Paper 049 Disc
Ecology Letters, (1999) 2 : 44±51
REPORT
C. Patrick Doncaster1 and Lars
Gustafsson2
1
Research Division of
Biodiversity and Ecology, School
of Biological Sciences, University
of Southampton, Southampton
SO16 7PX, U.K.
E-mail: [email protected]
2
Department of Zoology,
Uppsala University, VillavaÈgen 9,
S-752 36 Uppsala, Sweden.
Density dependence in resource exploitation:
empirical test of Levins' metapopulation model
Abstract
Levins' model of metapopulation dynamics is modified to incorporate variable
degrees of density dependence in the per capita exploitation of resource patches. We
demonstrate a simple means of testing for this density dependence in a sample of
metapopulations, each at its equilibrium balance of local colonization to extinction.
The fraction of habitable unoccupied patches equilibrates to a constant number under
the null model of density independent colonization, and to a constant proportion
under strong density dependence. We compare the null model to two density
dependent alternatives, using data on exploitation of nest boxes by collared
flycatchers Ficedula albicollis. The analysis shows how predicted trends in the
equilibrium unoccupied fraction are similar for both spatial interference and net
immigration. This needs to be recognized, since the null hypothesis of a constant
unused resource applies also to the dynamics of consumable resources, where it is
expressed in a constant stock of uneaten prey at the dynamic equilibrium of predators
to prey.
Ahed
Bhed
Ched
Dhed
Ref marker
Fig marker
Table marker
Ref end
Keywords
Collared flycatcher, exploitation competition, extinction threshold, interference,
metapopulation, population sink.
Ecology Letters (1999) 2 : 44±51
INTRODUCTION
THE EQUILIBRIUM MODEL
Interference competition has been one of the central
themes in ecology since the 1930s, with continuing debate
about its importance, whether in interspecific interactions
at the community level (Schoener 1985; Shorrocks 1993)
or intraspecific interactions at the individual level (e.g.
Hassell & Varley 1969; Sutherland 1996). The subject
remains of interest because the density dependent declines
in resource use typical of interference have proved
difficult to model in natural populations (e.g. GossCustard et al. 1995; Stillman et al. 1996; van der Meer &
Ens 1997). The aim of this paper is to draw on theoretical
developments by Lessells (1995) and Pagel & Payne
(1996), to demonstrate a simple graphical technique for
distinguishing density dependent effects on the exploitation of resources. An empirical data set on collared
flycatchers (Ficedula albicollis Temm.) highlights the
distinction that must be made between sources of density
dependence. These can involve spatial interference in
resource use, caused by resident behaviours such as
territoriality, but also can be generated by a constant
supply of immigrants from outside the system.
Resource exploitation
#1999 Blackwell Science Ltd/CNRS
Ref start
Users of a renewing resource may compete with each
other to exploit the resource input, if it is present in
limiting supply. Exploitation competition thus sets an
upper limit on their numbers, with dynamics that are
regulated by resource carrying capacity. This is simply
illustrated by a metapopulation model of the sort
introduced by Levins (1969, 1970), in which the
exploitation of limiting resources is modelled in terms
of the colonization of limiting habitat.
dNi
ˆ ‰c …Ki ÿ Ni † Ni Š ÿ ‰d Ni Š
dt
…1†
where Ki is the number of habitable resource patches in
metapopulation i, of which Ni are occupied at any one
time. The positive term in square brackets describes the
colonization of empty patches, at a rate per patch of c.Ni,
while the negative term describes the extinction of local
populations from these patches, at a rate per population of
d. Both c and d have values that are defined by life-history
Paper 049 Disc
Density dependence in resource exploitation 45
characteristics of the exploiter, and so may be considered
constant for a particular user-resource system across
metapopulations that vary in resource richness, Ki.
Dynamic equilibrium is achieved at the balance of
colonization and extinction. This steady state is characterized by the number of unoccupied patches being constant
regardless of the value of Ki, the total number in the
metapopulation. That is to say, from eqn 1, at dNi/dt = 0:
…Ki ÿ
Ni †
d
ˆ
c
…2†
where Ni* is the equilibrium number of occupied patches.
For example, the removal of some habitable patches from a
metapopulation, such that its carrying capacity, K1, drops
to a new lower K2, is predicted to leave unchanged the
number of unoccupied patches, at the user-specific value of
d/c, while the occupied number readjusts from N1* to a
lower equilibrium N2*. The same relation applies to
variation in Ki across space as over time. This equilibrium
condition can be graphed for variations in Ni* with Ki, as
shown in Fig. 1. It is a feature of the model that the
metapopulation will not be viable unless Ki is larger than d/c,
since this is the equilibrium number of unoccupied
patches. The constant d/c therefore presents itself as a
deterministic extinction threshold for the system (Lande
1987; dubbed the ``Levins rule'' for this type of
metapopulation system by Hanski et al. 1996). Its value
reflects the exploitation efficiency of the user. If a large
fraction of Ki is unoccupied at equilibrium, the users can be
said to be inefficient in exploiting the resource patches, and
the metapopulation is vulnerable to deterministic extinction from a small reduction in Ki (Doncaster et al. 1996).
Density-dependent exploitation
If users interfere with each other's access to a limiting
resource, the exploitation rate that each can achieve will
depend on the numbers present. The effect of mutual
interference is thus to introduce density dependence into
the per capita exploitation of the resource (Hassell &
Varley 1969). In the case of the metapopulation model
above, the rate equation can now be expressed as:
dNi
ˆ ‰c …Ki ÿ Ni † Ni1ÿm Š ÿ ‰d Ni Š
dt
…3†
where the coefficient m describes the strength of density
dependence. For m = 1, the colonization rate per patch is
constant rather than a multiple of Ni (i.e. it is arithmetic
rather than geometric). For any m, the equilibrium
number of unoccupied patches, Ki ± Ni*, may now vary
as an implicit function of carrying capacity, Ki, and as an
explicit function of the occupied number, Ni*. Thus, at
dNi/dt = 0:
1=m
c
…Ki ÿ Ni †
ÿ Ki ˆ 0;
…Ki ÿ Ni † ‡
d
or equivalently
…Ki ÿ Ni † ˆ
d
Nim
c
…4†
For m = 0, these equilibrium conditions reduce to eqn 2.
For m = 1, the equilibrium number of unoccupied
patches is a constant fraction of the total number, as is
the number of occupied patches:
…Ki ÿ Ni †
d
;
ˆ
Ki
c ‡d
or equivalently
Ni
c
ˆ
c ‡d
Ki
Figure 1 Number of occupied patches for the Levins metapopu-
lation model (eqn 1) at the equilibrium balance of local
colonization to extinction (eqn 2). The relation is obtained from
setting eqn 2 in terms of the number of occupied patches:
Ni* = Ki ± d/c. The extinction threshold for a metapopulation is
at Ki = d/c, when Ni* = 0.
…5†
For m 4 0, eqns 4 and 5 reveal that Ni* will always be
greater than zero for Ki 4 0, which means there is
theoretically no minimum viable population size (Hanski
et al. 1996). Regardless of m, however, Ni* will be less
than unity for any value of Ki 5 1 + d/c. The value of 1 +
d/c therefore represents an effective extinction threshold
for an exploitation system incorporating interference.
#1999 Blackwell Science Ltd/CNRS
Paper 049 Disc
46 C.P. Doncaster and L. Gustafsson
Under conditions of weak interference, this threshold can
be just as important a predictor of viability as it is for
interference-free systems. The effect of density dependence in colonization rate on the amount of unoccupied
resource is shown graphically in Fig. 2, for three values of
m 4 0. A relationship of unoccupied to total resource is
now present, where there was none under densityindependent colonization.
Density dependence in the exploitation of a resource is
not necessarily a product of spatial interference. It can also
arise from numerical supply, in the form of a constant
immigration of exploiters from outside the system.
Consider a metapopulation i with internal colonization
dynamics described by the classical Levins model (eqn 1),
that has a fraction m of its colonists supplied as a constant
influx of immigrants from an external source of E
populations. Equation 1 is then modified to reflect these
two sources of colonists:
dependent per capita colonization in a constant influx from E
external populations only (the ``propagule rain'' of Gotelli
1991; or the mainland-island dynamics of Hanski 1994). For
any m, the equilibrium number of unoccupied patches again
varies implicitly with Ki, and explicitly with the occupied
number. From eqn 6, at dNi/dt = 0:
…Ki ÿ Ni † …m E ‡ …1 ÿ m† …Ki ÿ …Ki ÿ Ni ††† d
ÿ ˆ 0;
Ki ÿ …Ki ÿ Ni †
c
or equivalently
…Ki ÿ Ni † ˆ
d
c
Ni
m E ‡ …1 ÿ m† Ni
…7†
A value of m = 0 models a per capita colonization rate that is
independent of Ni, and this reproduces Levins' model with
internal dynamics only. A value of m = 1 models density-
For m = 0, the equilibrium condition reduces to eqn 2;
for m = 1, it is equivalent to eqn 5 (with E taking an
arbitrary value of 1). Figure 3 shows how intermediate
values of m give a family of user-resource equilibria with a
similar generic form to those shown in Fig. 2.
Clearly it is of great interest to distinguish between the
nested models of exploitation only (eqn 1) and exploitation with density-dependent interference or supply (eqns 3
and 6). Figures 2 and 3 suggest a ready means of doing so,
given a sample of values for Ni* and Ki. Density
dependence is revealed in a significant trend-line on a
plot of the amount of unused resource against the total
Figure 2 Number of unoccupied patches for the modified
Figure 3 Number of unoccupied patches for the modified
dNi
ˆ ‰c …Ki ÿ Ni † …m E ‡ …1 ÿ m† Ni †Š ÿ ‰d Ni Š
dt
…6†
metapopulation model given by eqn 3 at the equilibrium balance
of local colonization to extinction (eqns 4 and 5). The broken
lines show the effects of incremental increases in m above zero,
modelling increments to the strength of mutual interference in
per capita colonization. The continuous line at m = 0 describes
equilibrium states for density-independent colonization, commensurate with Fig. 1. All lines pass through (1 + d/c, d/c), at
which point Ni* = 1.
#1999 Blackwell Science Ltd/CNRS
metapopulation model given by eqn 6 at the equilibrium balance
of local colonization to extinction (eqn 7). The broken lines
show the effects of incremental increases in m above zero,
modelling increments to the proportion of externally supplied
immigrants. Equilibrium conditions for m = 0 and m = 1 are
identical to those of Fig. 2.
Paper 049 Disc
Density dependence in resource exploitation 47
usable resource. This is a very general relationship, which
is expected for any dynamic system at an equilibrium of
resource exploitation to renewal. Although we have
described it in terms of a resource that is occupied by a
user, it applies also to a resource being eaten by a consumer.
Lessells (1995) has built density dependence into the
framework of behavioural interference between Ni* consumers aggregating at a resource patch. An equivalent
dynamic at the metapopulation level would be, for
example, a metapopulation of territory holders mutually
impeding each others' prospection for new territories.
Pagel & Payne (1996) have built density dependence into a
predator±prey system incorporating net immigration of
predators. An equivalent dynamic at the metapopulation
level would be a metapopulation ``sink'', in which habitat
patches or territories are recolonized by immigration from
external sources as well as (or instead of) internal
recruitment. Here we test the metapopulation models for
the two types of density dependence with empirical data on
the use of nest boxes by collared flycatchers. Our aim is to
measure the strength of density dependence in the use of
boxes, and to evaluate it against the theoretical predictions
of spatial interference and of net immigration.
EMPIRICAL TESTS FOR INTERFERENCE AND
IMMIGRATION
Methods and techniques
A published analysis of flycatcher breeding populations
contains the data required to differentiate density
dependent effects from density dependent exploitation in
the use of available nest sites. The values come from an
analysis by Doncaster et al. (1997; Table 1) of dispersal and
fecundity in breeding populations of collared flycatchers on
the island of Gotland, Sweden, recorded during a longterm study by Gustafsson et al. (e.g. 1995). Adult and
fledgling collared flycatchers had been ringed every year
between 1987 and 1992 at nest boxes distributed among 11
discrete woodlands. The woodlands varied in size from 6.6
to 73.7 ha, and in separation from 0.6 to 9.4 km. They
contained a total of 955 nest boxes, of which in any year
33%±45% were occupied by collared flycatchers, at least to
the start of incubation. Although some boxes were
occupied by other species (&20% of all used boxes by
Parus major L. and 15% by Parus caeruleus L., with some
variation between years), collared flycatchers occupied the
majority of those used by birds, and in the following
analyses we ignore other species. Few flycatchers in these
populations bred in natural nest cavities outside these boxes
(PaÈrt & Gustafsson 1989). The question of interest here is
whether or not density dependence is implicated in the
exploitation of this fixed resource of nest boxes by collared
flycatchers, manifested either in mutual interference or
in immigration.
The ``null'' model assumes that each of the 11
woodland sites contains a metapopulation of breeding
collared flycatchers that exploit the limiting resource of
nest boxes without mutual interference or net immigration. Equations 1 and 2 apply to the colonization
dynamics and equilibrium number of boxes occupied by
flycatchers. The alternative models allow for density
dependence in the per capita exploitation of nest boxes by
flycatchers. This may involve spatial interference in access
to the resource (e.g. mutually impeded prospection), in
which case eqns 3±5 apply to the colonization dynamics and
equilibrium number of occupied boxes. Or it may involve
numerical supply in the form of a net influx of immigrants
into the woodlands, in which case eqns 4±7 apply.
Only the null model returns a constant for the number of
unoccupied boxes, Ki ± Ni*, against the total number of
boxes, Ki. Both alternative models return a positive relation,
predicting that sites with more boxes will have more unoccupied by flycatchers. The models can therefore be evaluated by comparing their fits to a plot of unoccupied against
total numbers of boxes. The comparisons were made with a
chi-squared test of maximum likelihood ratios, applied to
the implicit functions of (Ki ±Ni*) to Ki in eqns 4 and 7.
All the models depend on the populations being at
equilibrium. This is likely for these data, as the number of
nest boxes remained constant over the 6 years, while
population sizes showed little year-to-year variation and
different sites achieved higher or lower average fecundities from one year to the next without a visible trend (Fig.
4 in Doncaster et al. 1997). The models further assume the
parameter estimates of c, d, and m obtained from the plots
are constant between sites, and representative of all
individuals at each site. First year and older birds, for
example, may have different unitary rates of colonization
and abandonment of boxes, though this was not
suggested by the analysis of Doncaster et al. (1997).
Whether c and d vary between woodlands according to
their quality is more problematical, though the equal
exchange of migrants between woodlands revealed by
Doncaster et al. (1997) suggests no more than a
covariation of c with d and thus a constant d/c.
RESULTS
Figure 4 reveals the outcome of the test for density
dependence in the occupancy of Ki habitable boxes by
collared flycatchers. Clear grounds for rejecting the null
model are provided by the maximum likelihood estimates
of m for both spatial and numerical sources of density
dependence (Table 1). They are close to unity, suggesting
that breeding populations may directly match numbers of
#1999 Blackwell Science Ltd/CNRS
Paper 049 Disc
48 C.P. Doncaster and L. Gustafsson
available nest sites, as predicted by eqn 5. For the model
of spatial interference, m is not significantly different from
unity, and estimates for individual years are in the range
0.739 4 m 4 1. For the model of net immigration, m is
closer to unity but does differ significantly from it, as do
all estimates for individual years (0.968 4 m 4 0.982, P
5 0.002). The maximum likelihood values of d/c are close
to unity, indicating that the birds have effectively no
extinction threshold for these boxes.
Although some 15%±25% of nest boxes were used by
other species such as blue and great tits, which generally
settle before flycatchers, we estimate that interspecific
interference has a negligible influence on the value of m.
Its effect on Ki is difficult to quantify in practice because
of year-to-year variation in numbers and timing. If we
assume that flycatchers are excluded from about the same
number of boxes in each woodland on average, the bias
from ignoring other species is towards underestimating m
in Table 1 by a small amount (i.e. its true value is closer to
unity). If the flycatchers were being excluded from a
constant proportion of boxes by other species, the bias
would be towards overestimating m in Table 1 but again
by an amount that is unlikely to be substantial (515% for
as much as a 40% reduction in K).
The model of net immigration has a slightly higher
maximum likelihood than the model of spatial interference
(Table 1). However, the detailed analysis of dispersal
between these woodland sites by Doncaster et al. (1997)
concluded that this system of woodlands did not have sink
dynamics. The flycatchers showed a high year-on-year
Figure 4 Total number of nest boxes and number unoccupied by
breeding collared flycatchers at 11 woodland sites. Points are site
means for 1987±92, within given ranges. Fitted lines are
maximum likelihood estimates of the test models from the 11
means. The continuous line shows the null model of densityindependent colonization; the broken lines show the models of
density-dependent colonization, given by eqn 4 for spatial
interference and eqn 7 for numerical supply, and for a value of m
set to unity. Parameter estimates are given in Table 1.
Model
m
d/c
Ln-likelihood
function
Immigration
(eqn 7, numbers)
0
0.983
1
0
0.840
1
0
0.841
1
66.184
2.770
1.588
66.184
2.865
1.588
23.764
2.370
1.539
±54.629
±37.756
±42.572
±54.629
±41.546
±42.572
±32.182
±27.940
±28.334
Interference
(eqn 4, numbers)
Interference
(eqn 4, density)
Table 1 Maximum likelihood
Chi-squared
d.f.
P
33.75
2
50.0001
9.63
26.17
1
2
50.002
50.0001
2.05
8.48
1
2
50.2
50.02
0.79
1
50.4
Likelihood was calculated by multiplying together the expected probabilities of all 11 site means,
each given by its relative frequency in a binomial distribution of Ki boxes, of which Ki ± Ni* were
unoccupied:
11
X
Ki
ki ÿNi
Ni
Ln-likelihood ˆ
ln
…1
ÿ
p
†
p
i
i
Ki ÿ Ni
iˆ1
where pi is the model estimate of the unoccupied proportion. The ln-likelihood for the set of 11
points was maximized over the two unknown constants m and d/c. Chi-squared values are equal to
twice the difference in ln-likelihoods between the full model (in bold) and the nested models of m
fixed at zero and unity. Degrees of freedom are given by the difference between models in the
number of their parameters, following Hilborn & Mangel (1997). P values test the null hypothesis
that the model for fixed m is no less likely than the full model.
#1999 Blackwell Science Ltd/CNRS
estimates of parameters m and d/c
for the model of net immigration
(eqn 7), and the model of spatial
interference (eqn 4) plotted in
Fig. 4, and for the interference
model applied to densities
plotted in Fig. 5.
Paper 049 Disc
Density dependence in resource exploitation 49
fidelity to their breeding woodland, with only 35% of the
1896 recorded breeding attempts involving a change of
woodland. These dispersals were of adults (38%) as well
as juveniles dispersing in their first year (62%). They were
characterized by an equal exchange of numbers migrating
between any two woodlands, regardless of their relative
population sizes. Analyses of fecundity furthermore
showed no positive relation of growth rate to population
size. These observations are contrary to the expectation of a
source-sink system in which small populations are expected
to vary temporally and to receive more immigrants than
they produce emigrants. The alternative explanation is that
dispersal propensity varies between sites so as to depend
inversely on population size. Conditional dispersal of this
kind can be an evolutionary stable strategy, even in
temporally constant environments (McPeek & Holt 1992).
In the case of the flycatchers it leads to a situation akin to an
ideal free distribution between sites, in which the birds
distribute themselves such that the consumption rate at
each cannot be bettered by switching to another site, with
``consumption'' here referring to colonization of a nest box.
Thus it appears unlikely that the observed relation in Fig. 4
can be explained by numerical supply in the form of a
constant influx of immigrants to the sites.
The observed density dependence is explained most
readily in terms of spatial interference. Since the model for
spatial interference given by eqn 3 simply involves putting
density dependence into the per capita colonization rate of
nest boxes, the observed m 4 0 might involve either
mutual repulsion between colonisers in access to nest
boxes, or density dependence in the number of colonisers
produced per nest box. Doncaster et al. (1997) did indeed
find that the number of young per reproductive event was
inversely related to population size, but the relation was
weak, with a 1.6-fold variation in this measure of
reproductive success for a 14.5-fold variation in population sizes between woodlands. It seems likely that this
effect is outweighed by mutual repulsion between collared
flycatchers, which takes the form of territorial behaviour
that may encompass more than one box. Adult males take
up territories immediately on arrival in Gotland from
over-wintering sites in Africa, with females arriving 1
week after the first males on average (PaÈrt & Gustafsson
1989; Cramp & Perrins 1993). Territorial behaviour is by
definition an expression of interference competition for
limiting resources, and these can include food, mates,
refuge sites, or breeding sites. The trend-line in Fig. 4
suggests that interference competition is implicated in
access to nest boxes for these collared flycatchers, and
therefore that the boxes are a limiting resource here
(doubtless in association with other factors).
Since territoriality is expressed in spatial segregation, it
is relevant to examine the equilibrium conditions in terms
of nest box density, in addition to numbers as in Fig. 4.
Figure 5 and Table 1 show that the value of m remains
unchanged, despite some larger woodlands containing
fewer boxes than smaller ones (Doncaster et al. 1997); the
ln-likelihood does increase, however, from ±41.5 to ±27.9.
DISCUSSION
Our analysis of flycatcher populations has allowed us to
estimate the strength of mutual interference in access to
the nest box resource. That we should find an interference
effect is not unexpected for this territorial species; the
novelty of our analysis is in the model used to measure it.
Theoretical works by Lessells (1995) and Pagel & Payne
(1996) have independently pointed to the equilibrium
unused resource as an indicator of density dependence in
exploitation dynamics, but only one other study has
attempted to confront the theory with empirical data
(Tregenza et al. 1996 for an ideal free system).
Metapopulation concepts provide a suitable framework
for analysing the use of breeding stations such as nest
boxes, because the station represents a fixed resource that
is exploited by being occupied and renewed by being
abandoned. Models of consumable, rather than fixed,
resources are inherently more complex than this, because
of the extra dynamic required to account for the
independent renewal of a prey resource that is being
eaten up by the consumer. They nevertheless predict
relationships at the equilibrium of consumption to
Figure 5 Total number of nest boxes and number unoccupied by
breeding collared flycatchers at 11 woodland sites, shown as a
density per 6.6 ha. This is the size of the smallest woodland, and
values of 1 + d/c therefore correspond to the minimum number
of boxes per woodland for any site to be occupied by flycatchers.
Points and lines are as for Fig. 4, parameter estimates are given
in Table 1.
#1999 Blackwell Science Ltd/CNRS
Paper 049 Disc
50 C.P. Doncaster and L. Gustafsson
renewal equivalent to those in Figs 1±3, if it can be
assumed that all predators obtain equal per capita
consumption rates across resource patches. At the
individual level this is achieved at a balance of arrivals
to departures in the redistribution of individuals between
resource patches (e.g. Lessells 1995); at the population
level it is achieved at a balance of births plus immigration
to deaths plus emigration in the growth and decline of
populations (e.g. Pagel & Payne 1996). In both cases the
level of density dependence in exploitation is given by
trends in the stock of uneaten resource, just as it is given
by trends in the number of unoccupied patches at the
metapopulation level (eqns 4, 7).
Lessells (1995) has modelled the effects of density
dependence on an ideal free distribution of consumers
exploiting patches of renewing resource with or without
interference. She notes that spatial interference at the
equilibrium balance of consumers to resources can be
revealed in a resource stock that varies between patches, as
in Fig. 2. Since the ideal free model concerns a fixed
number of consumers distributing freely between resource
patches, it does not consider the density-dependent effects
of net immigration. Pagel & Payne (1996) have modelled
the effects of density dependence on a metapopulation
under the influence of net immigration or emigration. They
used a Lotka±Volterra predator±prey model, drawing the
analogy between consumption of renewing prey and
consumption (i.e. colonization) of regenerating habitat.
In the presence of net immigration this model has
consumer dynamics analogous to eqn 6, and produces
equilibrium trends in unused resource equivalent to those
in Fig. 3. They suggest measuring such trends as a way of
detecting net immigration. This is a useful thing to do,
because metapopulations may be sustained at well above
their extinction threshold by net immigration; a metapopulation can therefore appear robust to extinction, while its
persistence in fact relies entirely on an outside source of
immigrants. However, their analysis assumes no spatial
interference. A metapopulation can experience interference
in the form of mutual repulsion as modelled here, and this
will affect the amount of unused resource in much the same
way as net immigration, as we demonstrate in Figs 4 and 5.
It is clear from the models and analyses presented here that
the variation in unused limiting resource cannot on its own
distinguish net immigration from spatial interference, and
this should be borne in mind when seeking to measure
density dependence directly by this means.
For nest box occupancy, we can have reasonable
confidence that the system is at dynamic equilibrium. Other
systems, however, may express equilibrium conditions only
rarely, or not at all. This is particularly true for consumers at
resource patches that have slower dynamics of recovery
from being eaten than the dynamics of movement by
#1999 Blackwell Science Ltd/CNRS
consumers between patches. At these levels of foraging
individuals, the strength of behavioural interference can
often be measured directly in reduced food intake. The
relationships presented in Fig. 2 are still of interest,
however, because any departure of empirical data from
predicted consumer-resource densities for a given level of
interference signals the strength and direction of departures
from dynamic equilibrium for the system (Doncaster, 1999).
ACKNOWLEDGEMENTS
This work was supported by grants GR8/03738 and GR8/
03680 to C.P.D. from the UK Natural Environment
Research Council. We are grateful to Dr W. Liu for
advising on statistical techniques, and to five anonymous
referees for vital corrections and suggestions.
REFERENCES
Cramp, S. & Perrins, C.M. (1993). The Birds of the Western
Palearctic VII. Oxford: Oxford University Press.
Doncaster, C.P. (1999). A useful phenomenological difference
between exploitation and interference in the distribution of
ideal free predators. J. Anim. Ecol., in press.
Doncaster, C.P., Micol, T. & Plesner Jensen, S. (1996).
Determining minimum habitat requirements in theory and
practice. Oikos, 75, 335±339.
Doncaster, C.P., Clobert, J., Doligez, B., Gustafsson, L. &
Danchin, E. (1997). Balanced dispersal between spatially
varying local populations: an alternative to the source-sink
model. Am. Naturalist, 150, 425±445.
Goss-Custard, J.D., Caldow, R.W.G., Clarke, R.T. & West,
A.D. (1995). Deriving population parameters from individual
variations in foraging behaviour. II. Model tests and
population parameters. J. Anim. Ecol., 64, 277±289.
Gotelli, N.J. (1991). Metapopulation models: the rescue effect,
the propagule rain, and the core-satellite hypothesis. Am.
Naturalist, 138, 768±776.
Gustafsson, L., Qvarnstrom, A. & Sheldon, B.C. (1995). Tradeoffs between life-history traits and a secondary sexual character in male collared flycatchers. Nature, London, 375, 311±313.
Hanski, I. (1994). A practical model of metapopulation
dynamics. J. Anim. Ecol., 63, 151±162.
Hanski, I., Moilanen, A. & Gyllenberg, M. (1996). Minimum
viable metapopulation size. Am. Naturalist, 147, 527±541.
Hassell, M.P. & Varley, G.C. (1969). New inductive population
model for insect parasites and its bearing on biological
control. Nature, London, 223, 1133±1136.
Hilborn, R. & Mangel, M. (1997). The ecological detective:
confronting models with data. Monographs in Population
Biology 28. Princeton: Princeton University Press.
Lande, R. (1987). Extinction thresholds in demographic models
of territorial populations. Am. Naturalist, 130, 624±635.
Lessells, C.M. (1995). Putting resource dynamics into continuous
input ideal free distribution models. Anim. Behav., 49, 487±494.
Levins, R. (1969). Some demographic and genetic consequences
of environmental heterogeneity for biological control. Bull.
Entomol. Soc. Am., 15, 237±240.
Paper 049 Disc
Density dependence in resource exploitation 51
Levins, R. (1970). Extinction. In Lectures on Mathematics in the
Life Sciences, vol. 2, ed. Gerstenhaber, M. Providence:
American Mathematical Society, pp. 77±107.
McPeek, M.A. & Holt, R.D. (1992). The evolution of dispersal
in spatially and temporally varying environments. Am.
Naturalist, 140, 1010±1027.
van der Meer, J. & Ens, B.J. (1997). Models of interference and
their consequences for the spatial distribution of ideal and free
predators. J. Anim. Ecol., 66, 846±858.
Pagel, M. & Payne, R.J.H. (1996). How migration affects
estimation of the extinction threshold. Oikos, 76, 323±329.
PaÈrt, T. & Gustafsson, L. (1989). Breeding dispersal in the
collared flycatcher (Ficedula albicollis): possible causes and
reproductive consequences. J. Anim. Ecol., 58, 305±320.
Schoener (1985). Some comments on Connell's and my reviews
of field experiments on interspecific competition. Am.
Naturalist, 125, 730±740.
Shorrocks, B. (1993). Trends in the Journal of Animal Ecology:
1932±92. J. Anim. Ecol., 62, 599±605.
Stillman, R.A., Goss-Custard, J.D., Clarke, R.T., dit Durell,
S.E.A. & Le, V. (1996). Shape of the interference function in
a foraging vertebrate. J. Anim. Ecol., 65, 813±824.
Sutherland, W.J. (1996). From Individual Behaviour to Population
Ecology. Oxford: Oxford University Press.
Tregenza, T., Shaw, J.J. & Thompson, D.J. (1996). An
experimental investigation of a new ideal free distribution
model. Evol. Ecol., 10, 45±49.
BIOSKETCH
C. Patrick Doncaster is interested in empirical and theoretical
studies of processes at the population and metapopulation level.
Editor, M. Lambrechts
Manuscript received 1 September 1998
First decision made 5 October 1998
Manuscript accepted 18 November 1998
#1999 Blackwell Science Ltd/CNRS