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Open Access
Peer-reviewed
Research Article
Dynamical Transitions in a Pollination-Herbivory Interaction: A Conflict between
Mutualism and Antagonism
* Tomás A. Revilla ,
* E-mail: [50]tomrevilla@gmail.com
Current address: Biology Centre, Academy of Sciences of the Czech Republic,
Ceské Budejovice, Czech Republic
Affiliation Centre for Biodiversity Theory and Modelling, Station d'Ecologie
Expérimentale du Centre National de la Recherche Scientifique ŕ Moulis,
Moulis, France
x
* Francisco Encinas-Viso
Current address: Centre for Australian National Biodiversity Research,
Commonwealth Scientific and Industrial Research Organisation, National
Facilities and Collections, Canberra, Australia
Affiliation Community and Conservation Ecology Group, Centre for Ecological
and Evolutionary Studies, University of Groningen, Groningen, The Netherlands
x
Dynamical Transitions in a Pollination-Herbivory Interaction: A Conflict between
Mutualism and Antagonism
* Tomás A. Revilla,
* Francisco Encinas-Viso
PLOS
x
* Published: February 20, 2015
* [51]https://doi.org/10.1371/journal.pone.0117964
*
* [52]Article
* [53]Authors
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* [57]Figures
Figures
Fig 1
Table 1
Fig 2
Fig 3
Fig 4
Fig 5
Abstract
Plant-pollinator associations are often seen as purely mutualistic, while in
reality they can be more complex. Indeed they may also display a diverse array of
antagonistic interactions, such as competition and victim-exploiter interactions.
In some cases mutualistic and antagonistic interactions are carried-out by the
same species but at different life-stages. As a consequence, population structure
affects the balance of inter-specific associations, a topic that is receiving
increased attention. In this paper, we developed a model that captures the basic
features of the interaction between a flowering plant and an insect with a larval
stage that feeds on the plant's vegetative tissues (e.g. leaves) and an adult
pollinator stage. Our model is able to display a rich set of dynamics, the most
remarkable of which involves victim-exploiter oscillations that allow plants to
attain abundances above their carrying capacities and the periodic alternation
between states dominated by mutualism or antagonism. Our study indicates that
changes in the insect's life cycle can modify the balance between mutualism and
antagonism, causing important qualitative changes in the interaction dynamics.
These changes in the life cycle could be caused by a variety of external drivers,
such as temperature, plant nutrients, pesticides and changes in the diet of adult
pollinators.
Citation: Revilla TA, Encinas-Viso F (2015) Dynamical Transitions in a
Pollination-Herbivory Interaction: A Conflict between Mutualism and Antagonism.
PLoS ONE 10(2): e0117964. https://doi.org/10.1371/journal.pone.0117964
Academic Editor: Jordi Garcia-Ojalvo, Universitat Pompeu Fabra, SPAIN
Received: October 14, 2014; Accepted: January 6, 2015; Published: February 20,
2015
Copyright: © 2015 Revilla, Encinas-Viso. This is an open access article
distributed under the terms of the [58]Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium,
provided the original author and source are credited
Data Availability: All relevant data consists of graphs generated by computer
scripts provided in the supplementary information file.
Funding: TAR was supported by the TULIP Laboratory of Excellence
(ANR-10-LABX-41). FEV was supported by the OCE postdoctoral fellowship at CSIRO.
The funders had no role in study design, data collection and analysis, decision
to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Il faut bien que je supporte deux ou trois chenilles si je veux connaître les
papillons
Le Petit Prince, Chapitre IX - Antoine de Saint-Exupéry
Mutualism can be broadly defined as cooperation between different species
[[59]1]. In mutualistic interactions typically there are benefits and costs, in
terms of resources, energy and time devoted to them, but the net outcome is (+,+)
in the final balance. However, there can be other kinds of costs, concerning
detrimental interactions that run in parallel with mutualism, such as predation,
parasitism or competition, involving the same parties. Moreover, some of these
antagonistic interactions (e.g. competition) seem to be important for the
evolution and stability of mutualism [[60]2]. In general, these costs have
important consequences at the population and community level because the net
outcome of an interspecific association can turn out beneficial or detrimental
and more interestingly, variable [[61]3]. Variable interactions challenge the
view that ecological communities are structured by well defined interactions at
the species level such as competition (-,-), victim-exploiter (-,+) or mutualism
(+,+).
Pollination is one of the most important mutualisms occurring between plants and
animals. This form of trading resources for services greatly explains the
evolutionary success of flowering plants in almost all terrestrial systems. It is
responsible for the well being of ecosystem services. During the larval stage of
many insect pollinators, such as Lepidopterans (butterflies and moths), the
larvae feed on plant leaves to mature and become adult pollinators [[62]4-[63]7].
These ontogenetic diet shifts [[64]8] are very common and important in
understanding the ecological and evolutionary dynamics of plant-animal
mutualisms. Interestingly, in some cases larvae feed on the same plant species
that they will pollinate as adults [[65]6, [66]9]. This shows that in several
cases mutualistic and antagonistic interactions are exerted by the same species,
and a potential conflict arises for the plant, between the benefits of mutualism
and the costs of herbivory. One of the best known examples is the interaction
between tobacco plants (Nicotiana attenuata) and the hawkmoth (Manduca sexta)
[[67]10, [68]11], whose larva is commonly called the tobacco hornworm. There are
other examples of this type of interaction in the genus Manduca (Sphingidae),
such as between the tomato plant (Lycopersicon esculentum) and the five-spotted
hawkmoth (Manduca quinquemaculata) [[69]12]. These larvae have received a lot of
attention due to their negative effects on agricultural crops [[70]13].
The interaction between Manduca sexta and Datura wrightii (Solanacea) [[71]6,
[72]14] is another good example illustrating the costs and benefits of
pollination mutualisms [[73]6]. D. wrightii provides high volumes of nectar and
seems to depend heavily on the pollination service by M. sexta adults [[74]14].
However, M. sexta larvae, which feed on D. wrightii vegetative tissue, can have
severe negative effects on plant fitness [[75]15, [76]16]. We could assume that
the benefits of pollination might outweigh the costs of herbivory for this
mutualism to be relatively viable. The question is what are the conditions, in
terms of benefits (pollination) and costs (herbivory), for this mutualistic
interaction to be stable?
In the pollination-herbivory cases mentioned previously the benefits and costs
for the plant are clearly differentiated. This is because the role of an insect
as a pollinator or herbivore depends on the stage in its life cycle [[77]17].
Thus, whether mutualism or herbivory dominates the interaction is dependent on
insect abundance and its population structure. In other words the cost:benefit
ratio must be positively related with the insect's larva:adult ratio. For a
hypothetical scenario in which the costs of herbivory (-) and the benefits of
pollination (+) are balanced for the plant (0), an increase in larval abundance
relative to adults should bias the relationship towards a victim-exploiter one
(-,+). Whereas an increase in adult abundance relative to larvae should bias the
relationship towards mutualism (+,+). Under equilibrium conditions, one would
expect transitions (bifurcations) from (-,+) to (0,+) to (+,+) and vice-versa as
relevant parameters affecting the plant and the insect life-histories vary, such
as flower production, mortalities or larvae maturation rates. However, under
dynamic scenarios the outcome may be more complex: a victim-exploiter state (-,+)
enhances larva development into pollinating adults, but this tips the interaction
into a mutualism (+,+), which in turn contributes greater production of larva
leading back to a victim-exploiter state (-,+). This raises the possibility of
feedback between the plant-insect interaction and insect population structure,
which can potentially lead to periodic alternation between mutualism and
herbivory. Thus, when non-equilibrium dynamics are involved, questions concerning
the overall nature (positive, neutral or negative) of mixed interactions may not
have simple answers.
In this article we study the feedback between insect population structure,
pollination and herbivory. We want to understand how the balance between costs
(herbivory) and benefits (pollination) affects the interaction between plants
(e.g. D. wrightii) and herbivore-pollinator insects (e.g. M. sexta)? Also what
role does insect development have in this balance and on the resulting dynamics?
We use a mathematical model which considers two different resources provided by
the same plant species, nectar and vegetative tissues. Nectar consumption
benefits the plant in the form of fertilized ovules, and consumption of
vegetative tissues by larvae causes a cost. Our model predicts that the balance
between mutualism and antagonism, and the long term stability of the plant-insect
association, can be greatly affected by changes in larval development rates, as
well as by changes in the diet of adult pollinators.
Methods
Our model concerns the dynamics of the interaction between a plant and an insect.
The insect life cycle comprises an adult phase that pollinates the flowers and a
larval phase that feed on non-reproductive tissues of the same plant. Adults
oviposit on the same species that they pollinate (e.g. D. wrightii - M. sexta
interaction). Let denote the biomass densities of the plant, the larva, and the
adult insect with P, L and A respectively. An additional variable, the total
biomass of flowers F, enables the mutualism by providing resources to the insect
(nectar), and by collecting services for the plant (pollination). The
relationship is facultative-obligatory. In the absence of pollination, plant
biomass persists by vegetative growth (e.g. root, stem and leaf biomass are being
constantly renewed). For the sake of simplicity and because we want to focus on
the plant-insect interaction, we describe vegetative growth using a logistic
growth rate, a choice that is empirically justified for tobacco plants [[78]18].
In the absence of the plant, however, the insect always goes extinct because
larval development relies exclusively on herbivory, even if adults pollinate
other plant species. This is based on the biology of M. sexta [[79]6]. The
mechanism of interaction between these four variables (P, L, A, F), as shown in
[80]Fig. 1, is described by the following system of ordinary differential
equations (ODE): [journal.pone.0117964.e001] (1) where r: plant intrinsic growth
rate, c: plant intra-specific self-regulation coefficient (also the inverse its
carrying capacity), a: pollination rate, b: herbivory rate, s: flower production
rate, w: flower decay rate, m, n: larva and adult mortality rates, s: plant
pollination efficiency ratio, e: adult consumption efficiency ratio. Like e,
parameter g is also a consumption efficiency ratio, but we will call it the
maturation rate for brevity since we will refer to it frequently. Our model
assumes that pollination leads to flower closure [[81]19], causing resource
limitation for adult insects. Parameter g represents a reproduction rate
resulting from the pollination of other plants species, which we do not model
explicitly. Most of our results are for g = 0.
[82]thumbnail
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Fig 1. Interaction mechanism between plants (P), flowers (F), larva (L), adult
insects (A) and associated biomass flows.
Clipart sources: [83]http://etc.usf.edu/clipart/
[84]https://doi.org/10.1371/journal.pone.0117964.g001
We now consider the fact that flowers are ephemeral compared with the life cycles
of plants and insects. In other words, some variables (P, L, A) have slower
dynamics, and others (F) are fast [[85]20]. Given the near constancy of plants
and animals in the flower equation of ([86]1), we can predict that flowers will
approach a quasi-steady-state (or quasi-equilibrium) biomass F ~= sP/(w + aA),
before P, L and A can vary appreciably. Substituting the quasi-steady-state
biomass in system ([87]1) we arrive at: [journal.pone.0117964.e002] (2)
In system ([88]2) the quantities in square brackets can be regarded as functional
responses. Plant benefits saturate with adult pollinator biomass, i.e.
pollination exhibits diminishing returns. The functional response for the insects
is linear in the plant biomass, but is affected by intraspecific competition
[[89]21] for mutualistic resources.
We non-dimensionalized this model to reduce the parameter space from 12 to 9
parameters, by casting biomasses with respect to the plant's carrying capacity
(1/c) and time in units of plant biomass renewal time (1/r). This results in a
PLA (plant, larva, adult) scaled model: [journal.pone.0117964.e003] (3)
[90]Table 1 lists the relevant transformations.
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Table 1. Variables and parameters.
[92]https://doi.org/10.1371/journal.pone.0117964.t001
There is an important clarification to make concerning the nature and scales of
the conversion efficiency ratios s, e involved in pollination, and g for
herbivory and maturation. This has to do with the fact that flowers per se are
not resources or services, but organs that enable the mutualism to take place,
and they mean different things in terms of biomass production for plants and
animals. For insects, the yield of pollination is thermodynamically constrained.
First of all, a given biomass F of flowers contains an amount of nectar that is
necessarily less than F. More importantly, part of this nectar is devoted to
survival, or wasted, leaving even less for reproduction. Similarly, not all the
biomass consumed by larvae will contribute to their maturation to adult. Ergo e <
1, g < 1. Regarding the returns from pollination for the plants, the situation is
very different. Each flower harbors a large number of ovules, thus a potentially
large number of seeds [[93]22], each of which will increase in biomass by
consuming resources not considered by our model (e.g. nutrients, light).
Consequently, a given biomass of pollinated flowers can produce a larger biomass
of mature plants, making s larger than 1.
The PLA model ([94]3) has many parameters. However, here we focus on herbivory
rates (b) and larvae maturation (g) because increasing b turns the net balance
interaction towards antagonism, whereas increasing g shifts insect population
structure towards the adult phase, turning the net balance towards mutualism.
Both parameters also relate to the state variables at equilibrium (i.e. z/y =
bgx/n in ([95]3) for dz/dt = 0). We studied the joint effects of varying b and g
numerically (parameter values in [96]Table 1) using XPPAUT [[97]23]. ODE were
integrated using Matlab [[98]24] or GNU/Octave [[99]25]. We also present a
simplified graphical analysis of our model, in order to explain how different
dynamics can arise, by varying other parameters. The source codes supporting
these results are provided as supplementary material ([100]S1 File).
Results
Numerical results
[101]Fig. 2 shows interaction outcomes of the PLA model, as a function of b and g
for specialist pollinators (phi = 0). This parameter space is divided by a
decreasing R[o] = 1 line that indicates whether or not insects can invade when
rare. R[o] is defined as (see derivation in [102]S1 File):
[journal.pone.0117964.e004] (4) and we call it the basic reproductive number,
according to the argument that follows. Consider the following in system
([103]3): if the plant is at carrying capacity (x = 1), and is invaded by a very
small number of adult insects (z ~= 0), the average number of larvae produced by
a single adult in a given instant is eax/(y+z) ~= ea/y, and during its life-time
(n^-1) it is ea/yn. Larvae die at the rate µ, or mature with a rate equal to gbx
= gb, per larva. Thus, the probability of larvae becoming adults rather than
dying is gb/(µ+gb). Multiplying the life-time contribution of an adult by this
probability gives the expected number of new adults replacing one adult per
generation during an invasion (R[o]). More formally, R[o] is the expected number
of adult-insect-grams replacing one adult-insect-gram per generation (assuming a
constant mass-per-individual ratio).
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Fig 2. Outcomes of the PLA model as a function of the larval maturation and
herbivory rates for specialist pollinators (phi = 0).
The rectangular region in the bottom left is analyzed with more detail in [105]S1
File.
[106]https://doi.org/10.1371/journal.pone.0117964.g002
Below the R[o] = 1 line, small insect populations cannot replace themselves (R[o]
< 1) and two outcomes are possible. If the maturation rate is too low, the plant
only equilibrium (x = 1, y = z = 0) is globally stable and plant-insect
coexistence is impossible for all initial conditions. If the maturation rate is
large enough, stable coexistence is possible, but only if the initial plant and
insect biomass are large enough. This is expected in models where at least one
species, here the insect, is an obligate mutualist. In this region of the space
of parameters, the growth of small insect populations increases with population
size, a phenomenon called the Allee effect [[107]26].
Above the R[o] = 1 line the plant only equilibrium is always unstable against the
invasion of small insect populations (R[o] > 1). Plants and insects can coexist
in a stable equilibrium or via limit cycles (stable oscillations). The zone of
limit cycles occurs for intermediate values of the maturation rate (g) and it
widens with rate of herbivory (b).
Plant equilibrium when coexisting with insects can be above or below the carrying
capacity (x = 1). When above carrying capacity the net result of the interaction
is a mutualism (+,+). While in the second case we have antagonism, more
specifically net herbivory (-,+). As it would be expected, increasing herbivory
rates (b) shifts this net balance towards antagonism (low plant biomass), while
decreasing it shifts the balance towards mutualism (high plant biomass). The
quantitative response to increases in the maturation rate (g) is more complex
however (see the bifurcation plot in [108]S1 File).
Given that there is herbivory, we encounter victim-exploiter oscillations.
However, the oscillations in the PLA model are special in the sense that the
plant can attain maximum biomasses above the carrying capacity (x > 1). For an
example see [109]Fig. 3. Instead of a stable balance between antagonism and
mutualism, we can say that the outcome in [110]Fig. 3 is a periodic alternation
of both cases. This is not seen in simple victim-exploiter models, where
oscillations are always below the victim's carrying capacity [[111]27, [112]28].
The relative position of the cycles along the plant axis is also affected by
herbivory: if b decreases (increases), plant maxima and minima will increase
(decrease) in [113]Fig. 3 (see bifurcation plot in [114]S1 File). In some cases
the entire plant cycle (maxima and minima) ends above the carrying capacity if b
is low enough (see [115]S1 File), but further decrease causes damped
oscillations. We also found examples in which coexistence can be stable or lead
to limit cycles depending on the initial conditions (see example in [116]S1
File), but this happens in a very restrictive region in the space of parameters
(see bifurcation plot in [117]S1 File). Limit cycles can also cross the plant's
carrying capacity under the original interaction mechanism ([118]1), which does
not assume the steady-state in the flowers (see [119]S1 File, using parameters in
the last column of [120]Table 1).
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Fig 3. Limit cycles in the PLA model ([122]3).
Plant biomass alternates above and below the carrying capacity (dotted line).
Parameters as in [123]Table 1, with g = 0.01, b = 10. Blue:plant, green:larva,
red:adult.
[124]https://doi.org/10.1371/journal.pone.0117964.g003
[125]Fig. 4 shows the b vs g parameter space of the model when the adults are
more generalist. The relative positions of the plant-only, Allee effect, and
coexistence regions are similar to the case of specialist pollinators ([126]Fig.
2). However, the region of limit cycles is much larger. The R[0] = 1 line is
closer to the origin, because the expression for R[0] is now (see derivation in
[127]S1 File): [journal.pone.0117964.e005] (5)
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Fig 4. Outcomes of the PLA model as a function of the larval maturation and
herbivory rates for generalist pollinators (phi = 1).
AeLc: intersection of the Allee effect and Limit cycle zones.
[129]https://doi.org/10.1371/journal.pone.0117964.g004
In other words, this means that the more generalist the adult pollinators (larger
phi ), the more likely they can invade when rare. There is also a small overlap
between the Allee effect and limit cycle regions, i.e. parameter combinations for
which the long term outcome could be insect extinction or plant-insect
oscillations, depending on the initial conditions.
Graphical analysis
The general features of the interaction can be studied by phase-plane analysis.
To make this easier, we collapsed the three-dimensional PLA model into a
two-dimensional plant-larva (PL) model, by assuming that adults are extremely
short lived compared with plants and larvae (see resulting ODE in [130]S1 File).
The closest realization of this assumption could be Manduca sexta, which has a
larval stage of approximately 20-25 days and adult stages of around 7 days
[[131]29, [132]30]. For a given parametrization ([133]Table 1), the PL model has
the same equilibria as the PLA model, but not the exact same global dynamics due
to the alteration of time scales. Yet, this simplification provides insights
about the outcomes displayed in Figs. [134]2 and [135]4.
[136]Fig. 5 shows representative examples of plant and larva isoclines (i.e.
non-trivial nullclines) and coexistence equilibria (intersections). Isocline
properties are analytically justified (see [137]S1 File and supplemented
[[138]31] worksheet). The local dynamics around equilibria depends on the
eigenvalues of the jacobian matrix of the PL model at the equilibrium. However,
the highly non-linear nature of the PL model (see [139]S1 File), makes it
pointless to try infer the signs of the eigenvalues by analytical means (except
for trivial and plant-only equilibrium). Thus, we propose to use to local
geometry of isocline intersections to infer local stability [[140]32]. Plant
isoclines take two main forms: [journal.pone.0117964.e006] (6) In both cases,
plants grow between the isocline and the axes, and decrease otherwise. Larva
isoclines are simpler, they start in the plant axis and bend towards the right
when insects tend towards specialization (phi < n), as shown by [141]Fig. 5. When
insects tend towards generalism (phi > n), their isoclines increase rapidly
upwards like the letter "J" (not shown here, see [142]S1 File). Insects grow
below and right of the larva isocline, and decrease otherwise.
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Fig 5. Dynamics of the simplified version of the PLA model.
Plant isoclines in green and larva isoclines in blue. Several trajectories are
shown (starting with *). The dotted line at x = 1 is the plant's carrying
capacity. When gsa/yn < 1 the plant's isocline always decreases, when gsa/yn > 1,
it bulges above the carrying capacity and displays a hump. (A) Damped
oscillations leading to globally stable coexistence dominated by antagonism
(victim-exploiter). (B) The isoclines intersect as a locally stable mutualistic
equilibrium and as a saddle point. Insects can coexist with the plant or go
extinct depending on the initial conditions. (C) This is similar to case (B),
however, a stable mutualism occurs only after damped oscillations or the insect
go extinct, depending on the initial conditions. (D) Here the system develops
oscillations approaching a limit cycle (thick loop), which creates a periodic
alternation between mutualism and antagonism. Common parameters in all panels are
b = 10, y = 0.1, µ = 1, phi = 0. For the other parameters; in (A): s = 3, e =
0.7, a = 3, g = 0.02, n = 2; in (B): s = 2.1, e = 0.21, a = 2, g = 0.05, n = 1.5;
in (C): s = 3.7, e = 0.2, a = 3, g = 0.02, n = 1.5; in (D): s = 5, e = 0.3, a =
5, g = 0.02, n = 2.
[144]https://doi.org/10.1371/journal.pone.0117964.g005
The gsa < yn case in [145]Fig. 5A illustrates scenarios in which pollination
rates (a), plant benefits (s), adult pollinator lifetimes (1/n) and
larva-to-adult transition rates (g) are low. The plant's isocline is a decreasing
curve crossing the plant's axis at its carrying capacity K (x = 1, y = 0). The
intersection with the larva isocline creates a globally stable equilibrium,
approached by oscillations of decreasing amplitude. The local stability of this
equilibrium can be explained partly by the geometry of the intersection:
[146]Fig. 5A shows that if plants increase (decrease) above (below) the
intersection point, while keeping the insect density fixed, they enter a zone of
negative (positive) growth; and the same behavior holds for the insects while
keeping the plants fixed. In ecological terms, both species are self-limited
around the equilibrium, a strong indication of stability [[147]32]. Together with
the fact that the trivial (x = 0, y = 0) and carrying capacity equilibrium (x =
1, y = 0) are saddle points, we conclude that plants and insects achieve a
globally stable equilibrium after a period of transient oscillations (provided
that insects are viable, e.g. b, g, e are large enough). This equilibrium is
demographically unfavorable for the plant because its biomass lies below the
carrying capacity (x < 1). Indeed, for extreme scenarios of negligible plant
pollination benefits (i.e. a and/or s tend to zero), the plant's isocline
approximates a straight line with a negative slope, like the isocline of a
logistic prey in a Lotka-Volterra model, which is well known to cause damped
oscillations [[148]32].
The gsa > yn case in [149]Figs. 5B,C,D cover scenarios in which pollination rates
(a), pollination benefits (s), adult pollinator lifetimes (1/n) and
larva-to-adult (harm-to-benefit) transition rates (g) are high. One part of the
plant's isocline lies above the carrying capacity, which means that coexistence
equilibria with plant biomass larger than the carrying capacity (x > 1) are
possible, and this is favorable for the plant. [150]Fig. 5B, shows and example
where the larva isocline intersects the plant's isocline twice above the carrying
capacity. One intersection is a locally stable coexistence equilibrium, whereas
the other intersection is a saddle point. The saddle point belongs to a boundary
that separates regions of initial conditions leading to insect persistence or
extinction. This can explain the Allee effect, i.e. insect growth rates increase
(go from negative to positive) with insect density when insect populations are
very small.
As the second inequality of ([151]6) widens (gsa >> yn), the plant's isocline
takes a mushroom-like shape (or "anvil" or letter "W*"), as in [152]Fig. 5C,D.
The plant's isocline displays a very prominent "hump", like in the prey isocline
of the Rosenzweig-MacArthur model [[153]27]. As a "rule of thumb", intersections
at the right of the hump would lead to damped oscillations, for the reasons
explained before ([154]Fig. 5A, for gsa < yn). Also as a "rule of thumb",
intersections at the left of the hump (like in [155]Fig. 5C,D) are expected to
result in reduced stability. This is because a small increase (decrease) along
the plant's axis leaves the plant at the growing (decreasing) side of its
isocline, promoting further increase (decrease). This means that plants do not
experience self-limitation, which is an indication of instability [[156]32], and
we infer that oscillations will not vanish. [157]Fig. 5D shows an example where
an intersection at the left of the hump causes instability, leading to limit
cycles. However, [158]Fig. 5C shows an exception of this prediction (the
intersection is stable). In both examples the intersection occurs above the
plants carrying capacity, thus revealing oscillations alternating above and below
the plant's carrying capacity. We want to stress one more time, that these
predictions based on isocline intersection configurations (left vs right of the
hump) must be taken as "rules of thumb".
[159]Fig. 5C also reveals an important consequence of the dual interaction
between the plant and the insect. As we can see, the presence of a saddle point
leads to the Allee effect explained before. But this figure also shows that large
larval densities can lead to insect extinction. This can be explained by the fact
that at large initial densities, the larva overexploits the plant, and this is
followed by an insect population crash from which it cannot recover due to the
Allee effect.
As g, s, a increase and/or y, n decrease more and more, the decreasing segment of
the plant isocline (the part at the right of the hump) approximates a decreasing
line (actually a straight asymptotic line, see [160]S1 File), while the rest of
the isocline is pushed closer and closer to the axes. In other words, when
pollination rates (a), benefits (s), adult lifetimes (1/n) and larva development
rates (g) increase, plant isoclines would resemble the isocline of a logistic
prey, with a "pseudo" carrying capacity (the rightmost extent of the isocline)
larger than the intrinsic carrying capacity (x = 1). [161]Fig. 5D is an example
of this. These conditions would promote stable coexistence with large plant
equilibrium biomasses.
Discussion
We developed a plant-insect model that considers two interaction types,
pollination and herbivory. Ours belongs to a class of models [[162]33, [163]34]
in which balances between costs and benefits cause continuous variation in
interaction strengths, as well as transitions among interaction types (mutualism,
predation, competition). In our particular case, interaction types depend on the
stage of the insect's life cycle, as inspired by the interaction between M. sexta
and D. wrightii [[164]6, [165]14] or between M. sexta and N. attenuata [[166]10].
There are many other examples of pollination-herbivory in Lepidopterans, where
adult butterflies pollinate the same plants exploited by their larvae [[167]5,
[168]7]. We assign antagonistic and mutualistic roles to larva and adult insect
stages respectively, which enable us to study the consequences of ontogenetic
changes on the dynamics of plant-insect associations, a topic that is receiving
increased attention [[169]8, [170]17]. Our model could be generalized to other
scenarios, in which drastic ontogenetic niche shifts cause the separation of
benefits and costs in time and space. However, it excludes cases like the
yucca/yucca moth interaction [[171]35] where adult pollinated ovules face larval
predation, i.e. benefits themselves are deducted.
Instead of using species biomasses as resource and service proxies [[172]34], we
consider a mechanism ([173]1) that treats resources more explicitly [[174]36]. We
use flowers as a direct proxy of resource availability, by assuming a uniform
volume of nectar per flower. Nectar consumption by insects is concomitant with
service exploitation by the plants (pollination), based on the assumption that
flowers contain uniform numbers of ovules. Pollination also leads to flower
closure [[175]19], making them limiting resources. Flowers are ephemeral compared
with plants and insects, so we consider that they attain a steady-state between
production and disappearance. As a result, the dynamics is stated only in terms
of plant, larva and adult populations, i.e. the PLA model ([176]3). The
feasibility of the results described by our analysis depends on several
parameters. The consumption, mortalities and growth rates, and the carrying
capacities (e.g. a, b, m, n and r, c in the fourth column of [177]Table 1), have
values close to the ranges considered by other models [[178]34, [179]37].
Oscillations, for example, require large herbivory rates, but this is usual for
M. sexta [[180]15].
Mutualism-antagonism cycles
The PLA model displays plant-insect coexistence for any combination of
(non-trivial) initial conditions where insects can invade when rare (R[o] > 1).
Coexistence is also possible where insects cannot invade when rare (R[o] < 1),
but this requires high initial biomasses of plants and insects (Allee effect).
Coexistence can take the form of a stable equilibrium, but it can also take the
form of stable oscillations, i.e. limit cycles.
Previous models combining mutualism and antagonism predict oscillations, but they
are transient ones [[181]35, [182]38], or the limit cycles occur entirely below
the plant's carrying capacity [[183]39]. We have good reasons to conclude that
the cycles are herbivory driven and not simply a consequence of the PLA model
having many variables and non-linearities. First of all, limit cycles require
herbivory rates (b) to be large enough. Second, given limit cycles, an increase
in the maturation rate (g) causes a transition to stable coexistence, and further
increase in herbivory is required to induce limit cycles again ([184]Fig. 2).
This makes sense because by speeding up the transition from larva to adult, the
total effect of herbivory on the plants is reduced, hence preventing a crash in
plant biomass followed by a crash in the insects. Third, when adult pollinators
have alternative food sources (phi > 1), the zone of limit cycles in the space of
parameters becomes larger ([185]Fig. 4). This also makes sense, because the total
effect of herbivory increases by an additional supply of larva (which is not
limited by the nectar of the plant considered), leading to a plant biomass crash
followed by insect decline.
The graphical analysis provides another indication that oscillations are
herbivory driven. On the one hand insect isoclines (or rather larva isoclines)
are always positively sloped, and insects only grow when plant biomass is large
enough (how large depends on insect's population size, due to intra-specific
competition). Plant isoclines, on the other hand, can display a hump ([186]Fig.
5B,C,D), and they grow (decrease) below (above) the hump. These two features of
insect and plant isoclines are associated with limit cycles in classical
victim-exploiter models [[187]27]. If there is no herbivory or another form of
antagonism (e.g. competition) but only mutualism, the plant's isocline would be a
positively sloped line, and plants would attain large populations in the presence
of large insect populations, without cycles. However, mutualism is still
essential for limit cycles: if mutualistic benefits are not large enough (gsa <
yn), plant isoclines do not have a hump ([188]Fig. 5A) and oscillations are
predicted to vanish. The effect of mutualism on stability is like the effect of
enrichment on the stability in pure victim-exploiter models [[189]28], by
allowing the plants to overcome the limits imposed by their intrinsic carrying
capacity.
There is a minor caveat regarding our graphical analysis: whereas a hump in the
plant's isocline is a requisite for oscillations to evolve into limit cycles,
this does not mean that isocline intersections at the left of the hump always
lead to limit cycles ([190]Fig. 5C). To our best knowledge, this always happens
only for quite specific conditions in pure victim-exploiter models [[191]40]. As
long as we cannot prove by analytical means that intersection geometry determines
local stability, the prediction of limit cycles remains a "rule of thumb", based
on extrapolating our knowledge about other victim-exploiter models.
Classification of outcomes: mutualism or herbivory?
Interactions can be classified according to the net effect of one species on the
abundance (biomass, density) of another (but see other schemes [[192]41]). This
classification scheme can be problematic in empirical contexts because reference
baselines such as carrying capacities are usually not known [[193]42].
Our PLA model illustrates the classification issue when non-equilibrium dynamics
are generated endogenously, i.e. not by external perturbations. Since plants are
facultative mutualists and insects are obligatory ones, one can say the outcome
is net mutualism (+,+) or net herbivory (-,+), if the coexistence is stable, and
the plant equilibrium ends up respectively above or below the carrying capacity
[[194]33, [195]34]. If coexistence is under non-equilibrium conditions and plant
oscillations are entirely below the carrying capacity (e.g. for large herbivory
rates), the outcome is detrimental for plants and hence there is net herbivory
(-,+); oscillations may in fact be considered irrelevant for this conclusion (or
may further support the case of herbivory, read below). However, when the plant
oscillation maximum is above carrying capacity and the minimum is below, like in
[196]Fig. 3, could we say that the system alternates periodically between states
of net mutualism and net herbivory? Here perhaps a time-based average over the
cycle can help up us decide. The situation could be more complicated if plant
oscillations lie entirely above the carrying capacity (see an example in [197]S1
File): one can say that the net outcome is a mutualism due to enlarged plant
biomasses, but the oscillations indicates that a victim-exploiter interaction
exists. As we can see, deciding upon the net outcome require consideration of
both equilibrium and dynamical aspects.
Factors that could cause dynamical transitions
Environmental factors.
The parameters in our analyses can change due to external factors. One of the
most important is temperature [[198]43]. It is well known, for example, that
climate warming can reduce the number of days needed by larvae to complete their
development [[199]44], making larvae maturation rates (g) higher. For insects
that display Allee effects, a cooling of the environment will cause the sudden
extinction of the insect and a catastrophic collapse of the mutualism, which
cannot be simply reverted by warming. By retarding larva development into adults,
cooling would increase the burden of herbivory over the benefits of pollination,
making the system less stable by promoting oscillations. Flowering, pollination,
herbivory, growth and mortality rates (e.g. s, a, b, r, m and n in equations
[200]1) are also temperature-dependent and they can increase or decrease with
warming depending on the thermal impacts on insect and plant metabolisms
[[201]45]. This makes general predictions more difficult. However, we get the
general picture that warming or cooling can change the balance between costs and
benefits impacting the stability of the plant-insect association.
Dynamical transitions can also be induced by changes in the chemical environment,
often as a consequence of human activity. Some pesticides, for example, are
hormone retarding agents [[202]46]. This means that their release can reduce
maturation rates, altering the balance of the interaction towards more herbivory
and less pollination and finally endangering pollination service [[203]47,
[204]48]. In other cases, the chemical changes are initiated by the plants: in
response to herbivory, many plants release predator attractants [[205]49], which
can increase larval mortality (µ). If the insect does nothing but harm, this is
always an advantage. If the insect is also a very effective pollinator, the abuse
of this strategy can cost the plant important pollination services because a dead
herbivore today is one less pollinator tomorrow.
Another factor that can increase or decrease larvae maturation rates, is the
level of nutrients present in the plant's vegetative tissue [[206]50, [207]51].
On the one hand, the use of fertilizers rich in phosphorus could increase larvae
maturation rates [[208]51]. On the other hand, under low protein consumption M.
sexta larvae could decrease maturation rate, although M. sexta larvae can
compensate this lack of proteins by increasing their herbivory levels (i.e.
compensatory consumption) [[209]50]. Thus, different external factors related to
plant nutrients could indirectly trigger different larvae maturation rates that
will potentially modify the interaction dynamics.
Pollinator's diet breadth.
An important factor that can affect the balance between mutualism and herbivory
is the diet breadth of pollinators. Alternative food sources for the adults could
lead to apparent competition [[210]52] mediated by pollination, as predicted for
the interaction between D. wrigthii (Solanacea) and M. sexta (Sphingidae) in the
presence of Agave palmieri (plant) [[211]6]: visitation of Agave by M. sexta does
not affect the pollination benefits received by D. wrightii, but it increases
oviposition rates on D. wrightii, increasing herbivory. As discussed before, such
an increase in herbivory could explain why oscillations are more widespread when
adult insects have alternative food sources (phi > 0) in our PLA model.
Although we did not explore this with our model, the diet breadth of the larva
could also have important consequences. In the empirical systems that inspired
our model, the larva can have alternative hosts [[212]14], spreading the costs of
herbivory over several species. The local extinction of such hosts could increase
herbivory on the remaining ones, promoting unstable dynamics. To explore these
issues properly, models like ours must be extended to consider larger community
modules or networks, taking into account that there is a positive correlation
between the diet breadths of larval and adult stages [[213]7].
From the perspective of the plant, the lack of alternative pollinators could also
lead to increased herbivory and loss of stability. The case of the tobacco plant
(N. attenuata) and M. sexta is illustrative. These moths are nocturnal
pollinators, and in response to herbivory by their larvae, the plants can change
their phenology by opening flowers during the morning instead. Thus, oviposition
and subsequent herbivory can be avoided, whereas pollination can still be
performed by hummingbirds [[214]11]. Although hummingbirds are thought to be less
reliable pollinators than moths for several reasons [[215]9], they are an
alternative with negligible costs. Thus, a decline of hummingbird populations
will render the herbivore avoidance strategy useless and plants would have no
alternative but to be pollinated by insects with herbivorous larvae that promote
oscillations.
Conclusions
Many insect pollinators are herbivores during their larval phases. If pollination
and herbivory targets the same plant (e.g. as between tobacco plants and
hawkmoths), the overall outcome of the association depends on the balance between
costs and benefits for the plant. As predicted by our plant-larva-adult (PLA)
model, this balance is affected by changes in insect development: the faster
larvae turns into adults the better for the plant and the interaction is more
stable; the slower this development the poorer the outcome for the plant and the
interaction is less stable (e.g. oscillations). Under plant-insect oscillations,
this balance can be dynamically complex (e.g. periodic alternation between
mutualism and antagonism). Since maturation rates play an essential role in long
term stability, we predict important qualitative changes in the dynamics due to
changes in environmental conditions, such as temperature and chemical compounds
(e.g. toxins, hormones, plant nutrients). The stability of these mixed
interactions can also be greatly affected by changes in the diet generalism of
the pollinators.
Supporting Information
[216]S1 File. Supplement containing the appendices cited in the main text.
[217]https://doi.org/10.1371/journal.pone.0117964.s001
(PDF)
[218]S1 Fig. Detail of the b vs g parameter space for specialist pollinators in the PLA
model.
The ellipse describes the joint variation of g and b taking place in the
bifurcation diagram in [219]S2 Fig.
[220]https://doi.org/10.1371/journal.pone.0117964.s002
(EPS)
[221]S2 Fig. Bifurcation diagram for the PLA model.
Parameters g and b vary along the elliptical path drawn in [222]S1 Fig, with
reference for each quarter of a rotation. Solid (broken) lines represent stable
(unstable) equilibria, black (white) circles represent limit cycle maxima and
minima. The x = 1 line corresponds to the plant carrying capacity. HB[super]:
super-critical and HB[sub]: sub-critical Hopf bifurcations, BP: branching point
(transcritical bifurcation), LP: limit point (fold bifurcation).
[223]https://doi.org/10.1371/journal.pone.0117964.s003
(EPS)
[224]S3 Fig. Main configurations of the plant isocline.
We only consider the O-K segment in the positive octant (hatched square). In A
the isocline lies below the plant's carrying capacity (i.e. left of K), in B
parts of the isocline lie above (i.e. right of K).
[225]https://doi.org/10.1371/journal.pone.0117964.s004
(EPS)
[226]S4 Fig. Shape of the plant's isocline.
(A) As g increases and y, n decrease, points P and Q move closer to the diagonal
asymptote (broken line), and the isocline eventually adopts the form of a
mushroom. (B) As b increases, O, P, Q and the diagonal asymptote move towards the
plant axis and the isocline is compressed vertically.
[227]https://doi.org/10.1371/journal.pone.0117964.s005
(EPS)
[228]S5 Fig. Main configurations of the larva isocline.
The isocline consists of three black lines, but only the segment in the positive
octant (hatched square) is biologically relevant. For A and B phi < n. For C and
D phi > n. The green parabola p(x) is the numerator of the isocline and the
circles indicate its roots, where x[0]: positive root. The red parabola q(x) is
the denominator of the isocline, which has two roots x = 0 and x = x[v], both of
which are also the vertical asymptotes of the isocline. The isocline also has an
horizontal asymptote y[h]. The alternative in part D can be dismissed because it
implies a detrimental effect of plants on insects.
[229]https://doi.org/10.1371/journal.pone.0117964.s006
(EPS)
[230]S6 Fig. Shape of the larva isocline.
(A) For phi < n the larva isocline moves closer to the larva axis and becomes
more shallow as g and b increase. (B) For phi > n the larva isocline becomes
closer to the larva axis.
[231]https://doi.org/10.1371/journal.pone.0117964.s007
(EPS)
[232]S7 Fig. Plant oscillating above their carrying capacity in the PLA model.
Blue:plant, green:larva, red:adult. The carrying capacity is indicated by the
dotted line.
[233]https://doi.org/10.1371/journal.pone.0117964.s008
(EPS)
[234]S8 Fig. Oscillations in the PLA model started with different initial conditions
(*).
The oscillations can dampen out (blue) or converge to a limit cycle (red).
[235]https://doi.org/10.1371/journal.pone.0117964.s009
(EPS)
[236]S9 Fig. Interaction dynamics when flowers are explicitly considered.
Blue:plant, green:larva, red:adult, black:flowers. The dotted line indicates the
plant's carryng capacity.
[237]https://doi.org/10.1371/journal.pone.0117964.s010
(EPS)
Acknowledgments
We thank Rampal Etienne for the discussions that inspired us to write this
article. We thank the comments and suggestions from our colleagues of the Centre
for Biodiversity Theory and Modelling in Moulis, France, the Centre for
Australian National Biodiversity and Research at CSIRO in Canberra, Australia,
and two anonymous reviewers.
Author Contributions
Conceived and designed the experiments: TAR FEV. Performed the experiments: TAR.
Analyzed the data: TAR. Contributed reagents/materials/analysis tools: TAR. Wrote
the paper: TAR FEV.
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413. https://journals.plos.org/plosone/article/figure/powerpoint?id=10.1371/journal.pone.0117964.g002
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415. https://journals.plos.org/plosone/article/figure/image?download&size=original&id=10.1371/journal.pone.0117964.g002
416. https://journals.plos.org/plosone/article/figure/powerpoint?id=10.1371/journal.pone.0117964.g003
417. https://journals.plos.org/plosone/article/figure/image?download&size=large&id=10.1371/journal.pone.0117964.g003
418. https://journals.plos.org/plosone/article/figure/image?download&size=original&id=10.1371/journal.pone.0117964.g003
419. https://journals.plos.org/plosone/article/figure/powerpoint?id=10.1371/journal.pone.0117964.g004
420. https://journals.plos.org/plosone/article/figure/image?download&size=large&id=10.1371/journal.pone.0117964.g004
421. https://journals.plos.org/plosone/article/figure/image?download&size=original&id=10.1371/journal.pone.0117964.g004
422. https://journals.plos.org/plosone/article/figure/powerpoint?id=10.1371/journal.pone.0117964.g005
423. https://journals.plos.org/plosone/article/figure/image?download&size=large&id=10.1371/journal.pone.0117964.g005
424. https://journals.plos.org/plosone/article/figure/image?download&size=original&id=10.1371/journal.pone.0117964.g005
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