Ecology, 90(8), 2009, pp. 2129–2138
Ó 2009 by the Ecological Society of America
Safe sites, seed supply, and the recruitment function
in plant populations
RICHARD P. DUNCAN,1 JEFFREY M. DIEZ, JON J. SULLIVAN, STEVEN WANGEN,
AND
ALICE L. MILLER2
Bio-Protection Research Centre, P.O. Box 84, Lincoln University, Lincoln 7647 New Zealand
Abstract. The extent to which plant populations are seed vs. establishment limited can be
understood by quantifying the recruitment function, describing the number of seedlings that
establish as a function of the number of seeds added. Here, we derive a general equation for
the recruitment function based on a mechanistic model describing how the availability of safe
sites (sites suitable for germination and establishment) interacts with the number and
distribution of seeds added to a plot to determine the number of recruits. The parameters of
this recruitment function have a direct biological interpretation that can provide insight into
the processes limiting recruitment in plant populations.
Key words: Beverton-Holt function; establishment limitation; Hieracium lepidulum; microsite; plant
recruitment; recruitment function; safe site availability; seed-addition experiments; seed limitation; selfthinning.
INTRODUCTION
The availability of seeds and the availability of sites
suitable for seedling establishment (safe sites; Harper et
al. 1961) are key determinants of recruitment in plant
populations (Harper et al. 1965, Crawley 1990, Eriksson
and Ehrlén 1992). At one extreme, populations are
establishment limited, meaning the rate of seed supply is
sufficient to ensure that all available safe sites are occupied and population size is constrained by the number
of safe sites. At the other extreme, populations are seed
limited, meaning the rate of seed supply is low relative to
the availability of safe sites so that many safe sites are
unoccupied and population size is constrained by seed
supply.
In reality, most plant populations are probably
limited by a combination of the two processes (Eriksson
and Ehrlén 1992). This is because, rather than being a
dichotomy, the extent of seed vs. establishment limitation falls along a gradient (Manning et al. 2004, Clark et
al. 2007, Kollmann et al. 2007) that depends on the rate
of seed supply relative to the availability of safe sites. A
series of plots may all show seed limitation, in that seed
addition results in new seedling recruitment (Turnbull et
al. 2000), but the magnitude of the recruitment response
could vary markedly between plots. For a given level of
seed addition, plots with more available safe sites should
have higher recruitment and appear more strongly seed
limited. To understand the factors limiting recruitment
Manuscript received 31 July 2008; revised 11 November
2008; accepted 20 November 2008. Corresponding Editor: J.
Weiner.
1 E-mail: [email protected]
2 Present address: National Park Service, Joshua Tree
National Park, 74485 National Park Drive, Twentynine
Palms, California 92277 USA.
in plant populations, this interplay between safe-site
availability and seed-supply rate is of key interest.
One way to understand this interplay is through seedaddition experiments that allow us to quantify the
recruitment function (Poulsen et al. 2007). This function
describes the relationship between the number of seeds
that are experimentally added to plots and the resulting
number of recruits, and is typically a monotonically increasing curve that reaches an asymptote at high levels
of seed addition. Poulsen et al. (2007) show that the
form of the recruitment function can be used to infer the
relative strength of seed vs. establishment limitation, and
to assess the relative importance of density-dependent
and density-independent processes in affecting population recruitment.
Central to this approach is quantifying the recruitment
function. Poulsen et al. (2007) used the Beverton-Holt
function, which has two parameters, one that estimates
the proportion of seeds that would recruit in the absence
of density-dependent effects, and a second that estimates
the asymptotic number of recruits at high levels of seed
addition under negative density dependence. While the
parameters of the Beverton-Holt function have biological interpretations, the function was chosen because it
often fits observed recruitment data well and the mechanistic underpinnings were not discussed.
Our aim in this paper is to bridge the gap between the
underlying processes hypothesized to control recruitment in plant populations, and the functions used to
quantify these relationships. To do this, we derive a
general recruitment function based on a mechanistic
model of how safe sites and seed supply interact to
determine the number of recruits. We build on a model of
recruitment in which there are a limited number of safe
sites capable of supporting a single seedling, with seeds
competing to occupy these safe sites. The relationship
2129
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RICHARD P. DUNCAN ET AL.
between seed supply and the number of seedlings that
recruit (the recruitment function) is then determined by
the availability of safe sites and the distribution of seeds
among those sites. We show that the Beverton-Holt
function is a special case of a more general recruitment
function, and discuss how a process-based approach
applied to seed-addition experiments can reveal the
mechanisms underlying recruitment patterns and links to
other processes.
DERIVATION OF A GENERAL MECHANISTIC
RECRUITMENT FUNCTION
Brännström and Sumpter (2005) describe a model for
the distribution of individuals among sites that can be
used to derive, from first principles, classic models of
single-species population growth. Here we adapt their
site-based model to the problem of estimating recruitment functions.
Consider a plot divided into n sites of equal area over
which s seeds are distributed and where, sometime later,
we count the number of seedlings that have recruited, r.
We assume the s seeds are the only ones added; there is no
seed input from other sources or recruitment from a seed
bank. We define pk to be the expected proportion of sites
containing k seeds after the seeds have been distributed,
with the expected number of seedlings that recruit from a
site with k seeds defined by an interaction function /(k)
(Brännström and Sumpter 2005). The expected number
of seedlings that recruit in a plot is then given by
s
X
pk /ðkÞ:
r¼n
ð1Þ
k¼0
First, consider the case where s seeds are randomly
distributed over a plot in such a way that each seed is
equally likely to land in any of the n sites. For large n, the
expected proportion of sites containing k seeds will
follow a Poisson distribution having expectation given by
the average density of seeds per site, s/n:
pk ¼
ðs=nÞk es=n
:
k!
ð2Þ
The expected number of seedlings that recruit in a plot is
then given by
r ¼ nes=n
s
X
ðs=nÞk
k¼0
k!
/ðkÞ:
ð3Þ
We define a site as an area capable of supporting just one
seedling regardless of how many seeds land at a site, so
that seeds compete for safe-site occupancy. This equates
to contest competition, in which successful individuals
obtain all the resources they need for survival but
unsuccessful individuals do not (Brännström and Sumpter 2005). Under this assumption the interaction function
is defined as follows:
Ecology, Vol. 90, No. 8
/ðkÞ ¼
1
0
if k 1
otherwise:
ð4Þ
Combining Eqs. 3 and 4, the expected number of
seedlings that recruit in a plot as a function of the
number of seeds added is
r ¼ nð1 es=n Þ:
ð5Þ
If s n (a large number of seeds are added to a plot
relative to the number of sites) then es/n ! 0 and r ! n,
so that the number of recruits saturates at the number of
sites. This assumes, however, that all sites are suitable
for germination and establishment. We can relax this
assumption by introducing an additional parameter, b,
specifying the proportion of sites in a plot that are safe
sites for seedling recruitment. All seeds that land in nonsafe sites fail to recruit. The interaction function is then
b if k 1
/ðkÞ ¼
ð6Þ
0 otherwise
and the expected number of seedlings that recruit in a
plot as a function of the number of seeds added is
r ¼ bnð1 es=n Þ:
ð7Þ
This is the Skellam function when applied to singlespecies population growth (Skellam 1951) and is similar
to a function derived by Geritz et al. (1984) in a related
context. The parameters of this Skellam function have a
direct biological interpretation: n is the number of sites,
b is the proportion of those sites that are safe sites
capable of supporting a single seedling, and for s n (a
large number of seeds are added to a plot relative to the
number of sites) the number of recruits saturates at the
number of safe sites, bn. If s n (few seeds are added
relative to the number of sites, so that many safe sites are
unoccupied and recruitment is strongly seed limited)
then es/n ! 1 s/n and r ! bs: the number of recruits
increases approximately in proportion to the number of
seeds added and a plot of r vs. s approaches a straight
line with slope equal to b.
The relationship between the number of recruits and
the number of seeds added can be generalized to
situations in which seeds are not randomly distributed
but are aggregated so that some sites receive a
disproportionate number of seeds. Following Brännström and Sumpter (2005) we can use a negative
binomial distribution to describe aggregation:
pk ¼
kk Cðk þ kÞ
ðs=nÞk
CðkÞ Cðk þ 1Þ ðk þ s=nÞkþk
ð8Þ
where k is a (positive) dispersion parameter with smaller
values of k indicating stronger aggregation within sites.
The Poisson distribution (Eq. 2) arises as a special case
when k ! ‘. Assuming that seeds are aggregated within
sites, so that pk is given by Eq. 8 with dispersion
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PLANT RECRUITMENT FUNCTIONS
2131
parameter k, and that only a proportion of sites are safe
sites, with the interaction function given by Eq. 6, then,
substituting these into Eq. 1, the expected number of
seedlings that recruit in a plot as a function of the
number of seeds added is as follows (Brännström and
Sumpter 2005):
"
#
kk
:
ð9Þ
r ¼ bn 1 ðk þ s=nÞk
When k ! ‘ (i.e., pk follows a Poisson distribution) Eq.
9 reduces to the Skellam function (Eq. 7). When seeds
are aggregated so that k ¼ 1, Eq. 9 reduces to the
Beverton-Holt function (Beverton and Holt 1957):
r¼
bs
:
1 þ s=n
ð10Þ
Hence, both the Skellam and Beverton-Holt functions
are special cases of the more general recruitment
function given by Eq. 9 (Fig. 1). The family of recruitment curves described by Eq. 9 all have an
asymptote at the number of safe sites, bn. They differ
in the rate at which they approach that asymptote
because with higher levels of aggregation more seeds
must be added to achieve occupancy of a given number
of safe sites.
INTERPRETING
THE
PARAMETERS
The general recruitment function has three biologically interpretable parameters: n is the number of sites, b
is the proportion of those sites that are safe sites, and k
is the degree to which seeds are aggregated across sites.
These parameters allow us to estimate the relative
importance of density-dependent and density-independent processes affecting recruitment (Poulsen et al.
2007). In the absence of density-dependent mortality,
the relationship between r (the number of seedlings
recruited) and s (the number of seeds distributed) will be
a straight line with slope equal to b: seeds either land in a
safe site with probability b and recruit, or fail to land in
a safe site and die, with the proportion of seeds lost to
density-independent mortality equal to 1 b. Density
dependence manifests itself as curvature away from a
straight line: in addition to mortality resulting from
failure to land in a safe site, there is mortality resulting
from multiple seeds occupying the same safe site from
which only one plant will recruit. For a given proportion
of safe sites and level of seed addition, the total number
of sites and the degree of aggregation (n and k) determine the degree of curvature away from the densityindependent straight-line relationship. Fewer sites or
greater aggregation mean a higher probability of more
than one seed landing in the same safe site and hence
stronger density-dependent mortality.
Eq. 9 was derived assuming the n sites in a plot were
of equal area. In these circumstances, aggregation within
sites could arise if seeds were distributed in such a way
FIG. 1. A family of recruitment curves given by the general
recruitment function (Eq. 9) with n ¼ 1000 sites, b ¼ 0.1, and
values of k, the dispersion parameter, ranging from 0.2 to ‘.
Black lines highlight the particular cases where k ! ‘ (Skellam
function; Skellam 1951) and k ¼ 1 (Beverton-Holt function;
Beverton and Holt 1957). Note that the lines for 5 and ‘ merge
to the right.
that they were spatially clumped as opposed to
randomly (or uniformly) dispersed across a plot. In
seed-addition studies the experimenter has some control
over this, and we suspect that attempts are usually made
to achieve a random (or even close to uniform) spread of
seeds across plots. While some degree of secondary
redistribution of seeds is likely, the experimenter at least
avoids a high degree of spatial clumping. Given the
model we have described, random seed distribution
would appear to provide a strong theoretical justification for using the Skellam function (Eq. 7) to model
recruitment in seed-addition experiments.
However, even if seeds are randomly (or uniformly)
distributed across a plot, variation in the size of sites
capable of supporting a single seedling will lead to
aggregation. Seeds distributed at random have a higher
probability of landing in larger sites, resulting in larger
sites having more, and smaller sites fewer, seeds than
expected under an equal-area model. Hence, when seeds
have been randomly distributed across plots, it may be
possible to relax the assumption that all sites are of
equal area and to interpret the dispersion parameter, k,
in Eq. 9 as a measure of the degree of size heterogeneity
among sites.
Safe sites capable of supporting a single seedling may
vary in size for several reasons. First, resource availability could vary within a plot and the area required to
supply sufficient resources to support a seedling may
consequently vary. Seedlings in low-resource patches
may require larger sites than those in high-resource
patches. Second, seedlings will differ in their initial size
due to factors such as genetic differences and variation
in the timing of germination, with initial differences in
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RICHARD P. DUNCAN ET AL.
plant size often exaggerated through asymmetric competition for resources (Weiner 1990). Because resource
use scales with plant size, larger plants will require larger
sites to meet their resource requirements. Heterogeneity
in site area may therefore arise through a combination
of variation in resource availability and initial plant size,
enhanced by competition for local resources.
To illustrate how the dispersion parameter in Eq. 9
can be interpreted as a measure of size heterogeneity
among sites, we simulated a seed-addition experiment,
fitted recruitment functions to the simulated data, and
explored how variability in site area within plots affected
parameter estimates. Our simulated seed-addition experiment consisted of plots each having 1000 sites, of
which a proportion of sites in each plot (set to 0.1) were
randomly chosen to be safe sites. The first ‘‘treatment’’
consisted of plots with sites that were all of equal area
(set to 10 units). We simulated randomly distributing
seeds across plots at five densities: 5, 25, 125, 625, 3125,
and 15 625 seeds per plot, with 10 replicate plots per
density. Following seed distribution, the number of
recruits in each plot was calculated as the number of safe
sites occupied by at least one seed. We fitted three
recruitment functions to these data, the Skellam
function (Eq. 7), the Beverton-Holt function (Eq. 10),
and the general recruitment function (Eq. 9), using
maximum likelihood assuming the number of recruits
per plot was distributed as negative binomial.
The protocols for the second, third, and fourth
‘‘treatments’’ were identical, except the sizes of sites
within plots were made unequal. All plots still comprised
1000 sites covering the same total area, but the degree of
between-site heterogeneity differed among treatments
(the sizes of sites were chosen arbitrarily such that the
variance in site area within plots in each of the four
treatments was 0, 20.0, 77.7, and 289.5, respectively).
With random dispersal of seeds across plots, our
prediction is that estimates of the dispersion parameter,
k, should be large in treatment 1, where there is no
variation in site area, with the general recruitment
function well approximated by the Skellam function.
The dispersion parameter should decline across treatments as the between-site area variance increases.
The results from a typical experimental run are shown
in Fig. 2: as predicted, k is large in treatment 1 and
declines across treatments. The Skellam function and
general recruitment function give a nearly identical fit
for treatment 1, as expected. The Beverton-Holt function provides a reasonable fit to treatment 3, where the
degree of between-site heterogeneity results in a maximum-likelihood estimate for k that is close to 1.
The results from single experimental runs vary
stochastically, so we ran 1000 experiments in order to
examine how, on average, the parameter estimates for
the three functions were affected by size heterogeneity
(Fig. 3). For the general recruitment function, the
dispersion parameter, k, declined across treatments as
the between-site area variance increased, although
Ecology, Vol. 90, No. 8
estimates varied substantially when the dispersion
parameter was large (treatments 1 and 2). Estimates of
b (proportion of sites that are safe) and the degree of
bias in these estimates differed by function and
treatment. The mean estimate of b for the general
recruitment function was close to the true value (0.1),
with no evident bias across treatments. In contrast,
estimates of b derived from the Skellam and Beverton
Holt functions were biased away from the true value
when the level of between-site heterogeneity was
substantially different from that assumed in the function. The trends in bias were consistent with expectations given the assumptions underlying each function.
The Beverton-Holt function, for example, was biased
towards higher estimates of b when sites were similar in
size (treatments 1 and 2) because here the Beverton-Holt
assumes greater clustering than exists. The Skellam
function, in contrast, was biased towards lower estimates of b in sites with more highly varying sizes
(treatments 3 and 4) because it assumes a random
distribution of seeds among sites. Estimates of n, the
number of sites, were generally centered close to the true
value across treatments for all three functions, although
the general recruitment function showed some bias
towards higher values of n at the highest level of
between-site heterogeneity (treatment 4).
APPLYING
THE
MODEL
To illustrate the application of a mechanistically
derived recruitment function, we consider a situation
in which plots have been measured on several occasions
following seed addition, providing data on the number
of seedlings that germinate and how seedling numbers
subsequently change through time. We can make three a
priori predictions about how the parameters of the
recruitment function should change as the seedling
cohort ages. First, we expect the number of sites, n, to
decline: as seedlings grow they will require a larger area
to meet their resource requirements, meaning the area of
a safe site capable of supporting a single plant will
increase, and the number of sites will decline. Second, if
there is ongoing density-independent mortality of
seedlings, we expect the proportion of safe sites, b, to
decline. Third, if there is competition for safe sites as
seedlings increase in size, leading to density-dependent
mortality, then initial differences in plant size may be
exaggerated through asymmetric competition (Weiner
1990), leading to an increase in the heterogeneity of the
area of safe sites, reflected in a decline in the dispersion
parameter k.
To explore these predictions, we analyze data from an
experiment in which seeds of Hieracium lepidulum
(Asteraceae) were added to square 0.09-m2 plots at five
levels (25, 125, 625, 3125, and 15 625 seeds/plot, equating
to 278, 1389, 6944, 34 722, and 173 611 seeds/m2) and the
number of recruits were counted at intervals from 10 to
48 months after seed addition. The data are taken from a
larger seed-addition experiment in which we aimed to
August 2009
PLANT RECRUITMENT FUNCTIONS
2133
FIG. 2. Results from a typical simulation of a seed-addition experiment in which seeds were added to plots at five densities (xaxis) and the number of recruits (the number of occupied safe sites) was recorded ( y-axis). The experiment comprised four
treatments (panels a–d) that differed in the degree of size heterogeneity among sites (1 ¼ no heterogeneity, 4 ¼ greatest
heterogeneity, with the area variance between sites shown for each treatment). Solid circles show the mean number of recruits from
10 replicate plots for each level of seed addition in each treatment. Lines show the fitted recruitment functions (black, general
recruitment function; dashed, Skellam function; gray, Beverton-Holt function). Also shown is the estimate for the dispersion
parameter, k, for each treatment.
quantify how the recruitment function for H. lepidulum
(an invasive exotic herbaceous perennial in New Zealand) differed among upland forest and alpine habitats in
Canterbury, New Zealand. For illustrative purposes we
present data from 90 of 1080 seed-addition plots that we
established; we will present a detailed analysis of the full
data set elsewhere. The 90 plots analyzed here were all
located in one habitat type (creek beds within forest) in a
randomized block design, comprising six randomly
located blocks, with each block having three replicates
of each of the five seed-addition levels. The numbers of
recruits present in each plot were counted at 10, 13, 20,
26, 37, and 48 months after the seed was added in March
2003 (autumn; we conducted counts prior to 10 months,
but seed continued to germinate up to 10 months after
seed addition (late spring/summer), so we used this as
our start date). To simplify the analysis here we ignore
the blocking structure.
Within 20 m around each block we removed any
naturally occurring H. lepidulum plants to minimize
natural seed input. In addition, we monitored 36 control
plots (six in each block) in which no seeds were added.
Seedlings occurred sporadically in these control plots and
to account for natural recruitment we included the control
plots in our analysis (as having 0 seeds added) along with
an intercept term, i, in the recruitment function that
2134
RICHARD P. DUNCAN ET AL.
Ecology, Vol. 90, No. 8
FIG. 3. Parameter estimates (solid circles) derived from fitting the three recruitment functions to data from 1000 simulated seedaddition experiments (see Interpreting the parameters) for each of four treatments, where the treatments differ in the degree of sizeheterogeneity among sites (1, no heterogeneity; 4, greatest heterogeneity). The bars show the quantiles around each mean, within
which 95% of parameter estimates fell, and different line types distinguish parameter estimates obtained using the three recruitment
functions (solid black, general recruitment function; dashed, Skellam function; solid gray, Beverton-Holt function). (a) Estimates
for the dispersion parameter, k; (b) estimates for the proportion of safe sites, b; (c) estimates for the number of sites, n. In all plots,
the true values of n ¼ 1000 sites and b ¼ 0.1 are shown as horizontal dashed lines.
estimates the number of naturally occurring seedlings per
plot based on the numbers in control plots. The general
recruitment function we fitted was therefore
"
#
kk
r ¼ bn 1 þ i:
ð11Þ
ðk þ s=nÞk
We fitted this function to the data for each month in
which plots were measured, along with equivalent
versions of the Skellam and Beverton-Holt functions,
in a Bayesian framework using OpenBugs version 2.10
called from the BRugs package using R version 2.6.2
(R Development Core Team 2005). We assumed the
numbers of seedlings per plot followed a negative
binomial distribution, and used noninformative prior
distributions to allow the data to drive parameter
estimation. Models were run with three chains for
100 000 iterations and thinned to retain every 10th
iteration, with a burn-in of 5000 iterations. Posterior
distributions of the parameter estimates for n and k were
highly skewed, so we used medians to characterize
central tendency.
We compared the fit of the three recruitment
functions to the data for each month using the deviance
information criterion (DIC; Spiegelhalter et al. 2002),
which is a Bayesian equivalent of Akaike’s information
criterion (AIC; Akaike 1973, Burnham and Anderson
2002). The general recruitment function fitted the data
best in all cases, although there was little difference in fit
among functions at 10, 13, and 26 months (Table 1; DIC
values differed by ,2). The general recruitment function
provided a better fit at 26 and 37 months (difference in
DIC .4) and a substantially better fit at 48 months
(difference in DIC about 60).
Panels a–f in Fig. 4A plot the number of seedlings as a
function of the number of seeds added at each of the six
measurements, along with the fitted general recruitment
functions, while panels g–i in Fig. 4B plot changes in the
parameter estimates for these functions. As predicted,
the number of sites, n, declines through time as the
cohort ages and seedlings increase in size. The propor-
August 2009
PLANT RECRUITMENT FUNCTIONS
TABLE 1. Deviance information criteria (DIC) values for the
three recruitment functions (general, Skellam, and BevertonHolt; see Eqs. 9, 7, and 10, respectively) fitted to the data for
each measurement.
No. months
after seed
addition
General
Skellam
Beverton-Holt
10
13
20
26
37
48
921.9
774.9
773.2
699.8
608.8
490.8
922.9
776.1
778.2
701.1
613.6
550.4
922.8
776.5
778.2
701.3
613.6
550.1
DIC values Lower DIC values for measurements in each month
indicate a better-fitting model, with the best-fitting models
shown in boldface type.
2135
tion of safe sites, b, also declines, implying ongoing
density-independent mortality. The decline is steepest
between months 10 and 13, reflecting high seedling
mortality in the summer following germination, with
ongoing density-independent mortality to month 37, at
which point the proportion of safe sites appears to
stabilize at around 0.04. This estimate for the proportion
of safe sites implies that the large majority of seeds and
seedlings (;96%) failed to germinate or died due to
density-independent processes (see also Clark et al.
2007). The importance of density-dependent mortality
appears to increase through time, as evidenced by the
increasing curvature of the recruitment function (Fig.
4A). Of those seeds and seedlings not lost to densityindependent processes, for example, the proportion lost
to density dependence at the highest level of seed
addition increases steadily from 0.57 to 0.93 between
FIG. 4. (A) Panels (a)–(f ) show the number of Hieracium lepidulum seedlings as a function of the number of seeds added at each
of the six measurements (months 10–48). The gray circles are the number of seedlings in replicate seed-addition plots; the solid
circles are the means for plots with a given level of seed addition, while the lines are the general recruitment functions fitted to the
data for each measurement using Eq. 11. (B) Panels (g)–(i) show changes in the parameter estimates (n, b, and k) from the fitted
recruitment functions over time (number of months after seed addition).
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RICHARD P. DUNCAN ET AL.
FIG. 5. Simulated self-thinning trajectories (solid lines) for
eight plots with different numbers of seeds added (ranging from 8
to 1024, in powers of 2) and for plant mass ranging from 1 to
1000 in arbitrary units. Each line graphs the change in the
number of plants per plot as plant mass increases through time,
using Eq. 12. The parameters were set to: b (the proportion of
sites that are safe sites)¼0.2, c¼1000 (i.e., there are 1000 sites per
plot for plants of mass ¼ 1), k (the dispersion parameter) ¼ 10,
and a (the scaling exponent) ¼ 2/3. The dashed line has slope
3/2.
months 10 and 48. This increase in the proportion of
seedlings suffering density-dependent mortality is paralleled by a decline in k, implying increasing heterogeneity
in safe-site area, consistent with the development of a
size hierarchy through asymmetric competition (Weiner
1990).
EXTENDING
THE
MODEL
Because it is based on a mechanistic model, a
mechanistically derived recruitment function may naturally suggest extensions to other situations. We illustrate
this by showing how the model can be extended to derive a
relationship between the number of plants and the change
in plant mass (the self-thinning relationship; Westoby
1984). As a cohort of seedlings of the same age increases in
size, the safe-site area should increase to match plant
resource requirements, and the number of sites per plot
will decline (Fig. 4g). The area occupied by a plant can be
described as a power function of individual plant mass, M,
such that area scales as M a (White 1981; where a is the
scaling exponent), which implies that the number of sites
per plot will be related to plant mass by the function n ¼
cMa, where c is a constant.
Substituting this into the general recruitment function, taking logs, and rearranging gives
(
"
#)
1
kk
:
logðMÞ ¼
logðbcÞ logðrÞ þ log 1 a
ðk þ s=nÞk
ð12Þ
Ecology, Vol. 90, No. 8
As average plant mass increases and the number of sites
declines, the right-hand term in Eq. 11 will tend toward
0 and the relationship between plant mass and number
of recruits on a log–log scale will approach an asymptote
with slope 1/a, assuming b remains constant (i.e., no
ongoing density-independent mortality). This is an
equation for a self-thinning asymptote, defining an
upper limit to the number of plants of given mass that
can occupy a plot. If the area required to meet a plant’s
resource requirements scales as M 2/3 (Yoda et al. 1963)
then the expected slope of the self-thinning asymptote is
3/2 (alternatively, if the scaling exponent is 3/4 then the
expected slope is 4/3; Enquist et al. 1998). The righthand term will approach 0 more rapidly for larger values
of s (the number of seeds added), so that plots at initially
higher density will converge on the self-thinning
asymptote at lower plant mass. The mechanistic model
can thus be expressed as a function that describes the
self-thinning trajectory of plots with different numbers
of seeds added (Fig. 5). Ongoing density-independent
mortality or the development of a size hierarchy (i.e.,
declines in b or k, or both) will tend to shift the
trajectory and produce a convex self-thinning asymptote. Such variation might partly account for observed
differences in the slopes of self-thinning lines (White
1981, Westoby 1984). Data from experiments in which
different numbers of seeds are added to plots and the
number and mass of recruits are followed through time
could be used to fit both the recruitment and selfthinning functions, and to interpret the parameters in
light of the theory and assumptions underpinning the
model.
DISCUSSION
The importance of recruitment in shaping population
and community dynamics, as well as ecosystem processes (Zeiter et al. 2006), highlights the need to understand
the mechanisms underlying plant-recruitment patterns.
In particular, the relative balance between seed limitation and establishment limitation has been a major focus
of research in plant ecology (Shaw and Antonovics 1986,
Turnbull et al. 2000, Zobel et al. 2000). Quantifying the
recruitment function, relating seed availability to seedling production, provides a means of teasing apart these
limitations (Poulsen et al. 2007).
We have derived a general equation for a recruitment
function that has a mechanistic underpinning and
parameters that have biological interpretation in terms
of the characteristics of safe sites for seedling establishment. The model makes simplifying assumptions, some
of which can be accommodated if required. We assume,
for example, that all seeds added to a plot are viable. If
we know the proportion of nonviable seed prior to
addition then we can adjust the recruitment function
accordingly. Similarly, we incorporated seed input from
the background seed rain into the model by including
control plots and an additional parameter in the
recruitment function (see also Poulsen et al. 2007).
August 2009
PLANT RECRUITMENT FUNCTIONS
Formulating a mechanistically derived recruitment
function should be helpful in several ways. First, as described by Poulsen et al. (2007), the recruitment function
can be used to quantify the degree to which plant
populations are seed vs. establishment limited. While
recruitment can be understood as a balance between
these two processes (Manning et al. 2004, Kollmann et
al. 2007), quantitative assessments of this balance are
needed.
Second, deriving a recruitment function with a
mechanistic underpinning adds insight into the processes
governing recruitment. In particular, using the general
function, and assuming that seeds are randomly distributed across plots, we obtain estimates of the numbers of
sites, the proportion of these that are safe sites, and the
degree of size heterogeneity among sites. Comparing
how these parameters vary among locations with different physical and biological conditions may reveal why
some populations are more strongly seed or safe site
limited than others.
Finally, the mechanistic underpinning may more
naturally suggest hypotheses that can be experimentally
tested. In our Hieracium lepidulum example, we could
hypothesize how the size and availability of safe sites
should change through time, leading to a priori predictions about how the form of the recruitment function
should vary. The ability to make clear predictions may
facilitate the design of experiments better equipped to test
these, and lead to the development of more refined models
on the basis of experimental data that fail to match those
predictions. Likewise, a mechanistic model may naturally
suggest extensions to other situations, such as the relationship between plant mass and seedling numbers that
we derived. Ultimately, the interplay between data and
models should lead to a more in-depth understanding of
the processes governing recruitment.
Of course the applicability of any model depends on
how well it captures the biology of the situation to which it
is applied. Other recruitment functions, derived from
models with different mechanistic underpinnings, may be
appropriate in other circumstances (Poulsen et al. 2007).
For example, where a primary cause of seed loss is predation but predators are satiated at high seed densities, a
model that incorporates positive density dependence may
be appropriate (Blundell and Peart 2004). The characteristics and availability of safe sites, however, are
considered key drivers of plant recruitment (Harper et
al. 1965, Grubb 1977), suggesting the recruitment
function derived here should have wide applicability.
ACKNOWLEDGMENTS
Thanks to Wendy Ruscoe for discussion and Phil Hulme and
two referees for helpful comments on the manuscript. This
work was funded by the Bio-Protection Research Centre and
the Miss E. L. Hellaby Indigenous Grasslands Research Trust.
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