Oikos 118: 1633Á1646, 2009 doi: 10.1111/j.1600-0706.2009.17437.x, # 2009 The Authors. Journal compilation # 2009 Oikos Subject Editor: Frank Van Veen. Accepted 7 May 2009
Modeling herbivore competition mediated by inducible changes in plant quality
Kurt E. Anderson, Brian D. Inouye and Nora Underwood
K. E. Anderson ([email protected]), B. D. Inouye and N. Underwood, Dept of Biological Science, Florida State Univ., Tallahassee, FL 32306-4295, USA. KEA also at: Dept of Biology, Univ. of California, Riverside, CA 92521, USA.
Competition between herbivorous insects often occurs as a trait mediated indirect effect mediated by inducible changes in plant quality rather than a direct effect mediated by plant biomass. While plant-mediated competition likely influences many herbivores, progress linking studies of plant-mediated competition in terrestrial phytophagous insects to longer- term consequences for herbivore communities has been elusive, and there is little relevant theory to guide this effort. We present simple models describing plant-mediated interactions between two herbivorous insects or other functionally equivalent organisms. These models consider general features of plant-mediated competition including specificity of elicitation by and effects on herbivores, positive and negative interactions among herbivores, competition independent of changes in plant biomass, and the existence of multiple relevant plant traits. Our analyses generate four important conclusions. First, herbivores competing strongly via only one plant quality phenotype exhibit a limited range of outcomes. These include coexistence and competitive exclusion of either herbivore, but do not include initial condition dependence. Second, when the outcome of competition is competitive exclusion, the herbivore that persists is the one that can do so under the highest inducible reductions in plant quality. Third, competition via more than one inducible phenotype can exhibit a wider range of outcomes including multiple equilibria and initial condition dependence. Finally, transient dynamics may not predict the eventual outcome of competition when changes in plant quality are slow relative to herbivore population growth, especially when herbivores compete through multiple phenotypes. We interpret our results in terms of competition outcomes reported in the literature, and suggest directions for the future empirical study of herbivore competition mediated by inducible changes in plant quality.
Trait mediated indirect interactions (or interaction mod- Baldwin 1997, Walling 2000). The seeming prevalence of ifications) are interactions where an indirect effect is induced changes in plant quality suggests that competition mediated by changes in organismal traits (physical or between phytophagous insects is often occurring as a trait- behavioral) rather than by changes in density. There is mediated indirect effect (Ohgushi 2005, Kessler and growing evidence of the prevalence of trait-mediated Halitschke 2007). Competition studies include numerous indirect interactions from a variety of taxa (reviewed by cases of herbivore species influencing each other’s perfor- Werner and Peacor 2003). While trait-mediated indirect mance through effects on the host plant, even when interactions are still rarely included in models of multi- herbivores occupy the plant at different times (Harrison species interactions, theoretical and empirical studies and Karban 1986, Viswanathan et al. 2007), or in different suggest their potential importance in both short-term locations such as roots versus leaves (Kaplan et al. 2008). A studies of communities and longer-term community dy- wide variety of plant defenses and other traits can be namics and structure (Bolker et al. 2003, Werner and influenced by herbivore damage, and plant responses may Peacor 2003). vary with the type of damage received (Denno et al. 1995, Competition between phytophagus insects is emerging Karban and Baldwin 1997, Walling 2000, Ohgushi 2005, as an example where trait-mediated indirect interactions Kessler and Halitschke 2007). Effects of one herbivore on may have potentially widespread community effects. Such another can be due to the induction (Stout et al. 1998, competition was once thought to be rare, but studies Viswanathan et al. 2005, 2007, Kaplan et al. 2008) or documenting negative interactions among phytophagus suppression (Walling 2000, Zarate et al. 2007) of plant insects are increasingly common (reviewed by Denno defensive traits, or due to changes in other traits such as et al. 1995, Kaplan and Denno 2007). Renewed interest nutritional content (Denno et al. 2000, Sandstrom et al. in competition between herbivorous insects has been fueled 2000). This complexity can result in specificity in the in part by growth in our understanding of changes in host- elicitation of plant responses (different herbivores cause plant quality induced by herbivore feeding (Karban and different changes in the plant) and specificity in the effect of
1633 induced plant changes on the herbivores (different herbi- functionally equivalent organisms as a trait-mediated vores may respond differently to the same plant traits). indirect interaction generated by inducible changes in plant A recent meta-analysis of studies addressing competition quality. These models consider general features of plant in phytophagous insects (Kaplan and Denno 2007) quality mediated competition that have arisen in the reinforces the argument that it differs markedly from empirical literature (e.g. specificity of elicitation and effects, competition described by traditional conceptual frame- positive and negative interactions, competition independent works. In particular, Kaplan and Denno (2007) found from resource biomass, and the existence of multiple that competition between phytophagous insects is 1) often relevant plant pathways or traits). These models should highly asymmetrical (and may include facilitation when not be taken as descriptions of any particular phytophagus mediated by plant quality), 2) likely to occur between insect system. Instead, we have chosen to begin with species that belong to different feeding guilds and that are minimally detailed models because this allows us to focus spatially and/or temporally separated, and 3) often of the on general features and point to potential future directions same magnitude regardless of the amount of plant biomass for the field as opposed to making system-specific predic- consumed. While these results deviate from expectations tions; we expect it will be interesting to add additional from traditional competition theory, they are consistent details about mechanisms of induction and/or herbivore with the idea that competition among phytophagous insects behavior to these basic models. With this study, we also is commonly a trait-mediated indirect effect of plant quality hope to provide a framework for future theoretical and rather than quantity. Thus, availability or access to plant empirical work so that researchers can move towards biomass may be largely unimportant in determining the predictions about the importance of plant quality-mediated outcome of competition, making the factors leading to the competition for coexistence or the relative abundance of structure and dynamics of phytophagous insect commu- phytophagus insect competitors. nities somewhat cryptic. Mathematical models of competition are abundant, and some of these have included trait-mediated indirect effects (reviewed by Bolker et al. 2003, Abrams and Nakajima Model motivations and formulations 2007). However, most modern competition theory Á We model the population dynamics of two species of including that investigating indirect effects Á describes competing herbivores, H and H , feeding on a collection competition as being mediated by changes in the density 1 2 of plants or plant parts possessing inducible changes in or availability of a dynamic resource or prey (Armstrong and quality. We define plant quality by modeling the levels of McGehee 1980, Tilman 1982, Abrams 2003, Amarasakare inducible phenotypes (or other changes in quality) I that 2003, Abrams and Nakajima 2007). In contrast, a focus on effect herbivore reproduction and survival. We assume that biomass dynamics in terrestrial plantÁinsect systems is potentially counterproductive given that the strength of populations are ‘well mixed’, i.e. we ignore within-popula- competition between phytophagous insects may frequently tion variation in induction (Edelstein-Keshet and Rausher be independent from changes in biomass (Kaplan and 1989, Lundberg et al. 1994) and herbivore densities (i.e. Denno 2007). Rather, inducible changes in terrestrial plant aggregation, Underwood et al. 2005). Following the results phenotypes should be viewed as changes in host plant quality of Kaplan and Denno (2007) discussed in the introduction, rather than changes in biomass quantity (Karban and we do not model plant demography or biomass, focusing Baldwin 1997, Agrawal 2005, Kaplan and Denno 2007), only on the average level that induced defenses or phenotype as has been done in limited modeling studies of effects of changes have achieved in the population. We revisit the induced plant changes on the dynamics of single species of potential implications of these assumptions in the discussion. phytophagus insect herbivores (Edelstein-Keshet and When two populations compete, both intra- and Rausher 1989, Lundberg et al. 1994, Morris and Dwyer interspecific competition contribute to their dynamics. 1997, Underwood 1999, Underwood et al. 2005). To our Many previous empirical studies have clearly established knowledge, no modeling study of competition between that phytophagous insects can have either positive or phytophagous insects has explicitly considered effects of negative effects on the densities of other species (Kaplan inducible terrestrial plant defenses. Changes in biomass and Denno 2007). Far fewer empirical studies have available for consumption that are mediated by inducible addressed both intra- and interspecific effects for competing defenses have been shown to have substantial effects on herbivores in a single system, but again these studies stability in models of aquatic food chains (Vos et al. 2004), indicate that interactions can range from simple (all intra- yet we argue that a qualitatively different model structure is and interspecific effects negative, Denno et al. 2000) to required to address inducible defenses in terrestrial plants, complex (both positive and negative intra- and interspecific where multiple individual insects compete on the same plant effects, Van Zandt and Agrawal 2004). Our models over time versus systems where individual predators eat represent three scenarios inspired by interactions that have entire algal or zooplankter individuals. Without a well been observed empirically, although a complete description developed body of theory to draw upon, it is difficult to of these scenarios is not yet available for any particular formulate specific predictions for particular systems (e.g. system. Because Kaplan and Denno (2007) report that ‘species A and B will suppress species B over time’), or to 32/243 of studies surveyed include facilitation mediated by design experiments appropriate for assessing the long-term inducible changes in plant quality, we include this feature in consequences of plant-quality mediated competition. two of our models. The qualitative forms of these models In this paper we present simple models describing are illustrated in Fig. 1 and their mathematical details are competition between two phytophagus insects or other given below.
1634 a) SPC: – – between two herbivore species is mediated indirectly by a single inducible defense or plant quality phenotype H H (Fig. 1a), 1 2 + dH K H f (I) – 1 r H 1 1 1 + – 1 1 dt K1 I dH K H f (I) 2 r H 2 2 2 dt 2 2 K – 2 dI r (I; H )r (I; H ) |{z}dI (1) dt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}1 1 2 2 b) SPF: – – elicitation by herbivory and self-inhibition induction decay The induced defense level could represent a single trait or H H 1 2 the combined effects of several traits that, because of cross- + + talk (Karban and Baldwin 1997) or shared resource pools, + – act in concert and function as a single quality phenotype. In I the absence of induced changes in plant quality, each herbivore population j grows logistically up to a carrying – capacity Kj that is set by external factors for which there is no interspecific competition (e.g. intraspecific competition for oviposition or foraging sites; predation or parasitism). c) MP: – – Induced defenses lower the population growth rate of species j with effect fj(I), and species j elicits increases in the induced defense (i.e. decreases in quality) at rate r (I, H ). H1 H2 j j – – Specificity of induction effect occurs when f1(I)"f2(I) and 1 specificity of elicitation occurs when r1(I; H1)H1 " + + + + 1 r2(I; H2)H2 : Changes in induced defenses exhibit self- 1rj I1 I2 limitation, B0; such as would occur due to nutrient 1I – – limitation or autotoxicity (Baldwin and Callahan 1993), and decays at a constant per unit rate d (e.g. as observed by Figure 1. Diagrams illustrating interactions assumed in the three Underwood 1998). models. Arrows indicate direct causal effects; signs indicate whether these are positive or negative effects. (a) In the single phenotype competition (SPC) model, two herbivore species H1 Single phenotype faciltation (SPF) model and H2 compete indirectly by inducing changes in a quality Several studies report positive plant-mediated interspecific phenotype I and each species has negative intraspecific competi- and intraspecific effects for some herbivores, often without tion. Denno et al. (2000) finds this pattern for two planthopper species feeding on Spartina alterniflora (intraspecific effects on describing the responsible mechanism (Karban and Baldwin both herbivores negative ( , ) and interspecific effects on both 1997, Stout et al. 1998, Van Zandt and Agrawal 2004, herbivores negative ( , )). (b) In the single phenotype facilita- Kaplan and Denno 2007), although facilitation by induced tion (SPF) model, a competitor herbivore H1 benefits from susceptibility to herbivory may be mediated by either induced changes in plant quality, while a facilitator herbivore chemical (Lu et al. 2004) or morphological changes H2 suffers. Viswanathan et al. (2005) finds a similar pattern for (Utsumi and Ohgushi 2008) in the plant and by orienting flea beetles Psylliodes affinis and tortoise beetles Plagiometriona behavior of the herbivore (Hitchner et al. 2008). Thus, we clavata feeding on Solanum dulcamara (intraspecific effects ,0 model an asymmetric interaction (Fig. 1b) where one and interspecific effects ,). (c) In the multiple phenotype herbivore species is facilitated with effect g(I) from induced (MP) model, each herbivore species elicits changes in independent changes in the plant phenotype, induced plant quality phenotypes I1 and I2. Herbivores benefit from increases in the phenotype they elicit, and suffer from dH1 K1 H1 g(I) increases in the phenotype elicited by their competitor. Van Zandt r1H1 (2) and Agrawal (2004) finds a similar pattern for weevils Rhyssomatus dt K1 lineaticollis and monarch larvae Danaus plexippus feeding on The dynamics of herbivore H2 and the induced quality level Asclepias syriaca (intraspecific effects , and interspecific effects I are as described in Eq. 1. Changes in plant quality caused , 0). For the three empirical studies cited here, a ‘’ indicates a by H have a negative effect on H , so we refer to H as the significant positive effect at pB0.05 (facilitation), a ‘ ’ indicates 1 2 1 a significant negative effect (competition) and ‘0’ indicates no competitor herbivore. Since changes in plant quality caused significant effect. by H2 have a positive effect on H1, we refer to H2 as the facilitator herbivore.
Single phenotype competition (SPC) model Multiple phenotype (MP) model We begin with a simple model of a reciprocally negative Plants may possess multiple induction pathways and quality plant-mediated competitive interaction (Fig. 1). Our model phenotypes (e.g. secondary compounds, Walling 2000), extends LotkaÁVolterra competition such that competition and these phenotypes may have different effects on different
1635 herbivore species (Karban and Baldwin 1997). For example, strengths of effect on herbivore growth rates but herbivores different herbivore species may induce alternative secondary do not differ in the strengths with which they induce compounds, and then respond differently to varying changes in plant quality, concentrations of these compounds. We therefore present a multiple phenotype model where we consider two species a (a bI)H for IB of herbivores that are each capable of eliciting the j b production of a separate phenotype (Fig. 1c). Our model f (I) g I and r (I; H ) j j j j a reflects an empirical scenario described by Van Zandt and 0 for I] Agrawal (2004; Fig. 1c) where each induced phenotype b change has a positive effect on the eliciting herbivore’s (4) density (i.e. facilitation) while having a negative effect on the density of the non-eliciting (competing) herbivore, Here, gj is the per unit reduction in the growth rate of herbivore j caused by induction of defense or quality dH K H f (I ) g (I ) 1 1 1 1 2 1 1 changes, a is the maximum per capita induced defense or r1H1 dt K1 quality change elicitation rate, and b is the per unit dH K H f (I ) g (I ) reduction in the elicitation rate due to plant self-limitation. 2 r H 2 2 2 1 2 2 dt 2 2 K This simplification increases the ease of analysis, yet our 2 qualitative results are general for models in the form of dI 1 r (I ; H ) d I Eq. 1 (Supplementary material Appendix 1). dt 1 1 1 1 1 We examine these three outcomes using the zero isoclines of the system. Fig. 2 presents examples of isoclines dI2 r2(I2; H2) d2I2 (3) for each observed outcome of competition; isoclines for dt each herbivore are presented unconventionally assuming We are assuming for simplicity that each herbivore can that it has equilibrated with induced changes in quality for only directly elicit changes in its own beneficial induced simplicity. Results inferred from isoclines are identical to phenotype and the induced phenotypes have no direct those yielded from traditional invasion analysis (Supple- effects on one another at the level of the plant population mentary material Appendix 1). (although ‘cross-talk’ could still occur at the level of Indirect density-dependence resulting from feedbacks individual plants). We also assume the same restrictions between induced changes in quality and the herbivore on functional forms as for Eq. 1 and 2, making this model a populations causes the isoclines to be concave towards the general model of which the SPC and SPF models are special origin. For comparison, two-species LotkaÁVolterra com- cases. The links between the MP model and SPC and SPF petition with linear competition coefficients possesses linear models will allow us to explore which combinations of isoclines, while similar equations with density-dependent competitive and facilitative interactions tend to lead to competition coefficients typically possess either concave or behavior described by the simpler SPC and SPF models. convex isoclines (Ayala et al. 1973). The isocline of each herbivore j intersects its own axis at its equilibrium in the absence of the other herbivore species i, Analyses and results + + 1 H j Kj gjI bKj agj d We analyze the three models above with a focus on j Hi 0 2b equilibrium outcomes in order to provide insights into qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi how competition occurring as a trait-mediated indirect 2 interaction resulting from induced changes in plant quality ( bKj agj d) 4dbKj (5) might influence long-term herbivore population dynamics and community structure. However, because of seasonal or and intersects the axis of the other herbivore i when stochastic environmental variation, ecological systems may often not attain equilibrium, and, even when they do, + + dKj following herbivore dynamics over the generations required Hj 0andHi (6) to approach equilibrium can be logistically difficult. There- agj bKj fore we also present transient dynamics, which are most relevant to the results of the short-term experiments or Equation 6 is true when agj bKj, i.e. induced changes in highly variable systems common in the literature. quality must possess the ability to drive Hj extinct (Fig. 2a). If agjBbKj induced changes in quality can not reach a high enough level to drive Hj extinct Á even when the competing Equilibrium outcomes in the single phenotype herbivore is held at infinitely large densities Á and the Hj competition (SPC) model isocline never crosses the competing herbivore species’ axis (Fig. 2b). The SPC model exhibits three potential outcomes: extinc- We use these results to delineate conditions that lead to tion of H1, extinction of H2, and coexistence (Supplemen- each possible outcome of the SPC model. When agj bKj tary material Appendix 1). We explore these outcomes for j 1,2, the conditions leading to the different outcomes using a simple form of Eq. 1 where induction has different are as follows:
1636 Zero isoclines Numerical simulations Induction 3 1.2 8 Coexistence Herbivore 1 Herbivore 1 Herbivore 2 1.0 Herbivore 2 6 2 0.8 0.6 4 1 0.4 Herbivore 2 2 Induction level
Herbivore density 0.2 0 0 0 0 1 2 3 0 2 4 6 8 10 Herbivore 1 Time
Coexistence 3 1.2 8 1.0 6 2 0.8 0.6 4 1 0.4 Herbivore 2 2 Induction level
Herbivore density 0.2 0 0 0 0 1 2 3 0 2 4 6 8 10 Herbivore 1 Time
Herbivore 1 3 1.2 8 1.0 wins 6 2 0.8 0.6 4 1 0.4 Herbivore 2
2 Induction level
Herbivore density 0.2 0 0 0 0 1 2 3 0 2 4 6 8 10 Herbivore 1 Time
Herbivore 2 3 1.2 8 1.0 wins 6 2 0.8 0.6 4 1 0.4 Herbivore 2
2 Induction level
Herbivore density 0.2 0 0 0 0 1 2 3 0 2 4 6 8 10 Herbivore 1 Time
Figure 2. Examples of coexistence and extinction outcomes for the SPC model. Plots on the left are isoclines that represent the
dI dHj combination of herbivore and levels of induced plant quality changes where the density of herbivore do not change (/ 0and 0; dt dt
/ j 1; 2): Plots on the right present numerical simulations of the outcome illustrated to its left. Parameters used for the first coexistence scenario where agj bKj; j 1; 2: a 10; b 1; d 0.75; r1 10; r2 10; K1 1; K2 2; g1 0.125; g2 0.3. Parameters used for the second coexistence scenario where agj BbKj; j 1; 2: same as above; g1 0.095; g2 0.199. Parameters used for H2 extinction scenario: same as first coexistence scenario; g2 0.6. Parameters used for H1 extinction scenario: same as first coexistence scenario; g1 0.27.
dK + dK + 1) Both herbivores coexist when 1 H j 3) H excludes H when 1 BH j and 2 H1 0 2 1 2 H1 0 ag1 bK1 ag1 bK1
dK + dK + and 2 H j 2 H j 1 H2 0 1 H2 0 ag2 bK2 ag2 bK2
dK1 + (7) 2) H excludes H when H j and 1 2 2 H1 0 ag1 bK1 Coexistence is always guaranteed when ag BbK and dK + 1 1 2 BH j ag BbK .Ifag BbK but ag bK , species i will be able 1 H2 0 2 2 j j i i ag2 bK2 to coexist with species j when
1637 dK + illustrate using an example in Supplementary material i H j (8) j Hi 0 Appendix 2. agi bKi While the results in Eq. 7 and 8 appear cumbersome, their ecological interpretation is straightforward. An herbivore Equilibrium outcomes in the single phenotype can be driven extinct by induced defenses or other changes facilitation (SPF) model in plant quality when the quality change required to do so is less than that generated by the competing herbivore at its Since the competitor species benefits from species specific equilibrium in isolation. The herbivore that wins in induced increases in plant quality, there are only two competition is able to persist at higher densities in the outcomes for the SPF model Eq. 2: coexistence, and presence of induced defenses or other changes in plant extinction of the facilitator species. quality. Unsurprisingly, these results are similar to the Consider Eq. 2 and 4 where the competitor receives a coexistence criteria from LotkaÁVolterra competition; linear benefit from species specific increases in plant quality, namely, that species must compete more strongly intra- specifically than inter-specifically in order to coexist. g(I) fI (9) In contrast to LotkaÁVolterra competition, the possible The isocline of the competitor intersects its own axis at the equilibrium outcomes in the SPC model are never equilibrium density dependent on initial conditions. Initial condition depen- + + 1 dence arises in LotkaÁVolterra competition when both H j K fI bK af d 1 H2 0 1 1 species compete inter-specifically more strongly than intra- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b specifically. These dynamics can not occur in the SPC ( )2 ( ) model because competition is mediated by dynamic feed- bK1 afd 4dbK1 10 backs between the two herbivore competitors and induction of changes in plant quality. At least one species therefore and trends away from the facilitator axis as the density of competes more strongly with itself over a range of densities: the facilitator increases (Fig. 3); the competitor has a higher as its own density goes up, the herbivore causes concomitant equilibrium density in the presence of the facilitator species increases in induced defenses or other changes in plant than in its absence. The intersection points of the quality. These results do not depend on the precise facilitator’s isocline are the same as in the SPC model. Thus, the facilitator will coexist with the competitor when functions defining the induction effect, fj, and induction elicitation, rj, in Eq. 1, as long as the signs of these dK2 + functions lead to the relationships pictured in Fig. 1. ag2 BbK2 or H j and ag b 1 H2 0 Changing the functional forms can however alter the 2 K2 relative frequency of the different outcomes, which we ag2 bK2; (11)
Zero isoclines Numerical simulations Induction 3 1.0 10 Competitor Competitor Facilitator Facilitator 2 0.5 5
Competitor 1 Herbivore density Herbivore 0 0 0 0 1 2 3 0 2 4 6 8 10 Facilitator Time
3 1.0 6
2 4 0.5
Competitor 1 2 Induction levelInduction level Induction Herbivore density Herbivore
0 0 0 0 1 2 3 0 2 4 6 810 Facilitator Time
Figure 3. Examples of coexistence and extinction outcomes for the facilitation model. Parameters used for the coexistence scenario: a 10, b 1, d 0.75, r1 1, r2 1, K1 0.1, K2 1, g1 0.1, g2 0.125. Parameters used for competitor extinction scenario: same as above, K2 0.4.
1638 and otherwise goes extinct. The interpretation of this Hj is zero or small has important consequences for criterion is similar to that of the competition model; the predicting the outcome of competition. Either herbivore facilitator is able to coexist when the induced change in Hj is guaranteed to persist when plant quality required to drive it extinct is larger than the change in plant quality elicited by the competitor when the diKj + aigj BbiKj or H j and facilitator is not present. These qualitative results again hold a g b K i Hi 0 for other versions of the SPF model as long as the signs of i j i j the induced change in plant quality and competition aigj biKj (14) functions are similar among models. These conditions have interpretations that are similar to those for the SPC and SPF models. In contrast with the SPC and SPF models, however, parameter combinations in Equilibrium outcomes in the multiple phenotype d K + (MP) model the region i j BH j when a g b K do not i Hi 0 i j i j aigj biKj The MP model exhibits a much broader array of dynamical always lead to extinction of Hj. Rather, the fate of the behaviors than the SPC and SPF models. Numerical herbivore population may be determined by initial condi- simulations demonstrate that the MP model possesses tions in this parameter range. When initial condition four possible behaviors: extinction of H , extinction of 1 dependence is possible, an herbivore Hj will persist if its H2, coexistence, and extinction or coexistence determined initial density is high enough to elicit a large enough level of by initial conditions. This last condition is not possible in its beneficial phenotype in the face of increases in the the SPC or SPF models. For ease of comparison, we adopt detrimental phenotype, /giIj BKi fiIi: This can occur the same functional forms as Eq. 4 and 9. even when H is capable of achieving equilibrium densities Isoclines of the MP model are no longer monotonic i that can drive Hj extinct. Because Hj can exhibit negative because of the complex feedbacks between the two population growth at low densities and positive growth at herbivores and induced plant quality phenotypes. The higher densities, initial condition dependence manifests as isoclines are composed of two branches that join at a an apparent Allee effect whose strength is dependent on singularity and are no longer monotonic when this point is both the initial density of the competing herbivore and the positive (Fig. 4). The shapes of these isoclines can allow for levels of induction already present. multiple crossing points that correspond to equilibria under As already stated, the initial condition dependence in many parameter values. Although these equilibria are not competition outcomes occurs when interspecific competi- necessarily all stable, more than one typically is, leading to tion is strong relative to intraspecific density dependence initial dependence in the outcome. This scenario shares (Fig. 5). This occurs, for example, when effects of inducible many similarities with the initial condition dependent changes in plant quality g or elicitation rates a of these outcome in simple LotkaÁVolterra competition, where changes are high, or when herbivore carrying capacities K or strong interspecific competition relative to intraspecific self-inhibition b or decay d in induced changes in plant competition de-stabilizes the coexistence equilibrium. How- quality are small (results for b and d not shown). When ever, unlike LotkaÁVolterra competition, all observable plant-mediated competition becomes strong, coexistence outcomes of competition Á extinction of H1 or H2 and outcomes are replaced by outcomes determined by initial coexistence Á can be possible under the exact same conditions. Under very strong plant-mediated competition, parameter values in the MP model as a result of the effects facilitation can no longer ‘rescue’ small populations in the of herbivore facilitation by induced changes in plant quality face of strong interspecific competition. As a result, initial (Fig. 4). condition dependent outcomes no longer include coex- Despite their complex shape, the herbivore isoclines still istence equilibria, meaning that one competitor will always intersect their own and their competitor’s axes at only one eventually win in competition. Increasing the carry capa- point each, cities of both herbivores weakens interspecific competition + + 1 relative to intraspecific processes and increases the potential Hj jH 0 Kj fjIj bjKj ajfj dj for coexistence. Unsurprisingly, strong asymmetries in i 2b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij model parameters that translate into strong asymmetries ( b K a f d )2 4d b K (12) in effects of induced changes in plant quality on inter- j j j j j j j j specific competition tend to promote competitive exclusion and of the herbivore more negatively affected by plant quality changes. + + d K The primary effect of facilitation by changes in plant H 0 when H i j and a g b K j i a g b K i j i j quality f is to promote coexistence. When the results of i j i j competition are dependent on initial conditions, facilitation (13) increases the combination of parameters where coexistence The intersection of Hj’s isocline with Hi’s axis is now outcomes co-occur with exclusion outcomes. This effect determined by the effects on Hj of the phenotype induced occurs even when interspecific competition is otherwise by Hi. The beneficial effects of Ij on Hj do not appear in highly asymmetric. Facilitation is not required for initial Eq. 13 because Hj does not elicit induced changes in plant condition dependence in the outcome of competition, nor quality Ij when at zero density. However, Hj will elicit Ij is it required for the existence of initial condition dependent even at small densities; this discrepancy in dynamics when outcomes that include both coexistence and exclusion
1639 a) 20 Herbivore 1 b) 20 Herbivore 2 H2 always wins
10 10 H1 always wins Herbivore 2 Herbivore 2
0 0 0 5 10 0 5 10 Herbivore 1 Herbivore 1
c) 20 d) 15 H1 & H2 always coexist H1 wins or H1 & H2 coexist
10 7.5 Herbivore 2 Herbivore 2
0 0 0 5 10 0 2.5 5 Herbivore 1 Herbivore 1
e) 15 f) 20 H2 wins or H1 & H2 coexist
7.5 10 H1 or H2 wins Herbivore 2 Herbivore 2
0 0 0 2.5 5 0 5 10 Herbivore 1 Herbivore 1
g) 15 H1 wins, H2 wins or H1 & H2 coexist
7.5 Herbivore 2
0 0 2.5 5 Herbivore 1
Figure 4. Examples of herbivore isoclines in the MP model. Closed squares (j) indicate stable equilibria while open squares (I) indicate unstable ones. Except where indicated, parameter values are a1 75, a2 100, b1 5, b2 3, d1 1, d2 1, r1 1, r2 1, K1 2, K2 9, g1 0.15, g2 0.65, 81 0.1, 82 0.3. (a) a1 90; a2 30. (b) a1 50, a2 70, K1 3. (c) a1 70, a2 40. (d) a1 90, a2 40. (e) a1 73, a2 53. (f) a1 180, a2 120, K1 2.7, K2 8.0. (g) a1 105, a2 62.
1640 No facilitation Facilitation 1 1
Y2 0.5 Y2 0.5
0 0 0 0.5 1 00.51 Y1 Y1
15 15
K K2 7.5 2 7.5
0 0 0 7.5 15 0 7.5 15
K1 K1
150 150
α α 2 75 2 75
0 0 0 75 150 0 75 150 α α 1 1
H1 always H2 always H1 & H2 H1 wins or H2 wins or H1 or H1 wins, wins wins always H1 & H2 H1 & H2 H2 wins H2 wins or coexist coexist coexist H1 & H2 coexist
Figure 5. Potential outcomes of competition as functions of model parameters. For results in the ‘No facilitation’ column, 81 82 0, while for those in the ‘Facilitation’ column, 81 0.3 and 82 0.1 for variation in g while 81 0.1 and 82 0.3 for variation in K and a. Other parameters are a1 75, a2 100, b1 5, b2 3, d1 1, d2 1, r1 1, r2 1, K1 2, K2 9, g1 0.15, g2 0.65, except where indicated. outcomes (Fig. 5). However, the latter scenario is much tered in empirical settings. Namely, we start all trajectories more likely when facilitation is present. with plants having expressed no inducible changes in plant quality (i.e. I 0) to mimic dynamics at the beginning of a growing season or experimental introductions on na¨ıve Transient dynamics plants. In addition, we present our results unconventionally using two-dimensional phase plots of the herbivores in the Transient dynamics of models like ours that contain SPC, SPF and MP models to mimic empirical scenarios multiple interacting populations can be highly irregular where herbivore densities, but not plant quality levels, are and depend strongly on initial conditions. Methods for observed or manipulated (Fig. 6). characterizing transient dynamics independent of initial Induced plant quality levels and herbivore populations in conditions exist (Neubert and Caswell 1997), yet the result the three models can ‘overshoot’ their equilibria, and these is a ‘worst case’ scenario that may only result from overshoots tend to occur when the dynamics of induced unrealistic parameter combinations. Instead, we opt to changes in plant quality range from slower to slightly faster illustrate transient dynamics using numerical simulations than the dynamics of the herbivores (Fig. 6aÁc, see also where initial conditions reflect scenarios typically encoun- Fig. 2Á3). Transient overshoots can lead to a substantial
1641 a) Slow induction d) Fast induction
1.5 1.5
1 1 SPC
0.5 0.5
0 0 0 0.5 1 1.5 0 0.5 1 1.5
b) e)
1.5 1.5
1 1 SPF
0.5 0.5 Herbivore 2
0 0 0 0.5 1 1.5 0 0.5 1 1.5
c) f)
12 12
8 8 MP 4 4
0 0 0 2 4 6 0 2 4 6 Herbivore 1
Figure 6. Phase plots illustrating transient dynamics of the herbivores proceeding on plants that possess no initial induced changes in plant quality, Ii,j(0) 0; because the system is viewed in two dimensions only, the dynamics of changes in plant quality are not shown. Dots (m) indicate initial herbivore densities, closed squares (j) stable equilibria, and open squares (I) unstable equilibria. (a, b, c) Plant quality and herbivore dynamics proceed at similar time scales. SPC model: a 10, b 1, d 1, r1 10, r2 10, K1 1, K2 2, g1 0.125, g2 0.3. SPF model: a 10, b 1, d 1, r1 10, r2 10, K1 0.1, K2 1, g1 0.1, g2 0.125. MP model: a1 75, a2 100, b1 5, b2 3, d1 1, d2 1, r1 10, r2 10, K1 2, K2 9, g1 0.15, g2 0.72, 81 0.3, 82 0.1. (d, e, f) The dynamics of inducible changes in plant quality are much faster than herbivore dynamics. SPC and SPF model: r1 0.01, r2 0.01, other parameters as in (a) and (b). MP: r1 1.0, r2 1.0, other parameters as in (c). mismatch between the initial and near-equilibrium trajec- if changes in plant quality are unknown: herbivore tories of the herbivore populations because of a transient dynamics may behave differently among observations even ‘escape’ from the slowly increasing effects of induced when the starting densities in each observation are the same. changes in plant quality; such transient ‘escapes’ are not While these results are not surprising from a dynamical possible if inducible changes in plant quality are much faster systems perspective, we use them to illustrate the potential than herbivore dynamics (Fig. 6dÁf). Furthermore, trajec- for competitive dynamics between herbivores that are tories originating from different initial herbivore densities mediated indirectly by inducible changes in plant quality may cross when induction dynamics are slow; these to appear variable or unpredictable when plant quality different trajectories proceed from points with similar dynamics are not anticipated. combinations of herbivore densities but different levels of Multiple equilibria in the MP model can elongate the plant quality. In practice, this could cause the effects of transient phase of dynamics relative to the SPC and SPF competition on herbivore densities to appear highly variable models and can cause herbivore densities to be quite
1642 variable before eventually approaching equilibrium (Fig. This is similar to our result that the herbivore that persists 6c). Unstable equilibria can attract herbivore population under the highest levels of negative inducible changes in densities to their vicinity before the trajectories proceed on plant quality wins when the outcome of competition is towards their final, stable equilibrium, a result that can also competitive exclusion. However, herbivore coexistence is be observed in traditional LotkaÁVolterra competition possible in our models because of self-limitation of (Hastings 2004). Again, this transient effect is strongest inducible changes in plant quality; such intraspecific when inducible changes in plant quality are slow. Extensive density-dependent regulation of the predator is typically simulations suggest that unstable equilibria have less effect not present in models of apparent competition. Our and initial condition dependence is less frequent when inclusion of plant self-limitation is justified by biologically induction responses are fast relative to herbivore dynamics reality, as physiological or morphological limits to inducible (Fig. 6f), although this pattern is not universal. changes in plant quality are likely widespread (Baldwin and Callahan 1993). In other contexts, inducible traits that reduce consumption tend to stabilize consumerÁresource Discussion interactions (Vos et al. 2004, but see Lundberg et al. 1994, Underwood 1999) while, as we show, multiple defensive Interactions among organisms may commonly be trait phenotypes can lead to multiple equilibria and initial mediated indirect interactions, i.e. mediated by changes in condition dependence in the outcome of competition behavior or other phenotypes rather than density. Recent (Ramos-Jiliberto et al. 2008). Facilitation by inducible research has shown that competition between phytophagous changes in plant quality strengthens this effect by increasing insects indirectly mediated by inducible changes in plant the region of parameter space corresponding to initial quality traits is common and could be an important factor condition dependence and adding the possibility of coex- in structuring herbivore communities (Denno et al. 1995, istence and competitive exclusion outcomes co-occurring Agrawal 2005, Ohgushi 2005, Kaplan and Denno 2007, under the same parameter combinations (Fig. 5). However, Kessler and Halitschke 2007). The majority of relevant it is the presence of multiple induced phenotypes, not studies to date have focused on documenting the occurrence facilitation, which leads to initial condition dependence of competition mediated by plant quality traits; explicit (Fig. 2, 3, 5). connections between short-term effects of trait-mediated Our results suggest that transient dynamics of herbivores indirect competition and the long-term population and competing via changes in plant quality may often not reflect community consequences are still lacking in terrestrial longer-term dynamics, and thus results of short-term plantÁherbivore systems (although links between inducible experiments may not reflect the eventual outcome of defenses and outcomes of competition have been demon- competition. Transient dynamics are increasingly recog- strated in aquatic systems, van der Stap 2008). As a step nized as important features of ecological interactions towards understanding the long-term consequences of (Hastings 2004) because ecological systems may rarely indirect plant-mediated competition, and in hopes of reach equilibrium due to temporal environmental variation providing guidance for future empirical studies, we have such as seasonality. Those mediated by inducible changes presented a set of models that encapsulate the basic forms of in plant quality are likely no exception. Inducible defenses plant-quality mediated competition documented in the in plants, for example, can take several hours to days to literature. Our analyses of these models generate four employ (Karban and Baldwin 1997, Underwood 1998). obvious results: (1) herbivores that compete via one plant While this is shorter than the typical herbivore generation quality phenotype exhibit a limited range of outcomes that time, delayed changes spanning several herbivore genera- does not include initial condition dependence; (2) when tions have also been documented (Haukioja 1990), and the outcome of competition is competitive exclusion, the results from our modeling suggest that competition out- herbivore that persists is the one that can do so under the comes in these types of systems may only be evident after highest inducible reductions in plant quality; (3) competi- multiple generations. Our MP model suggests that compe- tion via multiple phenotypes can exhibit a wide range of tition mediated by multiple independent plant quality traits outcomes which includes initial condition dependence is particularly likely to show mismatches between short- and when interspecific competition is strong relative to intras- long-term dynamics (Fig. 6). Experiments that are too short pecific density dependence; and (4) transient dynamics may to capture the full effects of induced changes in plant not predict the eventual outcome of competition when quality on competing herbivore populations could provide changes in plant quality are slow relative to herbivore misleading results, especially if herbivore populations population growth, especially when multiple equilibria are initially grow away from equilibrium outcomes that include present. low densities or exclusion of one of the herbivores. Our models of plant quality mediated competition share Experiments started with different induction levels could many similarities with more traditional models of con- also produce strikingly different short-term outcomes, even sumer-resource interactions in addition to LotkaÁVolterra if initiated with similar herbivore densities (Fig. 6). This competition theory. In particular, models of apparent result, as well as the potential for long transients, is a direct competition characterize interactions where prey species consequence of the high dimensionality of the system have negative reciprocal effects mediated through the introduced by including dynamic plant quality traits. While densities of a shared natural enemy (Holt et al. 1994). our results pertaining to transient dynamics are not The theory of apparent competition states that shared theoretically novel, they do highlight that models of predation only allows one prey species to persist whose phytophagous insect competition focused solely on density population can grow under the highest predation pressure. mediated competition, as opposed to including trait
1643 mediated indirect effects, exclude potentially important Despite their generality, the structure of our models dynamics. highlights three basic requirements for empirical experi- Field experiments have shown that the outcome of plant- ments to be able to address the longer-term consequences of mediated competition can differ dramatically among plant-mediated competition. First, and trivially, experiments replicates or years (Van Zandt and Agrawal 2004). Results must estimate effects of both competitors on each other of our models, particularly the MP model, suggest that this (reciprocal effects). Many competition experiments focus could be partially due to the internal dynamics of plant- on effects of one species on another (Kaplan and Denno mediated competition, which may be particularly sensitive 2007), but given the potential for feedbacks this is clearly to initial conditions. Sensitivity of plant quality phenotypes not sufficient for determining long term consequences for can strongly depend on the identity of the initial colonizing either species. Second, experiments should measure the herbivore, which in turn strongly effects the subsequent strength of competitive effects, not just the direction of distribution and abundance of other herbivore competitors statistically significant differences between competition and (Viswanathan et al. 2005, 2007). Thus, for plant responses no-competition treatments (Inouye 2001). For example, mediated by multiple phenotypes, relatively small differ- even in the simple case where competition is driven by a ences in herbivore densities at the beginning of each season single inducible plant phenotype (SPC model), with could lead to quite different results due to sensitivity to relatively fast changes in plant quality, competition can initial conditions. In addition, changes in herbivore growth lead to one of two outcomes: coexistence or eventual or plant quality could also change competitive dominance extinction of one of the herbivores, regardless of initial due to shifts in parameter values that change the number of densities (Fig. 2). Experiments that only demonstrate the potential outcomes, the effects of which could be exacer- sign of competitive effects cannot distinguish between these bated by environmental variability. Temporal variability outcomes as competition may lead to both species coexisting affecting plant demography, biomass, or quality could but at very low densities. Finally, it is important for hypothetically work to ‘reset’ the system with unpredictable experiments to measure intraspecific as well as interspecific starting densities of herbivores in different growing seasons, effects because intraspecific density dependence in per capita or cause herbivore growth rates and plant growing condi- growth rates caused by factors other than plant quality is a tions to change, while spatial variability in plant quantity or necessary condition for coexistence in all the models we have quality could create refuges for inferior competitors presented here. Very few published studies of plant- (Amarasekare 2003). mediated competition to date have measured combined Our models have been constructed to explore general intraspecific competition and reciprocal interspecific effects (Kaplan and Denno 2007) and none to our knowledge have features of indirect plant-quality mediated competition in included strengths of competition in addition to sign. phytophagous insects that have been extensively described We believe that methods are available to allow empiri- in the empirical literature. As such, they do not specify the cists to test the importance of plant-quality mediated exact mechanisms by which herbivores are affected by competition in phytophagous insects for longer-term com- inducible changes in plant quality. Plant quality may affect munity structure. Effects of competition Á even in systems herbivores in a number of ways (reviewed by Karban and with initial condition dependence Á can be detected using Baldwin 1997, Walling 2000), including changes in response surface experiments (Inouye 2001) that follow fecundity, changes in survival rates reflecting changes in herbivore populations started at several combinations of food quality, toxic compounds, or exposure to predators, initial densities. In addition, dissecting the chemical or and changes in development time due to reduced con- nutritional pathways underlying plant quality responses to sumption or assimilation of plant biomass. While our herbivores can determine whether one or multiple plant models highlight results that are relevant to many systems, phenotypes are involved (e.g. testing whether each herbivore generating predictions for any particular system influenced induces independent chemical or physical plant changes and by trait-mediated indirect interactions will require models the extent to which those changes affect either herbivore). that include greater mechanistic detail (Bolker et al. 2003). Determining the number of phenotypes mediating compe- Such detail may potentially include the biochemical and tition can identify whether competitive outcomes are likely biomass components that compose ‘plant quality’ as well as to be sensitive to the initial experimental conditions. the factors responsible for intraspecific density dependence Phenotypes could also be manipulated genetically or (i.e. the carrying capacity). Physiologically-structured mod- chemically (Thaler et al. 1996) to assess the degree to els can be used to quantify the effects of different plant- which plant-quality mediated competition is responsible for quality components and on herbivore growth, survival, and observed variation in herbivore community structure. reproduction in a way that links with population dynamics Assuming that long-term experiments are not possible, (Kooijman 2000). While most studies to date show predicting long-term outcomes will require parameterizing competition occurs without measurable effects of plant models with information from short-term experiments. biomass changes (Kaplan and Denno 2007), there are Doing this requires quantifying functional forms of undoubtedly situations where plant biomass does play an relationships between herbivore vital rates and plant quality important role (Morris 1997). Integrating inducible traits phenotypes. For example, one could relate herbivore that reduce herbivore consumption of plant biomass into growth, mortality, and recruitment rates to different levels our framework is straightforward, although their effects of inducible defenses, or, similarly, relate the strength of could be quite substantial if herbivore densities respond induction to different densities of herbivores. Given recent strongly to both plant biomass and plant quality (Abbott advances in our understanding of genetic, chemical, and et al. 2008). ecological factors that underlie plant quality phenotypes
1644 (Karban and Baldwin 1997, Walling 2000, Kessler and Kaplan, I. and Denno, R. F. 2007. Interspecific interactions in Halitschke 2007), such mechanistic studies should be phytophagous insects revisited: a quantitative assessment of increasingly possible. competition theory. Á Ecol. Lett. 10: 977Á994. Kaplan, I. et al. 2008. Physiological integration of roots and shoots in plant defense strategies links above- and belowground Acknowledgements Á The authors thank Peter Abrams and the herbivory. Á Ecol. Lett. 11: 841Á851. ecology reading group at Florida State University Á especially T. E. Karban, R. and Baldwin, I. T. 1997. Induced responses to X. Miller Á for helpful comments. KEA was funded by a USDA herbivory. Á Univ. of Chicago Press. CSREES Post-doctoral Fellowship, Award No. 2005-35302-1699, Kessler, A. and Halitschke, R. 2007. 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Supplementary material (available as Appendix O17437 at /
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