Density regulation of co-occurring herbivores via two indirect effects mediated by biomass and non-specific induced plant defenses

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 one-plant, two-herbivore system model was analyzed assuming the spatiotemporal co-occurrence of the herbivores and the plants’ non-specific 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 non-specific plant defense that is induced through both intra- and transgenerational responses.

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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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Correspondence to Atsushi Yamauchi.

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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

$$ {\lambda}^3+G{c}^2\overset{\sim }{\alpha }{\lambda}^2+\left(\frac{m_p}{\overset{\sim }{\alpha }}+G{c}^2{m}_p\overset{\sim }{\alpha}\right)\lambda +G{c}^2{m}_p=0 $$
(A1)

Since all coefficients are positive, the real parts of all possible solutions of λ are negative when

$$ G{c}^2\overset{\sim }{\alpha}\left(\frac{m_p}{\overset{\sim }{\alpha }}+G{c}^2{m}_p\overset{\sim }{\alpha}\right)-G{c}^2{m}_p={G}^2{c}^2{\overset{\sim }{\alpha}}^2{m}_p>0 $$
(A2)

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

  1. (i)

    In the absence of the plant’s induced defenses, \( {\overset{\sim }{x}}^{\ast }=0. \)

In this case, we examine the stability of equilibrium

$$ \left({p_1}^{\ast },{q_1}^{\ast },{s_1}^{\ast },{{\overset{\sim }{x}}_1}^{\ast}\right)=\left(\frac{r}{u},0,\frac{m_p}{a},0\right) $$
(B1)

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 amq / bmp > 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

$$ \frac{1}{p}\frac{dp}{dt}=\frac{d\ \log\ p}{dt}= as-{\mathrm{m}}_p\ \log\ p(T)-\log\ p(0)=a{\int}_0^Ts(t) dt-{m}_pT=0 $$
(B2a)

and

$$ \overline{s}=\frac{1}{T}{\int}_0^Ts(t) dt=\frac{m_p}{a},\mathrm{and}\kern0.5em \overline{p}=\frac{1}{T}{\int}_0^Tp(t) dt=\frac{r}{u}. $$
(B2b)

Thus,

$$ \log q(T)-\log q(0)=b{\int}_0^Ts(t) dt-{m}_qT>0,\mathrm{if}\ \overline{s}=\frac{m_p}{a}>\frac{m_q}{b} $$
(B2c)

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

$$ \left({p_2}^{\ast },{q_2}^{\ast },{s_2}^{\ast },{{\overset{\sim }{x}}_2}^{\ast}\right)=\left(\frac{1}{\overset{\sim }{\alpha }u}{e}^{\overset{\sim }{\alpha }r-1},0,\frac{m_p}{a}{e}^{\overset{\sim }{\alpha }r-1},r-\frac{1}{\overset{\sim }{\alpha }}\right) $$
(B3)

with full dynamics Eq. (1a-d) (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

$$ \overset{\sim }{\alpha }<\overset{\sim }{\beta }\ \mathrm{and}\ \frac{1}{\overset{\sim }{\alpha }}<r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log\ \left[\frac{a{m}_q}{b{m}_p}\right] $$
(B4a)

or

$$ \overset{\sim }{\alpha }>\overset{\sim }{\beta }\ \mathrm{and}\ \frac{1}{\overset{\sim }{\alpha }}>r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]. $$
(B4b)

Equilibrium with two herbivore species P and Q

A characteristic polynomial of the system at equilibrium (4) is

$$ {\lambda}^4+G\frac{1}{2}{c}^2\left\{\overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1-A\right)\right\}{\lambda}^3+\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\left\{{m}_q\left(1+G{c}^2{\overset{\sim }{\beta}}^2\right)A+{m}_p\left(1+G{c}^2{\overset{\sim }{\alpha}}^2\right)B\right\}{\lambda}^2\kern0.5em +\frac{1}{2}\frac{G}{\overset{\sim }{\upalpha}-\overset{\sim }{\beta }}{c}^2\left\{\overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1-A\right)\right\}\left({m}_qA+{m}_pB\right)\lambda +G{c}^2{m}_p{m}_qA\ B=0 $$
(B5a)

where

$$ A=\overset{\sim }{\alpha}\left\{\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right)-\frac{1}{\overset{\sim }{\alpha }}\right\} $$
(B5b)
$$ B=\overset{\sim }{\beta}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right)\right\} $$
(B5c)

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

$$ \frac{1}{2}G{c}^2\left\{\overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1-A\right)\right\}>0,\kern0.5em \frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}G{c}^2\left({m}_qA{\overset{\sim }{\beta}}^2+{m}_pB{\overset{\sim }{\alpha}}^2\right)>0\kern0.5em \frac{1}{2}G{c}^2\frac{\left\{\overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1-A\right)\right\}{\left({m}_qA\overset{\sim }{\beta }+{m}_pB\overset{\sim }{\alpha}\right)}^2}{\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left({m}_qA{\overset{\sim }{\beta}}^2+{m}_pB{\overset{\sim }{\alpha}}^2\right)}>0\kern0.5em \mathrm{and}\ G{c}^2 AB{m}_p{m}_q>0 $$
(B6)

which can hold under \( \overset{\sim }{\alpha}\left(1+B\right)+\overset{\sim }{\beta}\left(1-A\right)>0 \). 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 amq / bmp, 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 amq / bmp > 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

$$ \overset{\sim }{\alpha }<\overset{\sim }{\beta }\ \mathrm{and}\ \frac{1}{\overset{\sim }{\alpha }}>r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]>\frac{1}{\overset{\sim }{\beta }} $$
(B7a)

or

$$ \overset{\sim }{\alpha }>\overset{\sim }{\beta }\ \mathrm{and}\ \frac{1}{\overset{\sim }{\alpha }}<r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]<\frac{1}{\overset{\sim }{\beta }}\kern0.5em $$
(B7b)

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

$$ {s_3}^{\ast }-{s_1}^{\ast }=\frac{m_p}{a}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{m_p}{a}=\frac{m_p}{a}\left\{{\left(\frac{a{m}_q}{b{m}_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-1\right\} $$
(C1)

which is positive under condition (5a). Therefore, the plant biomass increases with the coexistence of herbivores P and Q (i.e., s3* > s1*) 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

$$ {p_3}^{\ast }-{p_1}^{\ast }=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right)\right\}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{r}{u}=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[k\right]\right)\right\}{k}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{r}{u} $$
(C2)

where k = amq / bmp. A differentiation of Eq. (C2) with respect to k is

$$ \frac{\partial }{\partial k}\left({p_3}^{\ast }-{p_1}^{\ast}\right)=\frac{1}{u{\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}^3}{k}^{\frac{\overset{\sim }{\beta }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}\left[\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right]\right] $$
(C3)

Here, 

$$ F=\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right] $$
(C4)

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

$$ \frac{\overset{\sim }{\alpha }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left(\frac{1}{\overset{\sim }{\alpha }}-r\right) $$
(C5)

at k = 1, which is negative under \( r<1/\overset{\sim }{\alpha } \) and condition (5a). Where p3*p1* is negative, the density of the subordinate herbivore P is generally decreased as a result of coexistence with the superior herbivore Q (p3* < p1*), 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

$$ {s_3}^{\ast }-{s_2}^{\ast }=\frac{m_p}{a}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{m_p}{a}{e}^{\overset{\sim }{\alpha }r-1}=\frac{m_p}{a}\left\{\exp \left[\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right]-\exp \left[\overset{\sim }{\alpha}\left(r-\frac{1}{\overset{\sim }{\alpha }}\right)\right]\right\} $$
(C6)

This is positive under condition (5a); therefore, the plant biomass increases where there is coexistence of herbivores P and Q (i.e., s3* > s1*).

However, the difference in the density of herbivore P between conditions with P only and coexistence of herbivores P and Q is

$$ {p_3}^{\ast }-{p_2}^{\ast }=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right)\right\}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{1}{\overset{\sim }{\alpha }u}{e}^{\overset{\sim }{\alpha }r-1}=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[k\right]\right)\right\}{k}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{1}{\overset{\sim }{\alpha }u}{e}^{\overset{\sim }{\alpha }r-1} $$
(C7)

where k = amq / bmp. The differentiation of Eq. (C7) with respect to k is

$$ \frac{\partial }{\partial k}\left({p_3}^{\ast }-{p_1}^{\ast}\right)=\frac{1}{u{\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}^3}{k}^{\frac{\overset{\sim }{\beta }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}\left[\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right]\right] $$
(C8)

Here,

$$ F=\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right] $$
(C9)

is an increasing function of log[k]. Due to \( \log \left[k\right]<\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\alpha}\right), \) 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 \( \log \left[k\right]=\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\alpha}\right) \), 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 \( k=\exp \left[\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\alpha}\right)\right] \), suggesting that Eq. (C7) is negative. Where p3*p2* is negative, the density of the subordinate herbivore P is generally decreased as a result of its coexistence with the superior herbivore Q (p3* < p2*).

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

$$ {{\overset{\sim }{x}}_3}^{\ast }-{{\overset{\sim }{x}}_2}^{\ast }=\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]-\left(r-\frac{1}{\overset{\sim }{\alpha }}\right) $$
(C10)

which is positive under condition (5a). Where x3*x2* 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

$$ {s_3}^{\ast }-{s_2}^{\ast }=\frac{m_p}{a}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{m_p}{a}{e}^{\overset{\sim }{\alpha }r-1}=\frac{m_p}{a}\left\{\exp \left[\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right]-\exp \left[\overset{\sim }{\alpha}\left(r-\frac{1}{\overset{\sim }{\alpha }}\right)\right]\right\} $$
(C11)

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., s3* < s2*).

On the other hand, a change of density of herbivore P between the state with P only and the coexistence with herbivore Q is

$$ {p_3}^{\ast }-{p_2}^{\ast }=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]\right)\right\}{\left(\frac{a{m}_q}{bm_p}\right)}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{1}{\overset{\sim }{\alpha }u}{e}^{\overset{\sim }{\alpha }r-1}=\frac{\overset{\sim }{\beta }}{u\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}\left\{\frac{1}{\overset{\sim }{\beta }}-\left(r-\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[k\right]\right)\right\}{k}^{\frac{\overset{\sim }{\alpha }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}-\frac{1}{\overset{\sim }{\alpha }u}{e}^{\overset{\sim }{\alpha }r-1} $$
(C12)

where k = amq / bmp. A differentiation of Eq. (C12) with respect to k is

$$ \frac{\partial }{\partial k}\left({p_3}^{\ast }-{p_1}^{\ast}\right)=\frac{1}{u{\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)}^3}{k}^{\frac{\overset{\sim }{\beta }}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}}\left[\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right]\right] $$
(C13)

Here,

$$ F=\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left\{-r\overset{\sim }{\alpha}\overset{\sim }{\beta }+\left(\overset{\sim }{\alpha }+\overset{\sim }{\beta}\right)\right\}+\overset{\sim }{\alpha}\overset{\sim }{\beta}\log \left[k\right] $$
(C14)

is an increasing function of log[k]. Due to \( \left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\beta}\right)<\log \left[k\right]<\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\alpha}\right) \) under condition (5b), Eq. (C14) is minimized as \( \overset{\sim }{\beta}\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right) \) at \( \log \left[k\right]=\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\beta}\right) \), 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 \( k=\exp \left[\left(\overset{\sim }{\alpha }-\overset{\sim }{\beta}\right)\left(r-1/\overset{\sim }{\alpha}\right)\right] \), suggesting that Eq. (C12) is negative. Where p3*p2* is negative, the density of the superior herbivore P is generally decreased as a result of its coexistence with the subordinate herbivore Q (p3* < p2*).

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

$$ {{\overset{\sim }{x}}_3}^{\ast }-{{\overset{\sim }{x}}_2}^{\ast }=\frac{1}{\overset{\sim }{\alpha }-\overset{\sim }{\beta }}\log \left[\frac{a{m}_q}{b{m}_p}\right]-\left(r-\frac{1}{\overset{\sim }{\alpha }}\right) $$
(C15)

which is negative under condition (5b). Where x3*x2* 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 co-occurring herbivores via two indirect effects mediated by biomass and non-specific induced plant defenses. Theor Ecol (2020). https://doi.org/10.1007/s12080-020-00479-2

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Keywords

  • Plant–herbivore interaction
  • Induced defense
  • Indirect effect
  • Population dynamics