Abstract
Two herbivorous species that share a single plant can interact indirectly with one another, even without direct interaction. One type of indirect interaction is exploitative resource competition, which results from a reduction in plant biomass; another type is that caused by changes in plant traits. These are referred to as indirect effects, mediated, respectively, by biomass and plant traits. The two indirect effect types often occur simultaneously, and they are difficult to partition. To investigate the roles of the two indirect effects on both herbivores, a dynamic oneplant, twoherbivore system model was analyzed assuming the spatiotemporal cooccurrence of the herbivores and the plants’ nonspecific induced defenses. Our analysis revealed that the densities of coexisting competitively superior and subordinate herbivores were regulated by negative indirect effects mediated by plant biomass and defense, respectively. This indicates that indirect effects mediated by plant biomass and plant traits can be important regulators of herbivore population size in equilibrium with herbivore coexistence. Our results could be generally applicable to plant–herbivore interactions with nonspecific plant defense that is induced through both intra and transgenerational responses.
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References
Abrams PA (1987) On classifying interactions between populations. Oecologia 73:272–281. https://doi.org/10.1007/Bf00377518
Agrawal AA (2002) Herbivory and maternal effects: mechanisms and consequences of transgenerational induced plant resistance. Ecology 83:3408–3415. https://doi.org/10.2307/3072089
Anderson MJ (1999) Distinguishing direct from indirect effects of grazers in intertidal estuarine assemblages. J Exp Mar Biol Ecol 234:199–218. https://doi.org/10.1016/S00220981(98)001592
Anderson KE, Inouye BD, Underwood N (2009) Modeling herbivore competition mediated by inducible changes in plant quality. Oikos 118:1633–1646. https://doi.org/10.1111/j.16000706.2009.17437.x
Bateman AW, Vos M, Anholt BR (2014) When to defend: antipredator defenses and the predation sequence. Am Nat 183:847–855. https://doi.org/10.1086/675903
Chase JM, Leibold MA (2003) Ecological niches: linking classical and contemporary approaches. University of Chicago Press, Chicago
Chen MS (2008) Inducible direct plant defense against insect herbivores: a review. Insect Sci 15:101–114. https://doi.org/10.1111/j.17447917.2008.00190.x
Denno RF, Mcclure MS, Ott JR (1995) Interspecific interactions in phytophagous insects  competition reexamined and resurrected. Annu Rev Entomol 40:297–331. https://doi.org/10.1146/annurev.en.40.010195.001501
Ellner SP, Becks L (2011) Rapid prey evolution and the dynamics of twopredator food webs. Theor EcolNeth 4:133–152. https://doi.org/10.1007/s1208001000967
Goudard A, Loreau M (2012) Integrating traitmediated effects and nontrophic interactions in the study of biodiversity and ecosystem functioning. In: Ohgushi T, Schmitz OJ, Holt RD (eds) Traitmediated indirect interactions: ecological and evolutionary perspectives. Cambridge University Press, Cambridge, pp 414–432
Hairston NG, Smith FE, Slobodkin LB (1960) Community structure, population control, and competition. Am Nat 94:421–425. https://doi.org/10.1086/282146
Holeski LM, Jander G, Agrawal AA (2012) Transgenerational defense induction and epigenetic inheritance in plants. Trends Ecol Evol 27:618–626. https://doi.org/10.1016/j.tree.2012.07.011
Holt RD, Barfield M (2012) Traitmediated effects, density dependence and the dynamic stability of ecological systems. In: Ohgushi T, Schmitz OJ, Holt RD (eds) Traitmediated indirect interactions: ecological and evolutionary perspectives. Cambridge University Press, Cambridge, pp 89–106
Ikegawa Y, Ezoe H, Namba T (2015) Effects of generalized and specialized adaptive defense by shared prey on intraguild predation. J Theor Biol 364:231–241. https://doi.org/10.1016/j.jtbi.2014.09.003
Jia SH, Wang XG, Yuan ZQ, Lin F, Ye J, Hao ZQ, Luskin MS (2018) Global signal of topdown control of terrestrial plant communities by herbivores. P Natl Acad Sci USA 115:6237–6242. https://doi.org/10.1073/pnas.1707984115
Kaplan I, Denno RF (2007) Interspecific interactions in phytophagous insects revisited: a quantitative assessment of competition theory. Ecol Lett 10:977–994. https://doi.org/10.1111/j.14610248.2007.01093.x
Kimbrell T, Holt RD, Lundberg P (2007) The influence of vigilance on intraguild predation. J Theor Biol 249:218–234. https://doi.org/10.1016/j.jtbi.2007.07.031
Matsuda H, Abrams PA, Hori H (1993) The effect of adaptive antipredator behavior on exploitative competition and mutualism between predators. Oikos 68:549–559. https://doi.org/10.2307/3544924
Matsuda H, Hori M, Abrams PA (1996) Effects of predatorspecific defence on biodiversity and community complexity in twotrophiclevel communities. Evol Ecol 10:13–28. https://doi.org/10.1007/Bf01239343
Ohgushi T (2005) Indirect interaction webs: herbivoreinduced effects through trait change in plants. Annu Rev Ecol Evol S 36:81–105. https://doi.org/10.1146/annurev.ecolsys.36.091704.175523
Ohgushi T (2007) Nontrophic, indirect interaction webs of herbivorous insects. In: Ohgushi T, Craig TP, Price PW (eds) Ecological communities: plant mediation in indirect interaction webs. Cambridge University Press, Cambridge, pp 221–245
Ohgushi T (2012) Communitylevel consequences of herbivoreinduced plant phenotypes: bottomup trophic cascades. In: Ohgushi T, Schmitz OJ, Holt RD (eds) Traitmediated indirect interactions: ecological and evolutionary perspectives. Cambridge University Press, Cambridge, UK, pp 161–185
Preisser EL, Bolnick DI (2008) When predators don't eat their prey: nonconsumptive predator effects on prey dynamics. Ecology 89:2414–2415. https://doi.org/10.1890/080522.1
Ramirez RA, Eubanks MD (2016) Herbivore density mediates the indirect effect of herbivores on plants via induced resistance and apparent competition. Ecosphere:7. https://doi.org/10.1002/ecs2.1218
Rasmann S, de Vos M, Casteel CL, Tian D, Halitschke R, Sun JY, Agrawal AA, Felton GW, Jander G (2012) Herbivory in the previous generation primes plants for enhanced insect resistance. Plant Physiol 158:854–863. https://doi.org/10.1104/pp.111.187831
Rinehart SA, Schroeter SC, Long JD (2017) Densitymediated indirect effects from active predators and narrow habitat domain prey. Ecology 98:2653–2661. https://doi.org/10.1002/ecy.1956
Schmitz OJ (2009) Indirect effects in communities and ecosystems: the role of trophic and nontrophic interactions. In: Levin SA (ed) The Princeton guide to ecology. Princeton University Press, Princeton, pp 289–295
Shiojiri K, Ozawa R, Kugimiya S, Uefune M, van Wijk M, Sabelis MW, Takabayashi J (2010) Herbivorespecific, densitydependent induction of plant volatiles: honest or “cry wolf” signals? PLoS One 5:e12161. https://doi.org/10.1371/journal.pone.0012161
Strauss SY, Rudgers JA, Lau JA, Irwin RE (2002) Direct and ecological costs of resistance to herbivory. Trends Ecol Evol 17:278–285. https://doi.org/10.1016/S01695347(02)024837
Tilman D (1982) Resource competition and community structure. Princeton University Press, Princeton
van Velzen E (2020) Predator coexistence through emergent fitness equalization. Ecology 101:e02995. https://doi.org/10.1002/ecy.2995
Volterra V (1928) Variations and fluctuations of the number of individuals in animal species living together. ICES J Mar Sci 3:3–51
Werner EE, Peacor SD (2003) A review of traitmediated indirect interactions in ecological communities. Ecology 84:1083–1100. https://doi.org/10.1890/00129658(2003)084[1083:Arotii]2.0.Co;2
Wootton JT (1994) The nature and consequences of indirect effects in ecological communities. Annu Rev Ecol Syst 25:443–466. https://doi.org/10.1146/annurev.es.25.110194.002303
Wootton JT (2002) Indirect effects in complex ecosystems: recent progress and future challenges. J Sea Res 48:157–172. https://doi.org/10.1016/S13851101(02)001491
Yamamichi M, Klauschies T, Miner BE, van Velzen E (2019) Modelling inducible defences in predatorprey interactions: assumptions and dynamical consequences of three distinct approaches. Ecol Lett 22:390–404. https://doi.org/10.1111/ele.13183
Zheng SJ, van Dijk JP, Bruinsma M, Dicke M (2007) Sensitivity and speed of induced defense of cabbage (Brassica oleracea L.): dynamics of BoLOX expression patterns during insect and pathogen attack. Mol PlantMicrobe Interact 20:1332–1345. https://doi.org/10.1094/Mpmi20111332
Acknowledgments
We thank the members of the Centre for Ecological Research for their comments and encouragement. We wish to thank two anonymous reviewers for their helpful comments and suggestions.
Funding
This research was supported financially by JSPS KAKENHI Grant Numbers 19K06851. This research was partly supported by the International Research Unit of Advanced Future Studies at Kyoto University.
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Appendices
Appendix 1. Local stability of equilibrium in a system without herbivore species Q
At the equilibrium (3) with \( {\overset{\sim }{x}}^{\ast }>0 \), a characteristic polynomial of dynamic equation (1a), (1c), and (1d) is
Since all coefficients are positive, the real parts of all possible solutions of λ are negative when
from the Routh–Hurwitz stability criterion. The condition holds explicitly; therefore, the equilibrium is always locally stable.
Appendix 2. Local stability of equilibrium in a system with both species P and Q
Equilibrium with a single herbivore species P

(i)
In the absence of the plant’s induced defenses, \( {\overset{\sim }{x}}^{\ast }=0. \)
In this case, we examine the stability of equilibrium
with full dynamics Eqs. (1a–d) (see Eq. (2)). The stability of equilibrium (B1) is relatively difficult to consider directly because the defense level x^{*} = 0 is a boundary equilibrium at which dx/dt should be negative, rather than 0, corresponding to 1/\( \overset{\sim }{\alpha } \) ≥ r. We analyzed the stability of equilibrium (B1), ignoring dynamics of x with under 1/\( \overset{\sim }{\alpha } \) ≥ r. According to the eigenvalues of the system without dynamics of x, it is neutrally stable when am_{q} / bm_{p} > 1.
When the equilibrium (B1) is neutrally stable, numerous oscillating orbits may arise around the equilibrium. We may also consider the orbits’ stability following Volterra’s (1928) analysis. For a periodic solution with period T, Eq. (1) can yield
and
Thus,
and q(t) increases along the periodic orbit over the period T. Therefore, any planar periodic orbit repels if the planar equilibrium repels.
(ii) In the presence of the plant’s induced defenses, \( {\overset{\sim }{x}}^{\ast }>0 \).
In this case, we examine the stability of equilibrium
with full dynamics Eq. (1ad) (see Eq. (3)). This is feasible only when 1/\( \overset{\sim }{\alpha } \) < r, which is necessary for \( {\overset{\sim }{x}}^{\ast }>0 \). The system has four eigenvalues at equilibrium, although three of these are identical to the eigenvalues of equilibrium without the dynamics of q (i.e., equilibrium (3)). The remaining eigenvalue determines the stability of equilibrium (B3) against an invasion of Q. When it is positive, species Q may increase at the equilibrium. The stability condition of Eq. (B3) is
or
Equilibrium with two herbivore species P and Q
A characteristic polynomial of the system at equilibrium (4) is
where
According to the feasibility of equilibrium (4), A < 0 and B < 0 under \( \overset{\sim }{\alpha }<\overset{\sim }{\beta } \), whereas A > 0 and B > 0 under \( \overset{\sim }{\alpha }>\overset{\sim }{\beta } \). Applying the Routh–Hurwitz stability criterion, the equilibrium is stable if
which can hold under \( \overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1A\right)>0 \). Combining the feasibility and stability conditions, we obtain condition (5).
Coexistence of two herbivores P and Q, and existence of either P or Q alone
In this section, we show that the stability and feasibility conditions of coexisting equilibria with both herbivores P and Q never overlap those of equilibria with either P or Q alone.
(i) a case with 1/\( \overset{\sim }{\alpha } \) ≥ r and 1/\( \overset{\sim }{\beta } \) ≥ r
In this case, one of two equilibria is feasible and stable depending on am_{q} / bm_{p}, where either herbivore species can remain alone in the plant colony without defenses. It should be noted that condition (5) cannot hold with 1/\( \overset{\sim }{\alpha } \) ≥ r and 1/\( \overset{\sim }{\beta } \) ≥ r simultaneously, implying that the coexistence of the two herbivores P and Q is impossible in this case.
(ii) a case with 1/\( \overset{\sim }{\alpha } \) < r and 1/\( \overset{\sim }{\beta } \) ≥ r (by which \( \overset{\sim }{\alpha }>\overset{\sim }{\beta } \))
In this case, both equilibria may be feasible (i.e., the equilibrium with herbivore species P alone in the presence of plant defenses and that with herbivore Q alone in the absence of plant defenses). These equilibria may be unstable in the case of violation of condition (B4b) and am_{q} / bm_{p} > 1, respectively. (Notably, it can be shown that these two equilibria never stabilize simultaneously under 1/\( \overset{\sim }{\alpha } \) < r). A combination of these instability conditions coincides with condition (5b). That is, a feasibility/stability condition of equilibrium with both P and Q corresponds with the unfeasibility/instability conditions of equilibria with either herbivore alone. Considering a symmetric relationship between the two species, we can also show a similar proof for a case where 1/\( \overset{\sim }{\alpha } \) ≥ r and 1/\( \overset{\sim }{\beta } \) < r.
(iii) a case with 1/\( \overset{\sim }{\alpha } \) < r and 1/\( \overset{\sim }{\beta } \) < r
In this case, both equilibria with either herbivore species P or Q alone in the presence of plant defenses may be simultaneously feasible (notably, it can be shown that these two equilibria never stabilize simultaneously under 1/\( \overset{\sim }{\alpha } \) < r and 1/\( \overset{\sim }{\beta } \) < r). Considering a symmetrical relationship between the two species, these equilibria destabilize simultaneously when
or
based on Eq. (B4). When this condition is combined with 1/\( \overset{\sim }{\alpha } \) < r and 1/\( \overset{\sim }{\beta } \) < r, it can coincide with condition (5). This indicates that the feasibility/stability conditions of equilibrium with both P and Q corresponds with the unfeasibility/instability conditions of both equilibria with either P or Q alone.
Appendix 3. Comparison of variables between a state with P alone and with the coexistence of P and Q
Case 1: P and Q are subordinate and superior species, respectively, as in condition (5a)
(i) a case where \( r<1/\overset{\sim }{\alpha } \)
When \( r\le 1/\overset{\sim }{\alpha } \), a state with infestation of P only is not accompanied by any plant defense (x = 0, see Eq. (2)). In this case, the difference in plant biomass between infestation by P only and by both P and Q is
which is positive under condition (5a). Therefore, the plant biomass increases with the coexistence of herbivores P and Q (i.e., s_{3}^{*} > s_{1}^{*}) under \( r<1/\overset{\sim }{\alpha } \).
However, a change in the density of herbivore P between the state with P only and its coexistence with herbivore Q is
where k = am_{q} / bm_{p}. A differentiation of Eq. (C2) with respect to k is
Here,
is an increasing function of k. When k < 1, under condition (5a), Eq. (C4) is maximized as \( \overset{\sim }{\alpha}\overset{\sim }{\beta}\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(1/\overset{\sim }{\beta }+1/\overset{\sim }{\alpha }r\right) \) at k = 1, which is negative under condition (5a). Accordingly, Eq. (C3) is always positive because \( {\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)}^3<0 \) in condition (5a). Since Eq. (C3) is positive, Eq. (C2) is an increasing function of k, which is maximized as
at k = 1, which is negative under \( r<1/\overset{\sim }{\alpha } \) and condition (5a). Where p_{3}^{*} – p_{1}^{*} is negative, the density of the subordinate herbivore P is generally decreased as a result of coexistence with the superior herbivore Q (p_{3}^{*} < p_{1}^{*}), where \( r<1/\overset{\sim }{\alpha } \).
In the coexistence equilibrium, the plant generally develops some level of induced defense (see Eq. (6)). Therefore, the plant’s defense increases where there is coexistence with the superior herbivore Q (\( {{\overset{\sim }{x}}_3}^{\ast }>{{\overset{\sim }{x}}_1}^{\ast } \)).
(ii) a case where \( r\ge 1/\overset{\sim }{\alpha } \)
When \( r\ge 1/\overset{\sim }{\alpha } \), conditions with infestation of P only are accompanied by some level of plant defense (x > 0, see Eq. (3)). In this case, the difference in plant biomass between conditions of infestation by P only and that with coexistence of both P and Q is
This is positive under condition (5a); therefore, the plant biomass increases where there is coexistence of herbivores P and Q (i.e., s_{3}^{*} > s_{1}^{*}).
However, the difference in the density of herbivore P between conditions with P only and coexistence of herbivores P and Q is
where k = am_{q} / bm_{p}. The differentiation of Eq. (C7) with respect to k is
Here,
is an increasing function of log[k]. Due to \( \log \left[k\right]<\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\alpha}\right), \) under condition (5a) and \( r\ge 1/\overset{\sim }{\alpha } \), Eq. (C9) is maximized as \( \overset{\sim }{\alpha}\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right) \) at \( \log \left[k\right]=\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\alpha}\right) \), which is negative under condition (5a). Since \( {\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)}^3 \) is also negative under condition (5a), Eq. (C8) is always positive. This indicates that Eq. (C7) is an increasing function, which is maximized as 0 at \( k=\exp \left[\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\alpha}\right)\right] \), suggesting that Eq. (C7) is negative. Where p_{3}^{*} – p_{2}^{*} is negative, the density of the subordinate herbivore P is generally decreased as a result of its coexistence with the superior herbivore Q (p_{3}^{*} < p_{2}^{*}).
The change in the plant defense \( \overset{\sim }{x} \) between conditions where subordinate herbivore P alone is present and conditions where it coexists with superior herbivore Q is expressed as
which is positive under condition (5a). Where x_{3}^{*} – x_{2}^{*} is positive, the plant defense increases in response to P’s coexistence with superior herbivore Q (\( {{\overset{\sim }{x}}_3}^{\ast }>{{\overset{\sim }{x}}_2}^{\ast } \)).
Case 2: P and Q are the superior and subordinate species, respectively, as in condition (5b)
(i) a case where \( r<1/\overset{\sim }{\alpha } \)
Since condition (5b) is inconsistent with \( r<1/\overset{\sim }{\alpha } \), the coexistence of superior species P and subordinate species Q is impossible where \( r<1/\overset{\sim }{\alpha } \). Therefore, this case may be ignored.
(ii) a case where \( r\ge 1/\overset{\sim }{\alpha } \)
In this case, conditions with the infestation of P only are accompanied by some level of plant defense (x > 0, see Eq. (3)). The difference in plant biomass between the infestation of P only and that of both P and Q is
This is negative under condition (5b); therefore, plant biomass decreases where herbivores P and Q there coexist after an additional infestation of subordinate herbivore Q (i.e., s_{3}^{*} < s_{2}^{*}).
On the other hand, a change of density of herbivore P between the state with P only and the coexistence with herbivore Q is
where k = am_{q} / bm_{p}. A differentiation of Eq. (C12) with respect to k is
Here,
is an increasing function of log[k]. Due to \( \left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\beta}\right)<\log \left[k\right]<\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\alpha}\right) \) under condition (5b), Eq. (C14) is minimized as \( \overset{\sim }{\beta}\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right) \) at \( \log \left[k\right]=\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\beta}\right) \), which is positive under condition (5b). Accordingly, Eq. (C13) is always positive because \( {\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)}^3>0 \) under condition (5b). This indicates that Eq. (C12) is an increasing function, which is maximized as 0 at \( k=\exp \left[\left(\overset{\sim }{\alpha }\overset{\sim }{\beta}\right)\left(r1/\overset{\sim }{\alpha}\right)\right] \), suggesting that Eq. (C12) is negative. Where p_{3}^{*} – p_{2}^{*} is negative, the density of the superior herbivore P is generally decreased as a result of its coexistence with the subordinate herbivore Q (p_{3}^{*} < p_{2}^{*}).
The change in plant defense \( \overset{\sim }{x} \) between conditions where there is presence of the superior herbivore P alone and those where there it coexists with the subordinate herbivore Q is
which is negative under condition (5b). Where x_{3}^{*} – x_{2}^{*} is negative, the plant defense decreases in response to coexistence with the subordinate herbivore Q (\( {{\overset{\sim }{x}}_3}^{\ast }<{{\overset{\sim }{x}}_2}^{\ast } \)).
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Yamauchi, A., Ikegawa, Y., Ohgushi, T. et al. Density regulation of cooccurring herbivores via two indirect effects mediated by biomass and nonspecific induced plant defenses. Theor Ecol (2020). https://doi.org/10.1007/s12080020004792
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Keywords
 Plant–herbivore interaction
 Induced defense
 Indirect effect
 Population dynamics