Proc. R. Soc. B (2012) 279, 480–488
doi:10.1098/rspb.2011.0885
Published online 22 June 2011
On valuing patches: estimating
contributions to metapopulation growth with
reverse-time capture–recapture modelling
Jamie S. Sanderlin1,*, Peter M. Waser1, James E. Hines2
and James D. Nichols2
1
2
Department of Biological Sciences, Purdue University, West Lafayette, IN 47907, USA
USGS Biological Resources Division, Patuxent Wildlife Research Center, Laurel, MD 20708, USA
Metapopulation ecology has historically been rich in theory, yet analytical approaches for inferring demographic relationships among local populations have been few. We show how reverse-time multi-state
capture– recapture models can be used to estimate the importance of local recruitment and interpopulation dispersal to metapopulation growth. We use ‘contribution metrics’ to infer demographic
connectedness among eight local populations of banner-tailed kangaroo rats, to assess their demographic
closure, and to investigate sources of variation in these contributions. Using a 7 year dataset, we show
that: (i) local populations are relatively independent demographically, and contributions to local population growth via dispersal within the system decline with distance; (ii) growth contributions via local
survival and recruitment are greater for adults than juveniles, while contributions involving dispersal
are greater for juveniles; (iii) central populations rely more on local recruitment and survival than peripheral populations; (iv) contributions involving dispersal are not clearly related to overall metapopulation
density; and (v) estimated contributions from outside the system are unexpectedly large. Our analytical
framework can classify metapopulations on a continuum between demographic independence and
panmixia, detect hidden population growth contributions, and make inference about other population linkage forms, including rescue effects and source–sink structures. Finally, we discuss differences between
demographic and genetic population linkage patterns for our system.
Keywords: contribution metrics; density effects; dispersal; kangaroo rat; seniority; source–sink
1. INTRODUCTION
The metapopulation concept has produced substantial
theory and discussion [1–3], and has been used as the
basis for management and conservation of various species
[4–6]. The concept includes local populations at least partially linked via dispersal. Local populations that comprise
a metapopulation typically exhibit variation in contributions they make to, and receive from, other system
components. Incidence function approaches to modelling
metapopulation dynamics focus on variation among local
populations with respect to sizes of the patches that they
inhabit and isolation of these patches from potential
sources of colonists [7]. Source–sink approaches focus
on internal dynamics of local populations, specifically,
their abilities to provide emigrants to other local populations and to sustain themselves in the absence of
immigration [8,9]. We propose a comprehensive framework that considers the relative contributions of each
local population to every other local population and to
overall metapopulation growth [9–11].
While the value of this comprehensive framework is
intuitive, we have little knowledge of the magnitude or
sources of variation in contributions among local
populations within any metapopulation. The ability to
estimate for each local population the relative contributions to adult population growth from both internal
processes (survival, in situ recruitment) and external processes (immigration from other sites and from outside the
metapopulation) is relevant in understanding and managing any metapopulation system. For example, the
ability to assign values to local populations with respect
to their contributions to other local populations
is relevant to patch selection for reserve design [5,6].
Estimating rates of immigration is also crucial for understanding the genetic structure of metapopulations [12].
For our study, we had specific questions involving hypothesized sources of variation in these contributions to
and from local populations, including local population
location (central versus peripheral; distance from other
local populations), overall metapopulation density and
animal age. These hypothesized sources of variation
include factors emphasized in both incidence function
and source– sink approaches to metapopulation modelling (e.g. density effects [13,14], distance between local
populations [3]).
Despite the appeal of this comprehensive framework
for metapopulation understanding and management, we
agree with Kawecki’s [9, p. 413] statement about metapopulations that ‘the development of theory has outpaced
the accumulation of empirical data’. Fortunately, recent
years have brought important methodological advances
for drawing inferences about metapopulation dynamics
* Author and address for correspondence: Department of
Mathematics and Statistics, University of Otago, PO Box 56,
Dunedin 9054, New Zealand (jamie.sanderlin@gmail.com).
Electronic supplementary material is available at http://dx.doi.org/
10.1098/rspb.2011.0885 or via http://rspb.royalsocietypublishing.org.
Received 7 May 2011
Accepted 2 June 2011
480
This journal is q 2011 The Royal Society
Reverse-time analysis of a metapopulation J. S. Sanderlin et al. 481
from field data (e.g. [15,16]). Estimates from these
methods can be used to assign values to local populations
with respect to their metapopulation contributions,
viewed either asymptotically or time-specifically. The
most useful asymptotic approach is the concept of spatial
reproductive value developed by Willekens & Rogers [17]
as an extension of Fisher’s [18] reproductive value.
Reproductive value quantifies the expected long-term
contribution of an individual from any local population
to metapopulation size at some time in the distant
future. This metric has been suggested as a way to
assess local population value in the context of sources
and sinks (e.g. [9,10]). Alternatively, Runge et al. [11]
present an approach for assigning value to local populations based on their contributions over specified
periods of ecological time.
In this paper, we apply the conceptual framework of
Runge et al. [11] in combination with reverse-time modelling [19] to draw inferences about a metapopulation of
banner-tailed kangaroo rats (Dipodomys spectabilis).
Specifically, we investigate the contributions of local
population dynamics and dispersal between local populations to metapopulation growth. Because studies of
real-world systems seldom include every possible local
population, we consider contributions from outside the
system, in addition to those from within. After defining
the relevant contribution metrics, we estimate them by
applying reverse-time models to 7 years of capture– recapture data. Although reverse-time methods were proposed
a decade ago, to our knowledge this is their first usage
within a metapopulation.
Reverse-time modelling focuses on the ‘seniority’ parameter, g, the probability that an individual in a
particular state on sampling occasion t was present in
that state on sampling occasion t 2 1. Pradel ([20]; cf.
[21]) discussed the relationship of seniority to recruitment in a single population with no age-specificity.
Nichols et al. ([19]; cf. [22]) generalized the idea of
seniority to multiple ‘strata’, where g becomes the probability that an individual present in a specific stratum
represents a transition from that or another stratum
(strata may be defined as age classes, local populations
or via other state variables) in the previous sampling
period. These authors noted that g’s can be used to compute proportional contributions of various components to
population growth rate, l, and the method is increasingly
used to examine the impact of variation in recruitment
and age-specific survival rates on growth of individual
populations [23,24]. Reverse-time modelling has equal
promise in describing the demographic importance of
population-specific vital rates (survival, in situ reproductive recruitment, dispersal) to specific local populations
and to overall metapopulation growth. This is the focal
application in our analyses.
Banner-tailed kangaroo rats build and live in conspicuous dirt mounds occupied by a single adult or female
with juveniles. This ensures high capture probability
and unambiguous location of individual residences
during sample periods. Patterns of within-population dispersal [25 –27] and between-population dispersal [28]
have been documented in this study system, as has been
some degree of genetic connectivity among the local
populations [12,29,30]. We previously investigated
impacts of age, sex, distance between populations and
Proc. R. Soc. B (2012)
5
2
4
1
7
8
3
0
6
1 km
Figure 1. Kangaroo rat metapopulation study site in Cochise
County, southeast Arizona, USA. Dots on the map indicate
kangaroo rat mounds that were used more than once
during the study. Populations are labelled with numbers corresponding to our description of the system in the text. Local
populations comparable in size with those we sampled
existed within 500 m of the study site (south of populations
3 and 6, northwest of population 4, north and east of population 5). During this period, we were aware of only two (to
the northwest and south) additional substantial populations
within 2 km of the study site. Source: ‘Rucker study site,’
318360 33.0600 N and 1098150 48.8400 W, Google Earth, 8
June, 2007. Access date: 23 August, 2010. USA map
source: National Atlas of the United States, 5 March 2003,
http://nationalatlas.gov.
the nature of the habitat matrix on interpopulation dispersal, and demonstrated the importance of age and
distance between populations within our system to interpopulation dispersal [28]. Within the study system,
mounds occur in eight clusters, each separated from
other clusters by hundreds of metres (median 600 m,
range 100– 1600 m) of unoccupied land. Some populations of similar size exist outside of the study system
(minimum distance 500 m, figure 1), and may influence
dynamics of peripheral populations within the system.
Because kangaroo rats can disperse between these
mound clusters, we refer to them as ‘local populations’
and the eight-cluster system as a ‘metapopulation’, without meaning to imply any particular demographic
relationship among populations other than they are
linked. During our study, the overall metapopulation density and local population abundances fluctuated, possibly
influencing local population contributions to the metapopulation. We expect that larger populations will
contribute more to metapopulation growth than smaller
populations.
We ask four general questions with the reverse-time
multi-state approach: (i) what are the relative contributions of survival, in situ recruitment and immigration
to adult population growth for local populations and
the entire metapopulation? (ii) how ‘closed’ is the set
of local populations—is there an important component of
immigration from outside? (iii) does the importance
of survival, in situ recruitment and immigration vary
substantially among local populations? and (iv) do the
following factors represent important sources of variation
among local populations in their contributions to each
other and the entire metapopulation: local population abundance, local population location (central versus peripheral;
distance between pairs of local populations), overall
482 J. S. Sanderlin et al. Reverse-time analysis of a metapopulation
metapopulation density (number of animals per unit area
within the metapopulation, i.e. the group of local populations) and animal age. Our approach provides a useful
way to detect hidden (possibly extra-system) contributions
to population growth, and to make inferences about a possible rescue effect [31], presence of sources and sinks [8] and
degree to which local populations are demographically
independent [11]. Finally, we comment on the differences
between demographic patterns of interpopulation connectivity revealed here and genetic patterns of connectivity
described elsewhere for this same system [12,29,30].
2. MATERIAL AND METHODS
(a) Field data collection
We applied our approach to data from annual trapping surveys (late July/early August) in desert grassland, Cochise
County, southeast Arizona, USA (318370 N, 1098150 W),
from 1994 through to 2000. We placed three Sherman live
traps at entrances to each active (based on signs of fresh excavation or sandbathing) mound, baited traps with birdseed in
late afternoon and checked them before dawn the following
morning. Each survey consisted of, on average, three trap
nights (range two to five). We marked each animal with
two individually numbered ear tags, and recorded trapping
location, sex and age (‘juvenile’ for animals born the same
year, ‘adult’ for animals that were at least 1 year old; juveniles
graduated to the adult age class during the winter following
their birth). Animals first captured as adults were identifiable
from their size (more than 110 g) and reproductive criteria
(visible nipples in females, descended testes in males).
Banner-tailed kangaroo rat mounds on our 2 3 km site
were mapped with a theodolite. We determined edge-to-edge
distances between each pair of local populations by measuring
the distance between the closest mounds in each member of
the pair. Since other mound clusters occurred outside the
metapopulation, we classified six of the eight local populations
as ‘peripheral’, because they were interposed between local
populations outside the study system and our two ‘central’
populations. Sizes of all local populations fluctuated from
year to year. Compared with trends since 1980, population
density was relatively high (two to three animals per hectare)
between 1994 and 1998, and lower (approx. one animal per
hectare) in 1999 and 2000 [32].
(b) Hypotheses and predictions
We compared models examining several hypotheses regarding sources of variation in relative contributions of local
populations to the metapopulation. First, components of
population growth might differ between central and peripheral local populations. Connectivity with other local
populations of the system should be higher for central populations [3], leading to the expectation of greater dispersal to
and from central sites relative to peripheral locations. Immigration from outside the study area might be more likely to
contribute to dynamics of peripheral populations. Also,
some attributes of our set of local populations suggest a possible source –sink structure: two peripheral populations are
adjacent to dirt roads, potentially subjecting kangaroo rats
to additional mortality. Another two peripheral populations
suffered periods of local extinction during the late 1980s
and early 1990s. Thus, it seems that peripheral populations
may contribute less to the overall system than central populations. Moreover, when we originally selected the two
Proc. R. Soc. B (2012)
central populations for study 30 years ago [25], we naturally
chose an area with a conspicuously high rate of mound occupancy, and thus might have chosen local populations that
contribute more to the overall system.
Second, we constructed models designed to ask whether
overall metapopulation density, defined by number of animals
per unit area within the metapopulation, influences patterns of
demographic interaction among local populations. Relative
contributions of immigration, local recruitment and survival
might depend on system-level density. Emigration from
high-density populations is often predicted to be greater
than from low-density populations [13,14], and if survival of
emigrants is density-independent, demographic contribution
of immigrants to a local population should be greater when
density is high within the metapopulation. Possible mechanisms of positive density-dependence of dispersal include
increased competition for resources in natal populations [33]
or increased social interactions [34]. However, juvenile
kangaroo rats that disperse within populations survive better
at low density [35], suggesting the opposite prediction:
immigration might be more important when overall metapopulation density is low [14]. Possible mechanisms for negative
density-dependence include increased aggression towards
immigrants [13] or conspecific attraction [36]. Hypotheses
about system-level population density effects were based on
our expectation of considerable synchrony among the local
populations [37]. Because our local populations are close to
each other, they experience similar exposure to environmental
variation and densities are highly correlated among local populations (median Spearman’s r among dyads of our eight local
populations 0.62, range 20.12 to 0.91). In addition to this
specific question about overall metapopulation density, our
analysis included local populations that differed greatly in
abundance. Without evidence of substantial variation among
local populations in rates of survival, in situ recruitment and
emigration [28], we hypothesized that larger local populations
would tend to contribute more to overall metapopulation
growth than smaller populations.
Third, our previous results included best forward-time
models of both survival and dispersal that incorporated age
[28], so we modelled seniority as a function of age (i.e. grs(1)
= grs(0)). Because juveniles and adults occur in similar numbers, and because juveniles disperse more frequently than
adults [28], we expected greater contributions of juveniles
than adults to other populations. However, we predicted
that overall contributions to the entire metapopulation
would typically be larger for adults, based on the tendency
for adult survival rates to be higher than those of juvenile animals [28]. Previous forward-time results also provided strong
evidence that dispersal between local populations depended
on distance between them [28]. Distance was thus incorporated into all of our models, providing direct inference
about the strength of the relationship between distance and
between-population contributions for both ages.
To assess whether centrality or density substantially influences demographic interaction patterns, we compared
models that included these factors with models that did
not. In all models, we allowed contributions to population
growth that incorporate dispersal to depend on age and distance [28], while contributions to growth from within the
same local population depended only on age (i.e. adults present could include both surviving adults from the same
population in the previous year, or recruited juveniles from
the same population the previous year).
Reverse-time analysis of a metapopulation J. S. Sanderlin et al. 483
(c) Inference methods
We first define a metric reflecting the contribution of each of
our local study populations to the metapopulation. This
metric is defined for the eight-population study system and
does not include emigrants that may contribute to extrasystem locations. We can, however, estimate contributions
to the system via immigration from outside it. We define
rsðlÞ
the seniority parameter gt as the probability that an adult
present in local population r at time t was an animal of age
l(0 ¼ juvenile, 1 ¼ adult) in local population s at time t 2 1.
Seniorities over all time periods, g^ rsðlÞ , provide estimates of
the contributions of juveniles and adults from every population to each focal population in the metapopulation.
Growth rate of the entire metapopulation, lSt , is:
P8
r
Ntþ1
lSt ¼ Pr¼1
;
ð2:1Þ
8
r
r¼1 Nt
where Ntr is the number of adults in local population r at
time t. The contribution of local population 1 to growth of
the system involves both individuals that remained in local
population 1 between t and t þ 1, with expectation
11ð1Þ
11ð0Þ
1
Ntþ1
ðgtþ1 þ gtþ1 Þ, and individuals that moved from local
population 1 to local populations 2 to 8 between t and t þ
P8
r1ð1Þ
r1ð0Þ
r
1, with expectation
r¼2 Ntþ1 ðgtþ1 þ gtþ1 Þ. The proportional contribution of local population 1 to the
numerator of equation (2.1), and hence to growth of the
metapopulation, is:
c1t
P8
¼
r1ð1Þ
r¼1
r1ð0Þ
r
Ntþ1
ðgtþ1 þ gtþ1 Þ
:
P8
r
r¼1 Ntþ1
ð2:2Þ
The numerator of equation (2.2) expresses the number of
animals in the system at time t þ 1 that were either surviving
adults from local population 1 that remained in the system
between times t and t þ 1, or new recruits at t þ 1 to all
other local populations that were produced in local population 1 at t. A similar expression can be written for the
relative contribution of each local population. More generally, ^c s estimates the average overall contribution of each
local population, s, to the growth of the entire metapopulation
over all time periods.
If the study system is closed (i.e. dispersers from any local
population emigrate to one of the eight component local
populations and no immigrants from outside the metapopulation contribute to any component local populations), then the
sum of expression (2.2) over all eight local populations equals
P
1, 8s¼1 cst ¼ 1. More generally, the proportional contribution
of immigrants from outside the study system (denoted with
superscript 0) to population growth between t and t þ 1 is:
c0t
P8 P8
¼1
s¼1
rsð1Þ
rsð0Þ
r
Ntþ1
ðgtþ1 þ gtþ1 Þ
:
P8
r
r¼1 Ntþ1
r¼1
ð2:3Þ
When c0t ¼ 0, the system is closed and all contributions to
growth come from system components. Similarly, for any
specific local population, r, proportional contribution to
growth of that local population attributed to immigration
from outside the study system is:
r0
gtþ1
¼ 1
8
X
rsð1Þ
rsð0Þ
ðgtþ1 þ gtþ1 Þ:
ð2:4Þ
s¼1
We obtained estimates required to compute the contribution metrics of expressions (2.2)–(2.4) from kangaroo
Proc. R. Soc. B (2012)
rat capture– recapture data using reverse-time multi-state
modelling [19], with state defined by both age and local
rsðlÞ
population. We estimated seniority parameters, gt , and
rðlÞ
time- and state-specific capture probabilities, pt , using
this approach. We used estimates of capture probability to
estimate time- and state-specific adult abundance:
rð1Þ
^ r ¼ nt ;
N
t
^prð1Þ
t
ð2:5Þ
rð1Þ
where nt is the number of adults captured at time t in local
population r.
All expressions are time-specific to this point, but we were
interested in overall metrics associated with contributions of
local populations over the entire study duration. Therefore,
we computed expressions identical to (2.2) and (2.3),
except that we substituted average estimated local population
^ r ) for time-specific abundance and timeabundance (N
constant seniority parameter estimates (^
g:rsðlÞ ) for time-specific
values. We computed average relative contribution of local
population s to growth of the metapopulation as:
^cs ¼
P8
r¼1
^ r ð^
g rsð1Þ þ g^ rsð0Þ Þ
N
;
P8 ^ r
r¼1 N
s ¼ 1; . . . ; 8;
ð2:6Þ
where the average relative contribution of immigrants from
outside the metapopulation to metapopulation growth is
estimated as:
^c0 ¼ 1 8
X
^cs :
ð2:7Þ
s¼1
We developed a nested set of a priori capture–recapture
models designed for inference about potential sources of
variation in the local population contributions. In all
models, we modelled capture probability p as a function of
age and time (year), based on previous modelling results
[28]. Because prior investigation showed few differences
between sexes in survival or dispersal patterns, and because
the absolute number of dispersers was small [28,38], we
combined sexes in all analyses. In prior forward-time modelling
[28], the best dispersal models incorporated edge-to-edge
distance between local populations (modelled in a linearlogistic fashion); therefore, we modelled seniority parameters
that incorporated dispersal the same way. Our most general
model allowed contributions to growth of survival and immigration to vary with age (A, adults versus juveniles), location
(C, central versus peripheral), distance between local populations (D) and overall metapopulation density (De, relatively
high versus low years). We denote this model as g rr(A C De), g rs(D þ A þ C þ De), where the notation g rr(A C De) indicates that we modelled effects of age, centrality and
density on transitions within a local population in a multiplicative fashion (all interactions included; i.e. a different parameter
for each age, location and density combination). For transitions
involving dispersal between local populations, notation g rs(D þ
A þ C þ De) indicates that we modelled effects of distance
between local populations, age, centrality and density in an
additive manner (without interaction terms). The additive
structure was dictated by sample size (dispersal rates are
small), but also seems reasonable on ecological grounds.
Inferences about the above hypotheses were obtained using a
model selection approach [39,40]. We assessed goodness-of-fit
by computing the corresponding variance inflation factor, ^c, for
the most general model in the model set. We used a parametric
484 J. S. Sanderlin et al. Reverse-time analysis of a metapopulation
Table 1. Candidate model set and model selection statistics for reverse-time analysis of our kangaroo rat metapopulation in
Cochise County, southeast Arizona, USA. We modelled capture probability p as a function of age and time (year) for all
models. Seniority parameters, representing the probability that an animal present in local population r at time t was in local
population s at time t 2 1, were modelled as functions of age class of kangaroo rats (A), centrality (C), overall
metapopulation density (De) and/or distance between local populations (D). For each model, we report the log-likelihood
(log lik), number of parameters (np), quasi-likelihood Akaike’s information criterion, adjusted for small sample size
(QAICc), DQAICc and QAICc weight.
model
log lik
np
QAICc
DQAICc
QAICc weight
g rr(A) g rs(D þ A)
g rr(A C ) g rs(D þ A)
g rr(A De) g rs(D þ A)
g rr(A C ) g rs(D þ A þ C)
g rr(A De) g rs(D þ A þ De)
g rr(A C De) g rs(D þ A þ C)
g rr(A C De) g rs(D þ A þ De)
g rr(A C De) g rs(D þ A þ C þ De)
21232.950
21230.210
21230.720
21229.650
21230.720
21227.110
21227.610
21226.990
53
55
55
56
56
60
60
61
1777.261
1777.949
1778.638
1779.393
1780.839
1784.804
1785.479
1786.862
0.000
0.688
1.377
2.132
3.578
7.542
8.218
9.600
0.361
0.256
0.181
0.124
0.060
0.008
0.006
0.003
bootstrap approach by generating data as random variables governed by parameter estimates from our analysis. We computed a
Pearson goodness-of-fit statistic for the actual dataset and for
each bootstrap replicate. We estimated ^c as the ratio of the fit
statistic computed for the actual dataset to the average fit statistic
computed over all bootstrap replicates (1000). The fit of the
most general model, g rr(A C De) g rs(D þ A þ C þ De),
provided some evidence of overdispersion, ^c ¼1.48. We used ^c
to compute quasi-likelihood Akaike’s information criterion,
adjusted for small sample size (QAICc) for model selection
and to inflate model-based variance estimates [40]. All modelfitting and model-specific estimation was accomplished using
software developed by J.E.H. to implement the multi-state,
robust design, reverse-time modelling.
Because of model selection uncertainty, we computed all
rsðlÞ
rðlÞ
estimates of gt and pt using model averaging across all
models in the model set [40]. We calculated estimates of
overall contributions, ^cs , using model-averaged estimates in
conjunction with equation (2.5). Finally, we approximated
s and cs
estimates of variances for derived parameters, N
using the delta method [41] but ignoring covariance terms.
3. RESULTS
Our effective sample size (total number of releases) was
2614. The low-QAICc model for seniority was the simplest in the model set, g rr(A) g rs(D þ A) (table 1).
Under this model, the contribution of a focal population
to itself was a function of age. The contributions to a focal
population involving dispersal were a function of age and
distance. Models incorporating centrality have considerable support (summed QAICc weight 0.40). Models
incorporating population density have weaker support
(summed QAICc weight 0.25), and those incorporating
both density and centrality have almost no support.
Model-averaged contribution estimates g^ rsðlÞ are presented for the two sets of years reflecting relatively high
(electronic supplementary material, table S1) and low
(electronic supplementary material, table S2) density
and together (figure 2). At both high and low overall
metapopulation densities, self-contributions to each
local population are greater for adults than juveniles
(^
g rrð1Þ . g^ rrð0Þ ). By contrast, contributions from each
local population to others are greater for juveniles than
adults (^
g rsð0Þ . g^ rsð1Þ ; r = s). Demographically, local
populations are largely independent of others within the
Proc. R. Soc. B (2012)
system: self-contributions are one to two orders of magnitude larger than contributions via dispersal.
Within each overall metapopulation density level and
for each age, estimated self-contributions are larger for
central populations 1 and 2, than for peripheral populations 3– 8. As a corollary, estimates of immigration
from outside the study system are always larger for peripheral (range 0.23 –0.29) than for central populations
(range 0.18 – 0.25) (figure 2b, electronic supplementary
material, tables S1 and S2). Estimates of slope parameters
(b^ ) associated with the relationship between distance and
g^ rsðlÞ are negative for all candidate models, reflecting
reduced contributions from dispersal with increasing distance; populations separated by more than 0.25 km
contribute essentially nothing to each other (figure 2c).
Estimates of relative self-contributions are larger for
both age classes in the central populations during years
of low overall metapopulation density than during years
of high density (figure 2a; electronic supplementary
material, tables S1 and S2). This is true of juveniles in
peripheral populations as well, but not for adults.
Point estimates of overall relative contributions of local
populations to metapopulation growth ranged from 0.02
to 0.19 (electronic supplementary material, table S3
and figure 3). Thus, some local populations contributed
roughly 10 times more animals to the metapopulation
than others. This variation among local populations was
positively associated with local population adult abundance (figure 3). Average adult abundance for the entire
system during relatively high-density years was 115.6
(d
s:e: 3:6) animals; abundance during low-density years
was 71.5 (d
s:e: 1:4) animals. Estimated extra-system contribution to metapopulation growth was larger during
higher density years, ^c0 ¼ 0:27ðd
s:e: 0:09), compared
with lower density years, ^c0 ¼ 0:23ðd
s:e: 0:05). However,
estimated contributions from within-system dispersal,
g^ rsðlÞ , were slightly larger during low-density years
(electronic supplementary material, tables S1 and S2).
4. DISCUSSION
The reverse-time approach provides a compact and complete description of the roles of recruitment and dispersal
within our kangaroo rat system, and has wide potential for
inferring demographic contributions of local populations
in metapopulations. Our capture–recapture models allowed
Reverse-time analysis of a metapopulation J. S. Sanderlin et al. 485
(b)
low
high
0.5
0.3
0.1
∧r0
γ
∧rr
γ
(a)
C
high
12345678 12345678
population
P
C
P
population
(c)
low
0.5
0.3
0.1
high
low
∧rs
γ
0.020
0.015
0.010
0.005
0.5
1.0
1.5
0.5
1.0
1.5
distance (km)
Figure 2. Model-averaged estimates (95% CI) of seniority parameters from our reverse-time analysis of a kangaroo rat metapopulation in Cochise County, southeast Arizona, USA for high and low overall metapopulation density years for juveniles
(grey diamonds) and adults (black diamonds). These represent estimated contributions to growth of each focal population
from (a) themselves or self-contributions, g^ rr , (b) outside system immigration, g^ r0 , and (c) non-focal populations, g^ rs , grouped
by distance from the non-focal populations. We classified local populations 1 and 2 as central (labelled as ‘C’) and local populations 3–8 as peripheral (labelled as ‘P’).
∧r
N
(a)
30
20
10
1
∧r
C
(b)
2
3
4
5
6
7
8
5 6 7
peripheral
8
0.20
0.15
0.10
0.05
1 2
central
3
4
Figure 3. Model-averaged parameter estimates (approx. 95%
CI) from our reverse-time analysis of a kangaroo rat metapopulation in Cochise County, southeast Arizona, USA. For
both high (black circles) and low (grey circles) overall metapopulation density years, we include: (a) abundance estimates
(95% CI) for each local population r, and (b) relative contributions of each local population r to growth of the entire
metapopulation (95% CI).
us to make inferences about important sources of variation
in the contributions of each local population to every local
population in the metapopulation, grsðlÞ . These estimates
are unique within the metapopulation literature, and provide important details about location-specific processes
contributing to this metapopulation. Inferences about
these processes and extra-system immigration can further
be used to make inference about the contributions of each
local population to the entire system. Contribution metrics
provide a currency by which values can be assigned to local
populations for conservation prioritization. Similarly, potential management actions directed at local populations can be
assessed by the degrees to which they are expected to
increase overall metapopulation growth. Finally, we believe
this decomposition of metapopulation change into
Proc. R. Soc. B (2012)
components associated with local populations represents
an important step towards developing a general theory of
metapopulation dynamics applicable to many taxa.
As predicted, estimates of adult self-contributions were
greater than those of juveniles, whereas estimates of contributions involving interpopulation dispersal were greater
for juveniles. Juveniles are more likely to disperse than
adults in many vertebrates [42], so asymmetry in agespecific contributions involving interpopulation dispersal
may be a general phenomenon. Estimated contributions
to the entire system were greater for adults. Contributions
via dispersal to the growth of each local population were
greater from other local populations that were nearby.
Although this result was expected and is widely claimed
in metapopulation ecology literature [3], our estimates
are among the first to quantify it.
Population ‘centrality’ was a relatively important
source of variation in the contribution metrics, as inferred
from model selection. Based on estimates of g^ r0 , central
populations relied less on immigration from outside the
study system than peripheral populations. Estimated contributions of central populations to the entire system
were large, but did not always exceed estimated contributions of peripheral populations. If centrality is an
important determinant of local population contributions
in other metapopulations, this feature would have direct
implications for reserve design [5,6], where central populations may be more important. We note that the ideas
tested here for a specific metapopulation could have implications for larger scales, such as species’ geographical
ranges. Populations located in the interior of a species
range are typically characterized by higher abundances
and lower probabilities of local extinction and turnover
than local populations near range edges (e.g. [43]).
Ranked contributions of each local population to
growth of the entire system, ^cr , were consistent with
ranked average abundances during both low and high
486 J. S. Sanderlin et al. Reverse-time analysis of a metapopulation
overall metapopulation density years. This commonsense
result, that larger local populations make larger contributions to the metapopulation, is predicted for this
system because local populations have similar survival
and reproductive rates. This result is likely to be a
common feature of metapopulations and has clear
implications for reserve design [5].
Surprisingly, our results do not suggest a strong
relationship between metapopulation density and contributions resulting from immigration, counter to other
species with density-dependent dispersal [44], and predictions that density-dependent dispersal should evolve
in many metapopulation systems [45]. Estimated contributions from within-system dispersal were slightly larger
in years of low overall metapopulation density, consistent
with our previous result that juvenile kangaroo rats which
disperse within local populations survive better at low
density [35]. However, point estimates of extra-system
contributions were greater in years of higher overall metapopulation density, and both effects of density were weak.
Small estimated contributions to growth of any local
population by other local populations within the study
system were consistent with the low rates of dispersal estimated for this system [28]. However, in the light of the
low dispersal rate and the inverse relationship between
dispersal probability and distance, we were surprised at
the large estimated contributions of immigration from
outside the study system. Four local populations exist
just outside our study area (figure 1), and contributions
from ‘outside’ are greater to peripheral than to central
populations (figure 2b; electronic supplementary material,
tables S1 and S2) by an amount comparable with that
expected if peripheral populations gain immigrants from
such populations. However, even central populations have
large extra-system contributions. In Skvarla et al. [28], we
found that fewer than one out of 100 animals dispersed
the greater than or equal to 1 km needed to reach a central
population from outside our study area.
We can think of two ways in which flawed assumptions
in our analysis might contribute to unexpectedly large
estimates of immigration from outside the study system.
First, dispersal from far outside the system might be
more common than that observed within the study site
[28,46]. But the distribution of active kangaroo rat
mounds outside our study site was not obviously different
from that within it, i.e. we located no obvious ‘sources’ in
the surrounding area. Estimates of immigration from
‘outside’ seem far too large (nearly half the magnitude
of contributions from juvenile recruitment in situ) to
be accounted for by long-distance dispersal from an
undiscovered source, and we believe that a second,
more cryptic flaw in our assumptions is responsible. We
assumed that all surviving young animals are available
for sampling in July –August. If some late-born young
are not independent and exposed to traps during this
sample period, then these late entries would be ‘assigned’
to extra-system immigration.
Although not explicitly described with our application
owing to ambiguity of the source of ‘outside immigration’,
our approach can also be used to detect sources and sinks
[8] in other ecological systems where extra-system contributions are negligible. For example, we can use a metric
C r, defined in Runge et al. [11], to identify local population r ¼ 1 as a source or a sink using contributions of
Proc. R. Soc. B (2012)
local population 1 to all eight local populations at time
t þ 1 and the population size of local population 1 at
time t:
Ct1
P8
¼
r¼1
r1ð1Þ
r1ð0Þ
r
Ntþ1
ðgtþ1 þ gtþ1 Þ
;
Nt1
ð4:1Þ
where C r , 1 indicates that local population r is a ‘sink’
and C r . 1 indicates a ‘source’. C rt can be viewed as the
per capita contribution of individuals in local population
r to the abundance of the metapopulation in the next
time period.
The main data requirements for our approach are
capture–recapture or re-observation data collected at
multiple local populations. Data from a single population
permit inference on the contribution to growth of that
population made by internal (survival, in situ recruitment)
versus external (immigration) processes [19]. However,
application to a single population does not permit inference about the value of locally produced individuals
that emigrate to other locations. Our example depended
on the ability to identify young as having been produced
in the local population of capture. In cases where this is
not possible, we might use data from other sources (e.g.
genetic assignment tests [47]) to determine local population of origin for juveniles [48]. Our approach requires
the ability to estimate juvenile capture probability,
which can be provided by the robust design [49,50].
Applications to some species (e.g. seabirds [51]) will
include juveniles that can be marked during the year of
hatching but do not return to the breeding colony for
possible re-observation until their first year as a breeder,
which may be several years later [52]. We should be
able to deal with this situation by expanding the state
space of the reverse-time multi-state modelling to include
breeding adults from the previous time step in each local
population, as well as new breeders hatched in different
years from the various local populations.
One potential limitation to our method is that our
contribution metric does not include emigrants that may
contribute to extra-system locations (see §2c). This could
lead to underestimation of the population’s contribution to metapopulation growth if there is more dispersal
to extra-system populations than to within-system populations, suggesting this approach is ideally applied to
metapopulations completely closed to extra-system immigration. Otherwise, contributions of populations located
peripherally within the metapopulation may be more
prone to underestimation than those of central populations.
Study design and scale are important considerations for use
of these and any methods. The relevance of extra-system
contributions can sometimes be inferred from ancillary data
(e.g. radio-telemetry, tag-resight, tag-recovery [41,53]). In
addition, inference about temporary emigration to extrasystem locations should be possible using our robust
design approach [54] and multi-state models that include
unobservable states (e.g. [55]). When contributions from
locations outside the study system are potentially important, such methodological extensions to our basic
approach deserve consideration.
This analytical approach has clear use for identifying
where a population lies on the ‘metapopulation continuum’ demographically, such that magnitudes of g^ rsðlÞ
provide information about the relative importance of
Reverse-time analysis of a metapopulation J. S. Sanderlin et al. 487
We received support from the National Science Foundation
(DEB 0816925). We thank numerous laboratory members
for help with field sampling and three anonymous reviewers
for comments on previous versions of the manuscript.
metapopulation
single
population
0
0.2
discrete
populations
0.4
0.6
0.8
1.0
∧
g rr
Figure 4. Population continuum using average seniority estimates from our central (unfilled triangle) and peripheral
(filled triangle) populations located in a kangaroo rat metapopulation in Cochise County, southeast Arizona, USA.
Average estimates from within-system contributions were
rescaled to sum to 1 to remove ambiguity from outside
system contributions. Self-contributions close to 1 indicate
that local populations, r, are discrete populations, whereas if
self and non-focal population contributions are approximately
equal, the entire system could be classified as a single
population. Intermediate values indicate metapopulation
structures.
dispersal among local populations as determinants of
population growth. For example, if g^ rrð1Þ þ g^ rrð0Þ ! 1,
then we would conclude that local population r is relatively independent and might be best viewed as a
discrete population, whereas g^ rr g^ rs ; r = s, would
suggest that the entire system be viewed as a single population. Situations intermediate between these extremes
would reflect more classic metapopulation structures.
Assuming that estimated extra-system contributions are
primarily owing to late-born young (thus rescaling all
within-system contributions to sum to 1), the kangaroo
rat system is much closer demographically to a set of independent populations than to a metapopulation (figure 4).
This latter result might appear to contrast with previously reported patterns of genetic connectivity in the
same system. Despite their demographic insignificance,
cases of dispersal are frequent enough to limit genetic
differentiation among local populations (0.01 , Fst ,
0.03 during the years of this study) and to prevent the
expected loss of alleles during demographic bottlenecks
[29,30]. This comparison supports the concept that
different levels of immigration are needed for meaningful
demographic connectivity compared with genetic connectivity [12]. More specifically, the number of immigrants
required to make important genetic contributions is
much smaller than required to make substantive demographic contributions. The genetic and demographic
consequences of dispersal are thus particularly likely to
differ in species such as this one where overall dispersal
rates are low. Ideally, we would combine genetic and
demographic information, as alluded to by Lowe &
Allendorf [12], to form an encompassing view of a metapopulation. Regardless of how a metapopulation is
ultimately defined, our reverse-time approach improves
the demographic assessment of connectivity and provides
a framework for testing hypotheses about demographic
structure.
All fieldwork was approved by the Purdue University Animal
Care and Use Committee (PACUC no. 01–083) and the
Arizona Game and Fish Department.
Proc. R. Soc. B (2012)
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