The Effects of Plant Compensatory Regrowth and Induced Resistance on Herbivore Population Dynamics

Transcription

1 vol. 187, no. 2 the american naturalist february 216 The Effects of Plant Compensatory Regrowth and Induced Resistance on Herbivore Population Dynamics Christopher R. Stieha, 1,2, * Karen C. Abbott, 1 and Katja Poveda 2 1. Department of Biology, Case Western Reserve University, Cleveland, Ohio 4416; 2. Department of Entomology, Cornell University, Ithaca, New York Submitted February 24, 215; Accepted August 24, 215; Electronically published December 1, 215 abstract: Outbreaks of herbivorous insects are detrimental to natural and agricultural systems, but the mechanisms driving outbreaks are not well understood. Plant responses to herbivory have the potential to produce outbreaks, but long-term effects of plant responses on herbivore dynamics are understudied. To quantify these effects, we analyze mathematical models of univoltine herbivores consuming annual plants with two responses: (1) compensatory regrowth, which affects herbivore survival in food-limited situations by increasing the amount of food available to the herbivore; and (2) induced resistance, which reduces herbivore survival proportional to the strength of the response. Compensatory regrowth includes tolerance, where plants replace some or all of the consumed biomass, and overcompensation, where plants produce more biomass than was consumed. We found that overcompensation can cause bounded fluctuations in the herbivore density (called outbreaks here) by itself, whereas neither tolerance nor induced resistance can cause an outbreak on its own. Food limitation and induced resistance can also drive outbreaks when they act simultaneously. Tolerance damps these outbreaks, but overcompensation, by contrast, qualitatively changes the conditions under which the outbreaks occur. Not properly accounting for these interactions may explain why it has been difficult to document plant-driven insect outbreaks and could undermine efforts to control herbivore populations in agricultural systems. Keywords: mathematical model, outbreaks, overcompensation, pests, plant defenses, tolerance. * Corresponding author; stieha@hotmail.com, christopher.stieha@case.edu. Am. Nat Vol. 187, pp q 215 by The University of Chicago /216/ $15.. All rights reserved. DOI: 1.186/ Introduction When outbreaks of insect herbivores occur, population densities can change over multiple orders of magnitude (Berryman 1987; Kendall et al. 1999), with damaging effects on natural and agricultural systems (Berryman 1987). For many insects, the mechanisms driving these outbreaks have not been definitively identified (Kendall et al. 1999; Turchin et al. 23; Barbosa et al. 212). Hypotheses generally include either top-down mechanisms, such as densitydependent predation, parasitism, or disease, or bottom-up mechanisms involving interactions with the host plant (Berryman 1986; Haukioja and Neuvonen 1987; Myers 1988; Turchin et al. 23). Historically, bottom-up mechanisms were largely dismissed due to several influential articles that used verbal or theoretical arguments to suggest that plantdriven herbivore outbreaks are not likely (Hairston et al. 196; Lawton and McNeill 1979; Price et al. 198; Edelstein- Keshet 1986; Edelstein-Keshet and Rausher 1989). We know that plant responses to herbivory such as changes in the amount and quality of edible biomass play an important role in density-dependent or delayed density-dependent processes in herbivores (Edwards and Wratten 1983; Bergelson et al. 1986; Haukioja and Neuvonen 1987; Silkstone 1987; Morris 1997; Underwood and Rausher 22; Turchin et al. 23; Underwood et al. 25). While the ultimate effects of these density-dependent plant-herbivore feedbacks on longterm population dynamics have not yet been clearly established empirically (Underwood and Rausher 2, 22), the feedbacks can theoretically be strong enough to result in large-amplitude herbivore population cycles under some circumstances (Allen et al. 1993; Abbott and Dwyer 27; Abbott et al. 28). While there is no universally accepted definition of an outbreak (Berryman 1987; Kessler et al. 212), we consider intrinsically generated fluctuations in herbivore density to be outbreaks, consistent with past theoretical work (Kendall et al. 1999, 25; Turchin et al. 23). In this way, plant-herbivore feedbacks could be an important driver of herbivore outbreak cycles in nature. When herbivores feed, they can affect both the quantity and quality of edible biomass on their host plant (Coley and Barone 1996; Karban and Baldwin 1997; Kessler and Baldwin 22), and either of these effects could feed back to influence herbivore population dynamics. If herbivores are food limited, their population growth will increase with the quantity of available edible plant material. When plant dynamics are tightly coupled to the dynamics of food-limited herbivores, classic consumer-resource cycles may result (Allen et al. 1993; Bonsall et al. 23; Buckley et al. 25; Abbott and Dwyer 27). Such cycles are due to a simple repeating pat-

2 168 The American Naturalist tern: significant plant biomass is removed by a large population of herbivores, which causes the herbivore population to decline due to food limitation. This release from herbivory allows the plant population to recover, which allows the herbivore population to grow until the cycle restarts. While this scenario is reasonable for many systems, some plants exhibit compensatory regrowth following herbivory, and this should interrupt the cycle. Two forms of compensatory plant regrowth are (1) tolerance, where some or all of the biomass removed by herbivores is replaced (before significant herbivore population decline), so net damage is reduced compared to a plant with no compensatory mechanisms (McNaughton 1983; Strauss and Agrawal 1999; Stowe et al. 2); and (2) overcompensation, where the biomass produced by the plant after herbivory is greater than the biomass removed by the herbivores (McNaughton 1983; Belsky 1986; Agrawal 2). Although compensatory responses are often measured in terms of plant fitness (Agrawal 2), they have also been commonly measured in terms of growth (McNaughton 1983; Lennartsson et al. 1998). Compensatory responses have been well documented in a range of agricultural and natural plant species (Alward and Joern 1993; Lennartsson et al. 1998; Ferraro and Oesterheld 22; Fornoni et al. 24; Leimu and Koricheva 26; Poveda et al. 21) and are predicted to affect herbivore population dynamics (Kessler et al. 212; Poveda et al. 212). Nonetheless, these responses are almost never included in models of herbivore population dynamics (but see Lebon et al. 212, 214), and their general role in promoting or damping herbivore outbreaks remains unknown. In addition to compensatory regrowth, plants may also respond to herbivory by changing the quality of their edible tissues (Kessler and Baldwin 22). Changes in nutritional content or digestibility or induction of chemical defenses that reduce the survival or fecundity of the herbivore can drive herbivore population cycles, but only under fairly restricted circumstances (Edelstein-Keshet 1986; Berryman 1987; Lundberg et al. 1994; Underwood 1999b; Abbott et al. 28; Anderson et al. 29). In time-series analyses, plant quality by itself could accurately explain observed outbreak cycles in the pine looper moth Bupalus piniarius (Kendall et al. 25), but parasitism was required to explain outbreak cycles in the larch budmoth Zeiraphera diniana (Turchin et al. 23). Although the long-term studies required to test the effects of plant quality on herbivore population dynamics are difficult to perform, short-term studies can supply parameter values for mathematical models that predict herbivore population dynamics (Underwood and Rausher 2). In soybean Glycine max, differences in plant quality and variation in plant responses to the Mexican bean beetle Epilachna varivestis are predicted to drive the herbivore population to extinction (Underwood and Rausher 22), maintain a constant herbivore density (Underwood and Rausher 22), or cause the population to cycle (Underwood and Rausher 2). While the effects of induced changes in plant quality in particular, induced chemical resistance on herbivore population dynamics have been well studied using empirical (Underwood and Rausher 2, 22; Agrawal 24), statistical (Turchin et al. 23; Kendall et al. 25), and theoretical (Edelstein-Keshet 1986; Edelstein-Keshet and Rausher 1989; Underwood 1999a, 1999b) tools, the effects of induced changes in plant quantity tolerance and overcompensation on the dynamics of food-limited herbivores have not garnered as much attention (Juenger and Lennartsson 2; Kessler et al. 212). This is a meaningful gap in our understanding of plant-herbivore dynamics, because unlike resistance traits that negatively affect herbivore populations (Underwood and Rausher 22), compensatory plant regrowth potentially produces more consumable biomass, which should positively influence herbivore populations (Tiffin 2; Kessler et al. 212). Furthermore, compensatory responses and resistance are typically studied in isolation, but plants have been shown to express both responses simultaneously (Leimu and Koricheva 26; Núñez- Farfán et al. 27). Since we expect these plant responses to change herbivore population sizes in different directions, we need a theoretical framework that allows us to predict their net effect. The only such framework that exists (Lebon et al. 212, 214) assumes overlapping herbivore generations and is thus applicable to only a subset of fluctuating herbivore populations. To understand the net effects of plant compensatory regrowth and induced resistance on the many insect herbivores with discrete (e.g., univoltine) generations, discrete-time models are needed (Abbott and Dwyer 27). To study the simultaneous effects of plant compensatory regrowth and induced resistance on the population dynamics of univoltine herbivores, we develop and analyze a suite of difference equation models that describe herbivore population dynamics as a function of plant responses to herbivory. Abbott et al. (28) proposed a model for studying the simultaneous effects of food limitation and induced resistance on herbivore dynamics for the case where plants do not exhibit compensatory regrowth. Here we extend their model for annual plants by adding tolerance or overcompensation, and we analyze the new models to determine whether compensatory regrowth promotes or dampens herbivore outbreaks. As predicted, we find that plant responses to herbivory can drive herbivore population dynamics and that compensatory regrowth can qualitatively alter the herbivore population dynamics. We also find that induced resistance and plant compensatory responses can interact to have complex effects on herbivore population dynamics and alter when herbivore outbreaks are expected to occur. Accurate predictions about the role of host plants in driving insect herbivore outbreaks will require careful consideration of how plants respond to herbivory.

3 Effects of Plant Response on Outbreaks 169 Model Our models represent a plant-herbivore system with discrete annual generations. We extend the model developed in Abbott et al. (28) by adding the possibility for plant compensatory regrowth following herbivory. In the model, adult herbivores emerge in the spring, consume plant material, produce juvenile offspring, and die. Adult consumption of the plant material affects the plant biomass available for consumption by the juveniles later in the growing season. Adult feeding may also elicit compensatory regrowth and/or induced resistance, the effects of which will be felt by the juveniles. The juveniles feed on the plants, overwinter, and emerge as the adult herbivore population in the next year. We assume that plants return to carrying capacity at the beginning of each year. Because of our assumptions, our model best represents annual plant-herbivore systems with no top-down control by predators, parasites, or disease; plant biomass is renewed every year, either naturally or by planting; and there are no year-to-year carryover effects of plant responses to herbivory. These systems can be found in nature but are most likely in agricultural systems. For systems that do not fit these assumptions, our model can serve a base on which to build more complicated models (as in Abbott et al. 28, which also considered a second set of models with interannual plant dynamics). The thorough understanding of the base model that we provide here will likely be valuable context for interpreting those more complicated models. We use a recursion equation to define the relationship between the population density of adult herbivores in the current generation to their density in the previous generation. If ~H t is the population density of adult herbivores in generation t, r is the maximum per capita population growth rate, and s t is the fraction of juveniles in generation t to survive to adulthood, then ~H t p rs t ~H t21. (1) The survivorship fraction, s t, incorporates all effects of food limitation, induced resistance, and plant compensatory regrowth. We use phenomenological functions to describe each of these features of the plant-herbivore interaction in terms of parameters that have straightforward biological interpretations. Below, we present each of these functions to make clear our assumptions about how plants accumulate herbivore damage and how plant responses to herbivory affect the next generation of herbivores. When these individual functions are combined to create a single recursion equation for the herbivore s population dynamics, it is possible to rescale the model so that the herbivore dynamics can be fully described in terms of just four composite parameters (table 1): q, which is proportional to the herbivore s maximum population growth rate; a, representing the strength of induced resistance; g, the strength of compensatory regrowth; and f, the strength of food limitation. Modeling Food Limitation and Compensatory Regrowth The amount of edible plant biomass available to juveniles of generation t is a function of the amount of herbivore damage, d t,inflicted by adults at the end of the preceding Table 1: Definitions of terms used in models Term conditions, symbol Definition Description Range of parameter values Before rescaling: ~H t... Density of adult herbivores at generation t... p... Density of edible plant biomass at start of year... a... Max. juvenile survival rate... b... Strength of decrease in juvenile survival due to food limitation... c... Density of adult herbivores that consume half of plant biomass p... q... Effect of damage d t on induced resistance... r... Fecundity of adult herbivore... v... Relationship between herbivore density and plant... compensatory response After rescaling: H ~H t =cp Herbivore density... q a r Herbivore max. population growth rate 2 1,5 a q/a Induced resistance scaled by constitutive resistance 1 g cv Compensatory effect scaled by the herbivore half-saturation constant 1# f brc Strength of food limitation 1# , Note: Rescaled terms used in equations presented in table 2.

4 Food Limitation no regrowth Fraction of plant biomass, 1 d t z t A Effect on plant biomass no regrowth Rescaled density of adult herbivores, H Survival of juveniles, s t B Effect on herbivore survival ϕ =.1 ϕ =.1 ϕ =.1 ϕ =.1 ϕ = Rescaled density of adult herbivores, H Food Limitation tolerance Fraction of plant biomass, 1 d t z t C γ =.1 γ =.1 γ = 1.1 γ = 1 no regrowth Rescaled density of adult herbivores, H Survival of juveniles, s t D γ =.1 γ =.1 γ = 1.1 no regrowth Rescaled density of adult herbivores, H Food Limitation overcompensation Fraction of plant biomass, 1 d t z t E γ =.1 γ =.1 γ = 1.1 γ = 1 no regrowth Rescaled density of adult herbivores, H Survival of juveniles, s t F γ =.1 γ =.1 γ = 1.1 no regrowth Rescaled density of adult herbivores, H Induced Resistance Survival of juveniles, s t G α = α =.1 α =.2 α =.3 α =.4 α =.5 α =.6 α =.7 α =.8 α =.9 α = Rescaled density of adult herbivores, H

5 Effects of Plant Response on Outbreaks 171 generation. We assume that herbivore damage is a saturating function of the ratio ( ~H t21 )=p, where p is the density of edible plant biomass at the start of each year, and there are ~H t21 adults feeding prior to the emergence of generation t juveniles. This gives d t p ~ H t21 =p c 1 ( ~H t21 =p), (2) (Abbott et al. 28), where c is the half-saturation constant, equal to the density of adult herbivores required to consume half of the initial plant biomass. A low value of c means that it takes relatively few herbivores to consume half of the available plant material, suggesting that mutual interference among feeding herbivores is weak and that damage to the plant accumulates quickly as the herbivore population increases from low density. In a related model, the qualitative dynamics were not sensitive to whether a simple saturating herbivory function, such as this, or a sigmoidal herbivory function was used (Abbott and Dwyer 27). If the plant is capable of responding to herbivory with compensatory regrowth, some biomass lost through damage by the adults may be recovered before the juveniles begin feeding. We model this as z t p v( ~H t21 =p) 1 y 1 1 ½v( ~H t21 =p) 1 yš 2. (3) The function z t represents the fraction of the biomass lost to adult herbivory (d t ) that remains missing when the juveniles begin to feed, such that the amount of food available for the juveniles is (1 2 z t d t )p. With no compensatory regrowth, z t p 1, and with perfect compensatory regrowth (all lost biomass is replaced), z t p. Equation (3) approaches 1 with increasing herbivore density at a rate that is governed by v. The parameter y determines whether the function z t describes tolerance (y p 1) or overcompensation (y p ). In the case of tolerance, z t is a simple saturating function. For overcompensation, z t becomes negative at low adultherbivore density, and juveniles have access to more food than was present before their parents fed. Equation (3) is phenomenological and was chosen for its ability to give shapes of z t that are realistic for describing both tolerance and overcompensation; y must equal or 1 to avoid the unrealistic feature that z t ( when there are no herbivores. When plants do not compensate for herbivory (v, so z t p 1 at every t), we describe the plants as having no regrowth. This situation was studied in Abbott et al. (28), and we use it as a baseline for understanding how plant compensatory regrowth affects herbivore population dynamics. If the amount of available food has no effect on juvenile survival, then we say that herbivores are not food limited; they may instead be limited by plant defenses, as described below in the Modeling Induced Resistance section. When herbivores are food limited, the juvenile survival rate, s t, is a saturating function of the density of juvenile herbivores (r ~H t21 ) relative to the amount of food ((1 2 z t d t )p) available to them, s t p a t 1 1 br ~H t21 =½(1 2 d t z t )pš, (4) where a t is the maximum survival fraction of the juveniles in generation t, which is a constant in equation (4) but will become dynamic when we discuss induced resistance below. The parameter b governs how steeply juvenile survival decreases from this maximum due to food limitation. Smaller b corresponds to weaker food limitation, and when b p, herbivores are not food limited. When we rescale the model (table 1), the composite parameter f is b (the strength of food limitation) scaled by r (herbivore maximum population growth rate) and c (the half-saturation constant of the adult damage function). For brevity and to highlight the effect of the key parameter b, we refer to f simply as the strength of food limitation. Figure 1A, 1B shows the effect of f on the plant biomass available for juveniles and on the juvenile survival rate as a function of the rescaled density of adult herbivores (table 1). Our rescaled parameter g, which we refer to as the strength of compensatory regrowth, is the product of v (the parameter that governs how steeply the plant s ability to compensate for damage declines with increasing herbivore biomass) and c. Figure1C 1F shows the effect of this quantity on plant biomass and herbivore survival. Figure 1: Effects of food limitation, plant compensatory regrowth, and induced resistance on plant biomass and herbivore survival. Dashed lines represent plants with no regrowth. Panels in the left column show the fraction of plant biomass available for consumption by juveniles after consumption by adults, 1 2 d t z t. An undamaged plant would have biomass (1 2 d t z t )p p p (A), a tolerant plant could have a biomass close to p (C), and an overcompensating plant could have a biomass greater than p (E). For panels A F, we remove the effect of induced resistance on herbivore survival (q p in eq. [5]) and show only the effects of food limitation. In panels D and F, the strength of food limitation f p.1, which prevents perfect juvenile survival (s t p 1) despite a strong compensatory response, g. The unlabeled lines on top of the noregrowth lines in D and F are g p 1. In panel G, herbivores are assumed not to be food limited (b p in eq. [4]), so s t includes only the effects of inducible resistance, where a is the strength of the induced resistance. For this figure, we assumed that a t p a p 1. All graphs show rescaled parameters (table 1).

6 172 The American Naturalist Modeling Induced Resistance Besides plants replacing biomass lost to herbivores, plants may also respond to herbivory with changes in plant quality. We are specifically interested in induced resistance, where plants mount a chemical or mechanical defense whose strength increases with the amount of damage received (see Baldwin and Schmelz 1994; Underwood 2). In our models, induced resistance decreases the juvenile survival rate such that a t p a 2 qd t (5) (Abbott et al. 28). When plants are capable of induced resistance (q 1 ), the maximum juvenile survival rate in generation t (i.e., the asymptote in eq. [4]), a t, decreases with the amount of damage received by the plant, d t. The parameter a is the baseline survival rate of the juvenile herbivores when there is no induced resistance or food limitation, and q is the rate at which damage induces the resistance response. The rescaled parameter a defines the strength of the induced resistance as q/a. As the strength of the induced resistance, a, increases, the survival of the juvenile herbivores, s t, decreases (fig. 1G). Modeling Herbivore Population Dynamics We substitute equations (2) (5) into equation (1) to produce a complete model for herbivore dynamics. We consider seven distinct versions of this model that differ in whether the herbivore is food limited (b and, thus, f are positive) or not (f p ); whether the plant exhibits induced resistance (q 1, and thus, a 1 ) or not (a p ); and whether the plant exhibits tolerance or overcompensation (v and, thus, g are finite, with y p 1 or, respectively) or no regrowth (g ). We consider all possible combinations of these features; the reason there are seven models instead of 2#2#3 p 12 models is that plant regrowth is only relevant to herbivore dynamics if herbivores are food limited. All seven rescaled models are shown in table 2. We used the computer algebra system Maxima (Maxima 213) for algebraic manipulation and simplification. Because we are broadly interested in drivers of herbivore outbreaks, we analyzed the models to determine which conditions allow outbreaks of the herbivore population density (which we define as sustained and bounded fluctuations around an unstable point equilibrium), as opposed to convergence of the herbivore density to a stable point equilibrium. We therefore first calculated the equilibrium herbivore density and then determined whether the equilibrium density was stable (and therefore led to constant herbivore densities) or unstable (and led to herbivore outbreaks) by linear stability analysis (see, e.g., Otto and Day 27). Because many of our equations are quartic and quintic polynomials (table 2), we numerically solved for the equilibrium herbivore densities using the rootsolve package (Soetaert 29; Soetaert and Herman 29) in R (R Core Team 213) for specific parameter combinations. To confirm that we had found all roots greater than or equal to zero, we modified the step size and search space for the rootsolve function, reran the search for a subset of parameters, and confirmed the initial results. We never found more than one positive equilibrium in a given model. Trivial equilibria, where the equilibrium density of herbivores is zero, are present in all models but were unstable for all the parameter values we examined, meaning the herbivore population is never expected to go deterministically extinct. Therefore, we focus only on the positive equilibrium in each model. For each nondimensionalized parameter in our models, we looked at a range of values that encompassed weak effects to strong effects (fig. 1). For the strength of induced resistance, a, we looked at the range from to 1 (i.e., the full range that ensures nonnegative herbivore survival, s t, for any amount of feeding damage). The rescaled herbivore population growth rate, q, was varied from 2 to 1,5. The strength of food limitation, f, was tested from 1#1 25 to 1,. The strength of tolerance or overcompensation, g, was tested for each order of magnitude from 1#1 24 to 1. The effects of these parameter values on the amount of plant biomass available to juvenile offspring, 1 2 z t d t, and herbivore survival, s t, are shown in figure 1. Results The Effects of Plant Compensatory Regrowth on Herbivore Population Dynamics Food limitation by itself, with no induced resistance or compensatory regrowth (eq. [6]; eqq. [6] [12] in table 2), leads to stable herbivore population dynamics (Abbott et al. 28). The addition of tolerance to the model (eq. [7]) also leads to stable herbivore populations across multiple orders of magnitudes of the strength of compensatory regrowth, from very strong tolerance when g p.1 to negligible tolerance when g p 1. While tolerance allows for stable herbivore populations, overcompensation (eq [8]) destabilizes the herbivore population and leads to outbreaks for certain parameter combinations (fig. 2). Higher herbivore growth rates (higher q) and stronger overcompensation (lower g) broaden the range of f values (strengths of food limitation) for which outbreaks occur. Although tolerance does not affect the herbivore dynamics qualitatively, as both models (6) and (7) are stable, tolerance does have a quantitative effect. Equilibrium herbivore densities on tolerant plants are larger than herbivore densities in systems with no regrowth (fig. 3), as expected from the increased food available to juveniles feeding on tolerant plants. Under conditions where overcompensation (model [8]) also

7 Table 2: Rescaled models Model description Equation Food limitation, H no regrowth plant a t p qh t21 fh t fh t Food limitation, tolerant plant H t p qg(g 1 1)H t q(g 1 1)H t qH t21 fg 2 H t fg(g 1 2)H t (g 1 1)(g 1 2f)H t (g 1 f 1 1)H t Food limitation, overcompensating plant H t p qg(g 1 1)H t qh t qh t21 fg 2 H t fg 2 H t (g 2 1 g 1 f)h t (f 1 1)H t Induced resistance b H t p qh t21½(1 2 a)h t21 1 1Š H t Food limitation and H induced resistance, t p no plant regrowth c (q 2 aq)h t qh t21 fh t fH t (f 1 1)H t Food limitation and induced resistance, tolerant plant H t p qg(g 1 1)(1 2 a)h t q½g(g 1 3) 2 2a(g 1 1) 1 2ŠH t q(g 2 a 1 2)H t qH t21 fg 2 H t fg(g 1 1)H t ½g 2 (f 1 1) 1 2f(2g 1 1) 1 gšh t ½g(g 1 2f 1 3) 1 4f 1 2ŠH t (g 1 f 1 2)H t Food limitation and induced resistance, overcompensating plant H t p qg(1 2 a)(g 1 1)H t q(g 2 1 g 2 a 1 1)H t q(2 2 a)h t qh t21 fg 2 H t fg 2 H t ½g 2 (f 1 1) 1 g 1 fšh t (g 2 1 g 1 2f 1 1)H t (f 1 2)H t a Model FL 1 from Abbott et al. (28). b Model RS 1 from Abbott et al. (28). c Model FL 1 RS 1 from Abbott et al. (28). (6) (7) (8) (9) (1) (11) (12)

8 174 The American Naturalist Figure 2: Stable and outbreaking herbivore populations with food limitation and overcompensation (eq. [8]). The white area represents q 2 f parameter combinations where the herbivore population is stable. The area colored in shades of grey shows the q 2 f combinations where herbivore outbreaks occur for various levels of overcompensation, g. As the overcompensatory regrowth decreases (as g increases), herbivore outbreaks occur only at high herbivore growth rates, q, and intermediate levels of food limitation, f. Larger values of f mean stronger food limitation. Note that the strength of food limitation, f, along the Y-axis is log 1 scaled. Herbivore growth rate was tested for all values between 2 and 5 at intervals of 2.5. The strength of food limitation was tested for all values from 1 25 to 1 3 by changing the exponent in step sizes of.2. leads to stable herbivore populations, those equilibrium densities are greater than systems with tolerance or no regrowth (fig. 3). The size of this difference depends on the strength of the overcompensation effect, where a greater capability for regrowth (smaller g) leads to a greater difference. The Effects of Induced Resistance on Herbivore Population Dynamics When herbivores are not food limited and induced resistance is weak (small a in model [9]), the herbivore population is not regulated by the host plant and will grow exponentially, since our model does not include any other limiting factors. Our models are clearly not appropriate for herbivores that are not limited by the host plant whatsoever, so in the absence of food limitation, we are only concerned with the herbivore dynamics when induced resistance is strong enough to prevent exponential growth. In these cases, the herbivore population is always stable (Abbott et al. 28). Combined Effects of Compensatory Regrowth and Induced Resistance on Herbivore Population Dynamics When herbivores are food limited and plants exhibit induced resistance (model [1]), herbivore outbreaks may occur even though neither food limitation, in the absence of plant compensatory regrowth, nor induced resistance in isolation leads to outbreaks (fig. 4; Abbott et al. 28). As the herbivore growth rate, q, increases, the breadth of f 2 a combinations that produce outbreaks increases (fig. 4, top row). Outbreaks occur at intermediate levels of food limitation, f, and high levels of induced resistance, a (fig. 4, top row). The addition of plant compensatory regrowth changes these results. With stronger tolerance to herbivory (as g decreases in model [11]), outbreaks occur over a smaller range of parameter combinations (fig. 4, second and third rows); therefore, tolerance decreases the occurrence of herbivore population outbreaks. When stable, the equilibrium herbivore population density given plant tolerance (model [11]) may be greater than the average or even the peak den-

9 Effects of Plant Response on Outbreaks 175 Herbivore density at equilibrium Low compensatory regrowth γ =1 ω=1 ω=5 ω=1 No regrowth Tolerance Overcompensation A Herbivore density at equilibrium ω=1 High compensatory regrowth γ =.1 No regrowth Tolerance Overcompensation B ω= Strength of food limitation, ϕ Strength of food limitation, ϕ Figure 3: Stable herbivore equilibrium densities for the food-limitation-only models (eqq. [6] [8]) across values of f (the strength of food limitation, with higher values representing stronger food limitation) and q (the herbivore population growth rate). Value g is the strength of compensatory regrowth. Note that the Y-axes are not on the same scale. A, g p 1, where plants have minimal tolerance or overcompensation such that the biomass available for consumption by juveniles is very similar to the biomass available to juveniles when plants have no regrowth; B, g p.1, where plants have strong compensatory regrowth after herbivory. sity of the herbivore population cycle given no regrowth (fig. 5A, 5B; model [1]). Overcompensation (model [12]) also changes the parameter values where outbreaks occur. When plants have low levels of overcompensation (g p 1.1), the range of f 2 a combinations where outbreaks occur in model (12) changes only slightly compared to the model where plants have no regrowth (model [1]). To see this change, compare the first and fourth rows in figure 4. In both rows, the outbreaks occur at similar intermediate strengths of food limitation, f, and high levels of induced resistance, a. As overcompensatory regrowth increases (g decreases), outbreaks still occur at intermediate strengths of food limitation, but the incidence of outbreaks becomes fairly independent of the induced defenses, a (see fig. 4, fifth row). Finally, at high levels of overcompensation (g p.1 or g p.1), herbivore outbreaks occur at low rather than high levels of induced resistance, a (see fig. 4, last row). Thus, while overcompensation does not strictly damp outbreaks, it does qualitatively alter the conditions under which we expect outbreaks to occur. Discussion With our mathematical models, we have shown that bottomup processes, specifically plant responses to herbivory, can cause herbivore population outbreaks but that the occurrence of outbreaks depends on the types and strengths of the plant responses. Overcompensation is the only response that can drive outbreaks on its own; tolerance or induced resistance cannot. On the other hand, the combination of compensatory regrowth and induced resistance can drive herbivore outbreaks under many sets of conditions (see summary in fig. 6). Herbivore outbreaks occur when plants exhibit strong induced resistance and no regrowth, weak tolerance, or weak overcompensation. Herbivore outbreaks also occur with weak or no induced resistance and strong overcompensation. We suggest that one reason it has been difficult to demonstrate plant-driven herbivore outbreaks is that induced resistance and compensatory regrowth have been studied in isolation, but it is the combined effect of the two responses that determines when and whether outbreaks occur. Because of these interactive effects of plant responses on herbivore population dynamics, application of our results to mitigate pest outbreaks in natural or agricultural systems requires considering multiple types of plant responses to herbivory. Our work stresses the importance of considering the influences of both the responses that affect plant quality (e.g., induced resistance) and the responses that affect plant quantity (e.g., compensatory regrowth). Future work to determine whether these qualitative conclusions still hold for systems that do not conform to the simplifying assumptions of our model would be useful. In agreement with our models, overcompensation is predicted to cause herbivore population outbreaks by changing

10 ω =5 ω = 1 ω = 2 Food limitation and induced resistance, no regrowth (model (1)) Food limitation and induced resistance, tolerance (model (11)) γ = 1.1, less tolerance γ =.1, more tolerance Food limitation and induced resistance, overcompensation (model (12)) γ = 1.1, less overcompensation γ =.1 γ =.1, more overcompensation

11 Effects of Plant Response on Outbreaks 177 the maximum herbivore density that the plants can sustain (Kessler et al. 212). Overcompensation creates a positive feedback loop with herbivores, where low levels of herbivory lead to more edible biomass, which supports a larger herbivore population. In our models, this positive feedback readily results in herbivore outbreaks when herbivores have a high population growth rate and food limitation is neither too weak (minimizing the positive response to added food) nor too strong (preventing herbivore populations from becoming large in the first place). In other mathematical models, overcompensation produces stable herbivore populations for most biologically realistic parameter values (Lebon et al. 212, 214) where the plant biomass with herbivores can be either greater than or less than the plant biomass without herbivores (Lebon et al. 214). The difference between Lebon et al. (212, 214) results and ours is easily explained by our use of discrete time and their use of continuous time; discrete-time systems have a much greater propensity to cycle (May 1974). Compensatory regrowth increases the consumable biomass available to herbivores, and this increase in resources could support a larger herbivore density than systems without compensatory responses (figs. 2, 5). In other models, an increase in resources and the subsequent increase in herbivore density has been shown to destabilize herbivore populations and lead to cycles (the paradox of enrichment; Rosenzweig 1971). We see this destabilizing effect of added resources only in the case of overcompensation but not with tolerance or if we simply increase the plant carrying capacity,p. This is because, in our model, only overcompensation allows herbivore density to feed back strongly on itself via plant responses, as described in the previous paragraph. In contrast, plant carrying capacity is a constant that simply scales out of the model (table 1) with no qualitative effect on the dynamics. In addition to the models we built from here, Abbott et al. (28) also analyzed models with true plant population dynamics and, thus, the possibility of additional feedbacks. They showed that, as expected, outbreaks were easier to achieve when plants had interannual dynamics; exploratory simulations confirm that this result still holds when we add compensatory regrowth (C. Stieha, unpublished results). Besides leading to the paradox of enrichment, compensatory regrowth also allows herbivore populations to be maintained when they would otherwise go extinct. Understanding these dynamics is especially important for systems where plants do not immediately return to carrying capacity after an outbreak (e.g., Underwood 1999b; Abbott and Dwyer 27; Abbott et al. 28). Despite the fact that there are no empirical studies illustrating the effects of compensatory regrowth on long-term herbivore population dynamics (Kessler et al. 212), we do know that compensatory regrowth, by increasing food availability, can affect life-history parameters of herbivores. These changes in life-history parameters can drive herbivore cycles and outbreaks. Herbivores, especially insect pests, with more food are often larger (Brown and Weis 1995; Morris 1997), and larger individuals may be more fecund (Honěk 1993; Brown and Weis 1995). For example, in an overcompensating potato variety that produced larger potatoes after damage (Poveda et al. 21), the pupal weight of the Guatemalan tuber moth, Tecia solanivora, was shown to increase with tuber size (Poveda et al. 212). The increase in fecundity, as well as the change in other life-history parameters such as survival, should scale up to affect herbivore population dynamics, as predicted by our models. Viewing multiple plant responses may help us accurately explain herbivore population dynamics, but lumping plant responses into broad categories may obscure their effect on herbivore population dynamics. In our models, tolerance and overcompensation are both compensatory regrowth, but overcompensation produces more biomass than was lost, whereas tolerance only replaces lost biomass. This difference produces qualitatively different effects on herbivore populations dynamics when coupled with induced resistance (fig. 6). This is significant, since plants are known to express both compensatory responses and resistance responses simultaneously (Fornoni et al. 24; Leimu and Koricheva 26; Núñez-Farfán et al. 27). If there is a trade-off between compensatory response and resistance, as we might expect in some natural systems (Leimu and Koricheva 26), then outbreaks may be particularly likely. In our models, the combination of weak induced resistance and strong tolerance leads to stable herbivore populations, but we see outbreaks for all other combinations (strong induced resistance with weak tolerance; strong induced resistance with weak overcompensation; and weak induced resistance with strong overcompensation; fig. 6). Our ability to understand herbivore population dynamics may also be obscured by focusing on mean damage, when damage and the associated plant responses are variable within and across natural populations (Poveda et al. 21; Castillo et al. 213). In an individual-based simula- Figure 4: Outbreaks and stable populations in herbivores exposed to food limitation, compensatory regrowth, and induced resistance. The strength of induced resistance, a, is along the X-axis, and the strength of food limitation, f, is along the Y-axis. Black areas are parameter combinations that lead to outbreaks, and gray areas show combinations that lead to stable herbivore populations. The white hatched area shows parameter combinations where the model with no plant regrowth has an outbreaking herbivore population (the black regions from the top row repeated in the subsequent rows) to aid model comparison.

12 178 The American Naturalist Herbivore density Herbivore density Overcompensation Tolerance No regrowth tion model where the amount of damage was positively related to the amount of induced resistance, the variation in induced resistance was negatively correlated with herbivore density (Underwood 1999b), and this interaction could A Time (generations) Overcompensation Tolerance No regrowth Strength of induced resistance, α Figure 5: Time series and bifurcation diagrams for models (eqq. [1] [12]) with q p 5, f p 1.1, and g p.1. A, Time series of the herbivore population dynamics a p.8. B, Bifurcation diagram using a, the strength of induced resistance, as the bifurcation parameter. Lines show the stable equilibrium population density for the herbivore in each model. To the right, where the no-regrowth model forks, the point equilibrium is unstable, and we instead show the upper and lower bounds of herbivore densities. Although plant compensatory regrowth stabilizes herbivore population dynamics, for many values of a, the stable herbivore density is greater than the maximum of the outbreaking herbivore population with no regrowth. B drive herbivore dynamics and produce herbivore aggregations and outbreaks (Underwood 1999a, 1999b; Underwood et al. 25). Whether the heterogeneity occurs due to fundamental differences in plant responses among individual plants or aggregation of herbivores creating variation in damage, heterogeneity could affect herbivore population dynamics. Because outbreaks could conceivably be triggered by a single plant with especially weak defenses or an especially strong regrowth response, we suspect that our models present a conservative view of the interaction between plant responses to herbivory and herbivore population dynamics. Future work that accounts for herbivore dispersal behavior among plants as well as heterogeneity in plant responses would likely be interesting and help to extend the population level patterns we present here to the scale of individual plants. Plant responses to herbivory can be used to explain herbivore population dynamics and can also be used to actively control pest dynamics in agricultural systems. Resistant varieties are often planted for pest control, and varieties that show compensatory responses are increasingly recommended to maintain yield despite pest pressure (Rausher 21; Núñez-Farfán et al. 27; Alyokhin 29). Using plant responses for pest control in agricultural systems requires understanding which mechanisms are important in a system. For example, in systems regulated by bottom-up mechanisms, when food limitation, but not induced resistance, affects herbivore population dynamics, our models predict that noncompensating plants would maintain a stable herbivore population. Tolerant plants would maintain a stable but larger herbivore population, whereas overcompensating plants could cause a pest outbreak. This could be important for pests such as the Colorado potato beetle, Leptinotarsa decemlineata, where predators cannot limit population growth (Alyokhin 29) and food quantity significantly affects survival while plant resistance has a negligible effect (Morris 1997), probably due to the beetle s ability to handle toxic chemicals in an inherently toxic Solanaceous system (Alyokhin 29). Agricultural systems, where plants are replanted at a constant density at the start of each growing season, are particularly well suited to the assumptions of our model. In these systems, planting varieties with stronger induced resistance could have unexpected consequences due to the interaction with plant compensatory regrowth. When herbivores are food limited but not regulated by predators and plants do not have compensatory regrowth, our model predicts that increasing the strength of induced resistance generally promotes outbreaks, whereas decreasing induced resistance damps outbreaks and leads to stable herbivore population dynamics. Adding tolerance to our model maintains these stable herbivore populations, but these stable populations can be at a higher density than the peak density of an out-

13 Effects of Plant Response on Outbreaks 179 Figure 6: Summary of the conditions for herbivore outbreaks in all models of food-limited herbivores feeding on plants with inducible resistance (eqq. [1] [12]). Black regions show where outbreaks occur, and gray regions show where the herbivore population is at a stable equilibrium point. We use the labels low, med (medium), and high rather than giving values, since the exact values depend (quantitatively, but not qualitatively) on q and f. breaking population on plants with no compensatory regrowth (fig. 5). In overcompensating plants, however, increasing plant resistance may not qualitatively affect the dynamics or may damp or promote outbreaks depending on the strength of overcompensation (fig. 4). Since overall levels of resistance within agricultural systems may be quite low (Leimu and Koricheva 26), increasing overcompensation could potentially and unintentionally promote pest outbreaks. We predict that to prevent outbreaks in overcompensating plants, both resistance and overcompensation would need to be increased. Our work shows that correctly determining whether tolerance or overcompensation occurs within a system and estimating the strength of induced resistance are vital for using plant responses to control herbivore population dynamics and to prevent unintended pest outbreaks. These bottom-up mechanisms also need be tested within a tritrophic system, where predators have the potential to interact with plant defenses to drastically change the herbivore dynamics compared to a system without predators (Vos et al. 24). Understanding the effects of bottom-up mechanisms on herbivore population dynamics and using these mechanisms to mitigate pest outbreaks and control herbivore populations requires care due to the interaction between plant responses, which can, in turn, be modified by densitydependent processes and life-history parameters of the herbivore population. As a start, we recommend measurements that encompass the biological concepts of our four key parameters: the maximum herbivore growth rate, the strength of food limitation, the type and strength of the plant compensatory regrowth, and the strength of induced resistance. Only by considering plant responses to herbivory that affect both its quantity and its quality can we begin to implicate bottom-up mechanisms as drivers of herbivore outbreaks. Future theoretical work should consider both plant quantity and plant quality with the myriad plant-herbivore interactions found in natural systems. Acknowledgments We thank S. Catella, A. Kessler, C. Moore, B. Nolting, and two anonymous reviewers for comments on an earlier version of this work and members of the Poveda Lab, Thaler Lab, and the Plant Interactions Group at Cornell Univer-

14 18 The American Naturalist sity for discussion. This research was partially supported by a Scholar Award in Complex Systems to K.C.A. from the James S. McDonnell Foundation and Multistate Hatch Grant NE1231 to K.P. from Cornell University Agricultural Experiment Station funding from the National Institute of Food and Agriculture (NIFA) of the United States Department of Agriculture (USDA). Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of NIFA or the USDA. Literature Cited Abbott, K. C., and G. Dwyer. 27. Food limitation and insect outbreaks: complex dynamics in plant-herbivore models. Journal of Animal Ecology 76: Abbott, K. C., W. F. Morris, and K. Gross. 28. Simultaneous effects of food limitation and inducible resistance on herbivore population dynamics. Theoretical Population Biology 73: Agrawal, A. A. 2. Overcompensation of plants in response to herbivory and the by-product benefits of mutualism. Trends in Plant Science 5: Plant defense and density dependence in the population growth of herbivores. American Naturalist 164: Allen, L. J., M. K. Hannigan, and M. J. Strauss Mathematical analysis of a model for a plant-herbivore system. Bulletin of Mathematical Biology 55: Alward, R. D., and A. Joern Plasticity and overcompensation in grass responses to herbivory. Oecologia (Berlin) 95: Alyokhin, A. 29. Colorado potato beetle management on potatoes: current challenges and future prospects. Fruit, Vegetable and Cereal Science and Biotechnology 3:1 19. Anderson, K. E., B. D. Inouye, and N. Underwood. 29. Modeling herbivore competition mediated by inducible changes in plant quality. Oikos 118: Baldwin, I. T., and E. A. Schmelz Constraints on an induced defense: the role of leaf area. Oecologia (Berlin) 97: Barbosa, P., D. Letourneau, and A. Agrawal, eds Insect outbreaks revisited. J. Wiley, New York. Belsky, A Does herbivory benefit plants? a review of the evidence. American Naturalist 127: Bergelson, J., S. Fowler, and S. Hartley The effects of foliage damage on casebearing moth larvae, Coleophora serratella, feeding on birch. Ecological Entomology 11: Berryman, A. A What causes population cycles of forest lepidoptera? Trends in Ecology and Evolution 11: The theory and classification of outbreaks. Pages 3 3 in P. Barboas and J. Schultz, eds. Insect outbreaks. Academic Press, San Diego, CA. Bonsall, M. B., E. Van der Meijden, and M. J. Crawley. 23. Contrasting dynamics in the same plant-herbivore interaction. Proceedings of the National Academy of Sciences 1: Brown, D. G., and A. E. Weis Direct and indirect effects of prior grazing of goldenrod upon the performance of a leaf beetle. Ecology 76: Buckley, Y. M., M. Rees, A. W. Sheppard, and M. J. Smyth. 25. Stable coexistence of an invasive plant and biocontrol agent: a parameterized coupled plant-herbivore model. Journal of Applied Ecology 42:7 79. Castillo, G., L. L. Cruz, J. Hernández-Cumplido, K. Oyama, C. M. Flores-Ortiz, J. Fornoni, P. L. Valverde, and J. Núñez-Farfán Geographic association and temporal variation of chemical and physical defense and leaf damage in Datura stramonium. Ecological Research 28: Coley, P. D., and J. A. Barone Herbivory and plant defenses in tropical forests. Annual Review of Ecology and Systematics 27: Edelstein-Keshet, L Mathematical theory for plant-herbivore systems. Journal of Mathematical Biology 24: Edelstein-Keshet, L., and M. D. Rausher The effects of inducible plant defenses on herbivore populations. 1. Mobile herbivores in continuous time. American Naturalist 133: Edwards, P., and S. Wratten Wound induced defences in plants and their consequences for patterns of insect grazing. Oecologia (Berlin) 59: Ferraro, D. O., and M. Oesterheld. 22. Effect of defoliation on grass growth: a quantitative review. Oikos 98: Fornoni, J., J. Núñez-Farfán, P. L. Valverde, and M. D. Rausher. 24. Evolution of mixed strategies of plant defense allocation against natural enemies. Evolution 58: Hairston, N. G., F. E. Smith, and L. B. Slobodkin Community structure, population control, and competition. American Naturalist 94: Haukioja, E., and S. Neuvonen Insect population dynamics and induction of plant resistance: the testing of hypotheses. Pages in P. Barbosa and J. Schultz, eds. Insect outbreaks. Academic Press, San Diego, CA. Honěk, A Intraspecific variation in body size and fecundity in insects: a general relationship. Oikos 66: Juenger, T., and T. Lennartsson. 2. Tolerance in plant ecology and evolution: toward a more unified theory of plant-herbivore interaction. Evolutionary Ecology 14: Karban, R., and I. T. Baldwin, eds Induced responses to herbivory. University of Chicago Press, Chicago. Kendall, B. E., C. J. Briggs, W. W. Murdoch, P. Turchin, S. P. Ellner, E. McCauley, R. M. Nisbet, and S. N. Wood Why do populations cycle? a synthesis of statistical and mechanistic modeling approaches. Ecology 8: Kendall, B. E., S. P. Ellner, E. McCauley, S. N. Wood, C. J. Briggs, W. W. Murdoch, and P. Turchin. 25. Population cycles in the pine looper moth: dynamical tests of mechanistic hypotheses. Ecological Monographs 75: Kessler, A., and I. Baldwin. 22. Plant responses to insect herbivory: the emerging molecular analysis. Annual Review of Plant Biology 53: Kessler, A., K. Poveda, and E. H. Poelman Plant-induced responses and herbivore population dynamics. Pages in P. Barbosa, D. Letourneau, and A. Agrawal, eds. Insect outbreaks revisited. J. Wiley, New York. Lawton, J. H., and S. McNeill Between the devil and the deep blue sea: on the problem of being a herbivore. Pages in R. M. Anderson, B. D. Turner, and L. R. Taylor, eds. Population dynamics. Blackwell Scientific, Oxford. Lebon, A., Y. Dumont, F. Grognard, and L. Mailleret Modelling plant compensatory effects in plant-insects dynamics. Pages in Proceedings of the 212 IEEE Fourth International Symposium on Plant Growth Modeling, Simulation, Visualization and Applications, Shanghai, October 31 November 3.