Omnivory does not preclude strong trophic cascades

Omnivory has been cited as an explanation for why trophic cascades are weak in many food webs, but empirical support for this prediction has been equivocal — compared to carnivores, documented indirect effects of top omnivore populations on primary producer biomass have ranged from beneficial, to non-existent, and negative. To gain intuition about the effects of omnivory on the strength of cascades, we analyzed models of fixed and flexible top omnivores, two foraging strategies that are supported by empirical observations. We identified regions of parameter space in which omnivores following a fixed foraging strategy non-intuitively generate larger cascades than predators that do not consume producers at all, but that are otherwise demographically identical: (i) high productivity relative to consumer mortality rates, and (ii) smaller discrepancies in producer versus consumer reward create conditions in which cascades are stronger with moderate omnivory. In contrast, flexible omnivores that attempt to optimize per capita growth rates during search never induce cascades that are stronger than the case of carnivores. Although we focus on simple models, the consistency of these general patterns together with prior empirical evidence suggests that omnivores should not be ruled out as agents of strong trophic cascades.


Introduction 1
Trophic cascades occur when predators indirectly effect change in biomass at lower trophic 2 levels by directly reducing populations of intermediate consumers (Paine, 1980;Strong, 1992; 3 Terborgh & Estes, 2013). A growing number of factors that control the strength of trophic 4 cascades continue to surface from model-based and experimental studies, and their identifi-5 cation has improved our understanding of processes that dampen or enhance indirect effects 6 in ecological networks, and ecosystem responses to disturbances (Pace et al., 1999;Shurin al., 2008;France, 2012) -implies that weak cascades may not be a necessary outcome of 23 omnivory, and that the strength of cascades generated by omnivores may depend on other 24 factors (Wootton, 2017). Despite this growing body of empirical work, the population-and 25 community-level features that predict when omnivores occupying top trophic positions will 26 generate strong or weak cascades remain unclear.

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Here we analyze mathematical models describing trophic interactions between basal 28 producers, intermediate consumers, and top omnivores to systematically evaluate the effects 29 of omnivory on the strength of trophic cascades. We consider two types of empirically-30 observed foraging behaviors, namely fixed (Diehl & Feißel, 2000) and flexible (Fahimipour 31 & Anderson, 2015) omnivory, and present a comparison between trophic cascades in these 32 systems and traditional ones induced by analogous carnivores. We have chosen to study 33 minimally detailed models to focus on coarse-grained system features that may point to po-34 tential future directions for experimental work as opposed to making predictions about the 35 behavior of a particular ecosystem (Anderson et al., 2009). We draw two primary conclu-36 sions based on numerical and analytical methods: stronger trophic cascades with omnivory 37 are possible in high productivity systems if omnivores forage according to a fixed strategy, 38 whereas cascades are never stronger when omnivores forage flexibly.

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Models were analyzed with a focus on equilibrium outcomes to gain insight into how dif-41 ferences in predator foraging strategies (i.e., carnivory versus fixed or adaptive omnivory) 42 influence long-term community structure as measured by the trophic cascade. We first 43 considered differences in trophic cascades between carnivores in food chains and fixed omni-44 vores that lack flexibility in their foraging strategies. We modeled the population dynamics 45 of three species: (i) basal producers, that are eaten by (ii) intermediate consumers and

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(iii) top omnivorous predators (omnivores hereinafter) that in turn consume both producers and consumers (Diehl & Feißel, 2000). Analyses of similar three-node trophic modules have 48 demonstrated how the coexistence of all species and community stability are sensitive to 49 variation in system primary productivity and the strength of omnivory (parameters ρ and ω 50 in eqs. 1 and 3 below; discussed extensively by McCann & Hastings, 1997;Diehl & Feißel, 51 2000Gellner & McCann, 2011). For this reason, a primary goal of our analysis was 52 to elucidate how primary productivity and omnivory strength interact to influence trophic 53 cascades in three species modules with and without true omnivory.

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Two omnivore foraging strategies with empirical support were considered. We refer to 55 the first as a fixed foraging strategy, indicating that foraging effort toward producers and 56 consumers comprise a constant proportion of the omnivores' total foraging effort (McCann 57 & Hastings, 1997;Diehl & Feißel, 2000). The second strategy, which we refer to as flexible 58 foraging, indicates that the effort apportioned toward either producers or consumers can 59 be represented as dynamical variables that depend on the relationships between resource 60 availability, reward, and omnivore fitness (Kondoh, 2003).

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Fixed foragers 62 We assume a linear (type I; Holling, 1959) functional response relating resource densities to 63 per capita consumption rates, so that the dynamics of species' biomasses are represented by 64 the system of equations where r, n, and p are the biomasses of producers, prey, and omnivores.

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Here, ρ and k are the producer productivity rate and carrying capacity, α is the con-67 sumer foraging rate, µ i is the per capita mortality rate of species i, and e i,j is the resource 68 i assimilation efficiency for consumer j. We assumed a total foraging rate β for omnivores, 69 that is apportioned toward consumers proportionately toω, whereω = 1 − ω. We therefore 70 interpret ω as a nondimensional parameter describing omnivory strength  ings, 1997); the system reduces to a food chain when ω = 0. See Table 1

Flexible foragers
Equations (1) can be modified to include flexible foraging behavior by the omnivore, by 75 substituting the omnivory strength parameter ω with the dynamical state variable Ω. Flex-76 ible foraging behavior was modeled using a replicator-like equation (Kondoh, 2003), which 77 provides a reasonable representation of flexible omnivory in real food webs (Fahimipour & 78 Anderson, 2015). The behavioral equation is whereΩ = 1 − Ω, γ = e r,p Ωβr + e n,pΩ βn is the flexible omnivore's instantaneous per capita 80 biomass production rate, and the constant v is a nondimensional ratio between the time 81 scales of foraging adaptation and omnivore population dynamics (Heckmann et al., 2012).

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Values of v > 1 represent behavioral changes that occur on faster time scales than omnivore 83 generations. This behavioral model implies that omnivores gradually adjust their foraging 84 strategy during search if behavioral changes yield a higher instantaneous per capita biomass 85 production rate than the current diet.

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Model nondimensionalizations and assumptions 87 The parameters in equations (1) and (2) were transformed into nondimensional parameters 88 using scaled quantities, reducing the total number of model parameters to those with values 89 having clear interpretations (Murray, 1993;Nisbet & Gurney, 2003). We use substitutions 90 similar to Amarasekare (2007):r = r/k,n = n/e r,n k,p = p/e r,p k,ρ = ρ/µ n ,α = αe r,n k/µ n , 91β = βe r,p k/µ n ,f = e r,n e n,p /e r,p ,δ = µ p /µ n , and τ = µ n t. After substituting into eqs. (1)   92 and (2), the hats were dropped for convenience giving the nondimensional system representing flexible foraging behavior. Scaled resource, consumer, and omnivore biomasses 94 are represented as x = (r, n, p). For fixed ω, the vector field which maps (r, n, p) to 95 (F (1) (r, n, p, ω), F (2) (r, n, p, ω), F (3) (r, n, p, ω)) is denoted by F O,ω : R 3 → R 3 , and the co-96 existence equilibrium of the fixed foraging model eq. (3) is denoted by . 97 We assume that the equilibrium is stable, satisfying . We assume that the flexible 106 model likewise has a coexistence equilibrium This measure κ O of the relative cascade strength is similar to the "proportional response"  3) and (4) (Pace et al., 1999;Shurin et al., 2010;Kratina et al., 2012;Wootton, 2017) held when primary 135 productivity, ρ, was below a critical value (Fig. 1, orange region). However, we identified 136 a critical transition in relative trophic cascade strengths depending on productivity, and we 137 refer to this value as ρ crit or the critical productivity for convenience (Fig. 1). above this value, cascades induced by omnivores are non-intuitively stronger compared to 141 carnivores ( Fig. 1, blue region). Note that the transition from weaker (κ O < 0) to stronger 142 (κ O > 0) cascades with productivity does not depend on omnivory strength. Instead, om-143 nivory strengths near the system's extinction boundaries attenuate the discrepency between 144 cascade strengths, such that omnivory cascades are weakest when productivity is low and 145 omnivory strength approaches values leading to omnivore exclusion (Fig. 1, region i), and 146 strongest when productivity is high and omnivores have nearly excluded consumers (Fig. 1,   147 region ii).

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To explain the non-intuitive result of stronger cascades with fixed omnivory, we exam-149 ined the relationship between system primary productivity and the optimal foraging effort 150 that would lead to the highest per capita growth rate by omnivores at equilibrium, Ω ⋆ F (Sup-

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plementary Table 1). The dotted grey curve in Fig. 1 illustrates Ω ⋆ F as a function of ρ; the 152 growth rate-maximizing strategy monotonically approaches pure herbivory with increasing 153 productivity. Precisely at ρ > ρ crit , the optimal strategy becomes unsustainable for fixed  Table 1). The curve is solid if the critical productivity lies in the 3-species coexistence region, and dashed otherwise. The light and dark grey shaded regions mark the extinction of consumers and omnivores respectively. Parameter values are the same as in Fig. 1 with is sensitive to other model parameters. In Fig. 2 we show how ρ crit (i.e., the location 158 of the vertical dashed line in Figure 1 along the x-axis) depends on the ratio of resource 159 profitabilities, f . Recall that larger f values represent systems in which consumers are much 160 more profitable to omnivores than producers. For large enough profitability ratios, the curve 161 of ρ crit versus f enters a region in which coexistence between all species cannot be achieved 162 (Fig. 2). Thus, for large enough f we would not expect to see strong cascades with omnivory, 163 regardless of other population-or community-level properties. In Supplementary Fig. 1 The ratio of flexible omnivory to linear chain trophic cascade strengths, As p ⋆ F > 0 and evidently f β 2 > 0, we must have ϕ > 0. Moreover, since 0 < Ω ⋆ F < 1, we 173 must also have δf ρ > βf ρ − αδ. That is, Combining (7)  libria, κ F < 0, since ϕ < 0 cannot be true for a biological system. Thus, consistent with 176 conceptual models of trophic cascades (Strong, 1992;Pace et al., 1999), cascades in systems 177 with flexibly foraging top omnivores will be bounded in strength by those in their analogous 178 food chains. Numerical results confirm these analytical expectations, and illustrate how in-179 creasing consumer reward (i.e., increasing f ) attenuates this result but does not alter the 180 qualitative relationship between κ F and ρ (Fig. 3).

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Intuition suggests that trophic cascade will not occur when top predators additionally feed 183 on primary producers (Polis & Strong, 1996;Pace et al., 1999;Duffy et al., 2007;Shurin et 184 al., 2010;Kratina et al., 2012;Wootton, 2017), but our results predict that strong cascades 185 will emerge under a wider range of foraging types than previously appreciated. We identi-186 fied many cases in which omnivores are indeed likely to generate weak cascades, although 187 we have shown that this should not be a uniform expectation for omnivory in food webs.

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Particularly, in high productivity systems in which forging reward does not strongly differ 189 between producers and consumers ( Fig. 1; Fig. 2), omnivores were capable of generating 190 stronger cascades than would be expected if they did not consume producers at all. This 191 result provides at least one general explanation for the weaker (Finke & Denno, 2005;Denno 192 & Finke, 2006), comparable or indistinguishable (Borer et al., 2005), and stronger (Okun et 193 al., 2008;France, 2012) 198 Comparisons of fixed and flexible models showed that omnivores were capable of generating strong cascades only when consuming an energetically suboptimal level of primary 200 producers (Fig. 1). This leads to the question: how common is this type of fixed foraging 201 in food webs? Empirical evidence for fixed foragers exists for groups as diverse as protists, 202 arthropods, and mammals (Clark, 1982;Mooney & Tillberg, 2005;Diehl & Feißel, 2001).

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Fixed omnivory may also manifest in other ways, for example when organisms forage in 204 a way that is suboptimal in terms of pure energetics but is otherwise required to maintain 205 nutritional or stoichiometric balances (Berthoud & Seeley, 1999;Remonti et al., 2016;Zhang 206 et al., 2018). Suboptimal foraging has also been observed in heavily disturbed or human-207 altered systems where consumer behaviors are not adapted to current resource conditions, or 208 when changes in habitat structure alter the ability to efficiently locate preferred food sources 209 (Walsh et al., 2006). across the tree of life has shown some association with organismal brain sizes and body 216 masses by proxy (Eisenberg & Wilson, 1978;Rooney et al., 2008). Body mass distributions 217 may also influence cascades that are induced by species with size-mediated ontogenetic shifts 218 from herbivory to carnivory (Pace et al., 1999)

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Conclusions 247 Omnivory has long been cited as a reason for why trophic cascades are less frequent or weaker 248 than expected, although empirical data on the role of omnivory has been equivocal (Borer 249 et al., 2005;Shurin et al., 2010;Kratina et al., 2012;Wootton, 2017  scaled consumer search rate, α, (b) scaled omnivore search rate, β, and (c) scaled omnivore mortality rate δ ( Table 1). The curve is solid if the critical productivity lies in the 3-species coexistence region, and dashed otherwise. The light and dark grey shaded regions mark the extinction of consumers and omnivores respectively. The parameter values are the same as in Fig. 1 with ω = 0.4.