Ecological Modelling 179 (2004) 77–90
On optimal choices in increase of patch area and reduction of
interpatch distance for metapopulation persistence
Rampal S. Etienne∗
Community and Conservation Ecology Group, University of Groningen, Box 14, 9750 AA Haren, The Netherlands
Received 24 September 2003; received in revised form 14 April 2004; accepted 3 May 2004
Abstract
Metapopulation theory teaches that the viability of metapopulations may be enlarged by decreasing the probability of extinction of local populations, or by increasing the colonization probability of empty habitat patches. In a metapopulation model
study it has recently been found that reducing the extinction probability of the least extinction-prone patch and increasing the
colonization probability between the two least extinction-prone patches are the best options to prolong the expected lifetime of
a metapopulation. In this article I examine with a more detailed model whether this translates into enlarging the largest patch
and reducing the interpatch distance between the largest patches. Using two measures of metapopulation persistence (longevity
and resilience) I found, firstly, that the largest patch should generally be enlarged if the choice is to enlarge a patch by a certain
percentage (relative change) and that the smallest patch should generally be enlarged if the choice is to enlarge a patch by a fixed
amount of area (absolute change), and secondly that indeed one should reduce the interpatch distance between the two largest
patches, if the choice is among all pairs of patches. The strength of these rules of thumb (particularly the second part of the
first rule) depends on the parameter values, particularly the amount of clustering of patches, the mean dispersal distance and the
number of dispersers. Also, the rules of thumb are less pronounced when resilience is chosen as a measure of metapopulation
persistence than when longevity is chosen.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Conservation; Metapopulation; Time to extinction; R0 ; SPOM
1. Introduction
Fragmentation of habitat is considered a major
threat to species persistence. Populations in the fragments are much more prone to extinction than in
large continuous habitat unless the fragments are so
well connected that fragments can be frequently recolonized from other fragments. The metapopulation
concept (Levins, 1969, 1970) describes the balance
of local extinctions and recolonizations whereby a
species can persist much longer in the entire network
∗ Tel.: +31 50 363 2230; fax: +31 50 363 2273.
E-mail address: [email protected] (R.S. Etienne).
of fragments (called patches) than in any single patch.
This insight has led to the attitude that fragmentation
should be counteracted by increasing connectivity,
for example by constructing corridors (ecoducts) or
stepping stone patches. However, increasing patch
size or quality is another option. The questions for
managers are where to increase connectivity and
where to increase patch size/quality. Etienne and
Heesterbeek (2001) showed that increasing the connectivity between the patches with the smallest extinction probabilities and decreasing the extinction
probability of the patch with the lowest extinction
probability are the two optimal strategies answering
these questions. However, their analysis was fully
0304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2004.05.003
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R.S. Etienne / Ecological Modelling 179 (2004) 77–90
in terms of extinction and colonization probabilities
which a nature manager can only influence indirectly
by altering landscape characteristics such as patch
area (or patch quality) and (the effective) interpatch
distance. In terms of these landscape characteristics, Etienne and Heesterbeek (2001) remark, matters
might be different, because a change in one such
landscape characteristic may affect both extinction
and colonization probabilities simultaneously, and
presumably to different extents. In particular, the local
extinction probabilities of patches of different sizes
may react differently to changes in patch size.
It is not obvious, however, how extinction and colonization probabilities are related to landscape characteristics, because a variety of processes may underlie
these relationships (e.g. demographic and environmental stochasticity in local population dynamics, random
and directed dispersal, see Etienne and Heesterbeek,
2000). Especially our knowledge of dispersal is still
rather limited which aggravates establishing these relationships. Furthermore, in taking action, the financial
picture is a very important, yet highly uncertain and
variable, aspect as well. Hence, practical questions,
such as “is it better to enlarge patches or decrease (the
effective) interpatch distance”, “which patch should be
enlarged”, and “which distance should be decreased”,
seem to allow for an answer only in single cases. General answers, rules of thumb, which are very useful if
time and money are limited, have not been provided.
In this paper I will try to fill this gap and answer
the last two practical questions by constructing artificial landscapes for which I determine the conservation strategy which is optimal for metapopulation persistence, employing two measures of metapopulation
persistence, one measuring the longevity of an existing metapopulation and one measuring the resilience
of a species in a network when only one patch is initially occupied. I will use the two above-mentioned
landscape characteristics (patch area and interpatch
distance), and I will link them to the local extinction
and colonization probabilities. I chose patch area and
interpatch distance, because these are thought to be
the most important landscape characteristics for the
metapopulation processes of local extinction and colonization (Hanski, 1999a). In my analysis, patch area
may also be interpreted as patch quality which is the
third major player in the metapopulation field (Thomas
et al., 2001), although the exact link with extinction
and colonization is less well studied, and although
patch quality and size are sometimes assumed to be
inversely proportional (Walters, 2001).
The first practical question, “is it better to enlarge patches or decrease (the effective) interpatch
distance”, is not addressed in this paper. I do recognize that it is a very important question, but the
question thus stated asks for a comparison of apples
and oranges. For example, one could compare the
consequences of a 10% increase in patch size with
the consequences of a 10% decrease in interpatch
distance, but the results are meaningless, unless there
is a “currency” exchange rate between changes in
patch size and in interpatch distance, or unless one
considers specific scenarios (Drechsler and Wissel,
1998). Etienne and Heesterbeek (2001) performed
such a comparison on the level of extinction and colonization probabilities, because they could use the
common currency of probability, but in fact the currencies are then only mathematically exchangeable,
rendering the meaning of their result still questionable. The required exchange rate can only follow
from a socio-politico-economical analysis which is
very case-specific and beyond the scope of this paper.
2. Methods
2.1. Generating artificial landscapes
Although there are many sophisticated ways of generating landscapes including fractals (Johnson et al.,
1992; Andrén, 1994; Hargis et al., 1998; Meisel and
Turner, 1998; Hokit et al., 2001), for my purposes
a simple algorithm suffices. I used a 128 ×128 grid
each cell of which can be the center of a circular
patch. I first chose the number of patches (n = 5)
and then assigned each patch a patch area according
to a lognormal probability distribution (Hanski and
Gyllenberg, 1997) with mean log Am and standard deviation log rA . I chose one combination, (log Am =
log 25, log rA = log 2.5) which yields patch areas
which are usually larger than the minimum patch area
(see below), but sufficiently small so that the metapopulation approach is warranted (local extinction probabilities should not be too small, see below). The first
patch is then placed randomly anywhere in the grid.
The second patch is placed in the grid according to a
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
Fig. 1. Examples of artificially generated landscapes. (A) Randomly distributed patches. (B) Extremely clustered patches
(σ = 10).
Gaussian probability distribution centered around (but
not in) the first patch. By changing the value of the
standard deviation, σ, of this distribution, the amount
of clustering of patches can be tuned; I picked 10, 20,
40 and ∞ (uniform). For the other patches this is repeated with the probability distribution being the normalized sum of contributions from all patches already
placed in the grid. For an example of the landscapes
generated in this way, see Fig. 1. For the resulting landscapes centroid distances and edge-to-edge distances
can easily be calculated. I used the latter which seems
biologically more realistic, but I believe that using the
former would not change the results qualitatively.
2.2. Model: metapopulation processes
I used the stochastic patch occupancy model
(SPOM) in discrete time with separation of extinction
and colonization phases (Akçakaya and Ginzburg,
1991; Day and Possingham, 1995) which is also
used in Etienne and Heesterbeek (2001). See also
79
Gyllenberg and Silvestrov (1994) who present a
similar discrete-time model, but without separation
of extinction and colonization phases; I found in a
brief investigation of this model that it behaves very
similarly. As these references provide a detailed description of the model, I only summarize its most
important properties here. Consider a single-species
metapopulation distributed over n patches, which
can be either occupied or empty. There is a discrete phase in which local population dynamics take
place, but no dispersal. After this “extinction phase”
there is a “colonization phase”. During the extinction
phase, the population in each occupied patch i has
an extinction probability ei and during the colonization phase, dispersers from all occupied patches can
colonize an empty patch i with colonization probability ci which depends on the occupied patches j.
All these probabilities are considered to be independent, that is, I assume that extinctions and colonizations are not correlated. I will say more on this in
Section 4.
Because every patch is either occupied (denoted by
1) or empty (denoted by 0), the metapopulation is in
any of 2n states. With the extinction and colonization
probabilities of all patches one can calculate the probabilities of the transitions between any two states. One
can show (see e.g. Halley and Iwasa, 1998) that the
second largest eigenvalue λ2 of this transition matrix
(M) is a measure of the expected extinction time of
the metapopulation:
Text =
1
.
1 − λ2
(1)
This extinction time is an average over the extinction
times of all initial states where each state is weighed
according to the so-called quasi-stationary distribution
(Darroch and Seneta, 1965; Gilpin and Taylor, 1994;
Gosselin, 1998; Pollett, 1999) which is the probability
distribution of states for a system in quasi-equilibrium.
For any probability distribution qi over the states i the
expected extinction time can be calculated as
−
→
(2)
qi ((I − RM )−1 I )i
Text (q) =
i
where RM is the matrix which results when the first
−
→
row and the first column are deleted from M and I
n
is a vector of length 2 − 1 whose entries are all equal
to 1.
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R.S. Etienne / Ecological Modelling 179 (2004) 77–90
The expected metapopulation extinction time Text is
one of the two measures of metapopulation persistence
used in this paper. The other measure follows from the
reproduction matrix G (Gyllenberg, 2004). This matrix consists of elements Gij which give the probability that occupancy of patch j is produced by patch i (if
occupied) after one time-step, where all other patches
are empty. Gii is then the probability of patch i not
going extinct: Gii = 1 − ei , and the Gij are the probabilities that patch j is colonized by patch i: Gij =
cj (i). The dominant eigenvalue of this matrix, R0 , is
called the basic reproduction ratio. It can be shown
(Gyllenberg, 2004) that if a “typical” local population
is placed in an otherwise empty set of patches, this
will lead to an initially growing metapopulation if and
only if R0 > 1. The left eigenvector corresponding to
R0 is the stationary type-distribution of newly colonized patches, and a “typical” local population means
a population sampled from this distribution. Because
n may be small and because colonization usually decreases with distance which causes clustering of occupied patches, even after only one time-step the patch
network no longer satisfies the conditions for initial
growth anymore. Hence, R0 may be of very limited
practical use. Yet, I include it, because it is a measure
of invasion rather than of longevity (such as Text ), so
the two measures together give a broader picture of
overall metapopulation persistence. A measure comparable to R0 is the metapopulation capacity (Hanski
and Ovaskainen, 2000; Ovaskainen and Hanski, 2001).
2.3. Model: relations with landscape characteristics
I now go further than Etienne and Heesterbeek
(2001), because at this point I establish relations between extinction and colonization probabilities on the
one hand, and patch area and interpatch distance on
the other hand. I assume first, fairly realistically, that
the carrying capacity of a patch Ki is proportional
to its area Ai with proportionality constant ρ. I furthermore assume, following Hanski (1999a), that the
local extinction probability ei in a patch of area Ai
takes the form
x A0
ei = min 1,
(3)
Ai
where A0 is the minimum required patch area and x is
a parameter which measures the strength of environ-
Table 1
Parameter values used in calculating the optimal conservation
strategy
Parameter (unit)
Values
Default
log Am
log rA
A0
σ (grid cell lengths)
x
α−1 (grid cell lengths)
Rlocal
0
bρA0
r
10
20
4
40
0.5
8
1.1
0.1
log 25
log 2.5
1
∞
1
16
1.5
1
0
1.5
32
2
10
0.5
2
64
4
1
The wide central column contains the default parameter values;
the other columns contain alternative values used in replacing one
default parameter value at a time. Most parameters are dimensionless; for those that are not, the unit is mentioned explicitly, unless
it is irrelevant (as in the case of A0 ).
mental stochasticity and demographic stochasticity:
the larger the value of x, the weaker is the environmental stochasticity. For pure demographic stochasticity
and sufficiently large Ai , one can show that the local
extinction probability actually declines exponentially
with area for both exponential growth with a ceiling and logistic growth (Foley, 1997, Andersson and
Djehiche, 1998), but this can be mimicked heuristically by taking large x. For environmental stochasticity or pure demographic stochasticity with small Ki
the power-law dependence (3) can indeed be derived
(Goel and Richter-Dyn, 1974; Foley, 1997, see also
Hanski, 1999b). The values of x that I will use are
listed in Table 1; they cover a realistic range of values
(Foley, 1997). Because only the value of Ai relative to
A0 matters, one can view their ratio as a new variable
or, equivalently, set A0 = 1. The latter interpretation
has the advantage that the grid resolution is precisely
such that one grid cell has area A0 .
Deriving expressions for the colonization probability is more difficult, because it involves the still poorly
understood dispersal behavior. Let us start with the
target patch i in which mi immigrants arrive. I define
a successful colonization as the event that the population reaches a certain critical level
Nc =
3
log Rlocal
0
,
(4)
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
where Rlocal
is the local (within-patch) basic reproduc0
tion ratio. The probability of this event equals (Goel
and Richter-Dyn, 1974)
mi
1
Ci (mi ) = 1 −
.
(5)
Rlocal
0,i
This formula assumes asexual reproduction. In sexual
reproduction, Allee effects in mate finding may reduce the probability of colonization for small m. This
is modelled phenomenologically by a sigmoidal curve
in the incidence function model (Hanski, 1994) or the
logistic growth model (e.g. Brassil, 2001). A mechanistic model along the lines of (5) incorporating the
Allee effect can be constructed, but it requires more
detailed assumptions (e.g. on monogamous or polygamous reproduction, the process of mate finding and
its relation to patch size). However, such a detailed
model is neither necessary nor warranted within the
scope of this paper (see Section 4). For simplicity I
also assume that Rlocal
0,i is the same for all patches, regardless of their size, which is not unreasonable because patch size is not a likely to be a limiting factor
for the initial growth of a population.
I will now derive a formula for the number of immigrants mi . Let us assume that the number of emigrants
is proportional to the carrying capacity, and thus to
the area of a patch with proportionality constant bρ. If
an emigrant disperses in a random direction, then the
probability that it heads for patch i with area Ai equals
the fraction of the horizon that this patch occupies
(the so-called pie-slice algorithm, see Etienne and
Heesterbeek, 2000. Assuming that patches do
not
√ overlap on the horizon, this fraction equals
( Ai /dij )π−3/2 . If the probability that dispersers actually disperse as far as dij declines exponentially
with distance, i.e. e−αdij with α−1 the average dispersal distance, then the average number of immigrants
arriving at patch i is
e−αdij
mi =
π−3/2 bρAj Ai
.
(6)
dij
j
In principle I should have used a joint probability distribution for all the mi and summed over all possible
distributions of the emigrants over the patches, but I
experimented numerically with such a model and I
seldom obtained metapopulation extinction times that
differ largely from those calculated with the model I
81
have just described. Table 1 lists the values of the parameters I introduced; they are chosen such that they
are both realistic and consistent with the values of the
other parameters and that they do not cause numerical
problems (singularities in calculation of eigenvalues).
In testing the robustness of the results I considered
one important change of the model structure to account for the possibility of the rescue effect (Hanski,
1982, 1994). I modified the extinction probability by
multiplying it with the probability of non-colonization
raised to a power r representing the strength of the
rescue effect:
ei = ei (1 − ci )r
(7)
The case r = 0 corresponds to the model without
rescue effect. To examine the impact of the rescue
effect on the results I studied the cases r = 0.5 and 1.
2.4. Model: Output
As I mentioned above, the two measures of
metapopulation persistence are Text and R0 . I will
evaluate these measures for each of 1000 landscapes
in which I increase the area of the largest patch by
10%. I then calculate the required changes in the areas
of the other patches which would result in the same
values of the measures (starting from the original
landscape, i.e. as it was prior to the enlargement of
the largest patch). Similarly, I evaluate these measures
for each landscape in which I decrease the interpatch
distance between the two largest patches by 10% and
compute the required changes in the distance between
all other pairs of patches which result in the same
values of the measures. As I stated above, the results
by Etienne and Heesterbeek (2001), in terms of extinction and colonization probabilities suggest that
the required changes will often be larger. However, in
terms of patch area and interpatch distance, matters
may be different, because, for instance, the extinction
probability is least sensitive to changes in patch area
for large patches according to the model (formula (3)).
In finding the required changes in patch area and
interpatch distance, I did not modify the landscape.
Evidently, changing patch area would realistically result in different edge-to-edge interpatch distances; it is
even conceivable that two or more patches merge into
a single large patch. Because merged patches lead to
a totally different landscape causing discontinuities in
82
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
the measures of metapopulation persistence which are
not necessarily realistic, I refrained from implementing this (although merging patches as a management
strategy deserves further study). Changes in patch area
should be interpreted as changes in the effective patch
area which is a result of geographical area (which remains unaltered) and patch quality. Similarly, changing interpatch distance would be realized by moving
patches, thus modifying other interpatch distances as
well. Moving patches is of course generally impossible, so a change in interpatch distance should be interpreted as a local change in α, or in other words,
in the effective interpatch distance. One may envisage
that corridors create such changes (although, admittedly, corridors may lead to local changes in α in other
places as well). I furthermore believe that the value of
10% is sufficiently large to be realistic and sufficiently
small that the above-mentioned landscape changes are
negligible in most cases; in the cases where they are
not, the required change is probably so large that it is
not realistic in itself.
3. Results
For the default parameter set the required relative
changes in patch area and interpatch distance observed
in the 1000 landscapes are shown in Figs. 2 and 3 respectively. The boxplots summarize the statistics in the
1000 landscapes: the endpoints denote the minimum
and maximum observed values, the top and bottom
represent the 2.5 and 97.5 percentiles, while the lines
inside the box represent the 25, 50 and 75 percentiles.
For example, according to panel A, enlarging the second largest patch is a better choice than enlarging the
largest patch in 25% of the landscapes, but this advantage is almost always less than 7-fold (2.5 percentile),
whereas it is a worse choice in 75% of the landscapes,
the disadvantage being 45-fold at the 97.5 percentile.
In Figs. 4 and 5 and Table 2 the required changes
in patch areas and interpatch distances are ranked according to their size for each landscape, to facilitate
comparisons between landscapes. For example, if the
required change in patch area is the smallest, the patch
receives rank 1, if it is the largest the patch receives
rank 5 (there are five patches in the landscape) and
similarly for the interpatch distance. At the same time,
patches are ordered according to their patch area and
Fig. 2. Boxplots of the required change in patch area (in %) to
obtain the same metapopulation extinction time (panel A) or the
same R0 (panel B) as when the largest patch is enlarged by 10%.
A boxplot is shown for each of the five patches in the landscape,
ordered according to their size: for the largest patch, the second
largest patch, etc. In each boxplot, the endpoints represent the
minimum and maximum required change observed in the 1000
landscapes; the top and bottom of the box represent the 2.5 and
97.5 percentiles and the lines inside the box represent the 25,
50 (median) and 75 percentiles. Evidently, for the largest patch
all these percentiles coincide, because the largest patch is always
enlarged by 10%. The dashed 10%-line is shown for convenience
in comparisons between patches.
interpatch distances are ordered according to the sum
of the sizes of the patches they connect. The mean
rank over 1000 landscapes for the largest patch, the
second largest patch etc. and for the interpatch distance between the two largest patches, the two second
largest patches etc. is calculated. The required changes
are ranked in two ways: relative (i.e. in percentages)
and absolute. Figs. 4 and 5 show the dependence of
the ranks on the parameter values, while Table 2 takes
a more detailed look at the results for the default parameter set (compare with Fig. 2).
The ranks are significantly different (Friedman test,
P < 0.001). Most of the mean ranks are statistically
different from the value expected from pure chance
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
83
Fig. 3. Boxplots of the required change in interpatch distance (in %) to obtain the same metapopulation extinction time (panel A) or the
same R0 (panel B) as when the distance between the two largest patches is reduced by 10%. A boxplot is shown for the distance between
each pair of patches in the landscape, ordered according to the sum of their patch sizes:.for the distance between the two patches with
the largest sum, the two patches with the second largest sum, etc. In each boxplot, the endpoints represent the minimum and maximum
required change observed in the 1000 landscapes; the top and bottom of the box represent the 2.5 and 97.5 percentiles and the lines inside
the box represent the 25, 50 (median) and 75 percentiles. Evidently, for the distance between the two largest patches all these percentiles
coincide, because this distance is always reduced by 10%. The dashed 10%-lineis shown for convenience in comparisons between patches.
(which equals 3 in Fig. 4 and 5.5 in Fig. 5), because the
95%-confidence intervals of the mean ranks are, respectively, [2.91, 3.09] and [5.32, 5.68]. Table 2 confirms this by showing the percentage of landscapes
in which enlarging the patch with size rank i is the
optimal choice for the default parameter settings and
the probability that such a percentage is obtained by
chance: this probability is very low.
Fig. 4 shows that for both measures enlarging the
largest patch is the best option for all parameter values, if relative area changes are compared. If absolute area changes are compared, however, the opposite
holds: enlarging the smallest patch is the best option
for many, but not all parameter values and for R0 the
discrepancies are minor. Enlarging the largest patch is
more strongly supported in landscapes with low de-
Table 2
Percentage of the landscapes (with default parameter settings) in which the ith largest patch is the best one to enlarge, for the two measures
of metapopulation persistence, and for both relative and absolute changes in patch size
Measure of metapopulation
persistence
Percentage of landscapes (i)
Probability
1
2
3
4
5
Text (relative)
R0 (relative)
Text (absolute)
R0 (absolute)
55.1
21.8
6.5
2.4
26.5
22.5
12.7
17.6
10.8
21.5
20.7
23.3
6.4
20.6
29.0
28.6
1.2
13.6
31.1
28.1
1.4 ×10−140
0.36
9.2 ×10−28
6.1 ×10−9
The last column contains the probability that the percentage for the largest (relative change) or smallest (absolute change) patch is obtained
by chance relative to the probability of the expected percentage, i.e. 20%.
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
R 0 (relative)
20
40
σ
60
0
80
3
2
60
0.5
1
x
1.5
3
2
2
0.5
1
x
1.5
2
1
3
2
1
0
16
32 −1 48
α
64
16
32 −1 48
α
2
1
3
2
1
0.1
1
bρA0
10
1
bρA0
2
3
2
0
1
2
local
3
0
4
1
2
local
3
3
2
1
32 −1 48
α
3
2
3
2
1
x
1.5
2
0
16
32 −1 48
α
4
3
2
1
64
4
3
2
1
1
bρA0
10
0.1
1
bρA0
10
5
4
3
2
4
3
2
1
1
2
3
4
0
local
1
2
R
local
3
0
5
4
3
2
1
0.5
r
0.5
2
64
5
4
0
0
3
R0
1
1
80
5
0
mean rank
mean rank
4
60
1
16
4
4
5
5
0.5
r
2
R0
R0
0
3
1
1
1
40
σ
4
2
5
4
20
5
0.1
mean rank
mean rank
3
1.5
4
10
5
4
1
x
1
0.1
5
0.5
5
4
0
1
0
mean rank
mean rank
3
2
64
5
4
3
1
0
5
80
5
4
2
5
0
mean rank
mean rank
3
60
4
2
5
4
40
σ
1
0
5
20
5
4
3
1
0
80
1
0
mean rank
40
σ
mean rank
4
1
mean rank
20
5
mean rank
mean rank
5
2
mean rank
0
4
3
1
1
1
mean rank
2
5
4
mean rank
2
3
5
mean rank
3
4
R 0 (absolute)
mean rank
mean rank
mean rank
4
mean rank
5
5
mean rank
T (absolute)
mean rank
T (relative)
mean rank
84
4
3
2
1
0
0.5
r
1
0
0.5
r
1
4
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
mographic stochasticity (small values of x) and low
patch clustering (large values of σ; for the purpose
of plotting the results, the parameter σ is given the
value 80 for the completely random case where σ is
actually infinite) and for species with small dispersal
distances (small α−1 ), low emigration (small bρA0 )
and low productivity (small Rlocal
), but note that there
0
are some subtleties in the relationship with dispersal
distance (α−1 ). This suggests that the largest patch is
especially important when the patches are relatively
isolated. This makes sense, because in this situation
colonization is infrequent and the metapopulation becomes extinct when the least extinction-prone patch
does.
These results can be illustrated by the following example. If one is considering adding, say, a hectare of
habitat to one existing patch (an absolute change), then
one should do this with the smallest patch. If the decision concerns improving patch quality in one patch (a
relative change), then the largest patch is a better candidate. Adding a strip of a certain width (e.g. 100 m)
of habitat around a patch is an act which lies between
an absolute and a relative addition, which suggests
that any patch can be chosen. From simulations for
the default set I found that this is close to the truth;
the largest patch prevails slightly, the mean rank of the
largest and smallest patches being, for Text , 2.734 and
3.433, and, for R0 , 2.769 and 3.243.
Fig. 5 shows that for both measures reducing the
interpatch distance between the two largest patches is
the most fruitful option, regardless of whether the required changes are compared in a relative or absolute
way, although in the latter case, the trend is less manifest. Reducing the interpatch distance between the two
largest patches is more strongly supported in the same
cases as listed above for changes in patch area, except
one: for species with large dispersal distance (large
α−1 ) the trend is now stronger instead of weaker.
As for the rescue effect, Figs. 4 and 5 clearly show
that the ranks hardly depend on the strength of the
rescue effect r. For R0 they do not depend on r at
85
all, because R0 is a measure for initial growth where
only one patch is occupied, and thus the rescue effect
plays no role yet. The robustness of the results to the
rescue effect does not mean that the rescue effect is not
important at all. The rescue effect is strongest when
many patches are occupied, so the results which are
obtained with a relatively small number of patches
should not be blindly extrapolated to larger numbers.
The figures show that R0 and Text behave similarly
in most cases. This suggests that R0 can be used as
a proxy for Text for this type of comparison between
changes in patch areas and interpatch distances. This
is particularly interesting for larger networks in which
computing Text is a difficult and time-consuming, if
not impossible task.
Thus, although some general rules of thumb can be
extracted from these findings, they are not as strongly
supported as those found by Etienne and Heesterbeek
(2001). Patch size does not fully determine the outcome of the type of comparisons I made, because the
largest patch was not always ranked first. There is,
however, not a simple characteristic of the landscape
that does better in this respect according to my examination of the results; connectivity of a patch evidently
does play a role, but this role is by no means decisive.
4. Discussion
The results suggest the following rules of thumb for
the best strategies in metapopulation management:
1A. Enlarge the largest patch if one must choose to
increase patch size by a fixed percentage (relative
change).
1B. Enlarge the smallest patch if one must choose
to increase patch size by a fixed amount of area
(absolute change).
2. Reduce the (effective) distance between the two
largest patches if one must choose among any
pair of patches.
Fig. 4. Mean rank of the patches versus each of the parameters of Table 1 for the largest patch (thin solid line), second largest patch
(solid line with filled circles), third largest patch (gray line), fourth largest patch (dotted line) and fifth largest (smallest) patch (thick solid
line). Rank 1 is given to a patch if the required change in patch area is smallest, rank 2 is assigned if the required change is the second
smallest, etc. The mean rank is the average over all 1000 landscapes. It is calculated for each of the two measures, Text and R0 , where
the required change is compared in two ways: relative (i.e. percentage of the area) and absolute (i.e. area size). The parameter σ is given
the (arbitrary) value 80 for the completely random case where σ is actually infinite to allow it to be plotted.
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
1
br A0
0
1
2
local
3
0
4
0.5
r
1
32 -1 48
a
1
br A0
1
10
9
8
7
6
5
4
3
2
1
0
0.5
r
1
mean rank
40
s
60
0.5
1
x
1.5
mean rank
0
16
32 -1 48
a
0.1
1
br A0
4
0
1
2
3
local
R0
10
9
8
7
6
5
4
3
2
1
0
0.5
r
1
40
s
60
80
0
0.5
1
x
1.5
2
0
16
32 -1 48
a
64
10
9
8
7
6
5
4
3
2
1
10
10
9
8
7
6
5
4
3
2
1
20
10
9
8
7
6
5
4
3
2
1
64
10
9
8
7
6
5
4
3
2
1
0
10
9
8
7
6
5
4
3
2
1
2
10
9
8
7
6
5
4
3
2
1
64
10
9
8
7
6
5
4
3
2
1
80
mean rank
0
10
2
3
local
R0
20
10
9
8
7
6
5
4
3
2
1
2
mean rank
mean rank
mean rank
0
1.5
10
9
8
7
6
5
4
3
2
1
R0
10
9
8
7
6
5
4
3
2
1
16
0.1
mean rank
0
1
x
10
9
8
7
6
5
4
3
2
1
10
10
9
8
7
6
5
4
3
2
1
0.5
10
9
8
7
6
5
4
3
2
1
64
0
mean rank
0
mean rank
0.1
10
9
8
7
6
5
4
3
2
1
2
10
9
8
7
6
5
4
3
2
1
80
mean rank
32 -1 48
a
60
0.1
mean rank
16
40
s
R 0 (absolute)
10
9
8
7
6
5
4
3
2
1
1
br A0
10
10
9
8
7
6
5
4
3
2
1
4
0
mean rank
1.5
20
mean rank
1
x
mean rank
0
mean rank
0.5
10
9
8
7
6
5
4
3
2
1
80
mean rank
mean rank
60
10
9
8
7
6
5
4
3
2
1
0
mean rank
40
s
10
9
8
7
6
5
4
3
2
1
0
mean rank
20
mean rank
mean rank
0
T (absolute)
R 0 (relative)
mean rank
mean rank
T (relative)
10
9
8
7
6
5
4
3
2
1
mean rank
86
1
2
3
local
R0
10
9
8
7
6
5
4
3
2
1
0
0.5
r
1
4
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
The strength of these rules of thumb depends on
the parameter values, particularly the amount of clustering of patches, the mean dispersal distance and the
number of dispersers. Also, the rules of thumb are
less pronounced when R0 is chosen as a measure of
metapopulation persistence than when Text is chosen.
In any case, Table 2 shows that the weaker negative
formulation (1A Do not enlarge the smallest patch if
one must choose to increase patch size by a fixed percentage, 1B Do not enlarge the largest patch if one
must choose to increase patch size by a fixed amount
of area) are very strongly supported.
Rules 1A and 2 may have been concluded from
the rules of thumb found by Etienne and Heesterbeek
(2001), but rule 1B shows that such a hasty interpretation of their results is incorrect, because their rules
only apply to the level of extinction and colonization
probabilities. The relationships between these probabilities and landscape characteristics as presented in
this paper make the difference. Rule 2 is in line with
Frank (1998) who finds that connecting patches may
be disadvantageous if emigrants are lost to patches of
no importance.
Although the results increase our insight into the
sensitivity of the metapopulation to alterations in its
configuration, a definite rule of thumb still requires
knowledge of the amount of effort (the cost) needed
to establish these alterations, as noted by Etienne and
Heesterbeek (2001). If this is fairly constant across all
patch sizes and interpatch distances, then the results
give an indication of where the emphasis should be put
in metapopulation management. If it is not, then an additional sensitivity analysis incorporating these costs
is required. Because this is usually case-specific, no
rules of thumb can be expected at this level. Yokomizo
et al. (2003) show how to combine economic costs of
conservation with ecological risks of non-conservation
by simply attaching a weight to the ecological risk indicating the (financial) value of the (meta)population.
87
The number of patches in the artificial landscapes
is small. Nevertheless, I expect the rules of thumb to
hold for larger numbers of patches provided the rescue
effect is not too strong, but evidently differences in
patch sizes will be smaller, suggesting a weakening
of the rules of thumb. In those cases we can lump
patches of similar size together and the rules of thumb
should be applied to any of the patches in a certain
size class. For example, “enlarge the largest patch”
becomes “enlarge a patch in the largest size class”.
Which patch to choose can then be determined by the
rules of thumb of Etienne and Heesterbeek (2001) for
systems with equal extinction probabilities.
The question which always arises in simulation
studies such as this, is whether artificially generated landscapes are representative of real ones. As
Tischendorf (2001) remarks, there is no general answer to this question, because there is no general
pattern in realistic landscapes. Nevertheless, he finds
that correlations between landscape indices and ecological response variables are similar in artificial and
real landscapes, implying that using artificial landscapes is meaningful. As I studied different degrees of
clustering, I aimed to cover a range of possible landscapes, and I believe that the conclusions therefore
hold quite generally.
Evidently, the model is a simplification of reality.
Nevertheless, Levins-type models, of which the model
used in this paper is just one example, are remarkably accurate for their simplicity, which justifies their
wide-spread use (Keeling, 2002). I realize however
that the submodels for the extinction and colonization
probabilities depend crucially on the assumptions and
specific choices of underlying mechanisms. This is
particularly true for the dispersal mechanism that determines the colonization probability. Other descriptions may be equally possible (e.g. density-dependent
emigration, different relationships with interpatch
distance, more detailed immigration). Yet, firstly, the
Fig. 5. Mean rank of the interpatch distances versus each of the parameters of Table 1 for the interpatch distance between the two largest
patches (thin solid line), the two second largest patches (solid line with filled circles), the two third largest patches (gray line), the two
third smallest patches (dotted line), the two second smallest patches (solid line with filled squares) and the two smallest patches (thick
solid line); the sum of the patch areas is used to define the “two . . . largest”. Rank 1 is given to an interpatch distance if the required
change in interpatch distance is smallest, rank 2 is assigned if the required change is the second smallest, etc. The mean rank is the
average over all 1000 landscapes. It is calculated for each of the two measures, Text and R0 , where the required change is compared in
two ways: relative (i.e. percentage of the distance) and absolute (i.e. distance itself). The parameter σ is given the (arbitrary) value 80 for
the completely random case where σ is actually infinite to allow it to be plotted.
88
R.S. Etienne / Ecological Modelling 179 (2004) 77–90
model captures the essential dependencies on the
areas of the patches of origin and destination and
their interpatch distances (Hanski, 1999a; Kindvall
and Petersson, 2000), and secondly, more detailed
models would no longer be in line with the full
metapopulation model (patch occupancy model, no
explicit local dynamics) and their predictive power is
questionable (Moilanen and Hanski, 1998; King and
With, 2002). Still, one aspect missing in the model
deserves mentioning. Correlated extinctions are potentially major threats to the robustness of the results
of metapopulation models (Harrison and Quinn, 1989;
Hanski, 1991; Wissel and Stöcker, 1991; Moilanen,
1999; Etienne and Heesterbeek, 2001; Matter, 2001;
Ovaskainen et al., 2002). Incorporating correlation
in a simulation model is easy, but calculating explicit expressions for quantities such as M41 in
the model is very difficult. One can proceed along
the lines of Etienne and Heesterbeek (2001) making the correlation matrix dependent on interpatch
distances. This method involves multi-dimensional
integrals which are computable to the required accuracy in a reasonable time only for homogeneous
networks (in which all patches and interpatch distances are equal). In any case, although there are
profound effects of correlations on metapopulation
persistence, I do not expect a substantial modification of the rules of thumb, because increasing patch
size and decreasing the effective interpatch distance
have no direct relationship with correlations in local
extinctions.
These considerations impel us to be careful in using the rule of thumb (as one should with all rules of
thumb). If in a particular case sufficient resources are
available for extensive research, this should always
be preferred. With ample information about model
parameters, one can then merge decision-theoretic
approaches (Possingham, 1996, 1997; Possingham
et al., 2001) with more detailed spatial models, ranging from individual-based models (Kean and Barlow,
2000; Keeling, 2002) to models invoking graph
theory (Bunn et al., 2000; Urban and Keitt, 2001;
Jordán et al., 2003). But even then one must be
aware that these models are not perfect (Lindenmayer
et al., 2003). If resources for extensive studies are
not available, the results suggest guidelines to conservationists as to how to proceed with at least some
underpinning.
Acknowledgements
I thank Hans Heesterbeek and an anonymous reviewer for helpful comments, and Saskia Burgers
for statistical advice. Part of the work for this article
was carried out at Biometris and the Tropical Nature
Conservation and Vertebrate Ecology Group, both
part of Wageningen University and Research Centre,
Wageningen, The Netherlands and at Environmental
Science, Faculty of Geosciences, Utrecht University,
Utrecht, The Netherlands.
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