as red and blue curves, and the corresponding stable and unstable equilibria for the mean field model as solid and unfilled circles, respectively. In A and B, the correlations r between the log maximal fecundities of the species are negative (r = 1) and positive (r = 0.95), respectively. r2 ¼ : s = Parameter values as in Fig. 1 with environmental variance i 0 05 and temporal autocorrelation 0. differences (i.e., l1 larger than l2) result in larger fluctu- Deterministic coexistence requires that the additive ations in their ratio (i.e., variance k1,t/k2,t equals effects of niche overlap and PFD need to be smaller ð ðl l Þþr2 þ r2Þð ðr2 þ r2Þ Þ = exp 2 1 2 1 2 exp 1 2 1 when r 0), than the ratio of maximal fecundities (see Eq. 4). When extinction risk is generally greater due to larger fecundity both competitors experience the same strength of PFD, differences rather than smaller fecundity differences the interactive effects of PFD and niche overlap are (blue region larger in Fig. 6B than in Fig. 6A). symmetric: a simultaneous increase in the strengths of PFD and niche overlap have a more negative impact on coexistence than increasing the strength of one more DISCUSSION than the other (see Eq. 5). Asymmetries in the strength Many competing species are likely to experience both of PFD, which are common (e.g., Appendix S1: negative and positive frequency-dependence. Positive fre- Table S1), can facilitate coexistence if the species more quency-dependence (PFD), in and of itself, does not vulnerable to PFD has the higher maximal fecundity. allow for coexistence and leads to alternative stable states This trade-off is affected by niche overlap as high- supporting only a single species (Amarasekare, 2002, lighted in the coexistence condition in expression 4. Fukami and Nakajima, 2011). In contrast, negative fre- For example, a twofold advantage in fecundity for one quency-dependence allows for stable coexistence (Adler species requires that the other species’ vulnerability to et al., 2007) but does not allow for alternative stable PFD must be more than 50% less for coexistence (i.e., states. For competing species experiencing both positive if k1 = 2k2, then b2 < 1/2–q whereas b1 < 2–q). The and negative frequency-dependent feedbacks, a new greater niche overlap, the stronger the trade-off needs dynamic emerges supporting alternative stable states to be. Given the empirical prevalence of asymmetries in including ones at which the species coexist (Fig. 1). This PFD for coexisting competitors (e.g., the asymmetries dynamic can occur when positive frequency-dependence in reproductive interference reported in Appendix S1: occurs at low species frequencies, and negative frequency- Table S1), our results suggest there may be a counter- dependence dominates at intermediate species frequen- vailing strong trade-off in the maximal fecundities or cies. More generally, this dynamic arises when there are sensitivities to competition (see below) for these species multiple changes in the sign (i.e., positive vs. negative) of pairs. frequency-dependence. When these conditions are met, Our deterministic analysis complements and extends we find that coexistence is determined by more complex earlier work by Kishi and Nakazawa (2013), who ana- interactions of both positive and negative frequency- lyzed the dynamics of two competing species experienc- dependence, rather than the “mutual invasibility crite- ing reproductive interference. Unlike our discrete-time rion” of modern coexistence theory (Chesson, 2000) or by model, which accounts for the simultaneous effects of species growth rates when rare (Hofbauer and Sigmund, PFD and competition on fecundity, Kishi and Naka- 1998, Schreiber, 2000). Our deterministic analysis high- zawa (2013)’s model is continuous time and assumes lights that niche overlap and PFD have negative synergis- that competition increases mortality rates while repro- tic effects on coexistence, yet trade-offs between PFD and ductive interference reduces birth rates. Thus, our model fecundity can facilitate deterministic coexistence. integrates more naturally into the framework of modern July 2019 POSITIVE FREQUENCY DEPENDENCE Article e02664; page 9

800 r = A 0.6 C 600

400 0.5

Minimum density 200 r 0 0 20406080 Time (yr) 0.0

r =−0.6

B Cross-correlation, 150

−0.5 100

Minimum density 50

020406080 0.0 0.2 0.4 0.6 0.8 1.0 Time (yr) Environmental variance, σ2

0.0 0.2 0.4 0.6 0.8 1.0 Probability of species loss

FIG. 4. Extinction increases with variation in environmental fluctuations and decreases with correlated species responses to the environment. In the left panels, plots of the minimum of the species densities (i.e., the smaller value of N1,t and N2,t) for multiple simulations of the stochastic model. In A, the species have positively correlated responses to the environment (cross-correlation coefficient r > 0). In B, the species have negatively correlated responses to the environment (r < 0). In panel C, a contour plot of the probability of the loss of a species within 50 yr with respect to the magnitude of the fluctuation r1 = r2 = r and the cross-correlation r. coexistence theory (Chesson, 2000, Adler et al., 2007) Just as strong positive density-dependence can make and is readily applicable to PFD in annual plants populations particularly vulnerable to environmental (Levine and HilleRisLambers, 2009, Godoy and Levine, fluctuations (Courchamp et al., 1999, Dennis, 2002, 2014, Hart et al., 2016) and insects (see also Ribeiro and Liebhold and Bascompte, 2003, Roth and Schreiber, Spielman 1986). Under the assumption of symmetry in 2014), environmental fluctuations make coexistence of reproductive interference, Kishi and Nakazawa (2013) competitors experiencing PFD more tenuous. Indeed, derived a similar coexistence condition to our condition our analysis reveals that asymptotic extinction is inevita- in expression (4) that holds for asymmetric, as well as ble when either of the competitors are repeatedly pushed symmetric, PFD. Kishi and Nakazawa (2013) numeri- over the critical threshold where positive frequency- cally demonstrated trade-offs between the strength of dependence kicks in. Despite this extinction only reproductive interference and sensitivity to competition, occurring asymptotically, the rare species decreases at a which determines niche overlap, could facilitate coexis- super-exponential rate (i.e., the species’ population size tence; a finding that complements our results about decays faster than ert for any r < 0) and, thus, would trade-offs between sensitivity to PFD and maximal rapidly go extinct due to demographic stochasticity. This fecundity. extinction risk is most severe for interacting species Article e02664; page 10 SEBASTIAN J. SCHREIBER ET AL. Ecology, Vol. 100, No. 7

0.2 A 1.0 B

0.8 0.8 0.3 0.6 0.6

0.4 0.4 Increasing 0.4 Increasing autocorrelation autocorrelation Invasion growth rate growth Invasion Temporal autocorrelation Temporal Persistence probability 0.1 0.2 0.2

0.0 0.0 0.0

−0.5 0.0 0.5 −0.5 0.0 0.5 Cross−correlation Cross−correlation

FIG. 5. The storage effect does not overcome the effects of PFD. (A) The competitive component of the invasion growth rate that measures the strength of the storage effect, and (B) the probability of extinction in 50 yr are shown for different values of the cross-correlation r and temporal autocorrelation τ. Parameter values: a11 = a12 = a21 = a22 = 1, k1 = k2 = 100, b1 = b2 = 0.01, and survivorships s1 = s2 = 0.5.

3.0 A) Probability of extinction B) Probability of extinction

2 of species 1 of species 2 λ 1 2.5 λ 1.5 2.0 Maximum fecundity ratio ratio Maximum fecundity 1.0 123456 123456

Relative strength of PFD b1 b2 Relative strength of PFD b1 b2

● ●

0.0 0.2 0.4 0.6 0.8 1.0 Probability of species loss

FIG. 6. Fluctuations, asymmetric PFD, and extinction risk. The probability of extinction of (A) species 1 or (B) species 2 within r2 ¼ r2 ¼ 100 yr. Species started at the stable coexistence equilibrium for the mean field model (i.e., deterministic model with 1 2 0) r2 ¼ r2 ¼ : = whenever it exists. Parameter values: 1 2 0 1, r 0, and remaining parameters as in Fig. 2D.

exhibiting opposing demographic responses to environ- in the relative frequencies of the species and, therefore, mental fluctuations e.g., one species producing more off- are more likely to drive one of them to sufficiently low spring in cooler years while the other species produces frequencies, at which point PFD kicks in. In sharp con- more offspring in warmer years. Specifically, these nega- trast, for species with highly positively correlated tively correlated responses result in greater fluctuations responses to environmental fluctuations, fluctuations in July 2019 POSITIVE FREQUENCY DEPENDENCE Article e02664; page 11 species frequencies are minimal and extinction risk is ACKNOWLEDGMENTS much smaller. These effects of species’ correlated responses to envi- We would like to thank two anonymous reviewers and R. Iri- ronmental conditions are in direct opposition to Ches- tani for their extensive and thoughtful comments. S. J. Schreiber son’s storage effect (Chesson and Warner, 1981, was supported by the US National Science Foundation Grants DMS-1313418, 1716803. M. Yamamichi was supported by the Chesson, 1994). The storage effect promotes coexistence Japan Society for the Promotion of Science (JSPS) Grant-in- when there are (1) species-specific responses to environ- Aid for Scientific Research (KAKENHI) 15H02642, 16K18618, mental conditions, (2) covariance between environmen- and 16H04846 and by John Mung Program and Hakubi Center tal conditions and the strength of competition, and (3) for Advanced Research of Kyoto University. S. Y. Strauss buffered population growth. The first ingredient is stron- acknowledges NSF DEB 13-42841 and the California Agricul- gest when the species exhibit negatively correlated tural Experiment Station. responses to the environment and the weakest when the species exhibit nearly perfectly correlated responses to LITERATURE CITED the environment. 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