Quantifying the Relative Importance of Competition, Predation, and Environmental Variation for Species Coexistence

Home , Storage effect

bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Quantifying the relative importance of competition, predation, and environmental variation for species coexistence

Lauren G. Shoemaker1, corresponding author email: [email protected], phone: (970)-691-0459, fax: NA 2Botany Department, University of Wyoming, Laramie, WY, 82071

Allison K. Barner2,3, corresponding author email: [email protected], fax: NA 2Department of Environmental Science, Policy, and Management University of California, Berkeley, Berkeley, CA, 94720 3Department of Biology, Colby College, Waterville, ME, 04901

Leonora S. Bittleston4,5 email: [email protected] 4Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139 5Department of Biological Sciences, Boise State University, Boise, ID, 83725

Ashley I. Teufel6,7 email: [email protected] 6Santa Fe Institute, Santa Fe, NM, 87501 7Department of Integrative Biology, The University of Texas at Austin, Austin, TX, 78712

Data accessibility statement: All code to replicate this study can currently be found on GitHub at https://github.com/lash1937/foodwebs_env_variation_coexistence. Upon acceptance, code will be archived on Zenodo.

Running Title: Competition and predation aﬀect coexistence

Keywords: Coexistence theory, ecological networks, species interactions stabilizing mechanisms, environmental ﬂuctuations, diamond model, storage eﬀect

1 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Type of Article: Letters

Number of Words: Abstract: 150, Main text: 4644, Text Box: NA

Number of References: 64

Number of Tables: 1

Number of Figures: 6

Number of Text Boxes: 0

Statement of authorship: LGS and AKB contributed equally to this manuscript and led the working group from which this project originated. All authors contributed to conceptual development, analyses, manuscript writing, and editing.

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1 1 Abstract

2 Coexistence theory and food web theory are two cornerstones of the longstanding eﬀort

3 to understand how species coexist. Although competition and predation are known to act

4 simultaneously in communities, theory and empirical study of the two processes continue to

5 be developed independently. Here, we integrate modern coexistence theory and food web

6 theory to simultaneously quantify the relative importance of predation, competition, and

7 environmental ﬂuctuations for species coexistence. We ﬁrst examine coexistence in a classic

8 multi-trophic model, adding complexity to the food web using a novel machine learning ap-

9 proach. We then apply our framework to a parameterized rocky intertidal food web model,

10 partitioning empirical coexistence dynamics. We ﬁnd that both environmental ﬂuctuation

11 and variation in predation contribute substantially to species coexistence. Unexpectedly, co-

12 variation in these two forces tends to destabilize coexistence, leading to new insights about

13 the role of bottom-up versus top-down forces in both theory and the rocky intertidal ecosys-

14 tem.

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15 2 Introduction

16 For many decades, community ecologists have developed two complementary theoretical

17 and empirical directions for studying the interactions among species and their dynamic

18 consequences: food web theory (Cohen & Stephens, 1978; McCann, 2011; Terborgh, 2015)

19 and coexistence theory (Chesson, 2000; Barab´as et al., 2018). Despite their long independent

20 histories, central to both approaches is a shared interest in explaining the mechanisms that

21 maintain biological diversity, coexistence, and the stability of ecological communities (Ives

22 et al., 2005; Godoy et al., 2018).

23 Food web theory focuses on consumptive interactions between species across diﬀerent

24 trophic levels (Fig. 1A), while competition between species is represented through consumer-

25 resource dynamics (e.g., exploitation competition, Godoy et al. (2018)). Empirical studies

26 of food webs tend to quantify the presence or absence of links between species at diﬀerent

27 trophic levels or the frequency of consumptive events among all species in the food web

28 (Berlow et al., 2004; Pascual et al., 2006). From theoretical studies of food webs, we have

29 gained the insight that diverse ecological communities are stabilized by weak interactions be-

30 tween species (McCann et al., 1998), negative feedbacks (May, 1973), and network structures

31 that minimize the frequency of widespread, generalist interactors (McCann, 2011).

32 In comparison, coexistence theory tends to focus on competitive interactions within a sin-

33 gle trophic level, exploring how multiple species competing for the same limiting resources,

34 space, or responding to environmental ﬂuctuations can coexist (Fig. 1B). Classic coexis-

35 tence work shows that diverse and stable communities can occur through three primary

36 mechanisms: (1) the partitioning of resources, where diﬀerent species are better able to take

37 advantage of diﬀerent limiting resources, such as nitrogen versus phosphorous in grasslands

38 (Tilman, 1982) (2) trait and demographic trade-oﬀs between species, such as if one species

39 has a higher dispersal rate while another is a superior competitor (Levins & Culver, 1971;

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40 Yu & Wilson, 2001), and (3) species partitioning environmental heterogeneity and inherent

41 landscape-level variation, stabilizing overall community dynamics (Chesson, 2000). Cur-

42 rent developments in coexistence theory tend to highlight how environmental variation and

43 niche partitioning can promote species coexistence via multiple mechanisms, the most widely

44 studied of which is the storage eﬀect (Chesson, 2000; Snyder & Adler, 2011; Barab´as et al.,

45 2018). Empirical applications of this coexistence framework—termed modern coexistence

46 theory (MCT)—tend to quantify demographic rates to examine the relative importance of

47 multiple coexistence mechanisms (Kraft et al., 2015; Germain et al., 2018).

48 Predation and competition are key forces structuring communities, and at their cores,

49 clear connections exist between these two robust theories for community ecology. Both

50 examine nearly identical questions such as: (1) Why do we observe diverse communities,

51 rather than having one or only a few species dominate? and (2) What properties make diverse

52 communities stable? Past work integrating food webs and coexistence generally fall into

53 three categories: (1) the inﬂuence of predators (or ”natural enemies”) on the diversity of prey

54 species (Jabot & Bascompte, 2012; Saleem et al., 2012; Griﬃn et al., 2013), (2) how predator

55 presence alters the strength of competition among prey species (Gurevitch et al., 2000), and

56 (3) coexistence of prey species in the context of predation (Holt, 1984; Holt et al., 1994;

57 Chesson & Kuang, 2008; McPeek, 2019; Klauschies & Gaedke, 2019). Despite the breadth

58 of previous work, few studies have incorporated known food web structure with realistic

59 complexity. The best known example, Brose (2008) implements a consumer-resource model

60 for a diverse simulated food web and analyzes the conditions for persistence in consumers

61 and resources, using an analog of Tilman’s R* theory (Tilman, 1982). While predation has

62 been explicitly incorporated into modern coexistence theory in select scenarios (Chesson &

63 Kuang, 2008; Kuang & Chesson, 2008, 2009, 2010), the approach relies on highly complex

64 underlying analytical derivations, making it diﬃcult to generally apply the framework across

65 more complex theoretical examples, much less empirical systems that require a new system-

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66 speciﬁc model and corresponding mathematical derivation (Ellner et al., 2019).

67 Here, we seek to explicitly combine food web theory and coexistence theory, integrating

68 predation and competition together in a conceptual and mathematical framework that is

69 generalizable, easy to use, and can be applied across diﬀerent systems (Fig. 1C). We extend a

70 recent conceptualization of modern coexistence theory (MCT) (Ellner et al., 2019) to examine

71 how ﬂuctuations in the environment, ﬂuctuations in predator abundance, their interaction,

72 and average ﬁtness diﬀerences between consumers can stabilize—or alternatively hinder—

73 coexistence. We ﬁrst decompose coexistence into its mechanistic components, in a simple

74 but highly studied diamond-shaped food web (McCann et al., 1998; Vasseur & Fox, 2007).

75 Doing so allows us to compare the relative importance of ﬂuctuations in the environment

76 versus predators for the coexistence of two competing species and quantify the strength of

77 top-down versus bottom-up ﬂuctuations. We then examine the generality of coexistence

78 partitioning across parameter space, extending this framework to incorporate additional

79 food web complexity via added consumers and predators. Finally, we apply our approach

80 to a classic empirical ecosystem–the rocky intertidal food web—highlighting its utility in

81 empirical scenarios and insights gained in both theoretical and empirical cases.

82 3 Methods

83 3.1 Partitioning Mechanisms that Promote Coexistence

84 To examine the relative importance of environmental ﬂuctuations versus predation in

85 maintaining coexistence, we build on the classic framework of MCT (Chesson, 2000). While

86 the framework of MCT traditionally focuses on primary producers—often in annual grass-

87 lands (e.g. Mayﬁeld & Stouﬀer (2017); Kraft et al. (2015))—recent advances that allow for

88 a simulation-based approach (Ellner et al., 2019), more easily facilitate application of MCT

89 across a range of applications. Here, we extend the approach of Ellner et al. (2019) to

90 incorporate ﬂuctuations in the environment and predation simultaneously.

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91 In brief, we examine coexistence using the mutual invasibility criterion, incorporating the

92 invader-resident comparison (Barab´as et al., 2018). Stable coexistence occurs if each species

93 can successfully invade when rare (i.e. at low-density). Each species’ growth rate when rare,

Ni,t+1 94 ri = ln , is calculated when the rest of the food web (minus the invader) is at its steady Ni,t

95 state abundance distribution. To incorporate the invader-resident comparison, we examine

96 average growth rates when rare as ri − rr where rr is the average growth rate of resident

97 species within the same trophic level. As the resident species are at steady state abundances

98 and there is a negligible eﬀect of the invader, rr ≈ 0, such that ri − rr ≈ ri. We include the

99 invader-resident comparison, however as it has important implications for the partitioning of

100 coexistence into its mechanistic components, as coexistence can be fostered by factors that

101 either beneﬁt the invader or harm the resident (Ellner et al., 2019; Hallett et al., 2019).

102 We next decompose each species’ growth rate when rare ri − rr into it’s mechanistic

103 components, following Ellner et al. (2019) (See Supplement 1 for the full derivation). The

104 decomposition yields four mechanistic components: the average growth rate with no ﬂuctu-

0 P 105 ations (∆i ), the eﬀect of variation in predator abundance (∆i ), the eﬀects of environmental

E 106 variation (∆i ), and the interaction eﬀect of both environmental and predator ﬂuctuations

EP 107 (∆i ) for each species’ growth rate when rare:

0 P E EP ri ≈ ri − rr = ∆i + ∆i + ∆i + ∆i . (1)

0 108 The ﬁrst term of the decomposition, ∆i , examines species’ ability to invade when rare if

109 both the environment and predator abundances are held constant at their means. We deﬁne

P 110 the second term, ∆i , as nonlinearity in predation, and it quantiﬁes the eﬀect of variability in

111 predator abundances. If the magnitude of consumptive eﬀects at high predator abundances

112 are less than population gains at low predator abundance (i.e. via saturating consumption),

E 113 nonlinearity in predator abundance can help stabilize coexistence. Similarly, we deﬁne ∆i

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114 as nonlinearity in response to the environment, and it quantiﬁes how underlying variability

EP 115 in environmental conditions can stabilize or destabilize coexistence. Finally, ∆i quantiﬁes

116 their interactive eﬀects not accounted for by each main eﬀect in isolation.

117 3.2 Applications of Coexistence Partitioning

118 3.2.1 Diamond model

119 Applying the framework above, we ﬁrst examined the relative importance of environmen-

120 tal ﬂuctuations versus ﬂuctuations in predator abundance using the four-species diamond

121 model (Fig. 1A). This classic model has a long history for analyses of trophic interactions

122 and species coexistence, including in identifying the stabilizing eﬀect of consumer asynchrony

123 in constant environments (McCann et al., 1998), with extensions explicitly incorporating en-

124 vironmental ﬂuctuations (Vasseur & Fox, 2007). Furthermore, in the model, competitors

125 share resources and predators, matching common empirical systems (Williams & Martinez,

126 2000) in a mathematically simpliﬁed and tractable manner.

127 The diamond model tracks abundance of a predator, P , two consumers, C1 and C2, and

128 a resource, R. Consumer 1 is the superior consumer, but is also the preferred prey species,

129 which maintains coexistence under a variety of parameterizations. In the model, dynamics

130 occur such that:

dP JP P [ΩPC1 C1 + (1 − ΩPC1 ) C2] = −MP P + (2) dt ΩPC1 C1 + (1 − ΩPC1 ) C2 + C0

dC1 ΩC1RJC1 C1R ΩPC1 JP PC1 = −MC1,t C1 + − (3) dt R + R01 ΩPC1 C1 + (1 − ΩPC1 ) C2 + C0

dC2 ΩC2RJC2 C2R (1 − ΩPC1 ) JP PC2 = −MC2,t C2 + − (4) dt R + R02 ΩPC1 C1 + (1 − ΩPC1 ) C2 + C0

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dR Ω J C R Ω R J C R = rR(1 − R/K) − C1R C1 1 − C2R C2 C2 2 . (5) dt R + R01 R + R02

131 where parameter deﬁnitions and values are given in Table 1. Disparities in consump-

132 tion of the resource and predator preference yields asymptotic dynamics where consumer

133 populations are highly asynchronous and both species co-occur (Vasseur & Fox, 2007).

ζs,t 134 Environmental variation alters consumer mortality rates, MCs,t = MCs,0 · e , where ζs,t

135 are random normally distributed environmental conditions. For each species, s, a time series

136 of environmental eﬀects ζ is calculated using the Cholesky factorization of the variance-

137 covariance matrix:

  σ2 ρσ2     . (6) ρσ2 σ2

138 where σ is the environmental eﬀect size and ρ is the cross-correlation of its eﬀect on

139 consumer species. Multiplying Eq. 6 by a 2 X T matrix of random numbers from a normal

140 distribution with mean 0 and unit variance, where T is the total number of timesteps to run

141 the model (T = 5000), yields ζ1,t and ζ2,t for each timestep t.

142 Applying MCT partitioning to the diamond model, we calculated growth rate when rare

143 for each consumer species and its mechanistic decomposition. We set σ = 0.55, represen-

144 tative of natural mortality rates from neotropical trees (Condit et al., 1995) and compared

145 scenarios with negative (ρ = −0.75), no (ρ = 0), and positive (ρ = 0.75) cross-correlation.

146 We then examined the relative importance of top-down versus bottom-up controls by varying

147 the predation preference for C1 from ΩPC1 = 0.5 (no preference) to ΩPC1 = 1.0 (full pref-

148 erence) in conjunction with varying the strength of environmental ﬂuctuations from σ = 0

149 to σ = 0.75. This spans the range of observed environmental ﬂuctuation eﬀects on mortal-

150 ity in both terrestrial and aquatic systems (Vasseur & Fox, 2007; Condit et al., 1995). For

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151 each parameter combination, we calculated each species’ growth rate when rare, determining

152 coexistence using the mutual invasibility criterion. Then, using an example with strong pre-

153 dation preference (ΩPC1 = 0.9), we decomposed coexistence into its mechanistic components

154 for low, intermediate, and high environmental variability (σ = 0.1, 0.4 and 0.7 respectively,

155 Supplement 2). All simulations and analyses were conducted in R (R Core Team, 2019).

156 3.2.2 Expanded diamond model

157 While the diamond model represents a natural starting-point for examining coexistence,

158 it is a highly simpliﬁed food web structure. We therefore expanded the model, ﬁrst by incor-

159 porating a third consumer (C3) (equation details in Supplement 3, Eq. S3.1). In including

160 a third consumer, we maintained the initial diamond model parameterization except for

161 predator preferences, and applied global optimization by diﬀerential evolution to estimate

162 the new parameters MC3 , JC3 , R03 ,ΩPC1 ,ΩPC2 , and ΩC2R and predator preferences (Table 1)

163 (Ardia et al., 2016). To ensure that populations were stable, we estimated parameters with-

164 out stochastic mortality of consumers. We created a scoring function based on counting the

165 number of times any of the populations fell below a value of 0.001 over the course of 5000

166 timesteps (excluding the ﬁrst 1000 ‘burn-in’ time points) and applied diﬀerential evolution

167 to minimize the scoring function using the R package DEoptim, with the DE / best / 1 /

168 bin with jitter (option 3), an initial population size of 100 individuals, and a 0.05 speed of

169 the crossover adaptation (Ardia et al., 2016). We ran the algorithm until the scoring func-

170 tion reached zero and recorded the parameter set from the ﬁrst member of the population.

171 We repeated this process to compare coexistence mechanisms under two alternative param-

172 eter combinations (Table 1). We visually conﬁrmed that these parameter sets resulted in

173 diﬀerent dynamics in the absence of environmental variation (Fig. S3.1, S3.2) and with en-

174 vironmental variation (Fig. S3.3, S3.4). Using these systems, we calculated coexistence and

175 its mechanistic decomposition when including environmental variation as described above.

176 To further investigate how complex systems are stabilized, we included a second predator

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177 in the model (Eq. S3.2). We followed the same method as above, using the parameter set

178 of replicate 1 from Table 1 to estimate the new parameters for the second predator. We

179 visually conﬁrmed that the estimated parameter set for the model containing two predators

180 resulted in stable dynamics (Fig. S3.5, Fig. S3.6). This methodology allows us to compare

181 coexistence mechanisms with the entire food web (two predators, three consumers, and one

182 resource) and a subset of species from the food web.

183 3.2.3 Empirical applications in an intertidal food web

184 Finally, we highlight the applicability of MCT partitioning in empirical systems, focusing

185 on the rocky intertidal ecosystem (Fig. 2), although the framework is applicable across

186 systems. The rocky intertidal communities of the Northeast Paciﬁc Ocean are well-studied

187 model systems for exploring the role of predation and environmental variation in species

188 coexistence (Connell, 1961; Dayton, 1971; Menge et al., 1997; Connolly & Roughgarden,

189 1999; Forde & Doak, 2004). In this system, a larger barnacle Balanus glandula competes

190 for space with the smaller barnacle Chthamalus dalli/ﬁssus and with herbivorous limpets

191 (Dayton, 1971). Whelks (predatory snails) and sea stars consume both barnacles, but are not

192 space-limited like barnacles and limpets. Of these ﬁve focal taxa, four have planktonic larvae

193 (sea stars, barnacles, limpets), which leads to a decoupling of local population abundance

194 and recruitment (Iwasa & Roughgarden, 1986). Thus, recruitment dynamics are ‘bottom-

195 up,’ primarily a function of variation in the environment and the availability of free space

196 (Menge, 2000). Coexistence among sessile, space occupying invertebrates in this system is

197 thought to be controlled both by ‘keystone’ predation (Paine, 1966, 1969) and bottom-up

198 variation in larval supply (Menge et al., 1997), though it remains unclear the degree to which

199 each mechanism contributes to coexistence.

200 Model. We modeled the rocky intertidal food web (Fig. 2) using a set of stochastic

201 diﬀerence equations, based closely on the model and parameterization of Forde & Doak

202 (2004). While here we summarize the model, the full set of equations (Eq. S4.1-S4.10) and

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203 parameterization (Table S4.1) can be found in Supplement 4. In brief, the model tracks

204 recruitment dynamics, competition for space, and predator-prey interactions through time.

205 First, the pelagic larval pool for each species (barnacles, limpets, and sea stars) is randomly

206 drawn from a lognormal distribution, based on the range of observed values for the system,

207 mimicking the spatial and temporal recruitment variation in coastal ecosystems (Menge

208 et al., 1997). Next, the realized recruitment from this larval pool to the local ecosystem

209 depends on the availability of free space. Following Shinnen & Navarrete (2014) and Forde

210 & Doak (2004), since no clear competitive hierarchy exists for the rocky intertidal (Menge,

211 2000), we model competition for free space using lottery competition, where the available

212 free space in the system is calculated at each time step (month) based on the individual size

213 and population abundances of the space-occupiers.

214 Both recruits and adults of all species are aﬀected by density-independent mortality, while

215 barnacles face additional mortality dependent on predator population size. Predators have

216 the same per-capita eﬀect on both barnacles. In this model, neither prey is preferred over

217 the other (Connolly & Roughgarden, 1999; Forde & Doak, 2004), though predator preference

218 for Balanus glandula has been observed (Dayton, 1971) and could be modeled in a future

219 study. Population growth of limpets, sea stars, and whelks is density-dependent, as their

220 predators are not explicitly modeled (Supplement 4).

221 We simulated the rocky intertidal model 500 times, each for 100 years, tracking population

222 size for each species per month (Forde & Doak, 2004). We examined coexistence across six

223 larval supply scenarios. First, we compared coexistence when all species with pelagic larvae

224 had ‘high’ versus ‘low’ mean supply rates, then subsequently each species was run individually

225 with ‘high’ supply, while all others were held at ‘low’ supply rates (see Table 1 in Forde &

226 Doak, 2004). Coexistence mechanisms were calculated following the procedure described in

227 Section 3.1. Invasion population size was set to 1 for all species, and invasion was simulated

228 after 50 years, such that resident species reached their steady state distribution. Here,

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229 variation in the environment manifests as variation in larval supply rather than mortality—

P 230 as in the diamond model, and nonlinearity in predation, ∆i , includes variation in predator

231 abundance and recruitment of predators.

232 4 Results

233 4.0.1 Diamond model

234 When decomposing coexistence into its mechanistic components for the classic diamond

235 model, we ﬁnd that both consumers are able to stably coexist, as both exhibit positive

236 growth rates when rare, ri − rr (Fig. 3). For the superior competitor (Fig. 3A), ﬂuctu-

237 ations in either the environment or the predator abundance matters little for coexistence,

0 238 as evidenced by the similarity in strength of ri − rr and ∆i , as well as the minimal ef-

E P EP 239 fects of ∆i , ∆i , and ∆i . However, for the inferior competitor (Fig. 3B), ﬂuctuations in

240 both predator abundance and the environment increase stability. High predator abundances

241 preferentially increase consumption and decrease the steady state-abundance of the superior

242 competitor, C1. This decrease in competition between C1 and C2 increases the stability of

243 coexistence for the inferior competitor (C2) via nonlinearity in predation. Similarly, non-

244 linearity in response to the environment increases stability for the inferior competitor, as

245 positive gains during good years outweigh losses during poor years. However, the interactive

246 effect between environmental fluctuations and fluctuations in predator abundance is destabi-

EP 247 lizing for C2. This destabilization of ∆i occurs because increased mortality of the resident

248 C1 decreases consumption and therefore population size of the predator. As C2 beneﬁts

249 from lower abundances of the superior competitor, the covariance between C1 mortality and

250 predator abundance destabilizes the growth rate when rare for the inferior competitor, but

251 not enough to hinder coexistence. These results are robust to changes in the cross-correlation

252 of environmental ﬂuctuations (Figs. S2.1, S2.2).

253 To examine the generality of these results, we calculated both consumer species’ growth

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254 rates when rare and coexistence when varying predation preference and the strength of

255 environmental variation. Predation preference had a stronger eﬀect on both coexistence

256 and growth rates when rare compared to the strength of environmental ﬂuctuations (Fig.

257 4). With no predation preference, C1 outcompetes the inferior consumer C2. As predation

258 preference increases, C2 then is able to outcompete C1. Only at high preference of the

259 predator for C1 do both species coexist, as the high predation preference yields oscillatory

260 dynamics that maintain coexistence.

261 However, while growth rates when rare for each species and overall coexistence depend

262 only moderately on the strength of environmental variation, the relative importance of co-

263 existence mechanisms changes substantially with increased environmental variation (Fig. 1).

0 P 264 With low environmental variation, coexistence of C2 with C1 is stabilized by ∆i and ∆i .

0 265 As the strength of environmental variation increases, ∆i becomes less important and even

EP 266 destabilizing at high environmental variation. Similarly, ∆i becomes increasingly desta-

P E 267 bilizing, while ∆i and ∆i become more stabilizing. The combination of these mechanisms

268 yields coexistence regardless of the amount of environmental variation, σζ , but highlights that

269 the relative importance of coexistence mechanisms changes with increasing environmental

270 variation and that their eﬀects often counteract one another.

271 4.1 Expanded diamond model

272 To assess how coexistence changes with increasing food web complexity, we ﬁrst examine

273 adding another competitor to the diamond model (Fig. 5A shows results for the ﬁrst param-

274 eter set). We ﬁnd that the inclusion of a third competitor causes a reduction in C1’s growth

275 rate when rare and destabilization by variation in predation and and environmental ﬂuctu-

276 ations (Fig. 5B). Further, our two parameter sets result in diﬀering stabilizing mechanisms

EP 277 for C2 (Fig. S3.7), where the eﬀect of ∆i can be stabilizing (Fig. S3.7 B) or destabilizing

278 (Fig. S3.7 E). These ﬁndings suggest that the stability of a food web is achieved not just

279 through its structure, but as a function of how the species interact with one another. Fur-

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280 thermore, as the classic diamond model represents a subset of this larger food web (with the

281 same parameterization), comparing Fig. 3 and Fig. 5 highlights the diﬀerent expectations

282 for coexistence when only considering part of the larger ecological community.

283 To compare how mechanisms of stabilization change when a second predator is included

284 (Fig. 5E) we decompose the coexistence mechanisms of this expanded system (Fig. 5 F-H).

285 The inclusion of a second predator leads to a further reduction in the growth rate when rare

286 of the superior competitor (C1, Fig. 5F). The second predator also decreases stabilization by

EP 287 nonlinearity in predation for C2 and causes the eﬀect of ∆i to switch from destabilizing to

288 slightly stabilizing (Fig. 5G). Again, we ﬁnd diﬀerent expectations for coexistence strength

289 and its mechanistic comparison when comparing the full model to ones that only consider

290 a subset of species interactions. In aggregate, our results from decomposing increasingly

291 complex food webs demonstrate that the nature of interactions as well the food web topology

292 impact the mechanism by which species coexist.

293 4.2 Empirical applications in an intertidal food web

294 Finally, we examined coexistence in a temperate rocky intertidal ecosystem, a classically

295 studied system in which both predation and environmental variation have been shown to

296 inﬂuence species coexistence. Here, variation in the environment drives variation in larval

297 supply rates of the three taxa that compete for space. In this model, under scenarios where

298 there is no variation in larval supply or predator abundance, the top competitor is the

299 smaller barnacle, Chthamalus dalli, though both barnacle species coexist (Fig. 6). When

300 larval supply is low across all planktonic taxa (barnacles, limpets, sea stars), this variation

301 in larval supply beneﬁts both barnacle taxa equally, and nonlinearity in response to the

302 environment is the strongest mechanism promoting coexistence. Nonlinearity in predation

303 also promotes coexistence of barnacle prey, though to a lesser extent than variation in the

304 environment. Higher positive invasion growth rates under predation suggests a potential

305 ‘hydra eﬀect’ of sea stars and whelks on their barnacle prey (Abrams, 2009). However,

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306 coexistence is strongly destabilized by covariation in predator abundance and environmental

307 conditions that promote barnacle growth, paralleling Fig. 3.

308 When larval supply is low, covariation in the environment (larval supply) and predation

EP 309 (∆i ) greatly beneﬁts limpets, who have no predators in this model. In other words, limpets

310 beneﬁt from the co-occurrence of high predator abundance and high larval supply, as high

311 predator abundance reduces the abundance of species that compete with limpets, though not

312 enough to allow limpets to ultimately coexist (Fig. 6). In fact, across all scenarios, limpets

E 313 do not have a positive growth rate when rare (Fig. S4.1). Variation in the environment (∆i ),

314 in particular, destabilizes limpet coexistence under low larval supplies enough to counteract

EP 315 the stabilizing eﬀect of ∆i .

316 5 Discussion

317 While coexistence theory and food web theory examine similar core questions, they do

318 so from traditionally disparate perspectives. Connections between the theories are becom-

319 ing more common (e.g. Kuang & Chesson (2008); Sommers & Chesson (2019)), including

320 recent developments that incorporate niche and ﬁtness diﬀerences (Godoy et al., 2018). A

321 uniﬁed framework that incorporates direct competition between species together with eﬀects

322 of predation is necessary for gaining a synthetic understanding of how biodiversity is main-

323 tained. Our extension of Ellner et al. (2019), builds directly on the framework of modern

324 coexistence theory (Chesson, 2000), including environmental ﬂuctuations through time and

325 space that can maintain coexistence among competitors via niche partitioning (Hallett et al.,

326 2019; Letten et al., 2018). Critically, our extension of MCT also incorporates predation and

327 ﬂuctuations in predator abundances, allowing both bottom-up and top-down mechanisms to

328 be incorporated simultaneously. The relative importance of both—as well as their interac-

EP 329 tions, as deﬁned by the term ∆i —can be examined simultaneously, yielding an extension

330 from the classic framework that allows for the examination of multiple ﬂuctuations across

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331 trophic levels. In this study we focus on comparing bottom-up ﬂuctuations in environmental

332 conditions that alter prey mortality rates or larval supply rates and top-down ﬂuctuations

333 in predator abundances and predator larval supply rates.

334 Applying this approach, it becomes apparent that ﬂuctuations are not always necessary

335 for stabilizing species’ growth rates when rare (e.g. Fig. 3A versus B), but rather individual

336 species may preferentially require ﬂuctuations. These results match insights from modern

337 coexistence theory, even when focusing on a single trophic level in isolation (Hallett et al.,

338 2019; Shoemaker & Melbourne, 2016). Indeed, the superior competitor in the diamond

339 model is only mildly aﬀected by ﬂuctuations in the environment and predator populations,

340 while both of these types of ﬂuctuations increase the stability of the inferior competitor’s

341 (C2) growth rate when rare. Extending our approach to more complex food web dynamics

342 highlights the importance of considering all key species in a system when quantifying coexis-

343 tence and corresponding diversity expectations. For example, C1 exhibited little response to

344 ﬂuctuations in the simple diamond model. Adding a new competitor to the model yielded a

345 destabilizing eﬀect of nonlinearity in response to the environment and predation (Fig. 5). In

EP 346 general, the interactive eﬀect of environmental ﬂuctuations and predator abundances, ∆i ,

347 are often destabilizing for the inferior competitor. This may be general across food webs

348 where consumers respond similarly to environmental ﬂuctuations (e.g. Kuang & Chesson

349 (2010)), as environmental conditions that promote consumer growth will correspondingly

350 yield increased predator abundances. Thus, the positive eﬀects of environmental variation

351 may be dampened by increasing predator abundances and therefore realized predation.

352 More generally, extending modern coexistence theory to examine multi-trophic systems

353 yields key insights into the relative importance of top-down versus bottom-up forces in al-

354 tering community composition and maintaining biodiversity (Gripenberg & Roslin, 2007;

355 Matson & Hunter, 1992). In many systems, both factors work simultaneously to stabilize

356 community dynamics. Our approach permits the partitioning of ﬂuctuations in bottom-up

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E P 357 (e.g. ∆i ) and top-down (e.g. ∆i ) dynamics to examine their relative importance, as well as

EP 358 considering their interaction (e.g. ∆i ). For example, for intertidal barnacle species Balanus

359 glandula and Chthamalus dalli, both top-down and bottom-up factors stabilize coexistence,

360 although bottom-up factors appear to be slightly more important, at least according to our

361 dynamical model formalization (Forde & Doak, 2004). Empirically, in intertidal systems,

362 the strength of bottom-up factors strongly covary with the strength of top-down processes

363 (Menge et al., 1997). This covariation is destabilizing, as environmental conditions that pro-

364 mote barnacle growth also increase predator abundances. More generally, a similar approach

365 could be applied across ecosystems to partition the relative importance of top-down and

366 bottom-up eﬀects operating simultaneously. For example, in grasslands, nutrient additions

367 often decrease plant biodiversity while herbivores provide a counteracting eﬀect, primarily

368 by reducing light limitation (Borer et al., 2014). Furthermore, in the aquatic detritus-based

369 ecosystems of carnivorous pitcher plants, nutrient additions tend to increase bacterial abun-

370 dance but not biodiversity, while top predators increase bacterial biodiversity, likely through

371 regulation of intermediate trophic levels (Kneitel & Miller, 2002).

372 Our approach is general across both theoretical and empirical contexts; however, it re-

373 quires a tight-coupling of demographic studies with interaction networks to yield dynamical

374 models of both consumer and predator abundances. Measurements of demographic rates and

375 food web interactions are usually made separately—often entire studies in their own rights

376 (e.g. Dibner et al. (2019); Gripenberg et al. (2019)), and thus all the necessary information

377 is diﬃcult to obtain for many systems. As such, we encourage future work to examine both

378 demographic rates along with trophic links and their corresponding strengths. Doing so

379 will allow for quantifying coexistence and stability, along with the baseline structure of food

380 webs (Pascual et al., 2006). In complex ecosystems, the number of equations and parameters

381 quickly grow with the number of species, so simplifying into functional groups or exploring

382 key species of interest may be necessary for computational tractability.

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383 While here we focus on how the underlying structure of food webs and species’ demogra-

384 phy may promote coexistence, a similar approach could additionally incorporate behavioral

385 dynamics. For example, recent work by Sommers and Chesson (2019) show that predator

386 avoidance behavior by prey species can alter coexistence via changes in the importance of

387 apparent competition relative to resource competition. If prey species partition resources,

388 these behavioral changes tend to promote coexistence; when prey species instead partition

389 predators, behavioral modiﬁcations undermine coexistence (Sommers & Chesson, 2019). In

390 multiple empirical systems, behavioral changes in prey species via fear-driven avoidance are

391 similarly well documented, such as with brown anoles, green anoles, and curly-tailed lizards

392 on tropical islands (Pringle et al., 2019). Additionally, predators have recently been shown

393 to alter their behavior due to fear-driven avoidance of humans, which positively impacts prey

394 species by increase their foraging ability (Suraci et al., 2019). Partitioning the mechanistic

395 eﬀects of such behavioral changes—including with multiple predator trophic levels—would

396 yield insight into the potential stabilizing eﬀects of behavior on coexistence of prey species.

397 A further extension of this work could be examining coexistence under global change,

398 with directional changes in environmental ﬂuctuation (Usinowicz & Levine, 2018). For in-

399 stance, as temperature increases so do metabolic and encounter rates, which likely have

400 important ramiﬁcations for coexistence, and in particular the contribution of relative non-

P 401 linearity in predation, ∆i (Moya-Lara˜no et al., 2014). Additionally, this framework could be

402 extended to examine eco-evolutionary dynamics. Classic single-trophic level applications of

403 modern coexistence theory show the ability for stabilizing coexistence mechanisms to evolve,

404 especially if one species has a greater evolutionary ability (Snyder & Adler, 2011). In con-

405 trast, multiple species can co-evolve in a manner that erodes the importance of stabilizing

406 coexistence mechanisms (Snyder & Adler, 2011). Extensions examining phenotypic varia-

407 tion (Gibert & DeLong, 2017) or eco-evolutionary dynamics in food webs may be critical,

408 as both can drastically alter food web dynamics, links, and species’ abundances, even under

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409 relatively small variation or selective pressures (Gibert & Yeakel, 2019). Applications to

410 eco-evolutionary food webs could provide insight into the evolutionary and environmental

411 factors impacting species coexistence and community stability.

412 6 Acknowledgements

413 We thank Artemy Kolchinsky, Andy Rominger, and Margaret Siple for conversations

414 in a Santa Fe Institute (SFI) Research Jam that led to the development of this project.

415 This work was supported by a James S. McDonnell Foundation (JSMF)-SFI Postdoctoral

416 Working Group grant to LGS and AKB (grant number: 357(14)). LGS, AKB, and LSB

417 were supported by JSMF Postdoctoral Fellowships, grants #220020513, #220020478, and

418 #220020477. AIT was supported by the Omidyar Program at the SFI.

419 7 Supporting Information

420 Appendix S1: Derivation of Coexistence Mechanisms

421 Appendix S2: The Classic Diamond Model

422 Appendix S3: Expanding the Diamond Model for Additional Complexity

423 Appendix S4: Rocky Intertidal Food Web Case Study

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Parameter Description Value 2 Consumers, 1 Predator Diamond Model r resource density independent growth rate 1.0 K resource carrying capacity 1.0

JC1 consumer 1 resource intake rate 0.8036

JC2 consumer 2 resource intake rate 0.7

JP predator intake rate 0.4

MC1,0 consumer 1 beginning mortality rate 0.4

MC2,0 consumer 2 beginning mortality rate 0.2

MP predator mortality rate 0.08

R01 half saturation constant; resource use by consumer 1 0.16129

R02 half saturation constant; resource use by consumer 2 0.9

C0 predator half saturation constant 0.5

ΩPC1 predator preference for consumer 1 0.92

ΩC1R consumer 1 preference for the resource 1.0

ΩC2R consumer 2 preference for the resource 0.98 3 Consumers, 1 Predator Extended Model

JC3 consumer 3 resource intake rate 0.670232, 0.917076

MC3,0 consumer 3 beginning mortality rate 0.097798, 0.142901

R03 half saturation constant; resource use by consumer 3 0.297716, 0.328642

ΩPC1 predator preference coeﬃcient for consumer 1 0.332333, 0.262048

ΩPC2 predator preference coeﬃcient for consumer 2 0.001743, 0.012992

ΩC3R consumer 3 preference for the resource 0.926310, 0.919742 3 Consumers, 2 Predators Extended Model

MP2 predator 2 mortality rate 0.061683

JP2 predator 2 intake rate 0.365359

ΩP2C1 predator 2 preference coeﬃcient for C1 0.384297

ΩP2C2 predator 2 preference coeﬃcient for C2 0.607627 Table 1: Parameters in the diamond model and its extensions to incorporate additional food web complexity. Parameterization of the classic diamond model matches that from Vasseur & Fox (2007). The mechanistic decomposition of the 2 consumers, 1 predator system is shown in Fig. 3. The bold values from the 3 consumers, 1 predator extended model are those of replicate 1. The mechanistic decomposition of the replicate 1 system are shown in Fig. 5B-D. The values of replicate 2 are given beside the estimates from replicate 1. The mechanistic decomposition of the replicate 2 system are shown in Fig. S3.7D-F. The bold parameters are subsequently used for the 2 predator extension. The mechanistic decomposition of 2 predator extension is shown in Fig. 5F-H. 28 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

A) Food web B) Coexistence theory C) Combination Legend: 0 Average growth rate P P P i (all variables set to averages) P Effect of variation in i predation rate E Effect of environmental C1 C2 C1 C2 C C i 1 2 fluctuations EP Interactive effect of varying i predation and environmental fluctuations R R R

Figure 1: Conceptual diamond model. (A) Food webs generally have links connecting trophic levels in direction of consumption (predator, P, eats consumers, C1 and C2, which share a resource, R). (B) Coexistence theory generally addresses only competition at one trophic level, integrating consumption of a resource into an interaction coefficient. (C) In this framework, we combine food webs and coexistence theory, using the mutual invasion criterion and an approach that allows for coexistence to be partitioned into species’ average growth rates 0 P E ∆i , predation variability ∆i , environmental fluctuations ∆i , and the interaction between EP predation and environmental fluctuations ∆i .

29 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Legend: 0 i P i E i EP sea star i whelks Pisaster ochraceus

limpets barnacle barnacle Balanus glandula Cthalamus dalli

space (shared resource)

Figure 2: Intertidal food web using conventions introduced in Figure 1C. Barnacles and limpets compete for space, and environmental fluctuations (pink) lead to variation in their larval supply. Sea stars and whelks influence barnacle dynamics through predation (blue). Limpets, however, are not consumed. The lines are not shown for the effect of variation in predation on limpets because the effect is negligible in our model; however, there is a strong indirect, interactive effect of predation and environmental variation (orange).

30 certified bypeerreview)istheauthor/funder,whohasgrantedbioRxivalicensetodisplaypreprintinperpetuity.Itmadeavailableunder bioRxiv preprint ircmeio A,wihi a eseetdb aito nete h niomn rthe or environment deviation the standard either and in mean show variation runs. Results by 500 (B). eﬀected across competitor inferior less the far to compared is predator which (A), growth competitor overall rior showing (∆ abundance (∆ model, predator conditions diamond in web ﬂuctuations of food fect classic the ¯ of rate Decomposition 3: Figure r i

− Growth Rate When Rare doi: r ¯ r https://doi.org/10.1101/797704

h rwhrt ihn niomna rpeao utain (∆ fluctuations predator or environmental no with rate growth the , -0.05 0.00 0.05 0.10 0.15 i E ,adtercmie ffcs(∆ effects combined their and ), A) r Effect ofvariationinpredationrates Low-density growthrate Interactive effectofvaryingpredation i r r Competitor 1 i 0 a i P CC-BY-NC-ND 4.0Internationallicense ; this versionpostedOctober8,2019. Mechanistic Partitioning i E C 1 R P i E P 31 C 2 andenvironmentalfluctuations i P ,teeeto utain nenvironmental in fluctuations of effect the ), i EP .Rslsaesonfrbt h supe- the both for shown are Results ). B) r i r The copyrightholderforthispreprint(whichwasnot r Effect of environmental fluctuations Effect ofenvironmental Growth rate under avg. conditions Growth rateunderavg. . Competitor 2 i 0 i P C i E 1 R P i E i 0 C P ,teef- the ), 2 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

A) Coexistence 0.75

Species 1 Wins 0.50 Species 2 Wins Coexistence Species 1 or 2 Wins Coexistence or Species 1 Wins 0.25 Environmental Variation 0.00 0.5 0.6 0.7 0.8 0.9 1.0 Predation Preference

B) Species 1 C) Species 2 Growth Rate When Rare

0.75 0.15 0.10 0.05 0.50 0.00 -0.05

0.25 -0.10 -0.15 Environmental Variation -0.20 0.00 0.50 0.75 1.00 0.50 0.75 1.00 Predation Preference Predation Preference

Figure 4: Comparison of the role of predation preference (i.e. top down eﬀects) versus variation in the environment (i.e. bottom up eﬀects) on coexistence of both competitors in the diamond model (A). Results are shown for 1, 760 runs (10 runs per each parameter combination). Coexistence requires that both species’ growth rates when rare are positive (panels B, C; orange and red colors). For species 1, increasing predation preference decreases its growth rate when rare initially, but then allows for coexistence via oscillatory dynamics. For species 2, increasing the predation preference for species 1 increases species 2’s growth rate when rare.

32 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

System Competitor 1 Competitor 2 Competitor 3 A) B) C) D) 0.2 P

C C C 1 2 3

R Growth Rate When Rare . . 0.1 0.0 − 0.1

0 P E EP 0 P E EP 0 P E EP ri −rr ∆i ∆i ∆i ∆i ri −rr ∆i ∆i ∆i ∆i ri −rr ∆i ∆i ∆i ∆i System Competitor 1 Competitor 2 Competitor 3 E) F) G) H) 0.2 P P

1 2 Growth Rate When Rare

C C C 1 2 3

R Growth Rate When Rare . . 0.1 0.0 − 0.1

0 P E EP 0 P E EP 0 P E EP ri −rr ∆i ∆i ∆i ∆i ri −rr ∆i ∆i ∆i ∆i ri −rr ∆i ∆i ∆i ∆i Mechanistic Partitioning Low-density growth rate Growth rate under avg. conditions Effect of variation in predation rates Effect of environmental fluctuations Interactive effect of varying predation and environmental fluctuations

Figure 5: Decomposition of the expanded diamond model, where parameters for Competitors 1 and 2 are the same as in Figure 3 and Competitor 3’s are the bold parameters in Table 1. (A) Illustration of food web with three competitors. (B-D) Mechanistic partitioning for each of the competitors in food web containing three competitors. (E) Illustration of food web with three competitors and two predators. (F-H) Mechanistic partitioning for each of the competitors in food web containing two predators and three competitors. Results of mechanistic partitioning are shown with mean and standard deviation calculated from 500 runs.

33 bioRxiv preprint doi: https://doi.org/10.1101/797704; this version posted October 8, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Figure 6: Application of coexistence partitioning to the empirical rocky intertidal foodweb, comparing two levels of larval supply (high and low). Results show the mean and standard error for 500 replicates, each run for 100 years. Both barnacles coexist, despite a destabilizing EP effect of the interaction between environment and predator fluctuations (∆i ). Limpets exhibit a slight negative growth rate when rare, suggesting competitive exclusion. Note that EP neither predators consume limpets, thus the effect of ∆i on limpet coexistence strength is an indirect effect mediated by predation on barnacles (see Fig. 2). Additional larval supply scenarios are presented in Fig. S4.1