Temporal scaling of episodic point estimates of seed predation to long-term predation rates

Authors


Correspondence author. E-mail: asdavis1@illinois.edu

Summary

1. Post-dispersal seed predation influences plant population dynamics in many terrestrial ecosystems, including weeds in arable systems, the focus of this study. Simulation models of management effects on weed demography require estimates of long-term seed predation as input, yet extremely rapid seed losses to predators force most measurements of weed seed predation rates to be made at a daily time-scale.

2. We compared several models for estimating inline image, the annual proportion of weed seeds consumed by granivores, from repeated short-term seed predation measurements. Competing models differed in duration of seed exposure to predators and weighting of short-term predation rates by concurrent seed rain. Verification data were obtained from field studies at experimental locations in Germany and the USA in which parallel measurements of short- and long-term weed seed predation were made in cereal crops.

3. Robust predictions of inline image were given by a model that exposed seeds to predators for two 48-h predation intervals, with no weighting of predation rates by seed rain. Model performance was consistent across locations, years, weed species and predator type. Resampling indicated that high temporal sampling intensity was critical to model performance and should be prioritised over high levels of spatial replication.

4. Estimates of inline image made from our experimental data and 10 published time series of short-term weed seed predation rates followed a normal distribution with μ = 0·52 and σ2 = 0·05. Additional work is necessary to assess applicability of the predictive model to seed predation time series from natural systems.

5.Synthesis and applications: Our results demonstrate that annual rates of weed seed predation may be reliably predicted from repeated short-term predation measurements. Predicted values may then be used in applied demographic models. At a more fundamental level, these results indicate that weed seed predation in arable fields is an episodic process, in which newly dispersed seeds are subjected to brief, intense predation, after which they become unavailable to granivores because of burial. Future efforts at managing agroecosystems to promote weed seed predation may benefit by viewing the process as dynamic and ephemeral.

Introduction

Wherever there are seeds, there are seed predators, helping to regulate plant population dynamics in a wide variety of terrestrial ecosystems (Crawley 2000). Although instantaneous rates of seed predation can reveal connections between transient environmental conditions and seed fate, it is the cumulative impact of seed predators on seed survival and plant regeneration that affects plant population growth rates (Crawley 2000), co-evolution between plants and their predators (Dalling et al. 2010) and restrains the growth of weed populations in arable systems (Liebman & Davis 2009).

Mounting evidence highlights post-dispersal predation of weed seeds as a key agroecosystem service (Heggenstaller et al. 2006; O’Rourke et al. 2006; Menalled et al. 2007; Westerman et al. 2008; Williams et al. 2009), yet quantifying granivory and its effects on weed demography remains problematic. Most measurements of seed predation in agricultural settings are made in short intervals lasting 1–7 days (numerous references within Lundgren (2009); but see Harrison, Regnier, & Schmoll (2003) and Williams et al. (2009), for exceptions). Such measurements are suitable for comparisons of relative effects of environmental variability on seed predation rates, but not as parameters for models of management effects on weed population dynamics, which usually require vital rates at an annual time step (Liebman & Davis 2009). Yet, direct measurements of long-term weed seed predation rates are difficult to obtain because seed removal from the soil surface, via consumption and burial, is so rapid that few, if any, weed seeds remain visible after several days (Westerman, Dixon, & Liebman 2009). Short-term point estimates repeated as a time series of sequential predation trials throughout the longer time period can illustrate variability of weed seed predation rates over time. However, there remains a pressing need for a means of accurate temporal scaling of point estimates of predation to long-term predation rates. The relationship between these quantities remains as yet unverified.

In contrast, the approach to measuring long-term post-dispersal seed predation in natural systems is direct, and comparatively simple: cumulative survivorship of marked seeds remaining on the soil surface is recorded at regular intervals over a period of several weeks to a year (Forget et al. 2006), linking short- and long-term seed removal rates in one survivorship curve. Although the results of the present study may be relevant to the study of seed predation in natural systems, we focus on the problem of temporal scaling from short- to long-term estimates of weed seed predation in arable systems, for which no simple solution currently exists.

When the scale at which inferences are to be made about a process differs from the scale at which it was measured, the measured quantity may be re-scaled. This practice requires explicit attention to measurement units associated with variables underlying the processes of interest, and knowledge of the relevant scales at which each of the processes operates (Schneider 2009). An early model for temporal scaling of weed seed predation rates (Mittelbach & Gross 1984) assumed that, at time-scales of several days or less, seed loss to predators follows a power function, with daily predation rates calculated as 1-Sint1/days, where Sint represents seed survival over a given predation interval. This model assumes constant seed exposure to predators and also assumes a constant predation rate. These may be reasonable assumptions at a daily time step, but newer field studies show that seed predation rates vary tremendously over the course of a single growing season, when viewed at a weekly time step (Menalled et al. 2007; Westerman et al. 2008; Baraibar, Westerman, & Recasens 2009; Davis & Raghu 2010).

A more recent predictive model described temporal scaling between short-term (weekly time step) and annual rates of seed predation (Westerman et al. 2003). This model was driven by three variables relevant to short-term predation rates: duration of seed exposure to predators (days), episodic survival of predation (seedssurviving seedsdispersed−1) and magnitude of seed rain (seedsdispersed). A key assumption of this model is seed predation varies with duration of seed exposure to predators and concurrent predator foraging activity. Empirical studies support the inclusion of seed exposure in this model: seed burial by bioturbation (Regnier et al. 2008), water and wind occurs rapidly (Westerman, Dixon, & Liebman 2009) and strongly influences predation rates (Westerman, Dixon, & Liebman 2009; Davis & Raghu 2010). Long-term seed predation dynamics predicted by this model arise from multiple seed rain events, surviving at cohort-specific rates.

Our aim was to develop a greater understanding of weed seed predation as a multi-scale temporal process, using the predictive model of Westerman et al. (2003) to explore the relationship between processes operating at short-term and seasonal time-scales. We had three main objectives, which are as follows: 1) evaluate the predictive strength of models for scaling short-term point estimates of seed predation to long-term predation rates, 2) determine the spatial and temporal sampling intensity necessary to support such models and 3) support future demographic modelling efforts by predicting long-term weed seed predation rates from published time series of weed seed predation point estimates. These objectives were framed by the following hypothesis: weed seed predation is an episodic process, such that pulses of weed seed rain undergo brief periods of intense predation, after which seeds become unavailable to predators because of burial by plant residues or soil.

Materials and methods

Temporal Scaling Methods

We tested the hypothesis that weed seed predation in arable fields is an episodic process using the predictive model of Westerman et al. (2003). This model (eqn 1) was developed to estimate the annual proportion of newly produced weed seed consumed by granivores, inline image, where inline image, with inline image the annual proportion of newly produced seeds that survive predation:

image(eqn 1)

Annual seed production Y is divided into multiple pulses (Yi) occurring within n periods of a specified length (in the current study, thirteen 2-day periods repeated on a weekly basis from July through October at Location 1, Urbana, IL, USA, and nine successive 2-day periods during October at Location 2, Rostock, Germany). In the present study, the Yi represents identical pulses of seeds offered to predators on a weekly basis and differs from actual seed rain (ri), which was measured concurrently in adjacent plots. Seeds from each pulse survive at the episodic rate Sj, where the duration k of seed exposure from pulse Yi to seed predators may be varied to simulate different rates of seed burial, with lower values of k indicating more rapid burial and less overall exposure to predators. In the current study, we subjected seeds to four levels of exposure: n predation intervals (long-term exposure: 24 days in Location 1; 18 days in Location 2), = 4 predation intervals (8 days exposure), = 2 predation intervals (4 days exposure) and = 1 predation interval (2 days exposure). These models will be referred to as models 1–4.

Estimates of Sj for eqn 1 were obtained from repeated 2-day estimates of seed predation from field experiments (see ‘VERIFICATION DATA’). Variability in the magnitude of seed rain from week to week throughout the study period, onset of weed seed rain through crop harvest, may have had density-dependent feedbacks to seed predation rate, as in the case of predator satiation because of high seed availability. To accommodate this possibility, we created a second version of the predictive model, in which Sj was positively weighted by concurrent seed rain ri measured in field studies (see ‘VERIFICATION DATA’)

image(eqn 2)

As with eqn 1, we varied k to result in several levels of seed exposure to predators, including long-term exposure, 8 days, 4 days or 2 days of exposure. This series of models will be referred to as models 5–8.

The predictive models in eqns 1 and 2 are flexible in describing how the fate of individual seed cohorts scales to inline image. At the smallest values of k, the seed exposure factor, both models treat seed predation as a completely episodic process (Fig. 1). Weed seed rain is followed by short, intense bursts of seed predation, after which seeds become unavailable to predators because of burial. As k increases, the short-term predation rates Sj are compounded for longer periods within the time series, resulting in exponential decline of newly produced seeds.

Figure 1.

 Newly dispersed pulses of weed seeds (P1–P4) may be exposed to predators for different amounts of time, depending upon seed burial rates, resulting in episodic (dotted line) or continuous (solid line) patterns of long-term seed predation.

Verification Data: Episodic Time Series and Long-Term Estimates of Weed Seed Predation

To evaluate the predictive strength of the various models for estimating inline image, we compared model predictions against data obtained from field experiments in which short-term and long-term estimates of weed seed predation rates were made concurrently within the same plots (Fig. 2). Time series of repeated short-term estimates of weed seed predation and direct estimates of long-term predation were made in Urbana, IL, USA, and Rostock, Germany, over two successive field seasons.

Figure 2.

 Episodic point estimates and long-term estimates of weed seed predation were made concurrently within the same field plots in Urbana, IL, USA, and Rostock, Germany. Solid and dotted arrows represent seed input and seed recovery, respectively, in seed predation arenas.

Field study location 1: Urbana, IL, USA

Studies were conducted in 2005 and 2006 near Urbana, IL, USA (40·048000°N, −88·236489°W), at the centre of the US maize (Zea mays L.) production region. Predation measurements were made in the maize phase of a maize-soyabean (Glycine max L.) rotation within four randomly located 9·2-m-long by 12·2-m-wide plots in which maize was at the BBCH 79/83 stage of development (early grain ripening). Measurements of seed predation in small plots may be highly heterogeneous because of rapid movement of predators across plot boundaries. However, because of the close spatial pairing of short- and long-term exclosures in this study, our results were unlikely to have been biased by plot size. Maize was planted with a no-till planter into undisturbed residue from the soybean phase. Hand labour was used to control weeds; no insecticides were used.

Short- and long-term seed predation rates (Fig. 2) were estimated for three weed species: Abutilon theophrasti Medik. (velvetleaf), Ambrosia trifida L. (giant ragweed) and Setaria faberi Herrm. (giant foxtail) (Davis & Raghu 2010). Within each experimental replicate, point estimates and long-term estimates of seed predation were made for each weed species using paired wire mesh exclosures spaced 1 m apart. Both exclosures in each pair gave one of three levels of predator access: control/no predators (1 mm by 1 mm openings), invertebrates only (1·5 cm by 1·5 cm openings) or invertebrates plus small vertebrates (8 × 15 cm openings). Each exclosure was set between maize rows within a 2·5-cm deep excavation in the soil, and the bottom panel was covered by soil to meet the surrounding soil surface.

Point estimates of seed predation were repeated weekly from mid-July through mid-October using seed cards (Westerman et al. 2003). Seed cards consisted of a 10-cm by 15-cm piece of coarse sandpaper to which 30 seeds of a single species (equivalent to 2000 seeds m−2) and a 0·5-cm layer of soil had been lightly adhered. Seed cards were placed in point estimate exclosures for a 48-h period of exposure to seed predators, after which they were removed and remaining intact seeds counted.

Long-term estimates of seed predation were made within 10 cm by 10 cm by 3 cm deep trays made of 1-mm wire mesh located within the long-term predation exclosures. Trays were filled with soil and buried to the level of the surrounding soil surface in the exclosure, leaving a 1-cm wire mesh lip protruding above the soil to prevent seed loss to water movement. Soil used to fill the trays was obtained from underneath a long-term grass sward with very low weed seedbank density (no seeds of the study species were found in 20 elutriated samples of this soil). Weed seeds of a single species were placed weekly on the soil surface of predation trays, using concurrent measurements of seed rain (Davis & Raghu 2010) to guide seed placement rate. Germinated seedlings were counted and removed weekly. At the end of the study period, trays were removed from the field, the soil washed through sieves and intact seeds enumerated.

Point estimates of Sj, the proportion of weed seeds surviving over a given predation interval, were calculated as (seeds recovered)/(30 seeds placed on card). Long-term, direct estimates of inline image, the proportion of seeds surviving the entire study period, were calculated as (seeds recovered + seeds germinated)/(total seeds placed in trays).

Field study location 2: Rostock, Germany

The study was carried out during October 2008 and 2009 in north-eastern Germany near the city of Rostock (55·066667°N, 12·116667°E). Two different sets of fields were used in each year. Predation measurements were carried out within the wheat (Triticum aestivum L.) phase of an oilseed rape (Brassica napus L.)-wheat-barley (Hordeum vulgare L.) crop rotation at the BBCH 13/15 stage of wheat development (3–5 leaves unfolded). Field sizes were similar to the average size of fields in the region, 75 ha.

Four adjacent 15 m by 15 m plots were selected in each field at least 50 m apart from the nearest field edge or semi-natural habitat. Point estimates of seed predation were made with seed cards as described for location 1. Twenty seed cards containing 40 boiled L. multiflorum L. seeds card−1 were placed in each plot (equivalent to 6400 seeds m−2). Seeds were boiled for 5 min to prevent seed losses to germination. Seed cards were randomly placed in the field at the same time as the trays. Cards were exposed in the field: (i) within a cage giving access to vertebrate predators only (10 per plot) and (ii) without a cage, allowed access to all predators (10 per plot). Exclosure cages were constructed from 1-cm-mesh metal screen and secured to the soil using nails. Locations and exclosure treatments were randomly assigned. Seeds remaining on the cards were counted after each of nine successive 2-day predation intervals beginning in mid-October and terminating at the end of October. Estimates of seed survival in trays and on seed cards were calculated as described for location 1. Seed rain was not estimated in Rostock.

For long-term measurements of seed predation, densities of 0 (Control), 1000 (Low), 2500 (Medium) and 5000 (High) seeds m−2 of Lolium multiflorum L. (annual ryegrass) were randomly assigned to each plot and manually added. To ensure a uniform coverage of the plot areas, pre-weighed amounts of seeds were applied to subsections of each plot. Because seeds were not completely evenly distributed and to limit the sampling error, estimates of long-term seed predation were determined as the percentage seeds removed in eight 25 cm by 40 cm by 5 cm deep trays (surface area 0·1 m2) to which exactly known numbers of seeds were added. Trays were randomly located in each plot, made of 1-cm metal mesh and had the bottom covered with a fine cloth mesh (0·56-mm mesh size) to prevent seed escape.

Each tray contained weed seed-free soil, obtained from beneath a long-term grass sward, and crop plants grown in a greenhouse to the same stage as those in the fields. Once in the field, trays were buried and levelled with the soil surface, leaving a 1-cm wire mesh lip protruding above the soil to limit seed loss because of water movement. Known numbers of seeds were then spread over each tray at the same density of the plot where they were placed in (i.e. 100, 250, 500 seeds tray−1). Trays were accessible to all predators, vertebrates and invertebrates. To assess invertebrate response to seed density, 12 small seed patches were created within each main plot. Each patch consisted of a 15 cm by 15 cm by 5 cm deep metal mesh tray (surface area 0·0225 m2) as previously described. Seed densities used were 1000, 2500 and 5000 seeds m−2. A metal mesh (8·7-mm mesh size) was fastened over small trays to exclude vertebrate predators. Seeds were exposed in the fields for 18 days, after which seeds were recovered from soil using sieves and flotation, and remaining intact seeds were counted.

Model Selection

Candidate models for predicting inline image from point estimate data were evaluated in several steps. First, model predictions were compared to observed values of inline image using a two-sample Kolmogorov–Smirnov (KS) test to determine whether they were sampled from the same distribution. Next, several goodness-of-fit criteria were calculated for each set of predicted and observed values, following the methods outlined by Willmott (1982). Mean bias error (MBE), the difference between the means of predicted and observed observations (inline image), indicates whether a given model systematically over- or under-predicts. The index of agreement, a robust measure for cross-comparison of models (Willmott 1982), was calculated as

image(eqn 3)

with unitless values of d ranging between 0 and 1 (where = 1 indicates perfect agreement between predicted and observed values). Root mean square error (RMSE), which summarises model performance in the original units, was calculated as

image(eqn 4)

where lower values of RMSE indicate better predictions. Finally, regression parameters for the least-squares fits between predicted and observed values were obtained using the base package of R v2.10.1 (R Development Core Team 2009).

Estimating Long-Term Seed Predation from Literature Values

Following model selection, estimates of inline image were made from published time series of episodic predation rates (see Appendix S1 for reference list) using the best performing predictive model. For a data set to be included, it needed to consist of at least five repeated point estimates of seed predation, made in independent sequential short-term trials. The data included in Appendix S1 are limited to field crop production environments because time series of sequential trials are not available in the literature for non-arable systems, where the norm is cumulative seed survivorship curves (over 120 articles spanning a 30-year period were surveyed; 21 studies with predation time series reported cumulative seed survivorship over time rather than independent, sequential predation trials).

Once a suitable data set was identified, we compiled seed survival rates (Sj) by digitally extracting x-y pairs from time-series figures (Digitize It ®). In the following example, we demonstrate the process of scaling from short-term Sj to an estimate of long-term seed survival of predation (inline image). The first step is to adjust seed exposure duration (k). If the Sj is measured over 2-day intervals and the desired duration of seed exposure for each predation interval is 4 days, then k = 2 intervals over which each Sj is compounded. Therefore, each element in the hypothetical series {Sj = 0·25, 0·35, 0·45, 0·30, 0·40} is compounded by the next element in the series to yield {Sj = 0·09, 0·16, 0·14, 0·12, 0·16}. Next, assume that each experimental seed pulse Yi was comprised of 100 seeds. The numerator of eqn 1 then becomes [(100*0·09) + (100*0·16) + (100*0·14) + (100*0·12) + (100*0·16)], and the denominator is (100 + 100 + 100 + 100 + 100), resulting in inline image = 0·13, and inline image = 0·87.

Results

Model Selection 1: IL, USA

Of the eight candidate models for estimating long-term weed seed predation rates (inline image) from time series of episodic predation rates in Urbana, IL, USA (Table 1), three were eliminated because of falsification by the KS test. Candidates discarded at this stage included models 1 and 2, with seed exposure intervals of 24 and 8 days, respectively, and no weighting of predation rates by concurrent estimates of weed seed rain. Next discarded was model 8, in which new seeds were exposed for 2 days, and episodic seed predation rates were weighted by weed seed rain.

Table 1.   Performance comparison of predictive models, varying in seed exposure to predators and weighting by seed rain, used to scale point estimates of seed predation to long-term predation (inline image) for Urbana, IL, USA, experiment
Model *Seed exposure (days)†Weighting by seed rainKS test
P > D
Mean bias error‡d§RMSE (seed seed−1 4 days−1)Regression line¶
β0β1
  1. KS, Kolmogorov-Smirnov test; RMSE, root mean square error.

  2. *Models 1–4 correspond to Westerman et al. (2003), eqn 1 in text. Models 5–8 correspond to a revised version of the Westerman et al. (2003) model, eqn 2 in text, in which seed predation intensity is weighted by seed rain during the predation period. The most parsimonious model (3) is shown in bold.

  3. †Duration of weed seed exposure to predators in models varied between 1 and 12 predation intervals (2-day predation interval−1).

  4. ‡Mean bias error (MBE) measures the average difference between predicted and observed values.

  5. §Willmott’s (1982) modelling index (d), or index of agreement, measures the degree of correspondence between observed values and values predicted by a given model. Values of d range between 0 and 1.

  6. ¶The parameters β0 and β1 are the intercept and slope, respectively, of the regression line between observed and predicted values.

124No0·0030·200·750·040−0·150·95
28No0·010·150·830·033−0·090·95
34No0·270·020·910·0230·010·96
42No0·51−0·100·890·0260·110·99
524Yes0·660·030·770·0420·410·52
68Yes0·810·0020·790·0400·390·56
74Yes0·18−0·090·820·0360·350·67
82Yes<0·001−0·240·750·0430·350·84

Goodness-of-fit for models 3 through 7 was then ranked according to their ability to maximise Willmott’s index of agreement, d, while minimising RMSE and MBE. Model 3, in which seeds were exposed for 4 days, with no weighting by seed rain, clearly outperformed the other models by these criteria (= 0·91, RMSE = 0·023, MBE = 0·02). The observed and predicted values of inline image follow the 1:1 line (Fig. 3), as underscored by the linear model of this relationship (β0 = 0·01, β1 = 0·96; R2 = 0·70; F1,58 = 137; P < 0·0001). The next best model, model 4, exposed seeds to predators for 2 days, with no weighting by seed rain, and was only marginally less suitable than model 3, in terms of RMSE and d. Whereas model 3 had a slightly positive MBE (0·02), indicating a small bias towards over-prediction, model 4, with shorter seed exposure had a slightly negative MBE (−0·10), indicating under-prediction.

Figure 3.

 Predicted and observed values of long-term weed seed predation (inline image) in Urbana, IL, USA.

Model selection on predictive models of inline image began with undifferentiated seed predation data from Urbana, Illinois, including variation in year, weed species and predator access, as described above. We then re-examined the data, sorting by these factors, to determine whether the best predictive model for inline image changed, either for a given factor as a whole, or by factor level. Regardless of whether the data were split by year, weed species or predator access, model 3 continued to show the best predictive performance. The index of agreement for models fit to subgroups of the data ranged from 0·79 to 0·97, indicating a high level of predictive power for all factor levels.

Resampling of the full data set via bootstrapped samples of varying size indicated that the predictive power of model 3 was quite insensitive to variation in spatial replication, provided that all episodic predation intervals were represented in each of the resampled data sets. As the proportion of spatial replicates was varied from 1 to 0·05, the bootstrapped value of d remained nearly constant, varying from 0·91 to 0·90 (Fig. 4). In contrast, reducing temporal sampling intensity substantially diminished the reliability of predictions of inline image from episodic predation estimates. The index of agreement, d, varied as a function of temporal sampling intensity (= 0·042*ln(P(data)) + 0·90. As temporal sampling intensity P(data) dropped from 1 to 0·60 of the weekly episodic estimates, d declined from 0·91 to 0·88. When the proportion of temporal sampling fell below 0·60, d declined rapidly, falling to 0·83 when <25% of the data were sampled.

Figure 4.

 Predictive strength of models to estimate long-term seed predation rates from short-term point estimates of predation made in Urbana, IL, USA (a), or Rostock, Germany (b), varied with the proportion of data sampled. Bootstrapped estimates of the index of model agreement, d, were affected more by reductions in temporal sampling intensity (solid circles) than by reductions in spatial sampling intensity (open squares).

The difference in sensitivity of the index of agreement for model 3, with respect to variation in spatial and temporal sampling intensity, was reflected in the spatial and temporal variances of seed predation at the Urbana, IL, USA location. Temporal variation was greater than spatial variation by more than a factor of seven (σ2time = 57·8, σ2space = 7·5).

Model Selection 2: Rostock, Germany

For location 2, only models 1–4 could be tested, because seed rain was not measured in Rostock. Model 3 (4 days seed exposure, with no scaling by seed rain) was the only member of this group of models that passed the first performance criterion, a KS test indicating that both predicted and observed values of inline image were sampled from the same distribution (Table 2). Other goodness-of-fit criteria, including d, RMSE and mean bias error, were also superior for model 3 compared to the other three models. The observed and predicted values grouped diffusely about the 1:1 line (Fig. 5), as indicated by the regression between predicted and observed values (β0 = 0·11, β1 = 0·96, R2 = 0·50, F1,26 = 26·0, P < 0·001).

Table 2.   Performance comparison of predictive models, varying in seed exposure to predators and weighting by seed rain, used to scale point estimates of seed predation to long-term predation (inline image) for Rostock, Germany, field experiment
Model*Seed exposure (days)†Weighting by seed rainKS test
P > D
Mean bias error‡d§RMSE (seed seed−1 4 days−1)Regression line¶
β0β1
  1. KS, Kolmogorov-Smirnov test; RMSE, root mean square error.

  2. *Models 1–4 correspond to Westerman et al. (2003), eqn 1 in text. Models 5–8 correspond to an additional version of the Westerman et al. (2003) model, eqn 2 in text, in which seed predation intensity is weighted by seed rain during the predation period. The most parsimonious model (3) is shown in bold.

  3. †Duration of weed seed exposure to predators in model varied between 1 and 12 predation intervals (2 days predation interval−1).

  4. ‡Mean bias error (MBE) measures the mean difference between predicted and observed values.

  5. §Willmott’s (1982) modelling index (d), or index of agreement, measures the degree of correspondence between observed values and values predicted by a given model. Values of d range between 0 and 1.

  6. ¶The parameters β0 and β1 are the intercept and slope, respectively, of the regression line between observed and predicted values.

114No<0·010·300·500·0750·590·27
28No<0·050·210·580·0600·320·40
34No0·210·040·810·0280·110·96
42No<0·001−0·130·710·0310·240·65
Figure 5.

 Predicted and observed values of long-term weed seed predation (inline image) in Rostock, Germany.

When predictive models for inline image were examined for goodness-of-fit to subgroups of the data (predator access: invertebrates only, invertebrates and vertebrates; subplot seed density), model 3 continued to be best supported by the data. The index of agreement for these models ranged from 0·73 to 0·93, indicating that model 3 resulted in robust estimates of inline image for all factor levels. Model 3 performed best in subplots with the highest seed densities (5000 seeds per subplot; = 0·93), which were closest to the seed density on seed cards (10,500 seeds m−2).

Bootstrapped predictions of inline image by model 3 remained stable for a wide range of sample sizes. The range of the modelling index of agreement, d, was narrow, peaking at 0·81 when the entire data set was sampled, and falling to 0·79 and 0·77, respectively, when spatial and temporal sampling intensity were reduced by 70% (Fig. 4). Spatial sampling intensity could be reduced by as much as 40% without decreasing the index of agreement below 0·81. In contrast, the index of agreement fell below 0·81 when temporal sampling intensity was reduced by more than 20%. Temporal variance in the data was greater than spatial variance by more than a factor of two (σ2time = 219·8, σ2space = 74·9).

Estimates of inline image from Published Time Series of Episodic Predation Rates

Given the apparent robust ability of model 3 (4 days seed exposure to predators, no scaling by weed seed rain) to predict inline image from episodic seed predation rates for several site-years in independent studies, we felt that it would be useful to apply this model to previously published time series of episodic estimates of seed predation (Table 3). Such information is generally lacking for agricultural weeds (for exceptions, see Harrison, Regnier, & Schmoll 2003; Williams et al. 2009) and is essential for clarifying seed predation impacts on weed demography (Liebman & Davis 2009).

Table 3.   Summary of source data for estimates of annual weed seed predation (inline image) from published time series of episodic seed predation rates
Number of countries*Number of weed species†Number of crop species‡Frequency of predator taxaSummary statistics on inline image§
InvertebratesVertebratesBothμσ2MinMax
  1. *Countries include Germany, the Netherlands, Spain and the USA (IA, IL, OH, MI), represented in 11 studies.

  2. †Weed species include Abutilon theophrasti, Ambrosia trifida, Capsella bursa-pastoris, Chenopodium album, Lolium multiflorum, Panicum dichotomiflorum, Polygonum persicaria, Setaria faberi, Stellaria media and Vicia villosa.

  3. ‡Crop species include cereal rye, lucerne, maize, soybean, spring wheat, sugar beet, triticale, winter wheat sole crop and winter wheat undersown with red clover.

  4. §Full data on estimates of inline image, and associated references for source data, are included in online Appendix S1. Predictions of long-term seed predation (inline image, seed consumed per seed produced per year) were made with the model of Westerman et al. (2003), eqn 1 in this paper, with an exposure period k of 4 days.

51096270·520·050·080·89

Estimates of long-term weed seed predation rates varied widely across crop, location and predator type (online Appendix S1), from 0·08 to 0·89. Across all studies, inline image followed a normal distribution with μ = 0·52 and σ2 = 0·05. Owing to the lack of replication in reported values, we did not statistically analyse treatment effects in the predicted long-term predation rates. However, some patterns were evident upon visual inspection of the data. First, variability in inline image was high within and among all locations and factor levels. Second, inline image tended to be higher in legumes and legume–cereal mixtures than in cereal crops alone. Finally, the proportion of total long-term seed consumption because of invertebrates or vertebrates varied by location and weed species, with one of the predator types tending to dominate under a given set of conditions.

Discussion

Despite differences in experimental approach, field study environment and cropping system, evaluations of model performance for the two verification data sets were quite similar. Model selection in both cases indicated peak predictive performance when seeds were exposed to predators for 4 days, without weighting short-term seed predation rates by seed rain. These results provide strong support for the hypothesis that weed seed rain is an episodic phenomenon, in which pulses of weed seed dispersal are followed by brief, intense periods of seed consumption by granivores, after which seeds are protected by burial. Previous empirical work demonstrated rapid burial of weed seeds by abiotic processes such as wind and rain (Westerman, Dixon, & Liebman 2009), with strong effects on episodic predation rates (Davis & Raghu 2010).

The predicted and measured values of inline image are in a similar range to those projected by Westerman et al. (2006), in which a simulation model was used to explore the relationship between seed rain, seed burial and granivore activity. They are also consistent with other direct measurements of inline image in field settings (Harrison, Regnier, & Schmoll 2003; Williams et al. 2009). In comparison, estimates of inline image made with a predictive model assuming continuous seed exposure throughout the season were excessive (inline image > 99%). Values of inline image reported in Appendix S1 are nonetheless sufficient to contribute to ecological weed management (Liebman & Davis 2009).

High variability in inline image predicted from literature values, within and among environments, was not unexpected. Granivory is finely tuned to many abiotic and biotic environmental variables, including temperature and precipitation, shelter from predators, predator population densities, seed supply, residue cover of the soil surface and seed burial rates, among others (Westerman et al. 2006; Westerman, Dixon, & Liebman 2009; Davis & Raghu 2010). A trend towards higher seed predation in legumes and legume–cereal mixtures has been explained as the result of high-quality habitat for seed predators in legume canopies, relative to the low cover provided by open or senesced cereal canopies (Davis, Dixon, & Liebman 2004; Heggenstaller et al. 2006). A widely observed, but as yet unexplained, outcome of the site-specific nature of weed seed predation is the tendency for local dominance of a given predator type: at some locations, invertebrates are dominant (Baraibar, Westerman, & Recasens 2009; Davis & Raghu 2010), whereas in other locations, vertebrate predators drive long-term seed losses (Westerman et al. 2011).

Our results may offer practical lessons for those contemplating future research on weed seed predation in agroecosystems. First, if the difficulty of measuring inline image directly is hindering research efforts, time series of episodic predation may be performed instead, and inline image estimated from these data. Second, in relatively uniform production systems, such as the maize and cereal fields in Urbana and Rostock, where temporal variation outweighs spatial variation, resources should be focused on a) adequate temporal sampling intensity to capture long-term variability in predation rates and b) comparisons of agronomic practices most likely to create variation in predator behaviour. Third, as examination of prediction strength by different subplot seed population densities in Rostock suggests, estimates of long-term seed predation are most accurate when time series of episodic predation rates are measured at a seed population density close to the ambient level. Density dependence in feeding by seed predators has been shown to be an important consideration for estimating predation rates, because of both predator satiation (negative density dependence) and aggregated feeding upon locally rich seed resources (positive density dependence) (Westerman et al. 2008; Davis & Raghu 2010). Finally, although we did not find support for a seed rain weighting factor in our predictive models, such a factor would be critically important in cases where seed rain and granivore activity show only partial temporal overlap (Westerman et al. 2011).

Viewing weed seed predation as an episodic process may inform agroecosystem design for increased weed seed destruction by granivores. For example, there should be a window of several days (and preferably, several weeks) between crop harvest and primary tillage to allow weed seeds dispersed by the harvest operation to be consumed. Another key consideration is how to support granivore activity that is more synchronous with weed seed dispersal. Temporal overlap between crop maturation and weed seed dispersal can have large impacts upon species-specific predation rates; diverse crop rotations are unlikely to leave any one weed species untouched by seed predation (Westerman et al. 2011). If opportunities for consuming seeds are fleeting events, then providing suitable habitat for seed predators at the time of maximum weed seed dispersal should be prioritised. Granivore activity during crop senescence may be hindered by increased vulnerability to predators (Davis & Raghu 2010) even as weed seed rain reaches its peak. Improved granivore habitat may therefore have to come in the form of adjacent or inter-penetrating non-crop vegetation. Linking understanding of weed seed predation from a plant-centric point of view to a better understanding of granivore behaviour should create rich opportunities for managing weed seed destruction by granivores as an agroecosystem service.

Acknowledgements

We gratefully acknowledge the contributions of numerous research assistants to this study. Funding for this work was provided by the United States Department of Agriculture and University of Rostock.

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