Population dynamics of Vulpia ciliata: regional, patch and local dynamics

Authors

  • Andrew R. Watkinson,

    1. Schools of Environmental and Biological Sciences, University of East Anglia, Norwich NR4 7TJ, UK; and *Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
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  • Robert P. Freckleton,

    1. Schools of Environmental and Biological Sciences, University of East Anglia, Norwich NR4 7TJ, UK; and *Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
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  • Lisa Forrester

    1. Schools of Environmental and Biological Sciences, University of East Anglia, Norwich NR4 7TJ, UK; and *Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK
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R.P. Freckleton (tel. +44 1865 271272; fax +44 1865 271249; e-mail r.freckleton@uea.ac.uk).

Summary

1 Data on the population dynamics of the annual grass Vulpia ciliata were collected at three levels, from the scale of the regional population down to small (10 × 10 cm) patches. We use these data to explore the degree to which fine scale processes influence large scale patterns of abundance.

2 Populations were characterized by their persistence, despite their small size. The mean half-life of populations was estimated to be around 45 years. Most populations are small (a few m2) in area, with only a few as large as a hectare in size.

3 Population regulation occurs as a consequence of density-dependent seedling recruitment. This reduces population growth by up to 87%. The nature of this density dependence appeared to be essentially the same across sites and years.

4 Interactions with perennial vegetation also significantly affected population dynamics, through reducing seedling recruitment and survival, and on average depressed population growth by a further 30% at one site and by up to 96% in another population.

5 Plants were aggregated and densities were positively spatially autocorrelated. This tends to buffer patches against extinction. Mean seed production per plant, was also significantly spatially autocorrelated; however, the strength of this was minor.

6 Data on small-scale extinction showed that disturbance is an important determinant of the distribution of numbers of plants within subplots. Comparison of the distribution of subplot densities with the results of a spatial simulation model suggested that disturbance at a relatively large scale (at least 20 × 20 cm) impacts on dynamics at the population scale.

7 An integro-difference equation model for patch expansion shows that populations are constrained to an area no larger than around 100 m2 on a time-scale relevant to the dynamics of this species (about 20 years).

8 We conclude that the most characteristic features of dynamics at the regional scale, namely the persistence and very small spatial size of individual populations, can be readily explained by processes operating at small spatial scales.

Introduction

Over the last two decades, a number of studies have explored how density-dependent and density-independent factors operate together at the local scale to determine plant population sizes (Symonides 1988; Watkinson 1996). In particular, the application of difference equation models (Watkinson 1980), in combination with data on survival, fecundity and dispersal, has allowed detailed analyses of how population size is determined in a range of annual plants, including Cakile edentula (Watkinson 1985), Erophila verna (Symonides et al. 1986), Salicornia europaea (Watkinson & Davy 1985), Sorghum intrans (Watkinson et al. 1989), Vulpia fasciculata (Watkinson & Davy 1985; Watkinson 1990), communities of annuals (Rees et al. 1996; Freckleton et al. 2000), as well as in a range of arable weeds (Cousens & Mortimer 1995).

Ecological studies have typically been restricted in terms of the spatial and temporal scales at which populations are monitored (May 1989) and, whilst numerous studies have studied the detailed demography of plant populations, this has generally been at a single site or over a short time period (Silvertown & Lovett Doust 1993; Cousens & Mortimer 1995). Consequently, the data collected have too often been insufficient to show how key demographic parameters vary between sites or years (Mack & Pyke 1983; Schmidt & Levin 1985), and hence how such variation determines patterns of occurrence. Nevertheless, simple population models have been successful in explaining varying patterns of abundance in Cakile edentula across a dune ridge (Watkinson 1985), in Salicornia europaea on high and low salt marshes (Watkinson & Davy 1985) and in Vulpia fasciculata on different dune systems (Watkinson & Harper 1978; Watkinson 1990).

A number of studies have explored population dynamics at a much more local level. Symonides (1983), for example, has described how the pattern of abundance of Erophila verna is related to the cover of perennial grasses and how the spatial and temporal flux of individuals may vary over very short distances. In multi-species systems, a number of studies have shown how spatial segregation within populations may be important in determining the outcome of their interactions with other species (e.g. Law et al. 1993; Rees et al. 1996; Pacala & Levin 1997), as well as the impacts of spatially discrete disturbances (Wu & Levin 1994). Data on the effects of temporal variability are less common and it is hard to rank the relative importance of spatial and temporal variability for most plant populations. Temporal variability is argued, for example, to be a key driver of population and community dynamics in desert annuals (Venable & Lawlor 1980; Venable et al. 1993), as well as for some, but not all, arable weeds (Freckleton & Watkinson 1998a,b; but see Lintell Smith et al. 1999).

Demographic analyses and models of plant populations have typically studied dynamics at the level of the individual population in both annual and perennial species (Alvarez-Buylla 1994; Silva Matos et al. 1999). Each population is, of course, part of a larger system of local populations which is in turn part of the total population of a species (Hanski & Simberloff 1997). To understand the population dynamics of a species in the widest sense entails understanding and linking its dynamics at all these different levels. The overall distribution of a species is then dependent upon the establishment, persistence and extinction of its constituent sub-populations (e.g. van der Meijden et al. 1992). Viewed in this way, it is clear that questions relating to the determinants of the dynamics of plants within a quadrat and the range and abundance of a species may be closely related. There is a hierarchy of population structures and it is often only scale that separates them.

The principle objective of this paper is to analyse the population dynamics of the winter annual grass Vulpia ciliata at a range of scales in an attempt to move away from the simple monitoring of populations at a single site to an integrated study of the temporal and spatial dynamics of populations from the local to regional scale. Previous analyses of the demography of this species were restricted to a single year and did not integrate information on the seed phase of the life cycle (Carey & Watkinson 1993) with that of the vegetative phase (Carey et al. 1995). Moreover, previous studies have provided no clear evidence for how the populations are regulated and how the dynamics of the perennial vegetation impacts upon the annual. In particular we intend to relate the dynamics of populations at the regional level to those at the patch and the more local level within a patch.

Methods

The species and study area

Vulpia ciliata Dumort. belongs to the Mediterranean-Atlantic element of the British flora (Preston & Hill 1997). It is widespread throughout southern and central Europe but subspecies V. ciliata ssp. ambigua (Le Gall) Stace & Auquier is restricted to England, Wales, the Channel Islands and a few locations on the neighbouring coasts of continental Europe (Watkinson et al. 1998). Within England, the distribution of V. ciliata is centred on the sandy heaths and warrens of west Norfolk and Suffolk. Although formally regarded as ‘scarce’ in a national context (Stewart et al. 1994), the plant may be locally abundant on roadsides and tracks in areas of disturbed sandy soil with a low cover of perennial grasses (Watkinson et al. 1998).

Distribution and abundance at the regional level

Between 1990 and 1996, all sites in Norfolk and Suffolk at which V. ciliata had been previously recorded were visited at least once, together with known extant populations and other possible locations in East Anglia. Our aim was to locate as many populations as possible. By checking previous records, where possible the first and last records for each population were recorded in order to determine how long populations persist. An estimate was made of the area of all extant populations by pacing across the population.

Population dynamics across the region

Five permanent quadrats (50 × 50 cm) were marked out at each of five sites (see Fig. 1a) in spring 1993: Holme next the Sea (National Grid reference TF 696439), Snettisham (TF 647325), Sandringham (TF 674274), Santon (TL 814884) and Mildenhall (TL 728758). These five sites spanned the north–south range of V. ciliata within East Anglia and included a range of habitat types: respectively, the rough grassland of a golf course at the edge of a sand dune system, maritime shingle, a roadside verge, the edge of a forest track on the border between a pine plantation and an area of heathland, and the sandy embankment of a river channel.

Figure 1.

Regional dynamics and distribution of V. ciliata. (a) The distribution of populations across East Anglia and the locations of all the sites visited. Detailed analyses were carried out at: 1, Mildenhall; 2, Santon; 3, Sandringham; 4, Snettisham; 5, Holme next the Sea. (b) The long-term probability of survival of these populations derived from data on population extinctions (ln ps = − 0.022 (± SE = 0.0001) t; d.f. = 5; P < 0.01). (c) The frequency distribution of the logarithm of the area covered by populations compared with a normal distribution.

The number of flowering plants was counted in each quadrat and fecundity estimated from the number of glumes per infructescence; there is typically one seed for each pair of glumes (Carey et al. 1995). The numbers of seedlings were counted in December; monitoring was then continued on the same basis until spring 1995. As the quadrats at Santon were destroyed by forestry operations in autumn 1993, five new plots were established in approximately the same position in June 1994. A new set of plots was also set up in an adjacent area.

Data on the dynamics of seeds at each of the five sites were obtained by sowing seeds from each of the respective sites into marked locations at the time of seed set in 1993 and then harvesting replicate soil samples at five times throughout the year to determine the fates of seeds using the technique of Carey & Watkinson (1993). On each sampling occasion the number of seeds that had germinated was counted and the seedlings removed. The number of seeds remaining on the surface of the soil was then counted, the soil sieved and the number of buried seeds counted. All of the seeds were examined under a microscope for signs of germination. Those seeds that had not germinated were placed in Petri dishes, kept moist and periodically checked. Any seeds germinating were noted and removed whilst those that did not germinate were tested for viability using tetrazolium chloride, allowing the seeds to be classified in a state of either enforced or innate dormancy.

Population dynamics at the local level

Monitoring of the spatial and temporal dynamics of population flux at the local level was carried out at two scales. At Sandringham, a major distinct section of the population was marked out in a grid of 30 × 5 m. In the central 20 × 20 cm of each 1-m2 grid cell, the number of seedlings was recorded in the winter (January) of each year, and the number of flowering plants in June; estimates were also made of the percentage cover of perennial vegetation through point quadrats. Plants were first counted in June 1993 and finally in June 1995.

At a finer scale, four 2 × 1 m permanent quadrats were set up at Mildenhall in June 1993; each plot was divided into a grid to create two hundred 10 × 10 cm subplots. Counts of the number of flowering plants within each subplot were made in June of each year; the numbers of seedlings were counted in December. Fecundity was estimated using the method described above. A point quadrat was used within each subplot to measure the percentage cover of vegetation.

Dispersal

Data on the dispersal of seeds from 100 individual infructescences at Mildenhall (see Carey & Watkinson 1993) were re-analysed. The dispersal of seeds from infructescences had been monitored by spraying them with aerosol paint and recording the distance moved by seed (data presented in Carey & Watkinson 1993). The distribution of distances moved by seeds from individual plants was quantified by fitting gamma distributions to the data using maximum likelihood parameter estimates. In addition, we fitted the gamma distribution to data taken from plants in which seed dispersal was increased by simulated disturbance (see Carey & Watkinson 1993).

Analytical methods

The analytical methods employed largely follow those developed by Pacala & Silander (1990) and Rees et al. (1996). The statistical background for the techniques employed are discussed at length by these authors.

Spatial analysis

The strength of spatial autocorrelations in the density and performance of plants in the subplots within each of the 2 × 1 m main plots (at Mildenhall) was quantified with Moran's I coefficient of spatial autocorrelation (e.g. Cliff & Ord 1981; Haining 1990) using differences in fecundity and density between a target plot and its eight contiguous neighbours. In order to assess the significance of the observed autocorrelation, the observed values of I were tested against the results of Monte-Carlo simulations in which the observed patch sizes were randomised (e.g. Haining 1990). Each subplot within a 20 × 10 grid was assigned one of the observed densities at random, and Moran's I determined for the whole grid. This procedure was repeated 1000 times to generate a mean I, as well as 95% and 99% confidence intervals for assessing statistical significance.

Density dependence

The annual changes in numbers of plants within subplots were found to be closely related to density. This density dependence was found, following exploratory analysis, to be well described by the relationship:

image(eqn 1)

Where λ is the finite rate of population increase and b characterizes the strength and nature of density dependence. At Sandringham, where we simply have counts of numbers of plants in successive years, we fitted equation 1 to the data on changes in numbers from one year to the next. Since the densities were based on direct counts of plants, and hence census error is negligible, we could fit this equation directly through regression analysis.

Since the data from Mildenhall included counts of seeds produced, in addition to counts of the numbers of plants, we were able to isolate the source of density dependence. The density-dependent population growth rate was primarily a consequence of the effect of subplot density on the proportion of seeds germinating and surviving to flowering. Following exploration of a range of models, it was found that this could be related to the density of flowering plants by the equation:

image(eqn 2)

where N is the number of flowering plants within a 10 × 10 cm subplot, f is the mean number of seeds produced per plant, pm is the density-independent probability of germination and survival to flowering, and b is a density-dependent feedback parameter. The local population growth rate is here divided by fecundity to take account of variations in fecundity between plots. Again, because the densities are counts of plants and the census error in the estimates of fecundity is therefore negligible, the effect of density on population growth could be analysed directly through regression analysis.

The imapcts of perennial cover on population growth

Analysis of the impact of perennial cover on the growth rate of populations censused at the seedling and flowering stages indicated that the negative effects of this cover occurred not through resource competition, but rather through seed survival through to germination, and that this could be described by an exponential function of the level of perennial cover. Equations 1 and 2 were modified, respectively, to include this effect in the following manner:

image(eqn 3)

for the data on rates of local population change from Sandringham, and for the data from Mildenhall:

image(eqn 4)

where α is a constant and ct is the percentage cover of perennial vegetation. Note that regression derived interspecific competition coefficients of the sort estimated in equations 3 and 4 are not expected to be biased by the spatial distributions of the interacting species (Freckleton & Watkinson 2000). 

Equations 3 and 4 were fit assuming an error distribution drawn from a continuous member of the exponential family (i.e. an exponential or gamma distribution). A Levenberg-Marquardt procedure was used to generate the best fit parameter estimates and asymptotic standard errors estimated from the variance-covariance matrix (e.g. Dobson 1990). We used the fitted models to estimate two summary measures of the effects of competition, the effective density dependence (β^) and the net effect of cover (φ^) which are the average proportions that local population growth is reduced by the effects of density dependence (i.e. 1 − (1 + N)b) and cover (i.e. 1 − eαc), respectively.

Local extinction

Localized extinction of a number of 10 × 10 cm subplots within the 2 × 1 m plots at Mildenhall occurred each year. We related this local subplot extinction to subplot density using logistic regression. The density-dependent probability of subplot survival (ps(N)) was related to subplot density (N) through two parameters, γ(0) and γ by the equation:

image(eqn 5)

The parameters γ(0) and γ measure, respectively, the asymptotic probability of extinction as densities become low, and the rate at which the probability of subplot extinction changes with increasing density. Equation 4 was fit using a Rosenbrock pattern search (Rosenbrock 1960) to generate the maximum likelihood estimates of γ(0) and γ; log-likelihood ratio tests were used to assess the statistical significance of the effects of density (Statistica 1990). We additionally used the fitted parameters to calculate γ(1), the probability that a subplot containing a single individual would survive to be occupied in the next year. We compared this estimate of γ(1) with the probability estimated by assuming that all subplot extinction resulted from demographic stochasticity, specifically to explore the hypothesis that immigration of seed into subplots from neighbours plays a role in determining extinction probabilities at this level. In this case, a Poisson distribution was used to generate an estimate of γ(1) for each subplot (note that using a binomial rather than Poisson distribution had little effect on these estimates of γ(1)). This was done by first using the best fit parameters of equation 4 to estimate Nt+1/Nt (= Nt+1 since Nt = 1) and then calculating the zero term of the Poission distribution, i.e.

inline image

Dynamics of perennial vegetation

Changes in the cover of perennial vegetation from one year to the next in subplots within the 2 × 1 m plots at Mildenhall were examined by relating the cover at time t + 1, ct+1 to that at time t, ct, in terms of deviation from the mean cover value (Hastings & Sugihara 1993). The change in cover was described by the equation

inline image

where ε is a random noise term with zero mean. The value of r was estimated through linear regression. The variance of ε was approximated by the residual variance from the regression, although it should be noted that this will slightly over-estimate the true value as measurement error in c cannot be accounted for.

Results

The size and persistence of populations across the region

Forty-six sites with extant or extinct populations were visited; the earliest records (vice-county recorder, West Norfolk) were from the 1940s. The probability of survival of populations ps(t) over time t could be described (linear regression of log proportion surviving on time constrained through the origin) by the equation:

inline image

in which a value of µ = 0.022 implies a half-life for populations of around 45 years (Fig. 1b). Only two of the previously recorded populations went extinct during the study period; one was extremely small and encroached upon by perennial vegetation, while the second larger population was destroyed by agricultural activities. The distribution of population sizes was approximately lognormal (Fig. 1c), implying a large number of small populations and only a few large ones. The largest population extended in a 10-m wide strip intermittently for c. 3 km along an exposed, disturbed shingle beach at Snettisham. More usually, populations ranged in size from one to a few thousand m2, with the area of most populations being c. 100 m2.

Population dynamics across the region

Between 8% and 30% of the initially sown seeds were not viable and a further proportion (from just a few percent to over 40%) were in a state of innate dormancy depending upon the population; the majority of seeds were in a state of enforced dormancy at the time of sowing (Fig. 2). Germination occurred over late summer and autumn. Loss of individuals was generally greatest during the autumn and early winter, resulting from either the death of seeds or young seedlings. Over the whole vegetative stage, from 5% to 30% of the seeds survived to flowering. Broadly the same patterns of seed dynamics were observed in each of the five populations that were studied (Fig. 2a–e), with the proportions dying during the period of study ranging from 60% to 80% and the proportion successfully germinating ranging from 15% to 30%. There was little evidence of a persistent pool of seed surviving for more than a year in the soil. This notion would appear to be backed up by the results of the sowing experiment of Carey & Watkinson (1993), who found that all seeds experimentally sown in late June had germinated by the end of March in the following year.

Figure 2.

The fates of seeds in the field at five sites (a–e) over an 11-month period, together with the combined data (f). See text for details of how the fates of seeds were determined and classified.

The pattern of population flux was generally similar across the five sites, although there were significant variations in detail (Fig. 3). The net rate of population change varied from almost 0 to 1.6 in 1993/94. The population crash on the verge of the forest track at Santon resulted from forestry operations: for the remaining sites the highest rate of population change was at the most northern site (Sandringham), while that at the other three sites varied from 0.7 to 1.2. For all sites except Santon (where conditions had changed dramatically as a result of tree felling), the net rate of population change in 1994/95 was < 1. This was owing to drought in spring 1995, resulting in higher mortality of plants during the vegetative stage than in the previous year. Variation in population size between sites was greater after the dry spring than at any other time. The net rate of population change across the five sites and 2 years was positively correlated with plant fecundity (r = 0.76, n = 10, P < 0.05).

Figure 3.

The flux of numbers of individuals within the five populations studied over a 2-year period.

Population dynamics at the patch level

The analysis of the dynamics of the population in the 30 × 5 m patch at Sandringham indicated that there was a strong temporal correlation in the density of plants (r = 0.64, n = 150, P < 0.001); at this broad scale, areas of high density tend to remain high, while areas of low density tend to remain low. The population structure was highly aggregated, with high values of skewness (2.35–3.59) and kurtosis (5.71–16.00), showing that the distribution of subplot sizes was extremely leptokurtically skewed (Table 1a). The per capita rate of population change was density dependent (Fig. 4a) in both years, although the strength of this and the relative effects of density dependence and perennial cover varied between the years. The best fit parameters of equation 1 are summarized in Table 1b. The effective density dependence, as measured by β^, was very low in 1993 and population dynamics were dominated by the effects of perennial cover. Conversely, in 1994, density dependence was much stronger and the effects of cover were weaker.

Table 1.   Summary and analysis of medium scale data from Sandringham. (a) Summary of mean and distribution of densities in 20 × 20 cm subquadrats within the central m2 of each 1 × 1 m quadrat within a 30 × 5 m quadrat. The mean, maximum density (Nmax), coefficient of variation (CV), skewness and kurtosis are presented, as well as the sample size (n). Figures given are for all 150 plots, with values for only those subquadrats containing plants (i.e. empty subquadrats are excluded) given in parentheses. (b) Best fit parameters of equation 3 describing the effects of density dependence and cover on population dynamics. λ′ is the theoretical maximum mean finite rate of increase estimated in the absence of an effect of perennial vegetation; λ is the value estimated after the effects of perennial cover are incorporated, i.e. the realized finite rate of population increase. α and b are, respectively, the per capita effects of perennial cover on population growth and the density-dependent parameter of equation 3. β● and φ● measure the proportions by which population growth is reduced by the effects of density dependence and perennial cover, respectively. Note that the numbers of plants were transformed to density per 10 × 10 cm prior to the fitting of equation 3 in order to allow comparison with the data from Mildenhall (Table 2; Fig. 6a, c & d). *  P < 0.05, **  P < 0.01, ***  P < 0.001 (a)
 MeanNmaxCVSkewnessKurtosisn
Jul 199329.5(59.0)289(289)1.89(1.14)2.38(1.44)5.71(1.64)150(75)
Jan 199424.7(44.7)338(338)2.05(1.37)2.39(2.52)13.84(7.36)150(83)
Jun 199415.8(33.9)257(257)2.28(1.38)3.59(2.38)16.00(6.94)150(70)
Jan 199532.6(56.2)295(295)1.79(1.2.0)2.35(1.62)5.11(1.81)150(87)
Jun 19958.5(21.3)126(126)2.26(1.2.0)3.47(2.14)14.72(5.41)150(60)
Yearλ′λαbMean coverβ^φ^
199323.2***1.14.11***0.43***0.740.010.96
19943.73***2.70.470.25***0.700.630.28
Figure 4.

Density-dependent population change in numbers (equation 3) of plants at Sandringham. (a) Observed changes in numbers for both years' data combined. (b) The average best fit curve for the data in (a) (solid line; see Table 1) with the average curves fitted for each of the four blocks at Mildenhall (see Table 2). (c) The average best fit curve for the data from Sandringham (solid line; see Table 1) with the average curves fitted for each of the 3 years data for all blocks combined at Mildenhall).

Population dynamics at the local level

The data from the four 2 × 1 m patches at Mildenhall indicated that fecundity differed between the main plots (Fig. 5a) and between years (Fig. 5b) but that the scale of the variation was trivial in comparison with the variation in fecundity within the plots. Density had no effect on mean fecundity, but variance in the mean fecundity per subplot declined with density (Fig. 5c). This pattern would be expected as a consequence of a decrease in sampling variance around the mean fecundity per subplot with increasing sample size, or density. For the purposes of modelling, we included this effect as follows: for subplots containing single isolated plants, the standard deviation of the logarithmically transformed mean plant fecundity was equal to 0.94; at a density N, mean plant fecundity, measured at the scale of the subplot, can be predicted by the standard error of mean plant fecundity, i.e. sf = 0.94/√N. This was found to give a good description of the decline in variance in mean fecundity with increasing density, which is unsurprising given that there is no biological basis for the decline, such as an effect of competition on the variance in mean plant fecundity.

Figure 5.

Analysis of data on mean plant fecundity (number of seeds produced per plant) from Mildenhall. (a) Average fecundity for each of the four blocks pooled over the 3 years' data. (b) Average fecundity for each of the 3 years, pooled across the four blocks. The differences between the blocks were not statistically significant (2 way anova on log + 1 transformed data: F3,6 = 3, P > 0.05), whereas there were significant differences between years (F2,6 = 28, P < 0.01). The year–block interaction was not evaluated as both factors are only pseudo-replicated each year. Note that whilst statistically significant differences exist, these are minor in relation to the scale of variation within years or blocks. (c) Mean number of seeds per plant plotted against subplot density.

Population growth was strongly density-dependent as a consequence of the effects of density on the recruitment of seeds (Fig. 6a) and this relationship was consistent between years and with the data on density dependence from Sandringham (see Fig. 4b,c). The parameter b, which measures the impacts of density on population growth in equation 3, was statistically significantly different from zero in seven out of eight cases (Table 2). This statistically significant density dependence translated into marked effects of density on population growth: the effective density dependence showed that population growth rates were reduced by up to 87% as a consequence of density dependence (in Table 2). There were significant effects of perennial cover in six of the eight fitted models (α in Table 2; Fig. 6b) and these reduced population growth rates by an average of 30% (in Table 2).

Figure 6.

Summary of fine scale dynamics at Mildenhall (see Table 2 for results broken down by year and block). (a) Density-dependent recruitment (per seed probability of germination and survival through until flowering) averaged across all data. The data are averaged across density classes for clarity. (b) Average effect of cover. The mean residuals (ln (observed/predicted)) are averaged for 4% perennial cover intervals. (c) Average probability of 10 × 10 cm patch extinction as a function of patch density. (d) Spatial autocorrelations of densities. The mean density of the eight contiguous neighbours is plotted as a function of the density of a target patch. The dashed line is the line y = x.

Table 2.   Analysis of the fine-scale population dynamics of Vulpia ciliata. Non-linear regression analysis was used to explore population growth as a function of emergence, cover plant fecundity, density dependence and extinction at the level of small (10 × 10 cm) subplots in the four main plots (A–D) at Mildenhall in 1993 and 1994. Mean = pooled average response for each year; All = pooled average response across all plots and both years. *  P < 0.05, **  P < 0.01, ***  P < 0.001, NS = not significant
PlotMaximal emergence1Fecundity2Population growth3Sub-plot extinction4Summary statistics5
pm± SEα± SEfσ2(f)λλb± SEγ(1)γN¯c¯β^φ^
  • 1pm and α are the parameters of equation 3 which relates the germination of seed to the density of flowering plants in the previous year. Pm and α and their standard errors were estimated through a non-linear modelling procedure (see text for details).

  • 2f, the mean plant fecundity, as well as its variance σ2(f), were estimated directly from the data and analysis of the spatial and temporal variability of this parameter is presented in Fig. 5. The finite rate of population increase was estimated by multiplying the per individual fecundity (f) by pm, the maximum mean per seed probability of germination.

  • 3 λ′ is the theoretical maximum mean finite rate of increase estimated in the absence of an effect of perennial vegetation; λ is the value estimated after the effects of perennial cover are incorporated, i.e. the realized finite rate of population increase. Log-likelihood ratio tests on the change in patch sizes were used to test whether there were significant variations in the value of λ between years and within years. The statistical significance of these tests is presented as superscripts indicating the significance level of including separate values of λ for the four blocks within years followed by that for between years (within years, between years). b models the density-dependent component of population growth in equation 3 and is estimated along with pm, and α (see footnote 1) using a non-linear modelling procedure. A value significantly different from zero indicates significant density dependence.

  • 4

    γ(1) and γ measure the effects of density on patch (i.e. 10 × 10 cm plot) extinction and are derived from logistic regressions. γ(1) is the estimated probability of a patch containing just a single individual surviving for a full year with the figure in brackets being the probability of survival predicted by demographic stochasticity, i.e. when the recruitment process is modelled as a Poisson process. γ is a measure of the effect of density. A significant positive value of γ indicates that there is a significant increase in the probability of patch survival with increasing density.

  • 5N¯ and c¯ are the mean density of Vulpia and the mean proportion of ground covered by perennial vegetation, respectively. β^ and φ^ are summary statistics measuring the proportion by which the population rate of change is reduced on average owing to the effects of density dependence and perennial cover.

1993
A0.7490.1790.008 NS0.23510.014.87.57.5NS,***0.595***0.0760.91(0.99)0.22*21.50.440.870.00
B1.2430.4011.302***0.3459.823.512.26.7NS ,*0.626***0.1120.82(0.99)0.44***12.30.460.850.45
C0.4330.1250.738**0.28710.856.64.73.2**, NS0.645***0.0910.93(0.87)0.01NS10.00.500.690.31
D0.7280.1890.630*0.32615.072.210.97.3NS,***0.806***0.0820.95(0.99)0.13**9.80.640.860.33
Mean0.7880.67011.441.88.86.20.6680.900.2013.40.510.820.30
1994
A0.5080.1570.969**0.35210.230.75.23.8NS,**0.557***0.0820.84(0.93)0.09**32.40.310.740.26
B1.0070.4811.049*0.4977.717.27.74.8NS, NS0.684***0.1000.66(0.94)0.35***23.50.490.790.38
C0.8040.2160.2620.28411.130.48.97.7**, NS0.690***0.0940.79(0.99)0.65***8.00.540.870.13
D0.1220.0240.261NS0.1129.825.81.21.0***,***0.112NS0.0770.53(0.62)0.07*5.80.570.030.14
Mean0.6100.6359.726.05.74.30.5110.720.2917.40.480.610.28
All0.6990.65210.533.97.35.30.589  15.40.490.710.26

Recruitment was consistently higher in 1993–94 than 1994–95 across all plots with statistically significant differences for five out of the eight estimates of λ, when compared with the other years' pooled estimate (Table 2).

There was one major disturbance on plot D by a mole which resulted in the extinction of all plants in 25% of the 10 × 10 cm subplots. Logistic regressions showed that the probability of subplot survival was significantly related to density in seven out of eight cases (γ in Table 2). In every case, the probability of survival of subplots containing a single individual (γ(1)) was considerably higher than that predicted from demographic stochasticity (Fig. 6c). Our interpretation of this is that mortality factors acting at the level of at least the scale of the whole 10 × 10 cm subplot must play a role in determining plant mortality and subplot extinction, but that this may be offset by immigration of seed into the subplot from neighbours. We explore this hypothesis in more detail below.

Spatial analysis showed that fecundity and density of plants at Mildenhall were spatially autocorrelated, and the distribution of subplot sizes was leptokurtically skewed, as at Sandringham (Table 3). Mean plant fecundity per subplot was also found to be significantly spatially autocorrelated. The scale of this spatial autocorrelation was minor, however, and unlikely to impact significantly on dynamics. The spatial structuring of the plants is however likely to be more important. As shown in Fig. 6(d), the pattern of spatial autocorrelation of plant densities is non-linear. This means that the neighbours of a low density patch will, on average, be of a higher density. The implication is that the spatial structure of the population introduces a form of stabilizing density dependence that will buffer the population against small-scale disturbances (see below).

Table 3.  Summary of fine-scale structuring within the main plots at Mildenhall and analysis of spatial autocorrelations. For each plot the mean plot density (N●), maximum plot density (Nmax), coefficient of variation (CV), skewness and kurtosis are given. Moran's I was used to analyse spatial correlations in mean plant fecundity (If) and plant density (IN). Mean values from randomizations (I *) , as well as 95% and 99% upper confidence intervals, are given for each
PlotYearN^NmaxCVSkewnessKurtosisIfI*95%99%INI*95%99%
A199321.5840.741.191.690.0045**< 0.00010.0020.00240.0673**− 0.00010.00870.0102
199432.41050.760.910.230.0063**< 0.00010.00220.00270.0972**0.00010.00930.0106
B199312.3450.771.020.830.00440.00010.00630.00740.0784**< 0.00010.00860.011
199423.51280.970.152.310.0120**0.00010.0040.00480.1671**− 0.00020.01180.0132
C199310611.092.306.410.0044*< 0.00010.00410.00470.1254**− 0.00040.01290.0161
19948521.052.407.780.0057**< 0.00010.0040.00450.1307**− 0.00020.01290.0161
D19939.8701.172.186.780.00390.00010.0050.00590.1261**0.00020.0150.0175
19945.8690.922.5611.440.0071**< 0.00010.00340.00460.0909**0.00040.01220.0154

Dispersal

The dispersal functions of seeds from individual plants were well described by gamma distributions (Table 4). There was considerable variation between the individual dispersal functions (Fig. 7a); this variation is of lesser significance, however, when compared with the impact of disturbance (a person walking through a stand) on the dispersal function (Fig. 7b).

Table 4.  Dispersal distributions for 20 groups of seeds marked by Carey & Watkinson (1993). The columns give the maximum likelihood estimates of the parameters of the gamma distribution: inline imagethe table shows the maximum likelihood estimates of the scale parameter θ and the shape parameter α, as well as the D statistic from a Kolmogorov Smirnov test on the cumulative frequency distribution vs. a gamma function and the sample size (n) for each group. Only one group (8) showed significant deviation from the fitted gamma distribution (**  P < 0.01)
GroupScale Parameter θShape Parameter αDn
10.4172.9870.13726
20.4291.8130.14213
30.2672.2500.2187
40.8943.9880.18635
50.5072.5110.21815
60.2831.3910.18045
70.9255.4650.21619
81.0404.9970.294**40
90.2580.6100.14653
100.2510.9860.68849
110.2031.6220.15440
120.1911.7120.19747
130.7014.2380.14127
140.1641.3240.09432
150.4332.2370.20225
160.2031.7680.20431
170.3322.1950.18431
180.5923.0190.08728
190.3332.0370.10619
200.3442.3070.09656
Mean0.438 (± 0.06)2.473 (± 0.29)  
Figure 7.

Dispersal of seeds and interactions with density dependence. (a) Best fit gamma distributions for dispersal from individual plants (see Table 4). (b) Mean dispersal curves for undisturbed plants (the average from (a)) and plants subject to a directional disturbance by a person walking through the population. (c) The probability of a seed recruiting as a function of density and distance from the parent plant. (d) The number of recruits per plant as a function of distance and density.

Combining the density-dependent recruitment function (equation 4) with the average dispersal function for undisturbed seeds showed both how the probability of seed recruitment varies as a function of distance from the parent plant (Fig. 7c) and that the effective dispersal function exhibits a progressive flattening with increasing density (Fig. 7d), owing to the recruitment-limited density dependence in this population.

Dynamics of perennial cover

The dynamics of the perennial cover were described well by a simple linear model (Fig. 8). The population growth parameter r(± SE) of equation 5 was estimated as 0.613 (± 0.022; n = 1600; P < 0.0001). The standard deviation of the noise term ε was estimated as 22.4 and was approximately normally distributed (Fig. 8b), allowing us to model the transitions in perennial cover as a stochastic density-dependent process.

Figure 8.

Density-dependent transitions in the perennial cover. (a) The deviation from the mean level of cover (c. 50%) is shown for successive years. The dashed line shows the line y = x, whilst the solid line is the fitted density-dependent relationship. (b) The distribution of residual values (ε in equation 5), measured at the scale of individual subplots, about the fitted relationship. The line shows the fitted normal distribution used for modelling.

Population models

We have used population models to explore two specific aspects concerning the dynamics and structure of populations, namely how population structures respond to disturbance and the degree to which spatial extent of populations are determined by restricted dispersal of seed from individual plants. In particular, we used a spatial simulation model to look at the determinants of patch sizes within populations (medium scale) and used integro-difference equations for population expansion to look at the determinants of the spatial extent of populations (large scale). We do not explore the models in detail, but rather use these to highlight the importance of the processes of disturbance and localized dispersal for population dynamics.

Spatial simulations

The spatial simulation model was based upon the set of models fitted to the data from monitoring the fate of plants within the four 2 × 1 m main plots at Mildenhall as we have been able to derive the most detailed description of the dynamics of these plots. Within the spatial model, the population dynamics are simulated in a habitat divided into 10 × 10 cm subplots. Individuals grow within these subplots, set seeds which are then dispersed across the grid, where they germinate according to the density-dependent function and the extent of perennial cover (equation 3).

The computer model, based on a 3 × 3 m main plot divided into 10 × 10 cm subplots, assumed:

1Fecundity The arithmetic mean of fecundity was assumed to be constant at the observed arithmetic mean value of 10.54 seeds per plant, while the variance in the logarithm of fecundity was assumed to decline with increasing density as described above. As the arithmetic mean fecundity is constant, the geometric mean declines according to GM = log AM − 0.5sinline image, where sf was calculated as described above. The mean fecundity per plot was then drawn randomly from a log-normal distribution specified by this geometric mean and variance.

2Dispersal The average dispersal curve for undisturbed seeds (Table 4; Fig. 7b) was used.

3Emergence Density-dependent seed emergence was incorporated into the model (equations 2 & 4). Recruitment was simulated explicitly by means of a series of Bernoulli trials (i.e. sampling without replacement) defined by this mean rate. These were assumed to incorporate all random variations in births and deaths throughout the life-cycle.

4Effects of cover The dynamics of perennial cover in each cell were simulated using a linear density-dependent model (Fig. 8a). The variance about this mean relationship was simulated by means of random numbers drawn from a normal distribution with mean zero and variance 22.37 (Fig. 8b).

A non-spatial model obtained from the fitted data above produced the same equilibrium densities as the spatial model. These were approximately 50% higher than those found in the field. Comparison of the observed frequency distribution of population sizes within 10 × 10 cm subplots with the output from the spatial model (Fig. 9a) indicated that the proportion of low density plots was greater in the field (Fig. 9c). We used the model to explore the hypothesis, proposed above, that disturbance at the scale of the 10 × 10 cm subplot impacts significantly on dynamics and, further, that this would account for the disparity between the simulated and the observed population structures. Disturbance, modelled through removing all individuals within blocks of grid cells of size 4 × 4 or 5 × 5 representing 20% of the habitat area, however, produced frequency distributions similar to those found in the field (Fig. 9b); this measure of disturbance is similar to that experienced by plot D in 1994. The impact of disturbance could be detected in the population for 3–5 generations. Small-scale disturbances (one cell) did not have an impact on population structure because, owing to the spatial structure of the population, cells were rapidly recolonized from the higher density populations in neighbouring patches.

Figure 9.

Simulated and observed densities in 10 × 10 cm subplots. (a) The simulation model (see text for details) includes only demographic stochasticity and predicts low numbers of low density patches. (b) When large (20 × 20 cm) disturbances that affect c. 20% of the habitat are incorporated into the model, the number of low density patches increases markedly. (c) The observed pattern of distribution of patch sizes.

Integro-difference equation model

This model considers population growth from a point source into an empty habitat. For convenience it is assumed that the habitat is linear. Using the simple framework we adopt here (i.e. an exponentially tailed distribution of dispersal into a homogeneous time-invariant habitat), we may use a simple scaling to translate linear population expansion into two-dimensions; in general the growth of the radius of the population in two dimensions is 2 √π times that in a single dimension (Fisher 1937; Skellam 1951; Okubo 1980). Two functions, f the density-dependent function (equations 24) and p the dispersal function (a gamma distribution; Table 4), are required to construct models that relate n(x), the local density of individuals at a distance x from the origin, at time t + 1 to that at time t. This is achieved through an integro-difference equation model (e.g. Hardin et al. 1990; Marinissen & van den Bosch 1992; Allen et al. 1996; Kot et al. 1996). Of the F seeds produced per plant, the number arriving a distance x from the origin is given by integrating across the habitat area, z:

inline image

The density-dependent function, f (equation 3) then operates upon this neighbourhood seed density to estimate recruitment at time t + 1. We used equation 6 to calculate d¯, the mean displacement of individuals from the origin, i.e. where N is total population size

inline image

As population growth proceeds populations tend to develop a ‘rectangular’ structure, with patches at the equilibrium density separated from patches at very low densities by a relatively short distance due to a sharp drop-off in densities (Fig. 10a). Hence a reasonable estimate of patch width in either direction 2d¯. The overall patch size is then given by 4d¯.

Figure 10.

Patterns of patch expansion. (a) Example of the travelling wave pattern of population growth predicted by the integro-difference equation model. The lines represent predicted densities each generation. (b) Mean displacement (in a linear habitat) using the function fitted to the data on dispersal from the 20 individual plants (Table 4). The dashed line shows the expansion of patch size when the dispersal function for disturbed seed is included in the model.

Figure 10(a) shows an example of the pattern of population expansion from a single individual predicted using this model. Populations grow as a travelling wave with the leading edge of the population moving at a constant rate, such that population dynamics at some point z + D at time t + 1 may be related to the density at a point z at time t by the simple recursive equation:

inline imageeqn 8

where D is the distance that the wave advances from one generation to the next and n is the local density of mature plants. This rate of growth means that, in the linear dimension, populations of this species will advance in length in the order of c. 1 m on a time-scale of 20 years or so (Fig. 10b). Note that whilst small numbers of seeds may disperse up to 30 cm or so during the course of a single generation, the tail of the gamma distribution has an exponential form and hence this small number contributes little to the overall rate of population expansion. In two dimensions, this rate of patch expansion would translate into a population establishing from a single individual occupying an area of 1 m2−10 m2 on the same time-scale of 20 years or so. When the dispersal function fitted to the group of disturbed seeds is used, this area increases by an order of magnitude to c. 10 m2−100 m2. In general terms, therefore, the scale of patch sizes predicted by this model is comparable with that seen at the regional scale (Fig. 1c).

Discussion

Vulpia ciliata is a scarce annual occurring at only a small number of sites on dry, infertile soils (Watkinson et al. 1998); in 1996 only 31 populations of the plants were recorded in west Norfolk and Suffolk, the core of the species' range within Great Britain. Most populations are small, but they may persist over a long period; the probability of population extinction does not appear to have changed over the last 40 years with populations having a half-life of approximately 30 years. Unfortunately, no data are available on patch colonization and the greater number of recorded patches now than in the past may reflect recorder effort rather colonization. In the very long-term, the distribution and persistence of V. ciliata will be a function of colonization rates because, despite their longevity, populations do go extinct. We have observed new populations in abandoned sand and gravel works and the colonization of a new patch on the edge of a golf course within the species' existing range. A new population has also recently arisen, near Norwich, over 20 miles from the nearest known population, as a consequence of transport of sand during works on a by-pass and new hospital. We have no records of populations becoming extinct and then being recolonized. Hence, we are uncertain of the degree to which the ensemble constituting the regional population may be considered as a metapopulation, rather than as a ‘shifting cloud’ of largely unconnected local populations. In general, the lack of specialized long-distance dispersal in this species, as well as of evidence of local population extinction followed by recolonization, would suggest the latter structure is more likely. We suspect that this may be the case for many plant populations.

At the large scale, the picture that emerges is one in which there is relatively low turnover of populations and where the dynamics of populations tend to be similar. The pattern of population flux at five sites monitored across the range was similar although there were also variations in detail. Inevitably, the picture that emerges of the long-term population dynamics at different sites is rather more complicated than that found in the 1 year of study by Carey et al. (1995).

Population regulation and interspecific competition

At the local level, we have had considerable success in unravelling population dynamics. Our results indicate that the dynamics of V. ciliata are stable and that, in contrast to previous suggestions (Carey 1991; Carey et al. 1995), intraspecific interactions are strong. Unlike many other annual populations studied (see Watkinson 1985), population regulation is not imposed through the effects of density on plant fecundity, but through recruitment; the spatial structure of the population also introduced a form of stabilizing density dependence. Spatial and temporal variations in fecundity nevertheless play an important role in determining population flux.

The mechanism for density dependence is likely to be a consequence of competition between plants for suitable microsites for germination (as predicted by the safe-sites hypothesis, Harper et al. 1965), density-dependent seed production, pathogen attack, or, potentially, germination inhibition through chemical inhibition of ungerminated seed by germinating seed (Linhart 1976). Whilst in Anisantha sterilis it has been speculated that density-dependent recruitment results from changes in red : far red light ratios under an establishing canopy (Lintell Smith et al. 1999), this is unlikely to be the case for V. ciliata as the canopy of Vulpia is sparse at any stage of the growing season.

The mechanism by which perennial cover affects the population dynamics of V. ciliata may be similar to the effect of perennial cover on seedling establishment in V. fasciculata. By means of a seed transplant experiment, Watkinson (1990) showed that the effect of perennials on V. fasciculata occurred not through resource competition, but rather through reducing the number of suitable microsites for germination and hence the establishment of seedlings, as was the case here. However, at least at Mildenhall, the effect of perennial plants on the population dynamics of V. ciliata is secondary to the impacts of density dependence. However their effect is generally stronger at Sandringham where the levels of perennial cover are noticeably higher. Watkinson (1990) found that the effects of perennial cover on populations of Vulpia fasciculata increased markedly as the perennial cover exceeded 50%. The levels of perennial cover at Sandringham averaged 72%, whilst those at Mildenhall averaged around 50%. The impacts of cover on V. cilata are therefore considerably weaker than on V. fasciculata because the population studied by Watkinson (1990) was hypothesized to have been driven to extinction by the presence of such levels of perennial cover, whilst there is no suggestion that either population of V. ciliata was likely to become extinct. This may reflect the smaller diaspore size of V. ciliata: as a consequence of its smaller size, the embryo may more easily become lodged in a crack in the sand or another microsite, such as a moss tussock, and hence be less susceptible to reductions in the number of available microsites by perennials.

Comparing our results with those of a recent study of population dynamics within a guild of annuals in a nutrient poor environment (Rees et al. 1996), we found similar magnitudes of density-dependent regulation but that interspecific effects were rather stronger in our system. This is likely to be a consequence of the different structuring of our community compared with that of Rees et al. They analysed competition within a guild of apparently similar species in which competitive interactions may well be strong (Rees et al. 1996). Under such conditions spatial segregation resulting from strong two-sided competition can result in weak net effects of competition (e.g. Pacala & Levin 1997; Pacala 1997). This distinction between the outcome of competition at the local and population levels was later confirmed experimentally by Turnbull et al. (1999). On the other hand, competition between annual and perennial species is typically asymmetric (Crawley & May 1987; Watkinson 1990; Rees & Long 1992). In particular, whilst spatial segregation of interacting species develops when interactions between species are more symmetric, thus leading to the depression of the average net impacts of interspecific interactions (e.g. Pacala 1997; Pacala & Levin 1997), this need not be the case when communities are more asymmetrically structured.

Linking dynamics across scales

At a large scale, the most characteristic features of populations of V. ciliata are their persistence and small area. These features result directly from the local scale characteristics of population dynamics. The small areas occupied by populations result, in part, from the highly restricted nature of dispersal from individual plants. The other component contributing to the small areas occupied of populations is the relatively low (λ = 1.0–7.7) realized finite rate of population increase (Table 2). Classical invasion theory predicts that populations grow in area in proportion to the square root of the finite rate of population increase (e.g. Fisher 1937; Skellam 1951; Okubo 1980). Hence, for example, in the absence of perennial cover when the theoretical finite rate of increase may be considerably higher (λ′ = 1.0–23.2), population areas may be as much as twice as large. Under Australian conditions in the absence of drought, the finite rate of increase of the closely related V. bromoides may range from 50 to 250 (Freckleton et al. 2000). Using these figures, patch sizes would therefore be predicted to be nearly an order of magnitude larger (i.e. up to 1 ha) on the same time-scale, even when using the highly restricted dispersal function fitted to the data of V. ciliata.

The estimates of patch persistence in the face of demographic stochasticity (Table 2) showed that isolated patches containing just a single individual were highly likely to persist (probability 0.62–0.99). This highlights one of the sources of persistence of the small populations. The other important source of persistence is the buffering nature of the spatial structure within populations. Whilst disturbance at the scale of at least a subplot (10 × 10 cm) was highlighted as being an important determinant of patch extinction, this is offset by density to some extent, and by spatial autocorrelations of densities. Furthermore, even low density patches may be buffered to some extent against extinction through the non-linear form of the spatial autocorrelations of density. Within a continuous population low density patches may therefore essentially be rescued by dispersal from neighbouring patches of higher density. The persistence of the patch in the longer term is also facilitated by the fact that gaps are always maintained within the perennial vegetation; succession does not lead to perennial dominance over the time-scale of 30–50 years. This probably reflects drought-induced mortality when perennial cover is high (Roques et al., in press).

The model analysis (Figs 9 & 10) also highlights the importance of dispersal and disturbance in structuring populations. Whilst the logistic regression analysis, through identifying systematic deviation from demographic stochasticity (Table 2), showed that disturbance was an important determinant of patch persistence, the comparison of observed and simulated frequency distributions showed that disturbances at a scale larger than a single patch were required to account for the observed population structure. The models of patch expansion showed that, even given an unlimited amount of available habitat, the restricted levels of local dispersal observed in this species would limit the spatial scale of populations to the order of 100s of square metres on a time-scale relevant to their persistence. These models then show how the large scale characteristics of the populations result from their local scale characteristics.

Concluding remarks

It is encouraging that we can link the characteristics of populations at a regional scale to the dynamics of small (10 × 10 cm) subplots. On reflection, however, one of the conclusions that must be drawn from this analysis is the likely unpredictability of population dynamics at the regional scale. This is because the regional population appears to lack any definable structure or organization. Whilst we can explain and accurately predict the characteristics of populations when they are formed, their creation and extinction would appear to be contingent on factors that lie unpredictably outside the species' autecology. An increase in building activities within the region could, for example, result in an increase in sand and gravel extraction and in the transport of sand, and hence the creation of new populations; alternatively, management favouring high perennial cover and low disturbance on heathlands could destroy populations. This contingency of larger-scale dynamics on such processes highlights the danger of ‘top-down’ approaches to studying regional dynamics and occurrence. As Lawton (1999) has argued, the role of contingency at this scale makes interpretation and prediction difficult, with the consequence that whilst we can explain regional patterns from local processes we cannot safely infer the latter from the former. Studies of the sort presented here that integrate scales of dynamics are required to accurately characterize regional patterns of occurrence.

Acknowledgements

We thank the Hunstanton Golf Club, Royal Society for the Protection of Birds, Crown Estate, Forestry Commission and National Rivers Authority for permission to work at Holme next the Sea, Snettisham, Sandringham, Santon Downham and Mildenhall, respectively. Thanks also to Gillian Beckett, Peter Carey and Peter Trist for their help in locating populations. A.R.W. thanks Mark Lonsdale for providing the space and the hospitality to enable him to write the first draft of the manuscript and for stimulating discussion. This work was funded by NERC grant GR3/8553A to ARW.

Received 19 October 1999 revision accepted 25 May 2000

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