Density-dependence at multiple scales in experimental and natural plant populations

Summary

• 1
  The survival and reproduction of individual plants may be related to the distribution of conspecific neighbours with which they interact.
• 2
  We used 2 years’ data from a field experiment to relate the reproduction and survival of focal Silene latifolia plants to the numbers of conspecific plants and flowers within a range of distances (0.16–150 m) from the focal plants. Thus we consider spatial scale as a continuously-varying component of population density.
• 3
  Because the data come from a single continuous population, we fitted regression models in two stages in order to remove spatial autocorrelation induced by environmental factors before modelling density effects. As well as single-scale models, we fitted multiple-scale models to explore interaction kernels. We statistically standardized plant size in some analyses to isolate pollination effects.
• 4
  Reproductive output per plant generally decreased with greater densities of plants and flowers; for example there was a negative effect of female flowers within a 70-m radius in the second year. However, at low female densities there was a positive effect of male flowers at a scale of 13 m in the first year. Seed-set per flower decreased with greater densities of flowers in the second year (at radii < 10 m). Seed predation was greater when more female flowers were present within 5–7 m. Plants’ survival probability increased with density within 0.28 m for all but the smallest plants.
• 5
  We also collected data on the pattern and reproductive success of S. latifolia plants in a natural population nearby and applied similar methodology. Here the negative effects of female flower densities occurred at finer scales and gave way to positive effects above 2 m, contrasting with some patterns from the experiment.
• 6
  Synthesis. In a sessile population interacting with mobile biota, the effects of density on individual fitness may vary with the way in which density is measured. This, together with the methodological challenges for analysing spatial data from continuous populations, should be considered when designing and interpreting studies of spatial density-dependence. Taking these issues into account, we may begin to explain the spatial patterns of natural populations.

Introduction

The variation of population densities in time and space is a central theme of population ecology. Early studies in population dynamics tended to assume spatially homogeneous populations that could be ascribed a single ‘mean-field’ density value. However, in a patchy population, local densities vary from place to place and with the extent of areas sampled (Smallwood & Schonewald 1996; Gaston et al. 1999; Pautasso & Weisberg 2008). Understanding population dynamics may therefore depend upon selecting appropriate spatial scales for investigation, especially in sessile organisms such as plants and many insect larvae (Rothman & Darling 1990; Ray & Hastings 1996; Gunton & Kunin 2007).

Although the essence of classical density-dependence is feedback in densities over time, this is often investigated by comparing the performance of different population patches in space (e.g. Walde & Murdoch 1988; Hopper et al. 1991; Hails & Crawley 1992). Associations between density and demographic rates among such patches constitute ‘spatial density-dependence’; further considerations may determine whether they entail classical demographic density-dependence (Dempster & Pollard 1986; Stewart-Oaten & Murdoch 1990). However, spatial density-dependence is often of interest in its own right (Cappuccino 1992; Bohan et al. 2000). In particular, if demographic density-dependence may regulate temporal fluctuations in population densities, one view of spatial density-dependence is that it shapes the spatial patterns of natural populations (Suzuki et al. 2003; Gascoigne et al. 2005; van de Koppel et al. 2005; Malkinson & Kadmon 2007).

A plant's fitness is affected by its neighbours through growth, survival and reproduction. Direct effects, such as competition for light, nutrients or water, are likely to act at fine spatial scales, whereas indirect effects, involving attraction or diversion of herbivores, pathogens, pollinators, etc., may involve coarser-scale densities. Direct competition may be species-non-specific, whereas specialized herbivores or pathogens may cause specific density-dependent (or density-vague: Strong 1986) mortality at coarse scales and low frequencies (e.g. Bucheli & Shykoff 1999; Elzinga et al. 2005). Herbivores may either increase (Janzen 1970; Harms et al. 2000) or decrease (the resource dilution effect, Otway et al. 2005) mortality as a species becomes locally more abundant. Success in sexual reproduction depends on conspecific neighbours, especially in obligate outbreeding species; local populations may go extinct via inverse density-dependence if they decline to the point of reproductive failure (Allee 1931; Stephens et al. 1999).

The best way to investigate spatial density effects is to manipulate local densities within a population and monitor responses. While this has been done for plants, in both natural populations (e.g. Fowler 1995; Shilo-Volin et al. 2005) and experimentally-planted ones (e.g. Kunin 1997; Elzinga et al. 2005), density is rarely manipulated with explicit regard to spatial scale (Lortie et al. 2005). Moreover, experiments that use simple, regular patterns of plants do not reflect the spatial heterogeneity of natural populations, so that neighbourhood densities tend to be similar for all individuals, across a range of spatial scales. A better approach may be to create a patchy pattern in which a diversity of scale-specific neighbourhood densities occurs and to use statistical analyses to differentiate scale-specific density effects on focal individuals.

This raises two challenges for experimental studies: to create more realistic, spatially-heterogeneous planting patterns, and to measure densities in a scale-specific, individual-focused manner. In this study, we used data from a population of Silene latifolia plants planted in fractal-like patterns to create diverse densities over spatial scales from 16 cm to 20 m in linear dimensions (0.026–400 m²). We surveyed flowering across the whole population and then calculated neighbourhood densities of plants, of flowering plants and of flowers for a set of focal individuals. Neighbourhoods were defined by circles of various radii, so that we obtained for each focal plant a unique set of scale-specific neighbourhood densities. We then collected fitness data from the focal plants and related these to the scale-specific densities in each of two seasons. Measuring the focal plants’ sizes enabled us to differentiate density effects on reproductive output due to density-dependent pollination from those that might involve differential growth rates. We also studied a natural population nearby for comparison. We expected scale-specific effects of plant or flower density on various fitness components: (i) a positive relationship between flower density and reproductive output at coarse scales through pollinator attraction to more concentrated resources, (ii) a positive relationship between flower density and seed predation at coarse scales because of the resource concentration effect (Root 1973), and (iii) some relationship between plant density and survival at very fine scales because of direct interactions (facilitation or competition) among neighbours.

Methods

species

Silene latifolia is a short-lived perennial herb. It is common in hedge banks and arable land in the UK, where it flowers between May and October, sometimes on the first year's growth. It is a dioecious species, pollinated by insects including syrphids and moths. Seed capsules develop over a period of 5–8 weeks following pollination and scatter seeds following dehiscence in dry conditions. Hadena moths (Lepidoptera: Geometridae) lay eggs in female Silene flowers and their larvae feed on the seeds. Under the heading ‘Gregariousness’, Baker (1947) describes S. latifolia as ‘usually solitary, but sometimes [growing] in small patches’ and says it ‘often occupies large areas, but never forms a continuous cover.’ In terms of scale-specific density, this suggests low densities at both finer and coarser spatial scales (Kunin 1998). By investigating the effects of density at a range of scales on the fitness of individuals, we may hope to explain such a pattern.

field experiment

The experiment was established in a 3.2-ha field at the University of Leeds Field Research Unit, near Tadcaster, West Yorkshire, UK (Ordnance Survey grid reference SE 449413) in April 2004. A total of 14 616 S. latifolia plants were planted in a complex layout that was designed to maximize the variation in densities at each of four spatial scales. These scales corresponded to nested squares with side-lengths of 16 cm, 80 cm, 4 m and 20 m (each square composed of a 5 × 5 block of finer-scale squares). The numbers of S. latifolia plants in these squares were powers of five; thus a 16-cm square could contain 1, 5 or 25 plants and could be situated in an 80-cm square containing a total of 1, 5, 25 or 125 plants, a 4-m square containing 1, 5, 25, 125 or 625 plants and a 20-m square containing 1, 5, 25, 125, 625 or 3125 plants. Combining these densities in all possible combinations gave 90 multi-scale density treatments, which we used to fill out three replicate blocks; the design is fully described elsewhere (Gunton & Kunin 2007). The plants were grown from seed and planted into short grass turf in April 2004, when they had four or five pairs of true leaves. It is not possible to distinguish the sexes at this stage, so a random assortment of males and females was likely.

A set of 249 focal plants was monitored to assess survival (one plant for each treatment in each replicate of the layout, minus 21 plants whose locations were lost). The dry masses of these plants were estimated 3 weeks after planting, from non-destructive measurements by means of a plant mass calibration formula parameterised from a separate set of experimental plants (Appendix S1 in Supporting Information). We checked these plants for survival at the end of April 2005. We also assessed reproduction of individual plants in two seasons and noted any evidence of seed herbivory by moth larvae (i.e. holes in capsules or broken capsule teeth). We attempted to collect one ripe capsule from a plant representing each density treatment in each replicate – i.e. 90 × 3 = 270 capsules. Where possible, capsules came from the focal plants used for the survival data. However, no female plants flowered in some replicate-treatments; thus only 169 capsules were collected from 147 plants in 2004, and 100 from 100 plants in 2005. Where a plant presented several capsules, one of median size among those on the plant was chosen. Where only empty capsules (post-dispersal) were found, the length and diameter of one of these were measured for estimating its seed production (see below). In this way, reproduction data were obtained from 209 plants in 2004 and from 158 plants in 2005. The total number of ripe capsules produced by each of these ‘seed plants’ in each season was calculated from the numbers retained on the plants plus numbers shed (estimated in year 1 by comparing numbers of open and closed capsules at each survey).

We chose to use the estimated total mass of seeds produced by each plant and the mean number of seeds per capsule for each plant as two quasi-independent response variables that would approximate reproductive output and pollination success, respectively. (Spearman's correlation coefficients between these variables were 0.51 in 2004 and 0.46 in 2005). When the capsules had opened, their seed contents were weighed and multiplied by the number of capsules of their parent plant to obtain total seed production data. Samples of 30 seeds from each capsule were also weighed to allow estimation of the total number of seeds in each capsule. To derive estimates of seed mass and number of seeds from the measurements of empty capsules, capsule calibration formulae (Appendix S1) were developed using data on the seed contents, lengths and diameters of a sample of 45 capsules. The dry masses of ‘seed plants’ plants were also estimated at the end of each season, using non-destructive measurements with further plant mass calibration formulae (Appendix S1).

Five surveys of the entire planting were conducted during summer 2004 to monitor flowering. In each of the occupied 80-cm squares, we counted the number of plants of each sex that were flowering and the number of flowers of each sex. These surveys took place on 10–11 June, 2–8 July, 26–30 July, 23–31 August and 27 September to 6 October. During this time the grass grew rapidly so that by July the S. latifolia plants were mostly obscured from sight. Approximately 43 % of the focal plants flowered in the first year and 45 % died over the winter (but flowering was not a useful predictor of mortality: Gunton & Kunin 2007). In June 2005, when the grass had again overtopped the S. latifolia plants, the field was mown and the cuttings removed. This stimulated a profusion of regrowth and flowering, while the grass remained short, and a single survey of flowering was conducted on 25–27 August 2005.

We also measured three sets of covariates in the vicinity of the focal plants: grass height (June 2004, at 469 locations), grass species composition (August 2004, at 668 locations) and soil characteristics (January 2005, at 12 locations). For the latter, 30-cm soil cores were analysed for NO₃, NH₄, P, K and Mg concentrations and pH by Phosyn plc (Pocklington, York YO42 1DN), and we measured water content in the laboratory. The field was grazed by sheep over winter to restore the sward, and the focal plants’ survival was assessed between 21 April and 11 May 2005.

natural population

We surveyed S. latifolia plants along three roads where they were present in verges and hedges within a circle of c. 800 m diameter, centred 1.2 km north of the experimental field (OS grid reference SE 450424) and constituting the nearest natural conglomeration of the species. We refer to these as our ‘natural population’ (although the roadside habitat obviously reflects human influence). The survey was conducted between 8 and 20 July 2005, when the plants in the field experiment had not resumed flowering since being cut in June. Ninety-four flowering plants were mapped to resolutions of 0.1 m with reference to the nearest roadside. The sex of each plant and the number of its flowers were recorded.

For 44 reproductive female plants we recorded the number of flowers, fruits (whether developing or already open) and past flowers, where all the petals had fallen but no fruit appeared to be set. The reproductive performance of each female plant as a proportion of its potential was estimated by dividing the number of fruits by the total number of fruits, flowers and aborted fruits. We refer to this quantity as the reproductive success ratio. Including the number of flowers in the denominator helps to account for the size of individuals and also for the fact that flowering continues more profusely when seedheads are not successfully produced (Wright & Meagher 2003). Adding the number of fruits to the denominator means that plants bearing mature fruits only are considered to have fulfilled their reproductive potential (reproductive success ratio = 1).

statistical methods: general

For the field experiment, the data for the estimated total masses of seeds and the mean numbers of seeds per capsule were analysed using linear regression models, and the data on seed herbivory and survival using logistic regression. For the natural population, the reproductive success ratios were modelled using generalized linear models with a binomial error distribution and the logarithmic link function.

Each analysis was based on modelling covariates first and then adding one or more scale-specific density terms, as described in the next section. The covariates were obtained as follows. A principal components analysis (PCA) of the soil data showed that much of the variability among the 12 data points was described by the pH value and the nitrate concentration. Estimates of these two variables at all the locations of seed plants were therefore obtained by interpolation, using ordinary kriging based on the spherical semi-variogram model (in the R package ‘geoR’, Ribeiro & Diggle 2007). The data on grass species composition (comprising 668 cases of presence/absence of each of four species) were summarized using the first two principal components extracted by a PCA. The first component was positively correlated with Poa and inversely with Dactylis, the second positively with Phleum. In this case, we simply used the values assigned to each location as factors in the subsequent regression models. Where no grass sample had been taken at the location of a seed plant, the value for the nearest sample in the same 4-m square was assigned to it, or the average from the nearest three or four samples if there were none from that square. The grass factors were quadratically transformed to reduce their skew and avoid them having undue weight in linear models.

We compared scale-specific regression models using Akaike's Information Criterion (AIC). This goodness-of-fit statistic offsets the log-likelihood of a model with a penalty for each explanatory variable, allowing identification of which model from a set is likely to be most capable of reproducing the observed data (Burnham & Anderson 2002). Using AIC provides a transparent, theoretically-grounded framework for comparing models according to their goodness-of-fit rather than by null-hypothesis testing (Stephens et al. 2005); we prefer this because of the large number of models we compare and the exploratory nature of our investigation. It also allows simultaneous comparison of multiple models, which can circumvent the problems of stepwise procedures for choosing a minimum adequate model (Whittingham et al. 2006). Lower AIC values indicate better fit; a margin (ΔAIC) of 2 units provides reasonable support for one model against another (Burnham & Anderson 2002), and a ΔAIC of 4 may indicate a better model with 95% confidence (Richards 2005).

For a more intuitive measure of goodness-of-fit, we present adjusted r² and partial-r² statistics for each model. For the logistic regressions, a statistic analogous to adjusted r² was based on the McFadden's (log-likelihood-ratio) r²:

• 

and a partial r² analogue calculated as

• 

where ln L_M, ln L₀ = log-likelihoods of the model of interest and of a model with only an intercept, respectively; k_M, k₀ = the numbers of coefficients in the model of interest and the null model, respectively, and ln L_M-j = log-likelihood of the model of interest with variable j deleted (Menard 2000; Shtatland et al. 2002).

All statistical analyses were performed using the software R (R Development Core Team 2007).

statistical methods: spatial regression models

Estimates of goodness-of-fit from ordinary least-squares regression are only comparable if the data points are independent. Environmental trends in space (exogenous processes) might cause data to be positively spatially autocorrelated (Wagner & Fortin 2005). However, density effects and other plant–plant interactions (endogenous processes, Dormann et al. 2007) may cause closer pairs of plants to show either more divergent or more similar performance. An analysis of density effects can only proceed safely once the effects of environmental autocorrelation have been accounted for, to prevent confounding spatial effects.

To address this problem, we fitted multiple-regression models in two stages. First, spatial covariate models were fitted. For the field experiment, these were linear regression models of each year's total seed masses and seed numbers and of inter-year survival against (i) the soil and grass variables and vegetation height, to control for environmental autocorrelation, and (ii) the X- and Y-coordinates of the plants and X², Y² and X × Y terms, to remove any coarse spatial trends. For the natural population, we modelled the reproductive success ratio against spatial covariates (the only ones available). For these we chose (i) the distance parallel to the road where 90 % of the plants were located and (ii) distance from the nearest roadside as coordinates. Next, the ‘all.regs’ method in the ‘hier.part’ package for R (Walsh & Mac Nally 2005) allowed us to select the most parsimonious models by calculating AIC values for the 1024 possible models that could be constructed with the field-data covariates and the 16 possible models (allowing squares of the two covariates) from the natural population. The best-fitting models are henceforth referred to as the ‘covariate models’. For the selected binomial model for the natural population, a ratio of residual deviance to degrees of freedom equalling 0.76 implied an adequate fit (Crawley 2007).

In the second stage, density-independent models were created and then density variables were added to them, to create both single-scale and multiple-scale models. Density-independent models were, for the continuous response variables, linear regressions of the residuals from the covariate models against the focal plants’ dry masses (log-transformed and centred), if this was supported by the AIC. There was negligible collinearity between dry mass and densities (minimum tolerance statistic for dry mass = 0.81). In the natural population, plant masses were not measured, and we also excluded plant dry mass from one analysis of the field experiment to see whether plant densities alone could explain total seed mass; in these cases the density-independent models consisted of an intercept only. For the binary response variables, residuals from the covariate models were not suitably distributed for regression, so we simply added the focal plants’ dry masses into the logistic covariate models, if doing so reduced the AIC value, and used these as density-independent models.

Density variables were added to the density-independent models as follows. For the experiment, three sets of neighbourhood-density variables were calculated. First, numbers of neighbouring plants around each focal plant were computed from the planting design, using 18 circles of exponentially-increasing radii (from 0.16 to 150 m, Table 1) around each focal plant. Mortality during the first summer was approximately 5%, so the original planting pattern gave a reasonable picture of plant densities throughout this season. Secondly, numbers of male and female flowering plants around each focal plant were computed for year 1 using the flower survey data, for 15 radii increasing from 0.54 to 150 m (Table 1); in year 2 this was not possible because individual plants could not be distinguished. Thirdly, numbers of flowers were computed, for the same 15 radii. For the year-1 data, mean numbers of flowers over the five surveys were used. Then, for the natural population, the numbers of male and female flowers were calculated within 19 radii from 0.11 to 150 m (Table 1). Nearest-neighbour distances were also calculated for the natural population, using geometric mean distances to the 1, 2, 3, 4 and 5 nearest neighbours.

Table 1.  Radii used for each predictor variable and the grouping of these radii into annuli for the multiple-scale models of total seed mass in 2005. The factor relating successive radii is 5^1/4 (approximately 1.495).

Effects of scale-specific density on each response variable were tested as follows, using AIC to select the best models. First, single-scale models were created by sequentially adding the numbers of flowers or plants within each radius into the appropriate density-independent model. For the analyses of seed mass, number and herbivory, we started by testing effects of flower densities and then, where these were supported, proceeded to look at effects of (flowering and overall) plant densities; for survival we were only interested in effects of overall plant densities. Where available, numbers of male and female plants or flowers were included as separate terms for one version of each model and their sum was used for another, to see whether models with a separate coefficient for each sex were supported. This gave a best-fitting scale-specific density model for each analysis (the ‘basic models’). We then improved the accuracy of the best scale for each basic model to ±5%. We did this by defining sequences of additional radii between the best-fitting radius and the next-smallest and the next-largest radii previously tested, then computing neighbourhood densities for these additional radii and comparing the additional models based on these density variables. Finally, we sought improvements in the resulting best models by adding any possible interaction terms between male and female densities and the mass of the focal plants, testing all combinations where several interactions were possible. F-tests on the full models including all possible interactions (or χ²-tests for the logistic models) indicated significant explanatory power with respect to null models (P < 0.05) for all analyses except the year-1 models for number of seeds and seed herbivory in the experiment.

Next we considered multiple-scale statistical models. Entering multiple scale-specific densities, defined as above, into the same model would result in collinearity between the variables for different scales. To avoid this, we decomposed the circular neighbourhoods used in the basic models into varying numbers of annuli, fitted multiple regression models with coefficients for each annulus and selected the model with the lowest AIC value. This was done for male and female flowers combined and for the sexes separately. We first used the largest radius that, among the basic models, gave a lower AIC value than the density-independent model, and then we tried the radius that gave the best-fitting basic model. Annuli were formed by grouping the pre-defined radii into roughly equal-sized ranges of (logarithmic) scales (Table 1). This investigation was only performed for the data on plant seed masses in year 2, as the other data would not support the estimation of more coefficients.

To examine how spatial autocorrelation in the data was accounted for first by exogenous factors (using the covariates) and then by endogenous factors (using density variables), we obtained semi-variograms of, respectively, the raw data, the residuals of the density-independent models and the residuals of the best single-scale neighbourhood-density models, using the R package ‘geoR’ (Ribeiro & Diggle 2007). The robust semi-variogram estimator (Cressie 1993) was used as it is less sensitive to outliers than the conventional estimator. We also performed Mantel tests for association between the pairwise spatial separations of the focal plants and the corresponding differences between their residuals in the models (Schabenberger & Gotway 2005). We used 1000 permutations to derive estimates of the one-tailed significance of any positive deviations of Mantel's R (based on ranking) from values expected under no association (using the R package ‘ecodist’, Goslee & Urban 2008).

Results

densities

In the field experiment the focal plants were subjected to a wide range of plant and flower densities. Densities declined with scale except at the finest scale for each variable (Fig. 1). Numbers of plants and flowers were highly correlated between consecutive scales, except between the finest (Fig. 2), and generally more so in year 2 (when floral densities were much higher). The correlation between numbers of male and female flowers increased from 0.6 at the finest scales to almost 1.0 at the coarsest (Fig. 3).

Figure 1. Box-and-whisker plots of scale-specific neighbourhood plant and flower densities, for the field experiment using (a) all plants as planted in year 1, (b) all flowers in year 1 (using the mean numbers from five surveys), (c) all flowers at the single year-2 survey; and (d) for the natural population, using all flowers surveyed in July 2005. The boxes span the interquartile range of values, with the median indicated by a horizontal line; the whiskers extend to the furthest value within 1.5 times the interquartile range, with outliers represented by dots. Densities are offset by one to allow the logarithmic axes – so 1 = no neighbourhood plants/flowers.

Figure 2. Spearman's rank correlation coefficients (ρ) between densities at each pair of scales in the ranges available, for: the field experiment using (a) all plants as planted in year 1, (b) all flowers in year 1 (using the mean counts from five surveys) and (c) all flowers at the single year-2 survey; and (d) for the natural population, using all flowers surveyed in July 2005. A separate line, starting at (x, 1.0), connects the sequence of coefficients for scale x against each successively-coarser scale. The dashed lines indicate ρ = 0.

Figure 3. Spearman's correlation coefficients between male and female flowers in the neighbourhoods of each scale used for density analyses (a) in the field experiment (year 1: solid circles; year 2: open circles) and (b) in the natural population (here solid circles indicate P < 0.05 for a null hypothesis of equality to zero).

In the natural population, the flower distributions showed less variation in densities (Fig. 1d), with stronger positive skew but similar patterns of correlation (Fig. 2d) between the densities at different scales as were found in the experimental population. The correlation between numbers of male and female plants was close to zero at fine scales in the natural population (Fig. 3b).

density effects: field experiment

All responses in the field experiment show scale-dependent density effects, with contrasting effects of male and female flowers in some cases (Table 2; Figs 4 and 5). Including the dry masses of the focal plants greatly improved the fit of the density-independent models for seed production and survival, but not those for seed herbivory. The coefficients for dry mass were positive in all cases and imply total seed masses increasing by factors of 1.3 (year 1) and 1.5 (year 2) for a doubling of plant dry mass, and numbers of seeds increasing by 20 (year 1) and 37 seeds (year 2) for a doubling of plant mass. In general, the year-2 reproduction data give better-fitting density models than the year-1 data. ΔAIC values in the following text indicate improvements in goodness-of-fit of various models when density is incorporated, relative to the relevant density-independent model (as specified under Statistical Methods: Spatial Regression Models).

Table 2.  Details of basic density models for various fitness components providing the best fits for each response variable. Separate coefficients for each sex were estimated where possible

Response variable	Year	Type of regression	N	Covariates (in covariate model only)* and their sign	Density measure†	Best radius	Coefficients ± SE†	Interactions supported*† and their sign	Improvement over density-independent model (AIC units)	Adj-r^2
* The covariates are indicated by: M = dry mass of focal plants at maturity; M1 = dry mass of focal plants at the start of the experiment; {X, X ^2, Y, Y ^2, XY} = spatial coordinates and products thereof; A = soil pH; N = soil nitrate concentration; {G1, G2} = factors summarizing grass species composition; V = vegetation height; H = level of herbivore damage. The role of the covariate model in our two-stage modelling procedure is explained in the text. † m, male; f, female; c, combined sexes. ‡Focal-plant dry mass was excluded from this model and the density-independent model against which it was tested, giving a much poorer fit in comparison with the following models. Partial-r^2 statistics can be compared in Figs 4 and 5. § The model for the 0.5-m scale was actually marginally better (by 0.2 AIC units), but the 7-m model is presented here for comparison with the result from 2005.
Seed mass per plant	2004	Linear	209	M+[Y+, Y^2–, N–, G2+V–]	(1) All plants	11 m	c: −0.000093 ± 0.000044		2.5	0.02‡
					(2) Flow’g plants	12 m	c: −0.00043 ± 0.00022		1.7	0.08
					(3) Flowers	13 m	m: +0.0073 ± 0.0046	m : f–, m : M+	7.7	0.13
							f: –0.026 ± 0.033	f : M–, m : f : M+
	2005	Linear	158	M+[X^2–, Y^2–, XY+, A+, G1+]	Flowers	70 m	m: +0.00033 ± 0.00018	f : M+
							f: −0.013 ± 0.004		24.5	0.42
Seeds per capsule	2004	Linear	209	M+[XY–, A–]	Flowers (m + f)	22 m	c: −0.20 ± 0.17		–0.7	0.02
	2005	Linear	158	M+[X^2–, XY+,Y–, G1+, G2+]	Flowers (m + f)	1.9 m	c: −0.39 ± 0.18		3.0	0.07
Seed herbivory	2004	Logistic	209	X–G1+	Flowers	7.2 m§	m: −0.078 ± 0.040		2.3	0.02
							f: +0.84 ± 0.35
	2005	Logistic	158	Flowers		6.5 m	m: −0.011 ± 0.004		3.2	0.03
							f: +0.135 ± 0.055
Survival	2005	Logistic	249	M1+, X+, XY–, G2+, H–	All plants	0.28 m	c: +0.081 ± 0.012	c : M1+	32.3	0.11

Figure 4. Effects of plant density (both sexes combined) on (a) over-winter survival and (b) seed production in year 1. Dots show partial r² for the density term while shading indicates estimated effects of plant density (error envelope of ±1 SE). The regression models summarised in (b) exclude the dry mass of the focal plants as a covariate (see text for explanation).

Figure 5. Effects of flower density on reproduction in year 1 (first column) and year 2 (second column): total mass of seeds per plant (a and b); number of seeds per capsule (c and d); and seed herbivory (e and f). Dots show partial r² for the density term(s) while shading indicates estimated effects of flower density (error envelope of ±1 SE). In a, b, e and f the darker shading indicates coefficient envelopes for male flowers and the paler those for female flowers (c and d are for both sexes combined). Data come from the basic models (i.e. single-scale models without interactions).

Survival of plants into the second year of the experiment generally increased with numbers of neighbouring plants within 0.28 m (Fig. 4a). However, there is a synergistic relationship between plant size and neighbourhood density (positive interaction), and model analysis shows that plants with dry masses below 80 mg in June 2004 were increasingly unlikely (and larger plants increasingly likely) to survive with increasing density. The critical plant size is small, as 80 mg is the eighth percentile of the dry mass values.

For total seed mass per plant, we first tested whether the reproductive output of focal plants in year 1 can be directly related to the numbers of neighbouring plants at any scale, by using plant densities according to the planting design and without the focal plants’ dry masses as a covariate. We found a negative effect on fitness of the number of neighbours at the scale of 11 m where goodness-of-fit peaks (Fig. 4b), although this is only marginally supported by AIC (Table 2). Clearer density effects were revealed when plant mass was controlled for and the effects of male and female flower densities were added, separately (Fig. 5a), together with interactions (see Table 2): now, at a scale of 13 m, a marginally positive effect of male flowers on seed production stands against a synergistic reduction by male and female flowers together (negative interaction), a negative effect of female flowers for larger plants only (negative interaction) and an additional synergistic increase from all three variables (positive three-way interaction). In year 2, overall plant density was not available but there are clear negative effects of female flowers, at a coarser scale of 70 m and becoming less negative for larger plants (positive interaction). There is also a secondary peak in goodness-of-fit at a scale of 1.2 m (Fig. 5b), where male flowers have negative effects and female flowers positive effects on total seed mass per plant.

For numbers of seeds per capsule (Fig. 5c,d), the coefficient for overall flower density is negative in year 2, with similarly good fits at scales of 1.9 and 3.8 m. Fits with male and female flowers separately are not supported, and no density-dependence was found in year 1.

Seed herbivory was observed on 14 (7%) of the seed plants from year 1 and 22 (14%) from year 2. For both years, the data support separate male and female effects (Fig. 5e,f). In year 1, two models are similarly good: they use flower densities within 0.5 and 7.2-m radii, respectively. Both have a positive coefficient for females and a negative one for males (Table 2). In year 2 there is a single best model with smaller coefficients of the same signs at a radius of 6.5 m.

The data for year-2 total seed mass per plant provide better fitting models than any of the other reproduction data and autocorrelation appeared to be successfully controlled (see below). These data were therefore used to explore models for density effects at multiple scales simultaneously. The largest radius available (150 m) was decomposed into 2, 3, 4, 5, 8 and 15 roughly-equal intervals (Table 1), providing a set of ‘annuli models’. The best fit comes from the 6-annuli model (ΔAIC = 17.5; adj. r² = 0.40; Fig. 6a). Using sex-specific annuli also leads to a 6-annuli model (ΔAIC = 16.7; adj. r² = 0.42; Fig. 6b) in which female effects predominate, altering from positive at fine scales to negative at coarser ones (in agreement with the findings for single-scale models above, in spite of large standard errors). When the best-fitting single radius (67 m) was decomposed into 2, 3, 4, 5 and 6 annuli, the fits improved further. The best combined-sex model uses just two annuli (ΔAIC = 21.7; adj. r² = 0.41; Fig. 6c), whereas the best separate-sex model uses three annuli (ΔAIC = 25.6; adj. r² = 0.43; Fig. 6d), with effects qualitatively similar to those reported for the 150-m models.

Figure 6. Scale- and sex- specific coefficients for effects of flower density at multiple scales on seed mass per plant in year 2, using the divisions of scale that give the lowest AIC values. The goodness-of-fit increases in the order (b) < (a) < (c) < (d). In (a) and (c) the blocks represent both sexes combined, whereas in (b) and (d), coefficients for male flowers are shown by shaded blocks, female effects by unshaded blocks. Vertical lines indicate ±1 SE. Note that there were 18 times as many male flowers as female in the whole field; this may help explain the smaller coefficients for male flowers.

density effects: natural population

For the natural population, the best covariate model uses only the square of distance from the roadside (soil and vegetation data were not available). The effect of varying scales in the density models is dramatic, and the best fits were obtained towards the coarsest scales we tested (Fig. 7). The best scale was estimated to be 19 m, giving a positive coefficient for female flowers and a negative coefficient for male flowers (ΔAIC = 22; adj. r² = 0.42), but there were a number of similar peaks in goodness-of-fit in the range 19–90 m. Below 20 m the male coefficient becomes marginally positive and below 2 m the female coefficient is negative; the best-fitting model in this region is for 0.7 m (ΔAIC = 0.8; adj. r² = 0.06). The nearest-neighbour models did not significantly improve on the density-independent model (data not shown).

Figure 7. Effects of flower density on seed production at varying spatial scales for the natural population. The dots show partial r² for the density terms while the shading indicates estimated effects of male (darker) and female (paler) flower densities (error envelope: ±1 SE).

autocorrelation analysis

Positive spatial autocorrelation in some of the response variables was reduced in the covariate models and further reduced in the models with biomass and density variables, with the semi-variogram graphs becoming flatter (Figs 8 and 9). Spatial independence of the residuals for the density models is suggested by Mantel tests (P > 0.1) in all but two cases. The model for seed herbivory in year 1 has autocorrelated residuals (Mantel's R = 0.19, P < 0.005) and the semi-variogram shows fluctuations (Fig. 8l). However, positive data are sparse here; the model is also similar to the model for year-2 herbivory (which shows some autocorrelation in its semi-variogram, Fig. 8u, but passed the Mantel test). There is also positive autocorrelation for the survival model (Mantel's R = 0.025, P < 0.05). Here, directional semi-variograms were calculated (Fig. 8c), showing autocorrelation to be positive within a range of 2 m along the longitudinal axis of the field and negative on the transverse axis. This could be due to the behaviour of the sheep that grazed the field over winter.

Figure 8. Semi-variograms of raw data (first column), residuals from covariate models (second column) and residuals from the best single-scale density models (third column) for survival from year 1 to year 2 (a–c); year-1 total mass of seeds per plant (d–f), number of seeds per capsule (g–i) and incidence of seed herbivory (j–l); and year-2 total mass of seeds per plant (m–o), number of seeds per capsule (p–r) and incidence of seed herbivory (s–u). In (c) and (u), the solid circles show directional semi-variograms for the longer axis of the field and the open circles those for the shorter axis.

Figure 9. Semi-variograms for the analysis of data from the natural population. (a) Raw data; (b) residuals from the covariate model; (c) residuals from the density model at the 13.6-m scale.

For the natural population, the semi-variogram plot shows clear positive autocorrelation at the finest scales (Fig. 9 c). The Mantel test is insensitive to this initial sharp increase in semi-variance, giving P > 0.1 for these residuals (and the Mantel statistic is actually negative), but we use these analyses tentatively. The local autocorrelation is probably due to the absence of environmental covariates in our models.

Discussion

Numerous studies (e.g. Kunin 1997; Mustajarvi et al. 2001; Elzinga et al. 2005) have used the size and internal density of population patches as two distinct predictors for aspects of plant fitness, but there may be uncertainty over whether the variously-sized patches function independently. Here we use individual-focused, multiple-scale modelling in a continuous population for an exploration of spatial density-dependence. This allows us to explore how spatial patterns may both cause and be caused by endogenous processes, acknowledging the inter-dependence of individuals. We discuss the single-scale models first, then multiple-scale models. After some methodological considerations, we attempt to integrate our findings.

density effects

The predominantly negative coarse-scale effects of flower density on seed production in the field experiment suggest competition for pollinator services (Otway et al. 2005). The marginally positive effects of male flowers in some cases may reveal a degree of pollen-limitation, but this facilitation is largely outweighed by the competitive effect (cf. Sih & Baltus 1987). In addition, the fact that reproductive output in year 1 was related to flower density more strongly than to total plant density suggests that density effects on total seed mass at the scales investigated here are largely due to density-related pollination.

Previous studies on flowers of separate sexes have reported positive effects of males (House 1993), or this together with negative effects of females (Aizen 1997) on female performance. However, many studies of hermaphrodite species report positive effects of density on both pollinator visitation rates and seed production (but see Spigler & Chang 2008). One of the earliest botanical studies to measure scale-specific density (Thomson 1981) found that pollinator visitation rates increased with flower density when measured at scales around 500–1000 m² (equivalent to radii of 13–18 m) (see also Johnson et al. 2003), and positive effects of plant density on reproduction have since been demonstrated in both natural populations (e.g. Kunin 1992; Moeller 2004; Wagenius 2006) and experimental ones (Kunin 1993, 1997; Aizen 1997). Such findings are generally attributed to the attraction of pollinators to larger floral displays, and increased pollen transfer per visit at high conspecific densities. However, the identity of the pollinators is important (Sih & Baltus 1987); our findings could reflect the nocturnal habit of the main pollinators (RMG observed almost no diurnal pollinator activity during the fieldwork). It is notable that we did not detect any decline in reproduction with decreasing flower density (classic Allee effects), even at the coarsest scales. Probably none of the experimental plants were sufficiently isolated to be noticeably pollen-limited, in view of the mobility of the large moths that may pollinate them.

Another study on seed herbivory in S. latifolia found that plants in larger experimental patches suffered less attack by Hadena bicruris than those in small patches, whereas fine-scale density had no effect (Elzinga et al. 2005). The patch sizes used in that study were 12 vs. 3-m squares – which might be interpreted as two levels of density at a coarse, 12-m scale. Our results, by contrast, suggest disproportional aggregation by Hadena towards high densities of flowers, as we predicted, but at fine scales.

There are two main incongruities in our results. First, between the two years, the density effects in pollination peak at different scales. This may reflect the different vegetational backgrounds in which flowering occurred; perhaps the long grass that obscured the plants in 2004 altered the community composition or foraging range of pollinators. The first-year results also have poorer statistical support, probably because of the low densities of flowers. Secondly, the effects of scale on the coefficients for reproductive success contrast between the experiment and the natural population (comparing seed mass per plant in 2005 with reproductive success ratio from the natural population). We suggest two possible explanations for the positive fine-scale effects of females in the experiment. One explanation is that they are related to the positive effects of density on seed herbivory at similar scales. Hadena moths are a significant pollen vector as well as seed herbivore for S. latifolia flowers (Westerbergh 2004), so if moths tended to seek out denser patches of female plants in the experiment, they may also have increased their pollination. Alternatively, the positive fine-scale effect of female flowers may be an artefact of the positive correlation between a focal plant's seed production and the number of flowers that it bore itself, because flowers on focal plants could not be reliably distinguished from other flowers in the surveys of the experiment. The fine-scale negative coefficients for male plants may then reflect resource competition among neighbours, and the lesser correlation between male and female flower densities in the natural population (where focal plants’ flowers also were not confounded with their neighbours’) perhaps allows more accurate resolution of their differential effects (see Methodology, below). There was also a problem of unrealistic leverage for some data points in the natural population regression models for fine scales, where the distributions of densities were positively skewed.

multiple scales

Having defined scale-specific densities by calculating numbers of plants or flowers within circles of a range of radii, we were able to decompose these circles into annular rings to explore the simultaneous effects of densities at a range of distances. This allowed us to simulate a continuously-varying ‘interaction kernel’ (i.e. a model of the additive effects of neighbours on focal plants with respect to their proximity). Some recent studies have modelled competition in this way (e.g. Schneider et al. 2006), but we believe this is the first attempt to elucidate a more general interaction kernel from field data (compare Hubbell et al. 2001; Stoll & Newbery 2005). Of course, we still make some assumptions. In particular, we chose to use a logarithmic division of scale-space, and we used the AIC values of the basic radial density models to suggest the maximum distance up to which neighbour effects should be sought.

Our interaction kernels show density effects mostly decaying with distance and show some similarity to the graphs of single-scale model coefficients against spatial scale (compare Figs 5b and 6d). Another study that compared the importance of two spatial scales for reproductive success found negative effects of plant density within 1 m and positive effects between 1 and 4 m (Spigler & Chang 2008); here the local negative effects were attributed to resource competition. However, observations of bumblebees on foxglove spikes showed that larger inflorescences (greater ‘fine-scale density’) reduced visitation rates per flower, yet increased the per-plant visitation rate, which increased further in patches of higher plant density (‘coarse-scale density’) (Grindeland et al. 2005). This, together with our results, would support the hypothesis that pollinators are attracted to denser floral displays at coarse scales but do not follow an ideal free distribution with respect to fine-scale floral density – possibly because they do not keep track of which flowers they have already visited (Ohashi & Yahara 2002).

methodology

Our two-stage modelling procedure was successful at removing spatial autocorrelation in most of the cases. We noticed that the final stage of refining the best scales to ±5% gave a pronounced improvement in flattening the semi-variograms. One hindrance to our methodology could be direct, mutually-reinforcing interactions among plants. For example, whereas size-symmetric competition simply reduces the growth of competing plants in relation to their density (possibly producing positive autocorrelation), size-asymmetric competition is an antagonistic interaction that tends to produce negative autocorrelation between pairs of neighbours (Weiner 1985). This latter effect will not be accounted for by density alone; if prevalent, such interactions may call for an autoregressive model (Dormann et al. 2007). However, the unexplained autocorrelation in our models (e.g. for survival) was mostly positive, and it is harder to think of positive mutually-reinforcing interactions that would couple survival or reproductive rates among neighbouring plants. Unmeasured environmental variables such as grazing pressure are perhaps more plausible. Overall, we believe that our attempt to account for exogenous autocorrelation before looking for density effects is conservative and an improvement over previous approaches in the literature (e.g. Hubbell et al. 2001; Stoll & Newbery 2005).

Another consideration for all studies similar to ours is that results are conditional on the patterns of densities observed. For example, a decline in coefficients of variation in densities with increasing scale gave finer scales more potential to explain variation, other factors being equal. Also, collinearity among variables reduces the precision of multiple regression coefficients (Quinn & Keough 2002), so the tight correlations between male and female flowers at scales above 4 m and the increased correlations among scale-specific plant densities following over-winter mortality in the field experiment made differential effects harder to detect. Indeed, a negative local association between the sexes has previously been reported for natural populations of this species (Iglesias & Bell 1989). A future experiment might aim to uncouple the distributions of the sexes as well as scale-specific densities overall.

Finally, it is interesting to compare the results for survival reported here with those we obtained from the same raw data by classifying plants into density treatments determined by the numbers of plants in pre-defined squares of various sizes (Gunton & Kunin 2007). Whereas here a 0.28-m radius was the best scale for explaining survival, our previous analysis yielded 0.8-m squares as the best scale, followed by 0.16-m squares. A circle of 0.28 m radius has the same area as a 0.5-m square, so there is broad agreement between our two methodologies.

synthesis and outlook

The analyses reported above relate the major fitness components of survival and reproduction to the densities of conspecific neighbours in certain S. latifolia populations. Amalgamating these to yield overall fitness functions suggests that higher densities of females, especially at scales around 5 m and above, can be detrimental to female fitness, through both reduced seed production and increased seed herbivory. Although we did not find a clear relationship between overall plant density and total seed production, such density-dependence is implied by the joint effects of plant size on seed production (positive) and of density on survival and growth rate. Survival increased with density within 0.28 m for larger plants (note the analogy between plant size and fine-scale density) and we previously found that growth rates decreased with increasing very fine-scale density (Gunton & Kunin 2007). Thus we might expect small but sparse clumps of S. latifolia to develop over time, so that, in loose terms, the spatial density-dependence we have observed could lead to the kind of gregariousness described by Baker (1947). Such patterns will also be heavily influenced by seed dispersal – which is predominantly within 2 m (R.M. Gunton, personal observation) and thus would contribute to fine-scale patchiness. There are also negative effects of fine-scale density (e.g. at 10 cm) on seedling survival when plants recruit at high enough densities (R.M. Gunton, unpublished data), and density-related seed germination could also regulate fine-scale densities in natural populations (Linhart 1976; Murray 1998). Future work should investigate the relative importance of density-dependence and pre-existing environmental heterogeneity in determining the spatial patterns of natural populations.

The analysis described here attempts to separate density effects from environmental factors and to compare the importance of different scales at which density may be measured in an experimental population. It also shows how findings may be sensitive to variations in methodology. We believe that this reveals challenges for future investigations, both where extensive manipulations are performed and also where experimental control is limited. Explaining the spatial patterns of natural populations with reference to density-dependence acting at multiple scales is an alluring challenge, and continuing conceptual and methodological advances should allow us to do so with greater confidence.

Acknowledgements

We thank Laura Harrison, James Rosindell and Doreen Gabriel for helpful discussion and methodological suggestions. Field assistance was provided by Estelle le Guerre and Bethany Hawkins. The work was funded by the Natural Environment Research Council.