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   Research Article

Pollinator Foraging Adaptation and Coexistence of Competing Plants

     * Tomás A. Revilla ,
       * E-mail: [50]tomrevilla@gmail.com
       Affiliation Institute of Entomology, Biology Center, Czech Academy of
       Sciences, Ceské Budejovice, Czech Republic
       x
     * Vlastimil Krivan
       Affiliations Institute of Entomology, Biology Center, Czech Academy of
       Sciences, Ceské Budejovice, Czech Republic, Department of Mathematics and
       Biomathematics, Faculty of Science, University of South Bohemia, Ceské
       Budejovice, Czech Republic
       x

Pollinator Foraging Adaptation and Coexistence of Competing Plants

     * Tomás A. Revilla,
     * Vlastimil Krivan

   PLOS
   x
     * Published: August 9, 2016
     * [51]https://doi.org/10.1371/journal.pone.0160076
     *

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     * [57]Figures

Figures

   Fig 1
   Table 1
   Fig 2
   Fig 3
   Fig 4
   Fig 5
   Fig 6
   Fig 7

Abstract

   We use the optimal foraging theory to study coexistence between two plant species
   and a generalist pollinator. We compare conditions for plant coexistence for
   non-adaptive vs. adaptive pollinators that adjust their foraging strategy to
   maximize fitness. When pollinators have fixed preferences, we show that plant
   coexistence typically requires both weak competition between plants for resources
   (e.g., space or nutrients) and pollinator preferences that are not too biased in
   favour of either plant. We also show how plant coexistence is promoted by
   indirect facilitation via the pollinator. When pollinators are adaptive foragers,
   pollinator's diet maximizes pollinator's fitness measured as the per capita
   population growth rate. Simulations show that this has two conflicting
   consequences for plant coexistence. On the one hand, when competition between
   pollinators is weak, adaptation favours pollinator specialization on the more
   profitable plant which increases asymmetries in plant competition and makes their
   coexistence less likely. On the other hand, when competition between pollinators
   is strong, adaptation promotes generalism, which facilitates plant coexistence.
   In addition, adaptive foraging allows pollinators to survive sudden loss of the
   preferred plant host, thus preventing further collapse of the entire community.

   Citation: Revilla TA, Krivan V (2016) Pollinator Foraging Adaptation and
   Coexistence of Competing Plants. PLoS ONE 11(8): e0160076.
   https://doi.org/10.1371/journal.pone.0160076

   Editor: Takeshi Miki, National Taiwan University, TAIWAN

   Received: April 4, 2016; Accepted: July 13, 2016; Published: August 9, 2016

   Copyright: © 2016 Revilla, Krivan. 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: The data used by figures was generated using computer code
   which is provided as supplementary information. The source code is available
   here: [59]https://github.com/tomrevilla/pollifor.

   Funding: This work was supported by Institute of Entomology RVO:60077344 awarded
   to VK and European Union's Horizon 2020 RISE grant 690817 awarded to VK.

   Competing interests: The authors have declared that no competing interests exist.

Introduction

     Et il se sentit très malheureux. Sa fleur lui avait raconté qu'elle était
     seule de son espèce dans l'univers. Et voici qu'il en était cinq mille, toutes
     semblables, dans un seul jardin!

     Le Petit Prince, Chapitre XX - Antoine de Saint-Exupéry

   The diversity and complexity of mutualistic networks motivate ecologists to
   investigate how they can remain stable and persistent over time. Mathematical
   models and simulations show that some properties of mutualistic networks (e.g.,
   low connectance and high nestedness) make them more resistant against cascading
   extinctions [[60]1], more likely to sustain large numbers of species [[61]2], and
   more stable demographically [[62]3]. However, simulations [[63]4, [64]5] also
   indicate that mutualism increases competitive asymmetries, causing complex
   communities to be less persistent. These studies consider large numbers of
   species, parameters and initial conditions, making it difficult to understand the
   interplay between mutualisms (e.g., between plant and animal guilds) and
   antagonisms (e.g., resource competition between plants). These questions are
   easier to study in the case of community modules consisting of a few species only
   [[65]6].

   In this article we consider a mutualistic module with two plant species and one
   pollinator species ([66]Fig 1a). This module combines several direct and indirect
   interactions that are either density- or trait-mediated (sensu [[67]7]). These
   include plant intra- and inter-specific competition (for e.g., space), plant
   competition for pollinator services, and pollinator intra-specific competition
   for plant resources (e.g., nectar). Some of these interactions depend on changes
   in population densities only (e.g., intra- and inter-specific plant competition),
   while the others depend also on individual morphological and behavioural traits.
   As some of them have positive and some of them negative effect on plant
   coexistence, it is difficult to predict their combined effects on species
   persistence and stability.
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   Fig 1. Community module consisting of plant 1 and 2, and pollinator A.

   (a) Plants affect each other directly (solid arrows) by competition for space or
   resources (c[1], c[2]), and indirectly (dashed arrows) via shared pollinator with
   plant preferences u[1] and u[2]. (b) When pollinator preferences are fixed and
   not too biased, a large density of plant 1 maintains a large pollinator density,
   which has an indirect positive effect on low density plant 2. In (c,d) pollinator
   preferences for plants are adaptive (dashed arrows change thickness). When
   pollinators are rare (c), preferences favour abundant plant 1, which results in a
   negative indirect effect on rare plant 2. When pollinators become abundant (d),
   competition between pollinators lead to balanced preferences, and the indirect
   effect on plant 2 becomes positive. The viability of plant 2 depends on the
   balance between indirect and direct effects. Image sources for panel (a) were
   taken from: [69]https://openclipart.org.

   [70]https://doi.org/10.1371/journal.pone.0160076.g001

   First, we will assume that pollinator preferences for plants are fixed. In this
   case, there is a negative effect of one plant on the other by direct competition
   and a positive indirect effect that is mediated by the shared pollinators, called
   facilitation [[71]8-[72]10]. As one plant population density increases,
   pollinator density increases too, which, in turn, increases pollination rate of
   the other plant ([73]Fig 1b). This is an indirect interaction between plants that
   is mediated by changes in abundance of the pollinator (i.e., density mediated
   indirect interaction). Because facilitation has the opposite effect to direct
   plant competition (see [74]Fig 1b) it is important to clarify under which
   situations the positive effect of facilitation prevails, and we study this
   question by using a mathematical model.

   Second, we will assume that pollinator preferences are adaptive. We will assume
   that pollinator fitness is defined as the per capita pollinator population growth
   rate that depends on plant (that produce resources for pollinators) as well as on
   pollinator densities. First, pollinators benefit from nectar quality and nectar
   abundance (which correlates with plant population density). Second, pollinators
   compete for resources. This competition will play an important effect when
   pollinator population densities are high. A game theoretical approach to
   determine the optimal pollinator strategy is the Ideal Free Distribution (IFD)
   [[75]11, [76]12]. This theory predicts that when pollinators are at low numbers,
   they will specialize on one plant only. As their population density will
   increase, they become generalists feeding on and pollinating both plants. This
   mechanism causes a negative effect of the preferred plant on the other plant,
   because when at low densities, pollinators will specialize on one plant only
   ([77]Fig 1c). This is an example of a positive feedback where "the rich becomes
   richer and the poor get poorer". Competition for pollinators is an example of a
   trait-mediated effect caused by pollinator behaviour. Pollinator specialization
   on one plant only is detrimental for the other plant. However, as pollinator
   population density will increase, competition for resources among pollinators
   will increase too [[78]13], and the IFD predicts that they become generalists,
   which promotes plant coexistence ([79]Fig 1d). Once again, combination of
   positive and negative effects between plants creates complicated feedbacks
   between population densities and pollinator behaviour that are impossible to
   disentangle without an appropriate mathematical model.

   Our main goal is to study how pollinator preferences and plant competition affect
   plant coexistence. First, we study the dynamics of the plant-pollinator module
   when pollinator preferences are fixed. Second, we calculate the pollinator's
   evolutionarily stable foraging strategy (ESS) at fixed plant and pollinator
   population densities, and we study plant coexistence assuming pollinators
   instantaneously track their ESS. This case corresponds to time scale separation
   where population dynamics operate on a slow time scale, while pollinator foraging
   preferences operate on a fast time scale. Finally, we consider the situation
   without time scale separation and we model preference dynamics with the
   replicator equation. Overall, we show that pollinator foraging adaptation has
   complex effects, sometimes equivocal, on plant coexistence. On the one hand
   pollinator adaptation increases competitive asymmetries among plants, promoting
   competitive exclusion. On the other hand competition for plant resources among
   pollinators promotes generalism over specialization, which can prevent the loss
   of pollination services for some plants and promote coexistence.

Methods

Mutualistic community model

   Let us consider two plant species P[i] (i = 1, 2) and one pollinator species A
   ([80]Fig 1a). Plants produce resources F[i] (i = 1, 2) such as nectar at a rate
   a[i] per plant. Resources not consumed by the pollinator decrease with rate w[i]
   (e.g., nectar can be re-absorbed, decay or evaporate). Resources are consumed by
   pollinators at rate b[i] per resource per pollinator. Pollinator's relative
   preferences for either plant are denoted by u[i] with u[1] + u[2] = 1. Plant
   birth rates are proportional to the rate of pollen transfer that is concomitant
   with resource consumption. Thus, we assume that plant birth rates are
   proportional to pollinator resource consumption rates (u[i] b[i] F[i] A)
   multiplied by conversion efficiency r[i]. Pollinator birth rates are proportional
   to resource consumption with corresponding conversion efficiency e[i]. Plants and
   pollinators die with the per capita mortality rate m[i] (i = 1, 2) and d,
   respectively.

   Assuming that plant resources equilibrate quickly with current plant and
   pollinator densities [[81]14], i.e., dF[i]/dt = 0, plants and pollinator
   population dynamics are described by the following model ([82]S1 Appendix)
   [journal.pone.0160076.e001] (1a) [journal.pone.0160076.e002] (1b)
   [journal.pone.0160076.e003] (1c) in which plant growth rates are regulated by
   competition for non-living resources (e.g., light, nutrients, space) according to
   the Lotka-Volterra competition model, where c[j] is the negative effect of plant
   j on plant i relative to the effect of plant i on itself (i.e., competition
   coefficient), and K[i] stands for the habitat carrying capacity [[83]4]. Notice
   that plant growth rates saturate with pollinator density (e.g.,
   [journal.pone.0160076.e004] ) and pollinator growth rates decrease due to
   intra-specific competition for plant resources (e.g., [journal.pone.0160076.e005]
   ) [[84]15]. In this model plants and pollinators are obligate mutualists, i.e.,
   without pollinators plants go extinct and without plants the pollinator goes
   extinct. We do not model facultative mutualism because this introduces additional
   factors (e.g. alternative pollinators, vegetative growth), which complicate the
   analysis of direct and indirect effects of the three species module.

Fixed pollinator preferences

   We start our analyses assuming that pollinator preferences for plants (u[1] and
   u[2] = 1 - u[2]) are fixed at particular values ranging from 0 to 1. This means
   that for u[1] = 1 or 0 pollinators are plant 1 or plant 2 specialists,
   respectively, while for 0 < u[1] < 1 they are generalists. Since model [85]Eq
   (1a) is non-linear, analytical formulas for interior equilibria and corresponding
   stability conditions are out of reach. However, it is possible to obtain
   coexistence conditions by means of invasibility analysis. First, we obtain
   conditions for stable coexistence of a single plant-pollinator subsystem at an
   equilibrium. Second, we ask under what conditions the missing plant species can
   invade when the resident plant-pollinator subsystem is at the equilibrium. In
   particular, we are interested in the situation where each plant species can
   invade the other one, because this suggests coexistence of both plants and
   pollinators. Derivation of invasion conditions are provided in [86]S1 Appendix.

   In general, invasibility does not guarantee coexistence [[87]16, [88]17]. Also, a
   failure to invade when rare does not rule out possibility of invasion success
   when the invading species is at large densities. For these reasons we complement
   our invasibility analysis by numerical bifurcation analysis using XPPAUT
   [[89]18], and parameter values given in [90]Table 1. While not empirical, the
   values fall within ranges typically employed by consumer-resource models (e.g.,
   [[91]19]). Plant-specific parameters are equal except for e[i] and u[i] (i = 1,
   2). We assume that e[1] > e[2], i.e., plant 1 provides pollinators with higher
   energy when compared to plant 2.
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   Table 1. Parameters of [93]Model (1a) and [94]Eq (4).

   [95]https://doi.org/10.1371/journal.pone.0160076.t001

Adaptive pollinator preferences

   When pollinators behave as adaptive foragers their plant preferences should
   maximize their fitness. The pay-off a pollinator gets when pollinating only plant
   i is defined as the per capita pollinator growth rate on that plant, i.e.,
   [journal.pone.0160076.e006] (2)

   We observe that these pay-offs depend both on plant and pollinator densities and
   on the pollinator distribution u[1], i.e., they are both density and frequency
   dependent. Now let us consider fitness of a generalist mutant pollinator with
   strategy [journal.pone.0160076.e007] . Its fitness is then defined as the average
   pay-off, i.e., [journal.pone.0160076.e008] (3)

   Using this fitness function we will calculate the evolutionarily stable strategy
   [[96]20, [97]21] of pollinator preferences at current plant and pollinator
   densities. When pollinators adjust their preferences very fast as compared to
   changes in population densities, we will use the ESS together with population
   dynamics [98]Eq (1a) to model effects of pollinator plasticity on population
   dynamics. This approach corresponds to time scale separation where population
   densities (plants and animals) change very slowly compared to pollinator
   adaptation. We are also interested in the situation when the two time scales are
   not separated, but pollinators foraging preferences still tend to the ESS. In
   these cases we use the replicator equation [[99]21] to model dynamics of
   pollinator preferences u[1] for plant 1 (u[2] = 1 - u[1])
   [journal.pone.0160076.e009] (4) where n >= 0 is the adaptation rate. [100]Eq (4)
   assumes that pollinator's preferences evolve towards a higher energy intake and
   its equilibrium coincides with the ESS. Thus, if pollinators obtain more energy
   when feeding on plant 1, preferences for plant 1 increases. When n >= 1,
   adaptation is as fast as population dynamics or faster. This describes plastic
   pollinators that track changing flower densities very quickly (i.e., within an
   individual life-span). This is the case when adaptation is a behavioural trait.
   In fact, for n tending to infinity pollinators adopt the ESS instantaneously.
   Adaptation can also involve morphological changes requiring several generations
   (i.e., evolution). In that case n < 1, and adaptation lags behind population
   dynamics (i.e., changes in preferences require more generations). And the n = 0
   case applies to non-adaptive pollinators. We remark that perfect specialization
   on plant 1 or plant 2 correspond to the equilibrium u[1] = 1 or u[1] = 0,
   respectively.

   Using [101]Model (1a) and replicator [102]Eq (4), we simulate the effects of
   pollinator adaptation and plant direct inter-specific competition on coexistence.
   We consider four common inter-specific competition coefficients: c[1] = c[2] = c
   = 0, 0.4, 0.8, and 1.2, and four adaptation rates: n = 0, 0.1, 1 and n = infty.
   Level n = 0 extends our analysis for non-adaptive pollinators (fixed preferences)
   beyond invasion conditions. Level n = 0.1 implies slow evolutionary adaptation,
   like in adaptive dynamics [[103]22]. At n = 1 adaptation is as fast as
   demography, i.e., pollinators adapt during their lifetime. For n = infty
   adaptation is infinitely fast when compared to population densities and
   preferences are given by the ESS.

   Community dynamics and the dynamics of pollinator preferences can be sensitive to
   initial conditions. There are four degrees of freedom for the initial conditions
   (P[1], P[2], A and u[1] at t = 0). We reduce this number to two degrees of
   freedom. First, we vary P[1](0) from 0 to K in 100 steps while P[2](0) = K -
   P[1](0), where K = K[1] = K[2] = 50 is the common carrying capacity. The choice K
   = 50 is high enough to avoid pollinator extinction due to the Allee effect in the
   majority of the simulations. Second, we consider two scenarios:
    1. Scenario I: Initial pollinator density A(0) varies from 0 to 50 in 100 steps
       and initial pollinator preference is equal to the ESS.
    2. Scenario II: Initial pollinator preference u[1](0) varies from 0.001 to 0.999
       in 100 steps [0.001, 0.01, 0.02, ..., 0.98, 0.99, 0.999] and initial
       pollinator density is kept at A(0) = 2.

   Scenario I assumes that pollinators preferences are at the ESS for given initial
   plant and pollinator densities, with an exception when the ESS is 0 or 1 in which
   case we perturb it to u[1] = 0.001 or u[1] = 0.999. This is necessary because the
   replicator [104]Eq (4) does not consider mutations that may allow specialists to
   evolve towards generalism.

   Scenarios I and II complement each other. In both of them initial plant
   composition (P[1]: P[2]) influences the outcome. For scenario II we also used
   A(0) = 50, but we did not find important qualitative differences. Thus, for both
   scenarios we simulate Models ([105]1a) and ([106]4) with 100 × 100 = 10^4
   different initial conditions. This systematic approach allows us to delineate
   boundaries between plant coexistence and extinction regions. Models ([107]1a) and
   ([108]4) is integrated (Runge-Kutta 4th, with Matlab [[109]23]) with the rest of
   the parameters taken from [110]Table 1. A plant is considered extinct if it
   attains a density less than 10^-6 after time t = 20000.

Results

Fixed preferences

   [111]System (1a) models obligatory mutualism between plants and pollinators.
   Plants cannot grow in absence of pollinators and pollinators cannot reproduce
   without plants. Thus, the trivial equilibrium at which all three species are
   absent (P[1] = P[2] = A = 0) is always locally asymptotically stable [[112]24,
   [113]25], because when at low population densities, pollinators cannot provide
   enough pollination services to plants that will die and, similarly, when at low
   densities, plants do not provide enough nectar to support pollinators.

   By setting dP[1]/dt = dA/dt = 0 with P[1] > 0, P[2] = 0, A > 0 in [114]Eq (1a)
   and (1c), non-trivial plant 1-pollinator equilibria are
   [journal.pone.0160076.e010] (5) where D[1] = -4b[1] de[1] K[1] m[1] r[1] u[1]
   w[1]+ (b[1] e[1] K[1](m[1] - a[1] r[1])u[1] + dr[1] w[1])^2. These two equilibria
   are feasible (positive) if a[1] r[1] > m[1] and D[1] > 0. The first is a growth
   requirement: if not met, even an infinite number of specialized pollinators (with
   u[1] = 1) cannot prevent plant 1 extinction. The second condition is met when
   pollinator preference for plant 1 (u[1]) is above a critical value
   [journal.pone.0160076.e011] (6)

   By symmetry, there are two non-trivial plant 2-pollinator equilibria (P[2±],
   A[2±]). They are feasible if a[2] r[2] > m[2] and D[2] > 0 (D[2] is like D[1]
   with interchanged sub-indices). The second condition is met when pollinator
   preferences for plant 2 are strong enough (i.e., preferences for plant 1 are weak
   enough) so that u[1] is below a critical value u[1b] [journal.pone.0160076.e012]
   (7) In both cases the equilibrium that is closer to the origin ((P[1-], A[1-])
   when plant 2 is missing and (P[2-], A[2-]) when plant 1 is missing) is unstable.
   This instability indicates critical threshold densities. When plant i and
   pollinator densities are above these thresholds, coexistence is possible.
   Otherwise, the system converges on the extinction equilibrium mentioned before.
   This is a mutualistic Allee effect [[115]26, [116]27].

   The equilibrium that is farther from the origin ((P[1+], A[1+]) when plant 2 is
   missing and (P[2+], A[2+]) when plant 1 is missing) will be called the resident
   equilibrium. Resident equilibria are stable with respect to small changes in
   resident plant and pollinator densities, but may be unstable against invasion of
   small densities of the missing plant species. In the case of the plant
   1-pollinator equilibrium (P[1+], A[1+]), plant 2 invades (i.e., achieves a
   positive growth rate when rare) if the competitive effect of plant 1 on plant 2
   (c[1]), is smaller than [journal.pone.0160076.e013] (8) whereas plant 1 invades
   the plant 2-pollinator equilibrium (P[2+],A[2+]) if the competitive effect of
   plant 2 on plant 1 (c[2]), is smaller than [journal.pone.0160076.e014] (9)

   Functions a(u[1]) and b(u[1]) are real when D[1] > 0 and D[2] > 0, respectively.
   In other words, invasibility only makes sense when the plant 1-pollinator
   resident equilibrium exists (u[1] > u[1a]) or, when the plant 2-pollinator
   resident equilibrium exists (u[1] < u[1b]), respectively. The graphs of Eqs
   ([117]6), ([118]7), ([119]8) and ([120]9) divide the pollinator
   preference-competition parameter space into several regions ([121]Fig 2 where c =
   c[1] = c[2]). Notice that because a and b are only feasible to the right of u[1a]
   and to the left of u[1b], respectively, their graphs may or may not intersect
   depending on the position of u[1a] and u[1b] (cf. panel a vs. b). We show this by
   setting a common plant carrying capacity K = K[1] = K[2] and making it larger or
   smaller than a critical value ([122]S1 Appendix) [journal.pone.0160076.e015] (10)
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   Fig 2. Interaction outcomes as a function of competition strength and fixed
   pollinator preferences.

   Solid lines (coloured, found analytically) determine regions where single plant
   equilibria exist and whether they can be invaded or not. Dashed lines (in black,
   found numerically, like g) determine outcomes that cannot be predicted by
   invasibility analysis. Plant 2 can invade Plant 1 in the region between the red
   vertical line u[1a] and the red curve a. Plant 1 can invade plant 2 in the region
   between the green curve b and the green vertical line u[1b]. The final
   composition of the community is indicated by P1 = plant 1 wins, P2 = plant 2
   wins, P1 + P2 = coexistence, EXT = extinction of all species; the "or" separator
   indicates that the outcome depends on the initial conditions. Parameters from
   [124]Table 1, with (a) K[i] = 60 and (b) K[i] = 35 (i.e., above and below
   critical K* = 37.5, see [125]Eq (10)). Representative dynamics for parameter
   combinations at a-o, b-o, c-o and d-o are illustrated in corresponding panels of
   [126]Fig 3.

   [127]https://doi.org/10.1371/journal.pone.0160076.g002

   Productive environments (K > K*) support coexistence of both plant-pollinator
   resident equilibria for intermediate pollinator preferences. This is not so in
   unproductive environments (K < K*), where resident equilibria occur within
   separated ranges of pollinator preferences (see below).

   First, we assume a high plant carrying capacity satisfying K > K*. Then u[1a] <
   u[1b], and a(u[1]) and b(u[1]) intersect like in [128]Fig 2a. This leads to
   several plant invasion scenarios. We start with preferences satisfying u[1a] <
   u[1] < u[1b]. Such intermediate pollinator preferences allow each species to
   coexist with the pollinator at a stable equilibrium. If competition is weak
   enough (see the region denoted as "P1 + P2" in [129]Fig 2a), the missing plant
   can invade the resident plant-pollinator equilibrium which leads to both plant
   coexistence. In contrast, if competition is strong enough (see the region denoted
   as "P1 or P2" in [130]Fig 2a), the missing plant cannot invade. Thus, either
   plant 1 or plant 2 wins depending on the initial conditions (i.e., the resident
   plant that establishes first wins). In between these two outcomes of mutual
   invasion and mutual exclusion, there are two wedge-shaped regions (see regions
   denoted as "P1", and "P2" in [131]Fig 2a). In the right (left) region plant 1
   (plant 2) invades and replaces plant 2 (plant 1) but not the other way around.
   The outcomes in the regions of [132]Fig 2a that are either to the left of u[1a],
   or to the right of u[1b] are very different, because whether the missing plant
   can invade or not when rare depends entirely on facilitation by the resident
   plant. Indeed, let us consider the region of the parameter space in [133]Fig 2a
   to the right of the vertical line u[1b] and below the curve a. In this region
   (denoted by "P1 + P2") pollinator preference for plant 2 is so low that plant 2
   alone cannot support pollinators at a positive density. It is only due to
   presence of plant 1 that allows plant 2 survival through facilitation ([134]Fig
   3a). Indeed, when plant 1 is resident, it increases pollinator densities to such
   levels that allow plant 2 to invade. In other words, plant facilitation due to
   shared pollinators widens plant niche measured as the range of pollinator
   preferences at which the plant can survive at positive densities. When
   inter-specific plant competition is too high (see the region above the curve a
   and to the right of u[1b]) plant 2 cannot invade. Similarly, when pollinator
   preferences for plant 1 are too low (i.e., to the left of the line u[1a]),
   coexistence relies on facilitation provided by plant 2 (resident) to plant 1
   (invader) and on plant competition being not too strong (below curve b); if
   competition is too strong (above b) plant 1 cannot invade.
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   Fig 3. [136]Model (1a) dynamics with fixed pollinator preferences.

   Population densities are represented by: green squares = plant 1, red diamonds =
   plant 2 and pink circles = pollinator. Panels (a) K[i] = 60, u[1] = 0.8, c[i] =
   0.4, (b) K[i] = 60, c[i] = 1.2, u[1] = 0.605, (c) K[i] = 60, c[i] = 1.2, u[1] =
   0.607 and (d) K[i] = 35, u[1] = 0.4, c[i] = 0.8 correspond to positions of points
   a-o, b-o, c-o and d-o in [137]Fig 2. Other parameters as in [138]Table 1.

   [139]https://doi.org/10.1371/journal.pone.0160076.g003

   Second, we assume low plant carrying capacity satisfying K < K*. Then u[1a] >
   u[1b], and a(u[1]) and b(u[1]) never intersect ([140]Fig 2b) in the positive
   quadrant of the parameter space. For intermediate pollinator preferences
   satisfying u[1b] < u[1] < u[1a] coexistence by invasion of the rare plant is not
   possible. The reason is that in this region neither plant 1, nor plant 2, can
   coexist with pollinators. So, there is no resident system consisting of one plant
   and pollinators that could be invaded by the missing rare plant. In regions to
   the left of u[1b] plant 2 coexists with pollinators and to the right of u[1a]
   plant 1 coexists with pollinators at a stable equilibrium and invasion conditions
   for the missing plant when rare are similar to the case where K > K*. Once again,
   in these regions coexistence of both plants can be achieved because of the
   resident plant facilitates the other plant invasion. The important prediction of
   this invasion analysis is that the density mediated indirect interaction between
   plants through the shared pollinator, i.e., plant facilitation, increases the set
   of parameter values for which coexistence of both plants is possible.

   Although invasion analysis proves to be very useful when analysing [141]Model
   (1a), it does not answer the question whether there are some other attractors
   that cannot be reached by invasion of the missing species when rare. Using
   numerical bifurcation software (XPPAUT [[142]18]), we found additional outcomes
   not predicted by invasibility analysis. When K > K* ([143]Fig 2a) invasibility
   analysis predicts that plant 2 cannot grow when rare for strong inter-specific
   plant competition when c > a. However, our numerical analysis shows that it is
   still possible for plant 2 to invade provided its initial population density is
   large enough. The community dynamics then either oscillate along a limit cycle
   ([144]Fig 3b), or converge to a stable equilibrium ([145]Fig 3c). Such behaviour
   was observed in the region denoted by "P1 or P1 + P2" of [146]Fig 2a. This shows
   that [147]Model (1a) has multiple attractors (including a limit cycle). The right
   dashed boundary of that region corresponds to a fold bifurcation where a locally
   stable interior equilibrium merges with an unstable equilibrium and disappears
   for higher values of u[1]. Between the two dashed curves there is another Hopf
   bifurcation curve (not shown in [148]Fig 2a) where the interior equilibrium
   looses its stability and a limit cycle emerges. As preference for plant 1
   decreases towards the left dashed boundary, the amplitude of the limit cycle
   tends to infinity.

   When K < K* we found a curve g(u[1]) that further divides the parameter space
   ([149]Fig 2b). For the intermediate pollinator preferences (u[1b] < u[1] < u[1a])
   where neither plant can be a resident, and below g curve ("P1 + P2 or EXT"),
   coexistence is achievable if both plants and the pollinator are initially at high
   enough densities. This is an extreme example of plant facilitation. If combined
   plant abundances are not large enough, then both plants and the pollinator go
   extinct as already predicted by the invasion analysis. Also, if one plant species
   is suddenly removed, extinction of pollinator and the other plant follows. Above
   the g curve, plant competition is too strong to allow any coexistence and the
   outcome is always global extinction ("EXT"). When preference for plant 1 is low
   (u[1] < u[1b]), the g curve is slightly above the b curve so that the possible
   coexistence region is slightly larger than the coexistence region obtained by
   invasion of the rare plant ("P2 or P1 + P2"). However, plant coexistence in the
   region between the two curves depends on the initial density of plant 1: if
   P[1](0) is very low, plant 2 wins as predicted by the invasion analysis, but if
   P[1](0) is large enough, plant 1 will invade and coexist at an interior
   equilibrium with plant 2. In the opposite situation, where preference for plant 1
   is very high (u[1] > u[1a]), g divides the region where plant 2 can invade
   (assuming c < a) as follows. Below g, competition is weak and plant 2 invasion is
   followed by stable coexistence thanks to resident facilitation. Above g,
   competition is strong and plant 2 invasion causes plant 1 extinction followed by
   plant 2 extinction. This is because pollinator preference for plant 1 is too
   strong (u[1] > u[1b]) which does not allow pollinators to survive on plant 2.
   Thus, invasion by plant 2 leads to global extinction ("EXT"). [150]Fig 3d shows
   an example of such global extinction caused by invasion. Once again, invasion of
   plant 2 is possible due to facilitation by plant 1. As plant 2 invades, it has
   also an indirect positive effect on plant 1 through facilitation. But this
   positive effect does not outweigh the direct negative effect plant 2 has on plant
   1 due to direct competition for resources. Apart from this case of global
   extinction caused by invasion, numerical analysis with parameters from [151]Table
   1 confirms predictions of our invasion analysis that in the case where one or
   both equilibria with one plant missing exist(s), the invasibility conditions c <
   a and c < b predict the existence of a locally stable interior equilibrium at
   which both plants coexist with the pollinator.

Adaptive preferences

Evolutionarily stable strategy and time scale.

   We calculate the evolutionarily stable strategy for fitness defined by [152]Eq
   (3). At the interior (i.e., generalist) behavioural equilibrium the two pay-offs
   [153]Eq (2) must be the same, i.e., V[1] = V[2], which yields
   [journal.pone.0160076.e016] (11) provided [journal.pone.0160076.e017] is between
   0 and 1. If V[1](u[1])>V[2](u[1]) for all u[1], the ESS is
   [journal.pone.0160076.e018] and if V[1](u[1]) e[2]), causing plant 2 to decline and to go extinct,
   eventually. However, as pollinator densities start to increase relative to plant
   densities, pollinators can become generalists which favours plant coexistence. We
   start with the assumption that pollinator preferences track instantaneously
   population numbers (panel a: n = infty), i.e., [journal.pone.0160076.e027] is
   given by the the ESS [168]Eq (11) (see the star-line -*-*- in [169]Fig 5a). We
   observe that as pollinator abundance increases, pollinators become generalists
   approximately at t ~= 3, which is fast enough to prevent plant 2 extinction, and
   population densities will tend to an interior equilibrium. When pollinators
   preference is described by the replicator equation (panel b: n = 1), we observe
   that pollinators will become generalists at a latter time (t ~= 11) due to the
   time lag with which pollinators preferences follow population abundances. Even
   with this delay, the decline of plant 2 stops and we obtain convergence to the
   same population and evolution equilibrium. However, when adaptation is yet
   slower, the pollinator preferences will follow changing population densities with
   a longer time delay (panel c: n = 0.25), and we get a qualitatively different
   outcome with plant 2 extinction. This is because when pollinators start to behave
   as generalists (t ~= 100), plant 2 abundance is already so low that it is more
   profitable for pollinators to switch back to pollinate plant 1 only. We obtain
   similar results as in [170]Fig 5 when e.g., c[1] = c[2] = 0.4, but coexistence
   becomes impossible when direct competition becomes too strong.
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   Fig 5. [172]Model (1a) dynamics with pollinator adaptive preferences.

   Population densities (units in left axes) are represented by: green squares =
   plant 1, red diamonds = plant 2, pink circles = pollinator. Pollinator preference
   for plant 1 (u[1], units in right axes) is represented by the black line. Initial
   population densities in all panels P[1](0) = P[2](0) = 25, A(0) = 1. Preferences
   in (a) are given by the ESS [173]Eq (11). Preferences in (b) and (c) are given by
   the replicator [174]Eq (4) with n = 1, 0.25 respectively and u[1](0) = 0.999.
   Parameters as in [175]Table 1, with K[i] = 50, c[i] = 0.

   [176]https://doi.org/10.1371/journal.pone.0160076.g005

   In the next section we study combined effects of initial conditions, plant
   competition for resources (c[i] > 0), and time scales on plant coexistence.

Scenario I (variation of initial plant and pollinator densities).

   Here we study the combined effects of population dynamics [177]Eq (1a) and
   adaptive pollinator preferences [178]Eq (4) on species coexistence. [179]Fig 6
   shows regions of coexistence (pink), exclusion of one plant species (red or
   green), and global extinction (both plants and the pollinator, white) for
   different initial plant and pollinator densities. For this scenario combined
   initial plant densities are fixed (P[1](0) + P[2](0) = 50). We contrast these
   predictions with the situation where population densities are fixed, i.e., when
   population dynamics are not considered and pollinator preferences are at the ESS.
   In this latter case the necessary condition for both plants to survive is that
   pollinators behave as generalists which corresponds to the region between the two
   curves d[0] and d[1] in [180]Fig 6. These are the curves along which the ESS
   predicts switching between specialist and generalist pollinator behaviour at
   initial population densities. These curves are found by solving [181]Eq (11) for
   A, when [journal.pone.0160076.e028] which yields [journal.pone.0160076.e029] (12)
   and when [journal.pone.0160076.e030] which yields [journal.pone.0160076.e031]
   (13)
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   Fig 6. Effects of foraging adaptation (n rows) and inter-specific competition (c
   columns) on plant coexistence under scenario I (variation of initial plant and
   pollinator densities).

   P[2](0) = 50 - P[1](0) and u[1](0) given by [183]Eq (11)). Pollinators begin as
   specialists on plant 1 to right of line d[1], on plant 2 to the left of line
   d[0], and generalists in between. Initial conditions in red and green result in
   extinction of plant 1 or 2, respectively. Initial conditions in pink and white
   result in coexistence or community extinction, respectively.

   [184]https://doi.org/10.1371/journal.pone.0160076.g006

   If population densities do not change, initial conditions to the left (right) of
   d[0] (d[1]) lead to exclusion of plant 1 (plant 2) because pollinators specialize
   on plant 2 (plant 1). Population and pollinator preference dynamics do change
   these predictions. The main pattern observed in the simulations is that plant
   coexistence becomes less likely as the plant competition coefficient (c)
   increases. This is not surprising because a higher inter-specific competition
   between plants decreases plant population abundance which makes coexistence of
   both plants less likely or impossible (panels c, d, g, h, k, l, o, p). In fact,
   when inter-specific plant competition is too strong and the pollinators do not
   adapt, plant densities can become so low that the system collapses due to
   mutualistic Allee effects (panel d, white region).

   The effect of pollinator adaptation rate (n) on coexistence is more complex, in
   particular when plant competition is moderate or weak (i.e., c  0 in [223]Eq (4)).

   We assumed that plant competition affects growth rather than death rates [[224]4,
   [225]30]. This assumption is sound when plants are mainly limited by space, or by
   resources whose access are linked to space, such as light. In such circumstances
   a plant could produce many seeds thanks to pollination, but space puts a limit on
   how many will recruit as adults. It remains to see how our results would change
   if competition is considered differently, when adult plant mortality is affected
   by competition (e.g., [[226]2]). This can be very important under scenarios of
   interference like allelopathy or apparent competition caused by herbivores
   [[227]31].

Adaptive preferences and population feedbacks

   The ESS [228]Eq (11) predicts that when at low densities, pollinators will
   pollinate only the plant that is most profitable, while at higher densities they
   will tend to pollinate both plants. This positive relationship between
   pollinator/consumer abundance and generalism was experimentally demonstrated for
   bumblebees [[229]13].

   Plant and pollinator densities are not static, they change within the limits
   imposed by several factors: e.g., space and nutrients in the case of plants, or
   plant resources such as nectar in the case of pollinators. On the other hand,
   plants and pollinators require minimal critical densities of each other in order
   to compensate for mortality. Thus, a given ESS at which one plant is excluded
   from the pollinator's diet will cause that plant to decrease in density, and,
   possibly, to go extinct. However, as population densities change the ESS can also
   change in ways that may favour coexistence. These outcomes will depend on the
   time scale of pollinator foraging adaptation. For this reason, we introduced the
   replicator [230]Eq (4) as a dynamic description of pollinator preferences and we
   coupled it with population dynamics. One of the main consequences of introducing
   replicator dynamics is the disappearance of complex dynamics such as limit cycles
   or global extinctions triggered by invasion ([231]Fig 3b and 3d). In contrast,
   the dynamics with pollinator adaptation are characterized by fewer stable
   outcomes (plant 1 only, plant 2 only, coexistence) with strong dependence on the
   initial conditions.

   Whether or not adaptive pollinator preferences promote plant coexistence depends
   critically on the strength of competition (i.e., competition coefficient) and the
   rate of pollinator adaptation (n). In our simulations, we determined the region
   of initial population densities and pollinator preferences leading towards plant
   coexistence, as a function of these two factors. The larger this region, the more
   likely plant coexistence. As competition strength increases, coexistence becomes
   less likely as expected from competition theory [[232]19]. As pollinator
   adaptation rates increase the pattern is more complex and sometimes equivocal, as
   adaptation can increase or decrease the likelihood of coexistence (Figs [233]6
   and [234]7). For example, when competition is weak or moderate in simulation
   scenario I, we see that the region of coexistence is generally wider when
   pollinator densities are initially large and more narrow when pollinators are
   initially rare ([235]Fig 6 for c  e[2]) while we keep all the other plant-specific parameters
   equal. We also ran simulations with plant 1 being better with respect to other
   plant-specific parameters (e.g., r[1] > r[2] or w[2] > w[1], keeping e[i] = 0.1
   and the rest of parameters as in [251]Table 1). In these simulations (not shown
   here) coexistence is generally more difficult to attain (e.g., coexistence
   regions as those in [252]Fig 6 get smaller). The reason is that in our model,
   plant rewards (e[i]) affect plants only indirectly, by influencing pollinator
   preferences. In contrast, other plant-specific parameters affect plant dynamics
   directly. Finally, the larger c[i] and K[i] the more likely plant i always win in
   competition, but this is a natural result expected in models derived from the
   Lotka-Volterra competitive equations.

   Some predictions from our model are in qualitative agreement with experiments.
   For example [[253]10] shows transitions from plant facilitation to competition
   for pollinators [[254]8] when one plant species (Raphanus raphanistrum) is
   exposed to increasing numbers of an alternative plant (Cirsium arvense). In the
   same study, the relative visitation frequency of a plant (Raphanus) declines
   faster than predicted by the decline in the relative proportion of its flowers
   [[255]10]. The ESS can explain this outcome as the superposition of a relative
   resource availability effect and a resource switching effect (i.e., first and
   second terms respectively, in the right-hand-side of [256]Eq (11)), as shown by
   [257]Fig 4 (compare it with Fig 6 in [[258]10]). The effect of resource
   competition on the relationship between pollinator density and generalism, was
   demonstrated by another experiment [[259]13]. Other studies show that invasive
   plant species can take advantage of changing pollinator preferences, increasing
   their chances to get included into native communities [[260]36]. Finally, one
   meta-analysis indicates that pollinators can be taken away by invasive plants,
   affecting native plants adversely [[261]28].

Inter-specific pollen transfer effects

   [262]Model (1a) considers only one single pollinator species. This makes
   pollinator generalism (i.e., u[1] strictly between 0 or 1) a requisite for
   coexistence. However, when pollinators are generalists, rare plants would
   experience decreasing pollination quality, due to the lack of constancy of
   individual pollinators delivering non-specific pollen or losing con-specific
   pollen [[263]37-[264]39]. We do not consider inter-specific pollen transfer
   effects (IPT) in this article. Modelling IPT effects requires additional
   assumptions about visitation probabilities [[265]30], pollen carry-over [[266]4]
   or pollinator structure [[267]35, [268]40]. Nevertheless, we simulated scenarios
   I and II again, but replacing our [269]Eq (1a) by a system of equations that
   considers IPT [[270]30]. We found that most of our results hold qualitatively,
   i.e., the coexistence regions display the same patterns like in Figs [271]6 and
   [272]7 (results not shown).

   There is no question that IPT affects pollination efficiency. However, the
   relative importance of IPT may also depend on the structure of the environment
   where interactions occur. A survey of field and laboratory results [[273]39]
   reports that in spite of strong IPT effects on plant reproduction for certain
   systems, many studies found little or no significant effects in other systems.
   One reason could be the scale of the system under study, which can influence the
   way mobile pollinators experience the resource landscape: fine grained or coarse
   grained, e.g., well mixed or patchy. Thus, if plant species are not totally
   intermingled, but also not isolated in clumps, the negative effects of IPT on
   seed set (a proxy for plant fitness) could be reduced [[274]41]. In addition,
   unless we consider a single flower per plant at any time, the resource is almost
   always patchy. This means that IPT effects in self-compatible plants would be
   stronger just after pollinator arrival, decreasing for the remaining flowers
   before the pollinator leaves the plant.

From modules to networks and from adaptation to co-evolution

   The scope of our work is limited to adaptation in a single pollinator species
   only. In real life settings adaptation can be affected by (i) competition among
   several pollinator species, and by (ii) plant-pollinator co-evolution.

   With respect to point (i), large community simulations [[275]30] indicate that
   inter-specific competition can force pollinators to change their preferences in
   order to minimise niche overlap. This can promote coexistence and specialization
   on rare plants at risk of competitive exclusion. Song and Feldman [[276]35]
   discovered a similar mechanism, with a polymorphic pollinator, i.e., consisting
   of specialist and generalist sub-populations. Thus, adding a second pollinator
   would be a next step to consider, in order to address inter-specific competition.

   Addressing point (ii) will require trade-offs in plant traits. We showed how
   differences in pollinator efficiencies (e[i]) indirectly affect plant dynamics
   [277]Eq (1a). However, pollinator efficiencies can depend on plant allocation
   patterns, which can affect their growth, mortality or competitive performance
   (r[i], m[i], c[i]). Plant adaptation likely happens over generations, so a
   replicator equation approach or adaptive dynamics [[278]22] will be useful to
   study plant-pollinator co-evolution.

   In spite of the complexity of real plant pollinator networks, small community
   modules will remain useful to tease apart the mechanisms that regulate their
   diversity. Models of intermediate complexity like [279]Eq (1a) can help us
   discover important results concerning interaction dynamics, pollinator foraging
   patterns (e.g., pollinator ESS) and the consequences of differences between
   ecological vs. adaptation time scales.

Supporting Information

[280]S1 Appendix. Analysis with fixed preferences.

   [281]https://doi.org/10.1371/journal.pone.0160076.s001

   (PDF)

[282]S1 File. Source codes used to generate figures and simulations.

   [283]https://doi.org/10.1371/journal.pone.0160076.s002

   (ZIP)

Author Contributions

    1. Conceived and designed the experiments: TAR VK.
    2. Performed the experiments: TAR VK.
    3. Analyzed the data: TAR VK.
    4. Contributed reagents/materials/analysis tools: TAR VK.
    5. Wrote the paper: TAR VK.

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