Context-dependent pollinator limitation in stochastic environments: can increased seed set overpower the cost of reproduction in an understorey herb?

Authors

  • Carol C. Horvitz,

    Corresponding author
    1. Department of Biology, University of Miami, PO Box 249118, Coral Gables, FL 33124, USA
      Correspondence author. E-mail: carolhorvitz@miami.edu
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  • Johan Ehrlén,

    1. Department of Botany, Stockholm University, S-106 91 Stockholm, Sweden
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  • David Matlaga

    1. Department of Biology, University of Miami, PO Box 249118, Coral Gables, FL 33124, USA
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    • Present address: United States Department of Agriculture – Agriculture Research Service, Invasive Weed Management Research, University of Illinois, 1102 S. Goodwin Ave., Urbana, IL 61801, USA.


Correspondence author. E-mail: carolhorvitz@miami.edu

Summary

1. In the understorey herb Lathyrus vernus seed production is pollen limited, but increased reproduction results in a lower probability of remaining reproductive. Putting these two results together, previous research reported that population growth rate λ was negatively impacted by high pollination.

2. Thus, costs and benefits have to be translated into the common currency of their respective effects on population dynamics to determine whether populations are truly pollen limited or whether they are already at an optimal level of pollination.

3. Also, when pollinators and demography vary from year to year we require a framework that examines reproductive benefits and demographic costs in the context of a variable environment. Whether or not additional pollination will increase the stochastic population growth rate λS depends upon the balance of stochastic elasticities of the costs and benefits.

4. In constant environment models, where seed survival, germination and seedling survival were increased, we found that the high cost of reproduction could be offset by improvements in seed survival and germination, but not by improvements of seedling survival.

5. In variable environment models, where changes in the sequence and frequencies of high- and low-pollination years mixed with occasional high-germination years were modelled, we found that increasing the frequency of high-germination conditions could offset the cost of reproduction, and the offset was even greater if high-germination years occurred after a high-pollination year or if high pollination was accompanied by high-germination conditions in the same year.

6. Both deterministic λ and stochastic λS were less sensitive to perturbation of reproduction than to perturbation of the probability for flowering plants to remain reproductive. In other words, a small change in the parameter which is related to the ‘cost’ of reproduction had a bigger effect than a small change in the parameter which is related to the ‘benefit’ of increased pollination for Lathyrus.

7.Synthesis. Stochastic environment-specific elasticities for reproduction and stasis of flowering plants differ in their response to environmental context. The cost–benefit relationships, the ultimate fitness consequences of supplemental pollen, are influenced by the frequency and sequence of years differing in pollen availability and recruitment conditions.

Introduction

Negative or positive correlations among different life-history parameters could affect the balance between current reproduction and future success. For example, reduced availability of pollen or an increase in the number of flowers damaged by herbivores may reduce seed production but may improve future performance, if saved resources can later be utilized. When plants have sufficient resources to produce more seeds than observed and the cause is insufficient pollen, seed production is said to be pollen limited. Pollen limitation is typically identified experimentally by comparing the reproductive output of experimental units receiving supplemental pollen to those receiving ambient pollen loads, and the most common currency used to evaluate pollinator limitation is fruit or seed production. However, increased reproduction induced by supplemental pollen may cause reallocation of resources within a plant (e.g. Zimmerman & Pyke 1988; Wesselingh 2007) or across years, sometimes resulting in future reduction of survival, growth or reproduction (e.g. Ehrlén & Eriksson 1995), and a more interesting currency of the effects of pollen supplementation is one that integrates across the life cycle. A few studies have analysed the effects of pollen supplementation on population growth rates by incorporating experimental results into population projection matrices (Bierzychudek 1981; Calvo & Horvitz 1990; Calvo 1993; Ehrlén & Eriksson 1995; Parker 1997; Garcia & Ehrlén 2002; Knight 2004; Price et al. 2008). There is a whole separate literature on experimental studies of the cost of reproduction (Sohn & Policansky 1977; Horvitz & Schemske 1988b; Primack, Miao & Becker 1994; Obeso 2002), but pollen limitation and cost of reproduction are intertwined and should be studied together and conceptually linked (Calvo & Horvitz 1990). This can be accomplished by considering the entire plant life cycle using a demographic approach.

Many authors mention that pollinators, resources, seed predators, herbivores and other environmental factors show temporal variation that is expected to influence the balance between allocation to pollinator attraction and allocation to production and survival of offspring. Haig & Westoby (1988) proposed that plants living in a constant environment should optimize allocation between the functions of pollinator attraction and seed maturation, and we should expect most plants to be at this equilibrium. According to this scenario, pollen supplementation should have no effect (but pollen deprivation should lower seed set). Nevertheless, reviews of empirical studies have found widespread evidence that seed production is enhanced by pollen supplementation [e.g. Burd 1994; Larson & Barrett 2000; Ashman et al. 2004; Knight et al. 2005; the latter two reporting on different aspects of the same large National Center for Ecological Analysis and Synthesis (NCEAS) data base on pollen limitation]. Ashman et al. (2004) reported that in the meta-analysis of the 85 cases where supplemental pollination was carried out at the whole-plant level, the average effect was a 42% increase in seeds per plant due to pollen supplementation, while Knight et al. (2005) reported that in the meta-analysis of all 482 cases, the average effect was a 75% increase in fruit set due to pollen supplementation. A likely explanation for these results is that plants are often not at Haig & Westoby’s ideal optimum because they do not live in constant environment (cf. Burd 2008). Burd (2008) found that pollen limitation was predicted to be common when stochasticity in ovule fertilization and resource availability were added to Haig & Westoby’s model. However, to our knowledge, no one has yet incorporated these concerns into stochastic population dynamic models focused on pollen limitation.

Here, we examine how demographic context determines the balance of trade-offs. We investigate the interplay between reproductive output and reproductive costs, highlighting the need to consider both the magnitude of the single-stage effect as well as elasticity of population growth rate to perturbations of the affected transitions. Maron, Horvitz & Williams (2010) emphasize that insights can be gained about the effects of plant–animal interactions on plant population dynamics by combining experiments with population projection matrices and argue that the effect of an interaction on a plant should be expressed in the currency of population growth rate (λ). Here we follow the same basic approach, but are specifically concerned with negatively correlated effects of pollen supplementation at different components of the life cycle (seed production increased but the probability for adults to remain reproductive decreased) and in different states of a variable environment.

In the system we study here, Lathyrus vernus and its biotic environment, previous work combining pollen supplementation experiments with population projection matrices had reported that there would be lower fitness with abundant pollination than with poor pollination due to a strong cost of reproduction (Ehrlén & Eriksson 1995). However, this particular experimental study had taken place in a year when there was a high rate of seed predation by a bruchid beetle, a low rate of pollination, and a low rate of germination, when viewed in the context of a broader array of data on this system (Ehrlén 1995; Ehrlén 2002). Thus, the biological questions remained: under what conditions (if any) could the cost of reproduction be offset enough to result in a situation where there could be selection on plant traits to increase their attractiveness to pollinators. Here, we addressed this question by looking at fitness defined in two ways: (i) as the asymptotic annual per capita population growth rate for a constant environment λ and (ii) as the long-run stochastic annual per capita population growth rate for a variable environment λS.

How do we translate into a single currency a proportional effect on one matrix element, representing the costs, and a proportional effect on another matrix element, representing the benefits? If the currency is population growth rate, the answer lies in elasticity. We used elasticities to guide and understand our simulations. One issue of interest is the elasticity of stochastic population growth to seed production in high-pollination years out of a sequence of both high- and low-pollination years. Our results are based on simulations of reasonable scenarios of combinations of demographic features of the environment and their principal value is to point to important qualitative insights.

In particular, our simulations were motivated by the following hypotheses. For constant environment models, we proposed that (i) removing predators, by increasing seed number, would alter the cost–benefit balance, making supplemental pollination beneficial, (ii) improving seed survival, seed germination or seedling survival would offset the cost of reproduction, making supplemental pollination beneficial. In variable environment models, we proposed that (i) for a two-state environment (low- and high-pollination years) an intermediate frequency of high-pollination environments would have the maximum stochastic growth rate, and (ii) adding high-germination years into the mix would improve the stochastic growth rate, offsetting the cost of reproduction and (3) the improvement would be heightened if high germination could be timed to come after or during a high-pollination year, rather than independent of current conditions.

Materials and methods

Study species and previous work

Lathyrus vernus (Fabaceae) is a long-lived perennial forest herb that ranges from central and northern Europe to Siberia. The work we present is based on data collected from populations found in deciduous forests characterized by Quercus robur, Fraxinus excelsior, Betula pendula near Tullgarnsnäset, SW of Stockholm, Sweden (58°6′ N, 17°4′ E). This is a very well-studied population with respect to its demography and plant–animal interactions including pollinators (Bombus spp.), seed predators (Bruchus atomarius) and herbivores (e.g. Ehrlén 1992; Ehrlén & Eriksson 1995; Ehrlén, Kack & Agren 2002). Populations can be characterized by seven stages: seeds in the seed bank, seedlings, three sizes of vegetative plants, one flowering stage and a dormant rhizome stage. We make use of several results of the previous empirical studies. Supplemental hand pollination resulted in an increase in per capita seed and seedling production of flowering plants, but the increase in reproductive effort exacted a cost in other demographic rates. Survival of reproductive plants was 100% for both hand- and open-pollinated plants, but when seed production increased due to supplemental pollination, the probability of a flowering individual to remain reproductive was negatively affected, while the probabilities of becoming a dormant rhizome or a smaller plant that was in a vegetative state increased. Ehrlén & Eriksson (1995) incorporated these experimental results into population projection matrix models

image(eqn1)

and found that the costs outweighed the benefits; supplemental pollination resulted in a decrease in asymptotic annual per capita population growth rate λ (given by the dominant eigenvalue of A). The elements of A (aij) are demographic transitions and contributions of each stage j to each other stage i over one time step.

Construction of matrices for each environmental state from data

We started with the population projection matrices for open- and hand-pollinated plants reported in Ehrlén & Eriksson (1995). These matrices are identical to one another except for the column listing the demographic rates and transitions of flowering plants (column 6 of the projection matrix). These matrix entries refer to the production of dormant seeds and seedlings, as well as the fates of large flowering plants, including whether they remain reproductive, change size, return to a vegetative state or become dormant rhizomes.

Although not a focus of the 1995 paper, bruchid beetles which were pre-dispersal predators, had attacked plants of both the hand- and open-pollinated treatments, with a predation rate of 0.53. Each beetle had occupied one developing seed. To calculate gross seed production, we added the number of beetles to the number of non-predated seeds, creating versions of both matrices that represent demography in a predator-free environment. The number of seeds increased (we turned beetles into seeds), but the cost remained the same as in the previously reported scenarios, since the production of each beetle took just as many resources as the production of each seed (J. Ehrlén, pers. observ.) (Appendix S1 in Supporting Information). These predator-free matrices (open pollination/predator free and hand pollination/predator free, Appendix S1 in Supporting Information) were the baseline scenarios for most of our analyses, with corresponding rates of seed and seedling production (Table 1).

Table 1.   Effects of hand pollination on reproduction (seeds produced at time t and seeds or seedlings counted at + 1 for each large flowering plant counted at time t) and on asymptotic population growth rate λ under various constant environment scenarios: naturally occurring seed predation, removal of seed predators, and simulated improvement in germination. Boldface indicates the baseline parameter values for most of the simulations in this paper
 With seed predatorsWithout seed predators
Observed germination rateObserved germination rateHigh germination rate
Open pollinatedHand pollinatedOpen pollinatedHand pollinatedOpen pollinatedHand pollinated
Seeds produced (t)3.165.846.7312.456.7312.45
Seeds (+ 1)1.833.393.917.220.300.55
Seedlings (+ 1)0.210.390.450.834.067.51
Germination0.1040.1040.1040.1040.9320.932
λ1.0311.0191.0631.0551.1571.164
Δλ = λhand−λopen −0.012 −0.007 +0.007

To test whether improvements in seed and seedling demographic rates could offset the cost of reproduction and result in an environment in which there would be an advantage to increased pollination, we modified these baseline-population projection matrices. We note that dormant seed and seedling production (matrix entries in the first and second rows of column 6) are the result of multiplying three parameters together, seed production × seed survival × seed dormancy for row 1, and seed production × seed survival × germination for row 2 (where dormancy and germination are conditional on survival). Similarly, the fates of seeds (matrix entries in the first and second rows of column 1) are the results of multiplying two parameters together, seed survival × seed dormancy for row 1, and seed survival × germination for row 2 (where dormancy and germination are conditional on survival). Modifying each parameter individually while holding the others constant, we gradually improved seed survival, seed germination and seedling survival, examining at which level, if any, there was reversal of the negative effect of hand pollination on λ. These parameters differed in how much ‘room for improvement’ there was; some were already quite favourable in the original matrices. For the effect of improving seed survival, we started with the observed value of 0.65 and increased it incrementally by factors of 1.1, 1.2, 1.3, 1.4 and 1.5, reaching a value of 0.97. To investigate the effect of improving seed germination, we started with the observed value of 0.10 and increased it incrementally by factors of 1, 2, 3, 4, 5, 6, 7, 8 and 9, reaching a value of 0.93. Although this may seem like an unrealistically large modification, germination rates were actually observed to vary between 0 and 1 (mean 0.28, SD 0.28) in 17 observations of six sites over three transition periods (one site–year combination was not studied) (Ehrlén 1995). For the effect of improving seedling survival, we started with the observed value of 0.91 and increased it incrementally by factors of 1.02, 1.04, 1.06 and 1.08, reaching a value of 0.99.

Population projection models for variable environments

We modelled population dynamics for a structured population in a variable environment as

image(eqn2)

which is similar to the model for a constant environment (equation 1), except here, instead of the single matrix A that projects the population from one time to the next, the matrix X(t) is a random variable that can take on one of several values,

image

each corresponding to one of K possible states of the environment. The elements of Aβ (aijβ) are demographic transitions and contributions of each stage j to each other stage i over one time step in each environment β. The long-run per capita annual population growth rate λS, the stochastic growth rate, depends upon the frequency and sequence of environments (Tuljapurkar 1982a,b, 1990, 1997). Probability rules determining which of the possible environmental states is chosen at each time are encapsulated in the × K matrix C with elements cαβ that give the probability of going from environmental state β to environmental state α over one time step. These rules determine the environmental dynamics, observable as sequences of environmental states along sample paths. Under commonly met assumptions about the process, over a very long time, sample paths will converge on an expected long-run stationary distribution of environments that is independent of the initial state, and the time average of cumulative population growth rate converges to the stochastic growth rate

image(eqn3)

which can be readily obtained by numerical simulation (Tuljapurkar 1990, 1997; Tuljapurkar, Horvitz & Pascarella 2003; Appendix S4 in Supporting Information has the MATLAB code). The sensitivity of λS to perturbations of variances and means of matrix elements as well as to perturbations of matrix elements in particular states of the environment provide diverse insights into what influences population growth in variable environments; these interesting stochastic elasticities, Eσ, Eμ and the Eβ, do not exist in the deterministic case (Tuljapurkar, Horvitz & Pascarella 2003). Of particular interest for this paper are the environment-specific (also called ‘habitat stage’) elasticities (Horvitz, Tuljapurkar & Pascarella 2005), which comprise a set of matrices, one for each state β of the environment. This parameter can be used, for example, to address the sensitivity of stochastic population growth to seed production in high-pollination years out of a sequence of both high- and low-pollination years. Eβ has elements eijβ that give the sensitivity of λS to proportional perturbations of the ijth demographic rate whenever the environment is in state β. Each eijβ value thus includes a weighting that depends upon how frequently the β state occurs as well as the expected sequence of states that will follow it. To un-weight it, and just focus on the ‘distilled’ sequence effect, it is thus of interest to determine the ‘per-occurrence’ normalized environment-specific elasticity, which is just Eβ divided by the frequency of state β in the long-run sequence (Horvitz, Tuljapurkar & Pascarella 2005). The Eβs are readily calculated by numerical simulation (Appendix S4 in Supporting Information has the MATLAB code).

Construction of the variable environment models

Four of the constant environment matrices were used to represent four different environmental states in models of variable environments. Environment 1 has low pollination, low germination (open-pollinated matrix, Appendix S1 in Supporting Information). Environment 2 has high pollination, low germination (hand-pollinated matrix, Appendix S1 in Supporting Information). Environment 3 has low pollination, high germination (open-pollinated matrix, Appendix S1 in Supporting Information, but with germination increased by a factor of 9). Environment 4 has high germination within the same year that there is high pollination (hand-pollinated matrix, Appendix S1 in Supporting Information, but with germination increased by a factor of 9) (see Table 1 for contrasts of relevant parameter values). To test whether a variable environment with some high- and some low-pollination years would offset the cost of reproduction seen in the constant high-pollination environment, we constructed two types of variable environments. In independent and identically distributed (IID) models, the probability of choosing the next environmental state each time depends only on the frequency of the states overall and not on the current state. In Markov models, the probability of choosing the next environmental state each time depends on the current state.

First, we investigated the hypothesis that in a variable environment comprised of two states, low- and high-pollination environments (environments 1 and 2), an intermediate frequency of high-pollination environments would have the maximum stochastic growth rate. The idea was that a variable environment would provide respite from the high cost and allow recovery of resources in a way that would not be possible in a constantly high-pollination environment. We employed 11 two-state IID models of environmental dynamics, increasing the frequency of high-pollination years from 0.05% to 99.5%.

Secondly, we wanted to investigate how adding a high-germination year to the mix could alter the stochastic growth rate. To do this, we needed to identify which of these two-state models was closest to the actual distribution of high- and low-pollination years to have a baseline scenario in which we could simulate changes in the frequency of good-germination conditions. We used data from Ehrlén, Kack & Agren (2002) on the proportion of flowers that set fruit in each of 24 plot-years in natural (open pollinated) conditions combined with data from two pollen supplementation field experiments. Fruit set varied from 2.8% to 15.5%. We set a threshold level by defining a ‘high’-pollination environment as one in which supplemental pollen does not significantly improve fruit set. At the level we set, which was based on the contrasting results of the two experiments, 29% of the 24 plot-years corresponded to low-pollination environments (Appendix S3 in Supporting Information). Thus, we choose the scenario with 30% low- and 70% high-pollination years as a baseline. We altered the frequency of good-germination years (from 0% to 21%), investigating the effect on the stochastic growth rate λS. We did this in three ways (i) replacement of a low-pollination, low-germination year (environment 1) by a low-pollination, good-germination year (environment 3) with probability that is independent of the current environment (a three-state IID model), (ii) replacement of a low-pollination, low-germination year (environment 1) by a low-pollination, good-germination year (environment 3) only after a high-pollination year (a three-state Markov chain model) and (iii) replacement of a high-pollination, low-germination year (environment 2) by a high-pollination, good-germination year (environment 4) with a probability that is independent of the current environment (a three-state IID model) (Table 2). The second of these simulates an environment with temporal autocorrelation between good pollinator years and good seedling conditions. One could envision such a sequence for forest herbs being the case if an opening in the forest occurs, which attracts pollinators, and subsequent early successional conditions favour recruitment from the seed bank. The third of these simulates an environment in which high germination happens in the same year as high pollination, which could also occur in a gap scenario.

Table 2.   Three-state environment models to investigate the effects of increasing the frequency of high-germination conditions within the context of an environment in which 70% of years have good pollinator services. Each matrix gives the probability of changing from one state of the environment to another at each time step. Environment (Env) 1 has low pollination, low germination. Environment 2 has high pollination, low germination. Environment 3 has low pollination, high germination. Environment 4 has high germination within the same year that there is high pollination. For each frequency of environment 3 there are two models specified, a model in which environment 3 can arise at any time (IID model) and a model in which environment 3 arises only after environment 2 (Markov model)
High germination in a different year from high pollinationHigh germination in same year as high pollination
Frequency of Env 3 IID modelMarkov modelFrequency of Env 4 IID model
Env at time t + 1Env at time tEnv at time tEnv at time + 1Env at time t
Env 1Env 2Env 3Env 1Env 2Env3Env 1Env 2Env 4
0Env 10.30.30.30.30.30.30Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.70.70.7
Env 3000000 Env 4000
0.035Env 10.2650.2650.2650.30.250.30.035Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.6650.6650.665
Env 30.0350.0350.03500.050 Env 40.0350.0350.035
0.07Env 10.230.230.230.30.20.30.07Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.630.630.63
Env 30.070.070.0700.10 Env 40.070.070.07
0.105Env 10.1950.1950.1950.30.150.30.105Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.5950.5950.595
Env 30.1050.1050.10500.150 Env 40.1050.1050.105
0.14Env 10.160.160.160.30.10.30.14Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.560.560.56
Env 30.140.140.1400.20 Env 40.140.140.14
0.175Env 10.1250.1250.1250.30.050.30.175Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.5250.5250.525
Env 30.1750.1750.17500.250 Env 40.1750.1750.175
0.21Env 10.090.090.090.300.30.21Env 10.30.30.3
Env 20.70.70.70.70.70.7 Env 20.490.490.49
Env 30.210.210.2100.30 Env 40.210.210.21

Results

Matrices for single environmental states

The demographic rates and transitions of flowering plants for the predator-free matrices illustrate the immediate benefits and costs of supplemental pollination in Lathyrus. Hand pollination increases seed and seedling production (Fig. 1a), but that effect is accompanied by a decrease in the probability of remaining reproductive, as well as increases in the probabilities of becoming smaller or going dormant (Fig. 1b). In the predator-free scenario, hand pollination improved seed production by 85% but only decreased the probability of remaining reproductive by 34%. Removing seed predators from the system meant higher reproduction and higher population growth rates for both hand- and open-pollinated plants compared to the previously reported scenarios for plants that had lost seeds to predators. Although the magnitude of the negative effect was reduced by removing the seed predator, supplemental pollination still had a negative effect on population growth rate. The Δλ for the difference between hand- and open-pollinated matrices was −0.012 in the presence of predators but only −0.007 when predators were removed from the system (Table 1).

Figure 1.

 Reproduction (a) and demographic transitions (a,b) of large flowering plants in open- and hand-pollinated treatments in the scenario without bruchid beetle seed predators. Note the positive effects on reproduction and the negative effects on future size and probability of reproducing as well as the increase in probability of becoming dormant. Data from Ehrlén & Eriksson (1995), modified to simulate the effect of removing seed predators from the system.

To understand why an 85% improvement in one parameter does not outweigh a 34% decrease in another parameter, we need to look at elasticity, which puts the two parameters into one currency. That currency is ‘percentage change in λ’. We focus on the without-predators scenario as it is the base line for the current paper and seed predators had relatively minor effects on elasticity structure. We note that flowering plants have the highest reproductive value (Fig. 2). The elasticity for seed production (summation of dormant seed production and seedling production) was much lower (e16 + e26 was around 5% in both pollination scenarios) than the elasticity for the probability of a plant to remain reproductive (e66 was 25.2% and 9.6%, respectively, in the open- and hand-pollinated scenarios, Appendix S2 in Supporting Information). If we add on the elasticities for plants to shrink or go dormant (e46 + e56 + e76 was 0.0427 and 0.0467, respectively), it does not change the picture much. In other words, a 1% change in the life-history parameter in which the cost is expressed has a bigger impact on λ than a 1% change in the life-history parameter in which the benefit is expressed.

Figure 2.

 Reproductive values for each stage in the open- and hand-pollinated treatments in the scenario without bruchid beetle seed predators.

Survival of seeds is a parameter underlying four matrix elements, the top two elements in column 1, which refer to the fates of seeds in the seed bank and the top two elements in column 6, which refer to the production of new seeds and seedlings counted at time + 1 for each flowering plant counted at time t. In the observed population, seed survival (a11 + a21) was 0.65. We proposed the idea that improvements in the survival of seeds could offset the cost of reproduction. However, we found that only at very high survival (0.97) did hand pollination have an advantage over open pollination (Fig. 3a).

Figure 3.

 Compensation for the cost of higher reproduction brought about by hand pollination. (a) Higher seed survival can compensate for cost; (b) higher germination can compensate for cost; but (c) higher seedling survival can NOT compensate for cost. Arrows indicate the level at which the cost of reproduction is compensated.

Seed germination [a21/(a11 + a21)] is a parameter that influences the same four matrix elements as seed survival, by determining which proportion of surviving seeds germinate and which remain dormant. In the observed population for this matrix, germination was quite low; only 0.10 of seeds that survived germinated in a given year and the rest went into or stayed in the seed bank. We proposed the idea that improvements in the germination rate (simulating a favourable recruitment window) could offset the cost of reproduction. We found that it took a germination rate about four times the observed to attain a situation in which hand pollination had a beneficial effect on λ, and that the magnitude of the benefit increased as germination rate increased above that (Fig. 3b). Improvement of the germination rate by nearly an order of magnitude resulted in an environment in which the Δλ for the difference between hand- and open-pollinated matrices was +0.007 (Table 1). The full range of simulated germination rates were within the range of natural variation observed by Ehrlén (1995) in his study of six populations over three transition periods.

Another parameter that could be important in translating increases in seed production to increases in population size is seedling survival. However in the observed population, it was already quite high (0.91), leaving little opportunity for simulated improvement. Over the limited range of possible improvements, increased seedling survival was not able to offset the cost of reproduction (Fig. 3c).

Models for variable environments

Instead of the predicted intermediate high stochastic growth rate, we found a monotonically decreasing one as the frequency of high-pollination years increased (Fig. 4a). Under this scenario, there was no combination of high- and low-pollination years that could overcome the cost of reproduction.

Figure 4.

 (a) Stochastic growth rate as a function of the frequency of high pollination years (at the baseline-germination rate). (b) Stochastic growth rate as a function of increasing the frequency of high-germination conditions within the context of an environment in which 70% of years have good pollinator services (see Table 2).

We started from a scenario in which 70% of the years are high-pollination years and 30% of the years are low-pollination years (with λS = 1.057, Fig. 4a). We found that a scenario in which just a few (3.5%) of the low-pollination years also include simulated high germination while most (26.5%) of the low-pollination years retain the observed ‘standard’ germination) resulted in a higher λS of 1.066 (Fig. 4b), which was higher than the pure low-pollination environment (λS = 1.062). An environment with many of these years (21% low pollination with high germination, 7% low pollination with standard germination 70% high-pollination years,) had a quite elevated λS of 1.095. High-germination years can help a population to ‘capitalize’ on high-pollination years in a way that would not be possible in a constant environment, thus they offset the cost of reproduction. We found that this was the case, especially at higher frequencies of high-germination years (Fig. 4b), and especially when high germination was timed to follow a high-pollination year. Not surprisingly, in light of the constant environment results, an even bigger advantage was seen when high-germination rates were added directly to high-pollination years (Fig. 4b).

In a variable environment, we are able to address how sensitive λS is to life-history transitions in the high-pollination environment in different dynamic situations. The high-pollination environment is the state that expresses the increased reproduction (the benefit) and the decreased probability of remaining reproductive (the cost). To translate these effects into the currency of λS, we employed Eβ, eijβ, the environment-specific elasticities. Specifically elasticity of λS to perturbations of seeds/seedlings produced in environment 2 is the sum e162 + e262 and elasticity of λS to perturbations of plants remaining reproductive in environment 2 is e662, where environment 2 is the high-pollination state of the environment.

First, population growth rate was more sensitive to the probability that flowering plants would remain reproductive (the parameter negatively affected by high pollination) than to reproduction (the parameter positively affected by high pollination) (Fig. 5). Secondly, environmental context affects these two elasticities differently. In the two-state models, we saw only a gradual, but steady increase in reproduction elasticity as the frequency of high-pollination years increased (Fig. 5a). In contrast, the stasis elasticity first rose rather rapidly and then began to level off. The maximum difference between these was at an intermediate frequency of high-pollination years. In the three-state models in which some of the low-pollination, low-germination years were replaced by low-pollination, high-germination years (Fig. 5b) there was a very gradual but steady increase in reproduction elasticity, which was nearly identical in the IID and Markov models as the frequency of high-germination years increased. In contrast, the stasis elasticity showed a decrease, with a slightly more marked decrease in the Markov models than the IID models. In the three-state models in which some of the high-pollination, low-germination years were replaced by high-pollination, high-germination years (Fig. 5c), there was a marked decrease in stasis elasticity and only a very slight decrease in reproduction elasticity as the frequency of high-germination years increased.

Figure 5.

 Environment-specific elasticity Eβ, total (a–c) and normalized (d–f) of λS to stasis of flowering plants and to reproduction in environment 2 in differing environmental contexts: (a, d) increasing frequency of environment 2 in a two-state model; (b, e) increasing frequency of environment 3 in a three-state model and (c, f) increasing frequency of environment 4 in a three-state model. Environment 1 has low pollination, low germination. Environment 2 has high pollination, low germination. Environment 3 has low pollination, high germination. Environment 4 has high germination within the same year that there is high pollination. Note the higher sensitivity of population dynamics to stasis of flowering plants (one of the parameters negatively affected by supplemental pollination) than to reproduction (which is positively affected by supplemental pollination). Environmental context affected these two elasticities differently.

In making comparisons among environments that differ in the frequencies of state β, it is also instructive to examine the normalized Eβ. These parameters, the normalized eijβs, factor out the weighting by environmental frequency and provide insights into how sensitive population dynamics is to a perturbation in a single occurrence of an environmental state. The importance to λS of stasis of flowering plants each time a high-pollination environment occurs steadily falls even as the frequency of high-pollination years increase (Fig. 5d). This is an excellent example of context-driven alteration of the cost–benefit ratio. It reflects that as the frequency of high-pollination years goes up, the expected sequence of environments that will come after environment 2 changes, which alters the significance of what happens in environment 2.

Discussion

Constant environments

We investigated several predator-free constant environments by increasing seed survival, germination and seedling survival; for each level of these parameters we constructed two matrices, the pollen-supplemented and the open-pollinated scenarios, and investigated the sign and magnitude of Δλ due to pollen supplementation. Freedom from predation in our study system lowered, but did not reverse, the cost of reproduction. Seed predators or herbivores did cancel out high seed set that results from increased pollen in other systems (e.g. Cariveau et al. 2004; Knight 2004). Horvitz & Schemske (1988a) found that hand pollination of a tropical forest understorey herb increased the number of initiated fruits by 24%, but a specialist herbivore of reproductive tissues voraciously consumed immature fruits. The effect of supplemental pollination on mean seed production of plants with a low level of herbivory was greater than the effect of supplemental pollination on mean seed production of plants with a high level of herbivory. Even though the variance was too large for the interaction (pollen supplementation × herbivory) to be statistically significant, the results suggested that there could be biotic windows of opportunity when supplemental pollen would matter more than it usually does.

In Lathyrus, the cost of reproduction could be offset by improvements in seed survival and germination, but not by improvements in seedling survival (which was already quite high). One important finding was that the effect of pollen supplementation on overall fitness depended on recruitment probability. In the environment originally studied, for which pollen supplementation was not advantageous, only about 10% of surviving seeds germinated, the rest remained dormant. Our simulations found that with 40% germination or higher, there was an advantage to pollen supplementation. The simulated levels of germination were well within the observed range of Ehrlén’s (1995) longer-term demographic study, suggesting that if the original pollen supplementation experiment had been carried out in a different year or site, the results might well have been opposite. This underscores the context dependency of the outcome of a given plant–animal interaction. The context we refer to here is other demographic rates in the life cycle. In a good-germination year, it is worth it to get better pollinators.

We note, however, that even with a very high (93%) germination rate, the improvement in population growth due to supplemental pollination was rather small (Δλ = 0.007). This effect at the population level is still much smaller than the 85% increase in seed production brought about by supplemental pollination, underscoring a second kind of context dependency. Even in this environment, the elasticity for seed or seedling production (the benefit parameter) is only 7%, one-third the elasticity for the probability of reproductive plants to remain reproductive (the cost parameter). The sensitivity structure of the population and especially the relative sensitivity to costs and benefits acting on different parts of the life cycle limit the translation of a stage-specific advantage to the population level.

Ashman et al. (2004) pointed out that for pollen supplementation to improve λ, pollen supplementation must not only result in increased seed production but also λ must be sensitive to changes in seed production. Our results support and expand on the same general point that the net effect of pollen supplementation depends not only on its effect on current seed production and future performance but also on how sensitive population growth rate is to changes in different fitness components. Ashman et al., in reviewing the literature, performed elasticity analyses for six populations where there was a positive effect of pollen supplementation on seed production. In only three of the six cases was there also a strong positive effect of pollen supplementation on population growth rate λ (for the study of Bierzychudek 1982 and for two of the populations studied by Parker 1997), and in two of these, elasticity for seed production and/or seedling survival was high. In two cases there was a negative, but non-significant effect of pollen supplementation (the studies of Ehrlén and Eriksson 1995; Garcia & Ehrlén 2002) and in one case there was a slightly positive, but non-significant effect (the study of Knight 2004). In all of these, elasticity of seed production was very low. Here, we take this idea one step further by considering negatively correlated effects of pollen supplementation. We emphasize the need to look at relative elasticities of the affected cost and benefit parameters; but what we really would like to do to be able to correctly quantify these effects would be to find out where and why the reversals in the outcome might occur. To do so, we need to know both the magnitude of the cost and benefit in stage-specific terms and the sensitivity of the stochastic growth rate to change in each stage transition that is affected. In the constant environment models, this interest translates into a concern about how the different life-history events contribute to differences between population growth rates of a high-pollination scenario and a low-pollination scenario. Proportional sensitivities are of interest here, one could use proportional effect sizes or logs of effect sizes [as in Davison et al. 2010 in their development of stochastic life table response (LTRE) analysis].

The key difference between most LTRE applications and the current issue is that here we are specifically concerned with the way the correlation structure and the sensitivity structure itself is affected by the treatments. In most LTREs, sensitivity is calculated for a reference matrix and the concern is with the effect size for the variability or co-variability of the matrix elements. Life table response analyses were used to look at pollen supplementation experiments (Knight 2004) and seed supplementation experiments (Price et al. 2008), but not in the same way as we propose. Price et al. (2008) examined whether increased seed production could push populations into density dependence. They added seeds to populations in the field and examined population response by parameterizing matrix models from these experimental field populations. They found that populations had higher growth rates when seeds were added at a rate corresponding to results of pollen supplementation experiments (twofold increase), but when seeds were added at a much higher rate (10-fold increase), population growth rates declined, indicating the potential for a long-term density-dependent response to a long-term release from pollen-limited seed production. Here, we propose combining sensitivity measures with effect size measures for co-varying parameters as a way to investigate trade-offs and identify potential thresholds. One issue to address in future work is whether it is appropriate or instructive for the current purpose to choose a mid-point matrix as the reference matrix for calculating sensitivity, as is done in most LTRE analyses. This is an issue because at the heart of our interest here is the change in the sensitivity structure itself under different conditions.

Variable environments

A conceptual framework of variable environments is a good match for the biological question that has been emerging from many studies of pollinator limitation. How does inter-annual variability in pollinators, predators, resources and other demographic features of the environment influence the ‘optimal’ levels of investment in seed production and pollinator attraction? In a variable environment, the metric that provides a measure of average fitness is the stochastic growth rate λS, which integrates over the variability in demographic rates which individuals and populations will experience over time. The stochastic growth rate depends on the frequency and sequence of environmental states as well as the life-history rates in each state of the environment. Its sensitivity to events in each of several environmental states is part of the picture. We do not look at a Δλ between two scenarios each of which lasts forever, but rather we look at the response of λS to changing the mix of environments experienced by populations. Furthermore, the importance of events occurring in a particular year depends not only on the current state of the environment but also on the history (expressed in the stage structure, which changes over time) and the expected future (expressed in the reproductive values of each stage, which also change over time). In stochastic demography it is not a problem, but rather an explicit consideration, that elasticity in one environmental state differs from another.

We look at the response of the elasticity structure itself to changing the mix of environments. Stochastic elasticity sums to 1 over the entire pattern of environments and life-history stages. There are (at least) two useful ways to decompose the sum (Haridas & Tuljapurkar 2005), one focuses on the summation of the elasticity for the means and deviations of matrix elements (as in Davison et al. 2010) and the other focuses on environment-specific elasticity. The latter is of interest in the current application because of our interest in the relative importance of life-history events in particular environmental states, especially the high-pollination environments.

We created variable environments for the Lathyrus system by simulating changes in the sequence and frequencies of high- and low-pollination years mixed with occasional high-germination years. Increasing the frequency of high-germination years offset the cost of reproduction seen in high-pollination years, and the offset was even greater if high-germination years occurred after a high-pollination year. Stochastic environment-specific elasticities (Eβ) for reproduction and stasis of flowering plants differed in their responses to this environmental context. Within the ranges of our simulations, the cost–benefit relationships and the ultimate fitness consequences of supplemental pollen were thus influenced by the frequency and sequence of years differing in pollen availability and recruitment conditions, but we never observed a higher sensitivity to the benefit parameter than to the cost parameter.

Conclusions

Summarizing over the constant and variable environment models, both deterministic λ and stochastic λS were less sensitive (elastic) to proportional perturbation of reproduction than to proportional perturbation of the probability for flowering plants to remain reproductive. In other words, a small change in a parameter that quantifies the cost has a bigger effect on population growth than a small change in a parameter that quantifies the benefit. Other species are likely to exhibit distinct elasticity structures and dynamics. The concepts we have introduced should be useful and can readily be applied to other systems, thus setting the stage for the next step in understanding the occurrence of demographic thresholds that alter the balance between benefits and costs of additional reproduction. We have provided insight into quantitatively addressing how inter-annual variability in pollinators, resources, predators, herbivores and other environmental factors can be combined to evaluate their effects on fitness and ultimately selection on traits involved in pollinator attraction. Future work will seek the appropriate ways to combine effect sizes with elasticities in the context of trade-offs in variable environments.

Acknowledgements

We are grateful for financial support to C.H. provided by a NSF OPUS grant DEB-0614457 and we thank Uppsala University and Stockholm University for facilitating and hosting a visit by C.H. to J.E.’s lab and Shripad Tuljapurkar, Ove Eriksson, Jon Agren, Rodolfo Dirzo and John Thompson for discussions. This is Contribution No. 662 of the University of Miami’s program in Tropical Ecology, Evolution and Behavior.

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