vol. 187, no. 2
the american naturalist
february 2016
The Effects of Plant Compensatory Regrowth and Induced
Resistance on Herbivore Population Dynamics
Christopher R. Stieha,1,2,* Karen C. Abbott,1 and Katja Poveda2
1. Department of Biology, Case Western Reserve University, Cleveland, Ohio 44106; 2. Department of Entomology, Cornell University,
Ithaca, New York 14853
Submitted February 24, 2015; Accepted August 24, 2015; Electronically published December 10, 2015
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.
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. 2003; Barbosa et al. 2012). Hypotheses generally include either top-down mechanisms, such as density* Corresponding author; e-mail: [email protected], christopher.stieha@case
.edu.
Am. Nat. 2016. Vol. 187, pp. 167–181. q 2015 by The University of Chicago.
0003-0147/2016/18702-56085$15.00. All rights reserved.
DOI: 10.1086/684522
dependent 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. 2003). 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.
1960; Lawton and McNeill 1979; Price et al. 1980; EdelsteinKeshet 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 2002; Turchin et al.
2003; Underwood et al. 2005). 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 2000, 2002),
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 2007;
Abbott et al. 2008). While there is no universally accepted
definition of an outbreak (Berryman 1987; Kessler et al.
2012), we consider intrinsically generated fluctuations in
herbivore density to be outbreaks, consistent with past theoretical work (Kendall et al. 1999, 2005; Turchin et al. 2003).
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 2002), 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. 2003; Buckley et al. 2005; Abbott and
Dwyer 2007). Such cycles are due to a simple repeating pat-
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. 2000); 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
2000). Although compensatory responses are often measured
in terms of plant fitness (Agrawal 2000), 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 2002; Fornoni et al. 2004; Leimu
and Koricheva 2006; Poveda et al. 2010) and are predicted
to affect herbivore population dynamics (Kessler et al. 2012;
Poveda et al. 2012). Nonetheless, these responses are almost never included in models of herbivore population dynamics (but see Lebon et al. 2012, 2014), 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 2002). 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. 2008; Anderson et al. 2009). In time-series analyses,
plant quality by itself could accurately explain observed
outbreak cycles in the pine looper moth Bupalus piniarius
(Kendall et al. 2005), but parasitism was required to explain
outbreak cycles in the larch budmoth Zeiraphera diniana
(Turchin et al. 2003). 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 2000). 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
2002), maintain a constant herbivore density (Underwood
and Rausher 2002), or cause the population to cycle (Underwood and Rausher 2000).
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 2000, 2002; Agrawal 2004),
statistical (Turchin et al. 2003; Kendall et al. 2005), 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 2000; Kessler et al. 2012). This is a meaningful
gap in our understanding of plant-herbivore dynamics, because unlike resistance traits that negatively affect herbivore
populations (Underwood and Rausher 2002), compensatory
plant regrowth potentially produces more consumable biomass, which should positively influence herbivore populations (Tiffin 2000; Kessler et al. 2012). Furthermore, compensatory responses and resistance are typically studied in
isolation, but plants have been shown to express both responses simultaneously (Leimu and Koricheva 2006; NúñezFarfán et al. 2007). 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. 2012, 2014) 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 2007).
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. (2008) 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.
Effects of Plant Response on Outbreaks
Model
Our models represent a plant-herbivore system with discrete annual generations. We extend the model developed
in Abbott et al. (2008) 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. 2008, 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
169
current generation to their density in the previous genera~ t is the population density of adult herbivores in
tion. If H
generation t, r is the maximum per capita population growth
rate, and st is the fraction of juveniles in generation t to survive to adulthood, then
~ t21 .
~ t p rst H
H
(1)
The survivorship fraction, st, 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, dt, inflicted by adults at the end of the preceding
Table 1: Definitions of terms used in models
Term conditions,
symbol
Definition
Description
Before rescaling:
~t
H
p
a0
b
c
q
r
v
...
...
...
...
...
...
...
...
Density of adult herbivores at generation t
Density of edible plant biomass at start of year
Max. juvenile survival rate
Strength of decrease in juvenile survival due to food limitation
Density of adult herbivores that consume half of plant biomass p
Effect of damage dt on induced resistance
Fecundity of adult herbivore
Relationship between herbivore density and plant
compensatory response
After rescaling:
H
q
a
g
f
~ t =cp
H
a0 r
q/a0
cv
brc
Herbivore density
Herbivore max. population growth rate
Induced resistance scaled by constitutive resistance
Compensatory effect scaled by the herbivore
half-saturation constant
Strength of food limitation
Note: Rescaled terms used in equations presented in table 2.
Range of
parameter values
...
...
...
...
...
...
...
...
...
2–1,500
0–1
1#1024 2 10
1#1025 2 1,000
Induced
Resistance
0.8
0.6
0.4
0.2
no regrowth
0
Survival of juveniles, st
1
A
1.2
1
B
ϕ = 0.00001
ϕ = 0.001
0.8
0.6
ϕ = 0.01
0.4
0.2
ϕ = 0.1
ϕ = 10
0
0
2
4
6
8
10
Rescaled density of adult herbivores, H
1.2
1.2
1
C
0.8
γ = 0.01
γ = 0.1
0.6
0.4
γ = 1.1
γ = 10
0.2
no regrowth
0
Survival of juveniles, st
0
2
4
6
8
10
Rescaled density of adult herbivores, H
1
D
0.8
0.6
γ = 0.01
0.4
γ = 0.1
γ = 1.1
0.2
no regrowth
0
0
2
4
6
8
10
Rescaled density of adult herbivores, H
0
2
4
6
8
10
Rescaled density of adult herbivores, H
1.2
1.2
1
E
γ = 0.01
γ = 0.1
0.8
0.6
0.4
0.2
0
γ = 1.1
γ = 10
no regrowth
Survival of juveniles, st
Fraction of plant biomass, 1 − dtzt
1.2
Effect on
herbivore survival
0
2
4
6
8
10
Rescaled density of adult herbivores, H
1
F
γ = 0.1
0.8
0.6
γ = 0.01
γ = 1.1
0.4
0.2
no regrowth
0
0
2
4
6
8
10
Rescaled density of adult herbivores, H
Survival of juveniles, st
Food
Limitation
overcompensation
Fraction of plant biomass, 1 − dtzt
Food
Limitation
tolerance
Fraction of plant biomass, 1 − dtzt
Food
Limitation
no regrowth
Effect on
plant biomass
1.2
1
0.8
0.6
0.4
0.2
G
α=0
α = 0.1
α = 0.2
α = 0.3
α = 0.4
α = 0.5
α = 0.6
α = 0.7
α = 0.8
α = 0.9
α = 1.0
0
0
2
4
6
8
10
Rescaled density of adult herbivores, H
Effects of Plant Response on Outbreaks
generation. We assume that herbivore damage is a saturat~ t21 )=p, where p is the density of
ing function of the ratio (H
edible plant biomass at the start of each year, and there are
~ t21 adults feeding prior to the emergence of generation t
H
juveniles. This gives
dt p
~ t21 =p
H
,
~ t21 =p)
c 1 (H
(2)
(Abbott et al. 2008), 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 2007).
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
zt p 1 2
~ t21 =p) 1 y
1 1 v(H
.
~
1 1 ½v(Ht21 =p) 1 y2
(3)
The function zt represents the fraction of the biomass lost
to adult herbivory (dt) that remains missing when the juveniles begin to feed, such that the amount of food available
for the juveniles is (1 2 zt dt )p. With no compensatory regrowth, zt p 1, and with perfect compensatory regrowth
(all lost biomass is replaced), zt p 0. Equation (3) approaches
1 with increasing herbivore density at a rate that is governed
by v. The parameter y determines whether the function zt
describes tolerance (y p 1) or overcompensation (y p 0).
In the case of tolerance, zt is a simple saturating function.
For overcompensation, zt becomes negative at low adultherbivore density, and juveniles have access to more food
than was present before their parents fed. Equation (3) is
171
phenomenological and was chosen for its ability to give
shapes of zt that are realistic for describing both tolerance
and overcompensation; y must equal 0 or 1 to avoid the unrealistic feature that zt ( 0 when there are no herbivores.
When plants do not compensate for herbivory (v → ∞, so
zt p 1 at every t), we describe the plants as having no regrowth. This situation was studied in Abbott et al. (2008),
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, st,
is a saturating function of the density of juvenile herbivores
~ t21 ) relative to the amount of food ((1 2 zt dt )p) available
(rH
to them,
st p
at
,
~
1 1 br Ht21 =½(1 2 dt zt )p
(4)
where at 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 0,
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. Figure 1C–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 dt zt . An undamaged plant would have biomass (1 2 dt zt )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 0 in eq. [5]) and show only the effects of food limitation. In panels D and F, the strength of food limitation
f p 0.1, which prevents perfect juvenile survival (st p 1) despite a strong compensatory response, g. The unlabeled lines on top of the noregrowth lines in D and F are g p 10. In panel G, herbivores are assumed not to be food limited (b p ∞ in eq. [4]), so st includes only
the effects of inducible resistance, where a is the strength of the induced resistance. For this figure, we assumed that at p a0 p 1. All graphs
show rescaled parameters (table 1).
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 2000). In our models,
induced resistance decreases the juvenile survival rate such
that
at p a0 2 qdt
(5)
(Abbott et al. 2008). When plants are capable of induced resistance (q 1 0), the maximum juvenile survival rate in generation t (i.e., the asymptote in eq. [4]), at, decreases with the
amount of damage received by the plant, dt. The parameter
a0 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/a0. As the strength of the induced resistance, a, increases, the survival of the juvenile herbivores,
st, 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 0); whether the plant exhibits induced resistance (q 1 0, and thus, a 1 0) or not (a p 0); and whether
the plant exhibits tolerance or overcompensation (v and,
thus, g are finite, with y p 1 or 0, 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 2013) 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 2007). Because many of our equations are quartic and quintic poly-
nomials (table 2), we numerically solved for the equilibrium
herbivore densities using the rootSolve package (Soetaert
2009; Soetaert and Herman 2009) in R (R Core Team 2013)
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 0 to 1 (i.e., the
full range that ensures nonnegative herbivore survival, st,
for any amount of feeding damage). The rescaled herbivore
population growth rate, q, was varied from 2 to 1,500. The
strength of food limitation, f, was tested from 1#1025 to
1,000. The strength of tolerance or overcompensation, g,
was tested for each order of magnitude from 1#1024 to 10.
The effects of these parameter values on the amount of
plant biomass available to juvenile offspring, 1 2 zt dt , and
herbivore survival, st, 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.
2008). 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 0.0001 to negligible
tolerance when g p 10. 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
Ht p
Food limitation,
overcompensating plant
c
b
a
Ht p
Ht p
Ht p
Model FL1 from Abbott et al. (2008).
Model RS1 from Abbott et al. (2008).
Model FL 1 RS1 from Abbott et al. (2008).
Food limitation and
induced resistance,
overcompensating plant
Food limitation and
induced resistance,
tolerant plant
Food limitation and
induced resistance,
no plant regrowthc
Ht p
Ht p
Food limitation,
tolerant plant
Induced resistanceb
Ht p
Food limitation,
no regrowth planta
Model description
Table 2: Rescaled models
(8)
3
2
qg(g 1 1)Ht21
1 qHt21
1 qHt21
3
2
2
2
1 fg Ht21 1 (g 1 g 1 f)Ht21
1 (f 1 1)Ht21 1 1
fg H
2
5
t21
(11)
(12)
4
3
2
qg(1 2 a)(g 1 1)Ht21
1 q(g2 1 g 2 a 1 1)Ht21
1 q(2 2 a)Ht21
1 qHt21
3
2
2 4
2
2
1 2fg Ht21 1 ½g (f 1 1) 1 g 1 fHt21 1 (g 1 g 1 2f 1 1)Ht21 1 (f 1 2)Ht21 1 1
(10)
4
3
2
qg(g 1 1)(1 2 a)Ht21
1 q½g(g 1 3) 2 2a(g 1 1) 1 2Ht21
1 2q(g 2 a 1 2)Ht21
1 2qHt21
3
4
2
2
1 2fg(g 1 1)Ht21 1 ½g (f 1 1) 1 2f(2g 1 1) 1 gHt21 1 ½g(g 1 2f 1 3) 1 4f 1 2Ht21
1 2(g 1 f 1 2)Ht21 1 2
2
(q 2 aq)Ht21
1 qHt21
2
1 2fHt21 1 (f 1 1)Ht21 1 1
5
t21
fg H
2
fH
3
t21
(9)
(7)
(6)
3
2
qg(g 1 1)Ht21
1 2q(g 1 1)Ht21
1 2qHt21
3
2
1 fg(g 1 2)Ht21 1 (g 1 1)(g 1 2f)Ht21
1 2(g 1 f 1 1)Ht21 1 2
qHt21 ½(1 2 a)Ht21 1 1
Ht21 1 1
4
t21
fg H
2
fg2 H
4
t21
qHt21
2
fHt21
1 fHt21 1 1
Equation
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 log10 scaled. Herbivore growth rate was tested for all values between 2 and
500 at intervals of 2.5. The strength of food limitation was tested for all values from 1025 to 103 by changing the exponent in step sizes of 0.02.
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. 2008).
Combined Effects of Compensatory Regrowth and Induced
Resistance on Herbivore Population Dynamics
When herbivores are food limited and plants exhibit induced resistance (model [10]), 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. 2008). 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-
Effects of Plant Response on Outbreaks
Low compensatory regrowth
γ = 10
High compensatory regrowth
γ = 0 .1
A
8
No regrowth
Tolerance
Overcompensation
6
4
ω =100
2
ω =50
ω =10
0
20
Herbivore density at equilibrium
Herbivore density at equilibrium
10
10
15
20
25
Strength of food limitation, ϕ
B
15
No regrowth
Tolerance
Overcompensation
10
5
ω =100
0
5
175
ω =10
5
10
15
20
25
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 10, 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 0.1, where plants have strong compensatory regrowth after herbivory.
sity of the herbivore population cycle given no regrowth
(fig. 5A, 5B; model [10]).
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 [10]). 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 0.01 or g p 0.0001), 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 occur-
rence 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
ω = 50
ω = 100
Food limitation and induced resistance, no regrowth (model (10))
Food limitation and induced resistance, tolerance (model (11))
γ = 1.1,
less tolerance
γ = 0.1,
more tolerance
Food limitation and induced resistance, overcompensation (model (12))
γ = 1.1,
less
overcompensation
γ = 0.1
γ = 0.01,
more
overcompensation
ω = 200
Effects of Plant Response on Outbreaks
the maximum herbivore density that the plants can sustain
(Kessler et al. 2012). 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. 2012, 2014) where the plant biomass
with herbivores can be either greater than or less than the
plant biomass without herbivores (Lebon et al. 2014). The
difference between Lebon et al. (2012, 2014) 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. (2008) 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
177
carrying capacity after an outbreak (e.g., Underwood 1999b;
Abbott and Dwyer 2007; Abbott et al. 2008).
Despite the fact that there are no empirical studies illustrating the effects of compensatory regrowth on long-term
herbivore population dynamics (Kessler et al. 2012), 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. 2010), the pupal weight of the Guatemalan tuber moth, Tecia solanivora, was shown to increase with tuber
size (Poveda et al. 2012). 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. 2004; Leimu and
Koricheva 2006; Núñez-Farfán et al. 2007). If there is a
trade-off between compensatory response and resistance,
as we might expect in some natural systems (Leimu and
Koricheva 2006), 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.
2010; Castillo et al. 2013). 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.
178
The American Naturalist
15
A
Herbivore density
Overcompensation
10
Tolerance
5
No regrowth
0
2
4
6
8
10
Time (generations)
25
B
Overcompensation
Herbivore density
20
Tolerance
15
10
5
No regrowth
0
0
0.2
0.4
0.6
0.8
1
Strength of induced resistance, α
Figure 5: Time series and bifurcation diagrams for models (eqq. [10]–
[12]) with q p 50, f p 1.1, and g p 0.1. A, Time series of the herbivore population dynamics a p 0.8. 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.
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
drive herbivore dynamics and produce herbivore aggregations and outbreaks (Underwood 1999a, 1999b; Underwood et al. 2005). 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
2001; Núñez-Farfán et al. 2007; Alyokhin 2009). 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 2009) 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
2009).
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-
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. [10]–[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 2006), 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. 2004).
Understanding the effects of bottom-up mechanisms on
herbivore population dynamics and using these mecha-
nisms 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-
180
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.
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Associate Editor: Jürgen Groeneveld
Editor: Judith L. Bronstein