Ecological Modelling 128 (2000) 211 – 220
www.elsevier.com/locate/ecolmodel
A reliability-theory approach to corridor design
Ferenc Jordán *
Department of Genetics, Eöt6ös Uni6ersity, Múzeum krt. 4 /A, H-1088, Budapest, Hungary
Received 21 June 1999; received in revised form 28 September 1999; accepted 16 November 1999
Abstract
Natural habitats are small islands in the sea of human environment. Area loss and fragmentation affect seriously
the survival of species, for small, isolated populations are exposed to several risks. As the area of suitable habitats
decreases, one way to escape local extinction is to migrate between habitats through ecological corridors, and utilize
the whole metapopulation landscape. Thus, natural and designed corridors can be key elements for survival. Here, we
present a method to study corridor pattern from a reliability-theory viewpoint. We analyze the probability of
successful migration in a metapopulation landscape network of habitats, stepping stones and corridors. We examine
the situation when individuals of a local population must migrate from a disturbed, critical habitat to others. In the
landscape graph, points represent habitats and stepping stones, while edges represent corridors. If corridors can be
destroyed, migration probability depends on the pattern of permeable corridors. Engineered corridors can enhance
the reliability of migration, depending on their position in the network. We present some general rules for designing
reliable landscape patterns (e.g. ‘necklace’ arrangement is less reliable for migration than ‘loop’). Then, we illustrate
our viewpoint by presenting two hypothetical landscape networks and comparing the possibilities for designing
reliable corridor topologies by creating one engineered corridor. Further, we determine the preferred topology for the
engineered corridor. Our hope is that this reliability-theory analysis will stimulate further development of the method
and in the fieldwork. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Reliability engineering; Landscape structure; Ecological corridor; Metapopulation
1. Introduction
Humans are causing the largest mass extinction
ever experienced by the biosphere so far (Wilson,
1992). The ecological effects of extinctions are
strongly amplified by the fragmentation of natural
habitats (Soulé, 1991; Simberloff, 1992). Small
fragments are not suitable for survival, at least the
* Tel.: +36-1-2661296; fax: + 36-1-2662694.
E-mail address: [email protected] (F. Jordán)
majority of species have to live on the brink of
extinction there (Wolf, 1987). Many factors
threaten populations living in small, isolated habitats (Kruess and Tscharntke, 1994; Turner and
Corlett, 1996; Bascompte and Solé, 1998; Didham
et al., 1998), and even relatively isolated communities are more invadable (Pimm, 1991). The selectivity of fragmentation is still debated (see, for
example, Gilpin and Diamond, 1981; Kareiva,
1987; Mikkelson, 1993), however, the chance of
survival often depends on, for example, habitat-
0304-3800/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 0 ) 0 0 1 9 7 - 6
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F. Jordán / Ecological Modelling 128 (2000) 211–220
preference abilities and migratory properties of
non-sessile organisms (Tilman et al., 1994).
Populations can be nearly totally isolated or
strongly mixed by the exchange of individuals
(Polis et al., 1997). If migration among local
populations through corridors is possible, a metapopulation landscape network emerges (Hanski,
1998). We use the term landscape as the mosaic of
habitat patches and corridors (Dunning et al.,
1992). Let ecological corridor mean an area,
which connects patches (habitats and stepping
stones) and makes migration possible for a given
species between them (long-term survival is not
possible in corridors). Ecological corridors are
basic components of metapopulation landscapes,
for they connect local populations and can reduce
extinction rate (for an experimental test, see
Gilbert et al., 1998). However, they may also have
disadvantageous effects, e.g. spreading diseases.
In addition, a too complex corridor pattern may
disturb animals in effective migration and orientation (Boswell et al., 1998). We define stepping
stones as relatively small, helping migration but
not being suitable for long-time survival, and
habitats as the places where local populations
may reproduce and survive for a long time.
As fragmentation and area loss proceed, ecological models considering spatial parameters become more interesting and important (Czárán and
Bartha, 1992; Kareiva, 1994). Corridors not only
increase the persistence of populations by landscape compositional effect, but their precise spatial pattern also matters; it is a physiognomic
effect (Dunning et al., 1992; Pickett and
Cadenasso, 1995). Landscape structure also affects strongly community structure through food
web subsidization (Polis et al., 1997).
The environment of local populations in a
metapopulation network often differs to a large
extent. Some local populations behave as sinks
and are maintained by immigration from source
populations. Sinks can be described by the b B d
and eB i relations, where b = birth, d = death,
e = emigration and i= immigration (BIDE-model,
Pulliam, 1988). Reverse relations correspond to
sources. Immigration may keep sink populations
alive (Dias, 1996). In certain cases, higher density
characterizes sink habitats rather than sources,
and sometimes the majority of habitats are sinks
maintained from only a few source habitats
(Stoner, 1992). The source/sink metapopulation
structure and, in general, migration can be common in undisturbed environments, as well. For
example, some species without dispersal can be
characterized by abnormal demography parameters. In these cases, normal populations need a
sink habitat and dispersers have safety functions
against overcrowding (Tamarin, 1980). Since migration is important for the majority of species in
both disturbed and undisturbed environments, the
study of corridors is useful both for understanding ecological processes and for designing corridors for conservation purposes (Wilson and
Willis, 1975).
Here, we present a method for studying the
physiognomic effects (Dunning et al., 1992) of
corridor pattern in a metapopulation network.
We analyze the probability of emigration from a
given, disturbed habitat to other habitats through
stepping stones and corridors. The probability of
successful emigration is called the reliability of the
metapopulation network. We present some corridor patterns advantageous for reliable migration
and illustrate our approach by analyzing two
hypothetical landscapes.
2. Methods
Reliability engineering is a technical science
(see, for example, Harr, 1987; Aggarwal, 1993).
Its mathematical basis, reliability theory, discusses
how to study, measure and enhance the reliability
of a given system (von Neumann, 1956; Barlow
and Proschan, 1965). Reliability is the probability
of successful operation, during a given time, under given conditions. It can be increased by reliability-enhancing
factors,
for
example,
redundancy, repair, storage or combinatorial optimization (Molnár, 1994). The reliability-theory
approach to biological problems is useful, for
reliability is obviously related to successful survival. Some biological applications have already
been presented (Oster and Wilson, 1978; Naeem
and Li, 1997; Naeem, 1998; Jordán and Molnár,
1999; Jordán et al., 1999).
F. Jordán / Ecological Modelling 128 (2000) 211–220
In this study, we show some possibilities for
combinatorial optimization in corridor design. We
study the reliability of migration in metapopulation landscape networks, where corridors can be
destroyed. Reliability is the probability that a
critical habitat (which is disturbed and must be
abandoned) and any other habitats are connected.
Corridor pattern influences this probability, for
some network design elements are advantageous
for reliable migration (Figs. 1 and 2). We seek for
the most advantageous topology for inserting an
engineered corridor into a hypothetical landscape
network. It is emphasized that we analyze metapopulation landscape networks only in a topological sense, not taking into account the properties
of corridors (e.g. length, width, even cost of de-
Fig. 1. Two simple metapopulation landscape networks.
Empty areas and lines represent habitats and corridors, respectively. The graph on the left contains four serially arranged
habitats connected by three corridors (this graph is a tree).
The graph on the right contains four habitats again, but they
are maximally connected (this is a complete graph). If corridors are destroyed in the first case, the probability of isolation
increases and the possibility for migration decreases. The
second network can lose one or two corridors (even three in
certain combinations) and still remains connected. If migration
is important, a pattern of corridors and habitats arranged like
the graph on the right is advantageous. The structure of the
metapopulation network graph influences the probability of
migration if corridors can be destroyed.
213
Fig. 2. Some examples for network design elements. White
areas represent habitats. An asterisk marks the critical habitat,
where disturbance is too high for survival. Letters mark habitats suitable for long-term survival. Black areas represent
stepping stones; they help migration. Lines represent corridors.
Corridors are destroyed with the probability p =0.2; thus, they
are permeable with q= 0.8. The reliability of successful migration from the critical habitat to habitat ‘X’ is R(X). The
metapopulation networks ‘D’, ‘A’ and ‘E’ contain serially
arranged spatial elements resulting in lower reliability. ‘C’, ‘F’
and ‘G’ contain spatial elements arranged in parallel. ‘B’
represents a bridge system. The habitat ‘H’ is multiply connected to the critical habitat. The reliability of the emigration
from the critical habitat depends strongly on the pattern of
corridors and stepping stones.
signed corridors). Furthermore, the elements of a
metapopulation landscape (i.e. habitats, stepping
stones and corridors) are strongly species-specific.
For example, the Panama Canal is a corridor for
fish but also a barrier for terrestrial animals.
There are no universal corridors or habitats,
hence a designed corridor might be useful only for
one or a few species.
3. The reliability-theory approach
We consider a network of habitats, stepping
stones and corridors. Long-term survival is possible only in habitats; corridors and stepping stones
only can help migration. We assume that one of
the habitats (called critical habitat) is strongly
disturbed, where survival becomes impossible and
the local population has to emigrate to other
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F. Jordán / Ecological Modelling 128 (2000) 211–220
habitats. Other habitats and stepping stones are
available from the critical habitat, if they are
connected by corridors. We assume that the pattern of habitats and stepping stones is constant,
but corridors are deleted with some probability.
We further assume that corridors can also be
designed and created and these are functionally
equivalent to natural corridors. Thus, the reality
of the method presented depends strongly on
time-scaling problems (migration time compared
to changes in the landscape pattern, migration
time compared to the duration of the disturbance, time required for corridor design and
construction, etc.).
We are interested in the probability of successful emigration from the critical habitat (i.e.
escaping local extinction), which depends on the
pattern and deletion probabilities of corridors.
We search for the best topology of a designed
corridor in a disturbed metapopulation network.
The best network design results in the highest
reliability of successful emigration from the disturbed critical habitat. The best designed topology may differ for different critical habitats.
In the metapopulation landscape graph, points
represent habitats and stepping stones. They
cannot be deleted. The properties of patches
(habitats or stepping stones) are not taken into
account; i.e. they are of equal quality. Edges
represent corridors and can be deleted independently with an optional probability, p = 0.2;
thus, they are permeable with q = 0.8. Corridor
deletion probabilities are assumed to be equal,
which is a strong simplification (some corridors
are surely more permeable and more safety).
This is open to further development: a measure
of permeability could be created by considering
the length and the width of the corridor and the
level of perturbations affecting the corridor.
Nevertheless, corridor pattern can be studied
only by considering equal corridors, even if it is
a simplification.
Let ‘basic failure’ (p) mean the deletion of a
corridor. Some combinations of basic failures
result in cuts disconnecting the critical point
from other habitat points (if the critical habitat
is connected to only stepping stones, successful
migration is considered to be impossible). Let
‘final failure’ (P) mean this disconnection. The
reliability of the metapopulation landscape
graph is R= 1− P:
L
R= 1− %
i=1
L
K
·p i·q (L − i)·
i
L
i
where p is the probability of basic failure (a
corridor has been destroyed); q is the probability of permeability through a corridor; L is the
number of corridors in the landscape; and K is
the number of cuts containing i edges and connecting the critical habitat to others.
In the simplest model, we consider an undirected graph. If the metapopulation is definitely
characterizable by a source/sink pattern of habitats, the edges of the graph are directed, according to the source and sink nature of the
connected patches. This makes no serious difference, only the number of possible migration
routes (permeable edge pattern) decreases.
A key corridor is represented by an edge,
whose single deletion results in a cut separating
the critical point. Key corridors are the most
important network elements, because in the case
of their deletion, a local population becomes immediately isolated and the risk of its extinction
may increase rapidly.
We present two illustrative hypothetical landscapes for analyzing the reliability of migration
in metapopulation networks. There is a single
critical habitat in each case, from which the local population is constrained to migrate, for disturbance is intolerably high. Migration is
possible through corridors and stepping stones.
Other habitats are suitable for survival if they
can be reached.
We analyze the reliability-increasing effect of
a single designed corridor. If a landscape graph
contains L edges (there are L corridors) and N
points (either stepping stones or habitats), there
are
N(N− 1)
− L− i
2
F. Jordán / Ecological Modelling 128 (2000) 211–220
different possibilities for inserting a novel edge
into the graph (constructing a corridor), where
i + 1 is the number of isomorphic graphs (containing the same number of both points and edges
and having the same topology). If we do not
exclude isomorphic graphs, then
N(N−1)
−L
2
is the number of edges in the complementer
graph. For simplicity, let i include not only isomorphic graphs, but also graphs where the ‘A’
and ‘B’ habitats are connected directly (in these
graphs, the engineered corridor does not affect
reliability, as discussed later).
We present the possible topologies of corridor
design for each landscape and compare the reliability of migration in the engineered metapopulation networks. Some engineered corridors result
in a reliable and in a less reliable subgraph (e.g.
bridge system and series-arranged patches, see
Fig. 4f). Others make the reliability of subnetworks more similar (see Fig. 4d). The probabilities of successful migration from the critical
habitat to the habitat ‘A’,‘B’ or to any of them
are R(a), R(b) and R(ab), respectively.
When calculating reliability we do not take
into account each edge, only those connecting
the considered points (it makes no sense to calculate the deletion probabilities of edges not
components of any routes between the critical
habitat and any of the considered habitats). For
example, in Fig. 4e we consider only five edges
calculating R(b). Similarly, in Fig. 3f we consider only four edges calculating R(ab). Here,
one edge connects a single stepping stone to the
critical habitat, this corridor is out of our interest.
If the routes to ‘A’ and to ‘B’ have no common
edges (i.e. we have two separate subnetworks),
then
(1 − R(a))(1 −R(b)) =1 − R(ab)
because the probabilities of unsuccessful migrations (P(a)=1−R(a) and P(b) =1 − R(b)) are
independent events (P(ab) =1 −R(ab)).
215
4. Results and discussion
We have determined what is the best topology
for an engineered corridor in a metapopulation
landscape. Engineered corridors inserted into various places enhance the reliability of emigration
from the disturbed critical habitat differently. We
compare these cases and present some general
rules (for the generality of rules, see Lawton,
1999).
The first metapopulation landscape (Fig. 3a)
concerns a metapopulation structure consisting of
four corridors, two stepping stones and three
habitats (one of them is disturbed, critical). An
engineered corridor can be designed according to
five different topologies.
The migration probability to habitat ‘A’ increases in two cases (Fig. 3e, f), to the same
extent. In these cases, ‘A’ is connected to one of
the stepping stones. If ‘A’ is connected directly to
‘B’, we do not consider this case to be different
from that of the unengineered landscape.
The migration probability to habitat ‘B’ does
not increase in Fig. 3e, f. However, it can be
increased in three ways. A corridor between the
stepping stones enhances it slightly (Fig. 3c), a
corridor between the unconnected stepping stone
and ‘B’ is much better (Fig. 3b), but the best
solution is to connect ‘B’ directly to the critical
habitat (Fig. 3d). Here, the reliability of the emigration depends strongly on corridor topology.
The reliability of the emigration to any of the
habitats suitable for survival increases in each
cases. The second solution is the weakest (Fig.
3c), for a single (key) corridor leads to both of the
habitats, the reliability of migration is the lowest.
It is better to create parallel routes containing
corridors connecting stepping stones and habitats
(Fig. 3b, e, f). In this way, one of the habitats can
be reached from the critical habitat through a
subnetwork, which contains no single cut (no key
corridor). Cuts containing a single edge make the
network flow less reliable (Fig. 3c). In general,
series-organized systems, like ‘biogeographic umbilici’ (Diamond and Gilpin, 1983) or ‘necklaces’
(see Fig. 5(a), cited in Pickett and Cadenasso,
1995) are the least reliable ones. The optimal
topology is the direct connection between the
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F. Jordán / Ecological Modelling 128 (2000) 211–220
critical habitat and ‘B’ (or any of the habitats, but
‘A’ is already connected directly). This is the third
case (Fig. 3d), where the reliability of emigration
is the highest.
The second metapopulation landscape outlines
a more complex problem (Fig. 4a). This metapopulation network graph contains six corridors,
three stepping stones and three habitats (one of
Fig. 3. The first metapopulation landscape, where the local population living in the disturbed, critical habitat has to migrate either
to habitat ‘A’ or ‘B’ (symbols as in Fig. 2). Corridors are destroyed with the probability p = 0.2, while they are permeable with
q=0.8. The reliabilities of successful migration from the critical habitat to habitat ‘A’, ‘B’ or any of them are R(a), R(b) and R(ab),
respectively. A novel corridor can be designed according to five different topologies. These engineered metapopulation networks
differ in the probability of the successful emigration from the disturbed habitat (either to ‘A’, to ‘B’ or to any of them).
F. Jordán / Ecological Modelling 128 (2000) 211–220
217
Fig. 4. The second metapopulation landscape. Symbols, corridor deletion probabilities and calculations are similar to those in Fig.
3. Note that in the sixth case (g), migration to ‘B’ is possible through the same subnetwork as in the ‘G’ case in Fig. 2.
218
F. Jordán / Ecological Modelling 128 (2000) 211–220
them is disturbed, again). An engineered corridor
can be placed in six different topologies.
The migration topology to ‘A’ does not increase
in three cases (Fig. 4e – g). (We do not consider the
sixth case to be different from the unengineered
one, because migration through this engineered
corridor requires the successful migration to habitat ‘B’. It can be disregarded if we analyze the
migration exclusively to ‘A’). A slight increase in
reliability can be performed by designing a corridor
connecting the stepping stones (Fig. 4c), but it is
better to connect ‘A’ to any of the stepping stones
(e.g. Fig. 4d). The best topology is when ‘A’ is
connected directly to the critical habitat (Fig. 4b).
The reliability of migration to ‘B’ does not
increase in two cases (Fig. 4b, d). It can be slightly
enhanced by increasing parallelism partly (Fig. 4c)
or by creating a bridge system (Fig. 4f). It is much
better to increase parallelism along the whole
length of the route from the critical habitat to ‘B’
(Fig. 4g). The best topology, again, is to connect
the critical habitat directly to ‘B’ (Fig. 4e).
The reliability of emigration to either ‘A’ or ‘B’
always increases. Partial parallelism (Fig. 4c),
bridge system (Fig. 4f), and longer parallel routes
(Fig. 4d, g) enhance it more or less to the same
extent. Direct habitat-to-habitat corridors are
equally the best solutions, again (Fig. 4b, e).
5. Conclusion
We have shown that some considerations of
reliability engineering may be useful and should be
applied in designing ecological corridors. The number and area of patches are not the only important
data for conservation projects. What also is very
interesting is the precise pattern of corridors, i.e. the
detailed structure of the metapopulation network.
Some arrangements are prone to damages resulting
in isolation of habitats. Other networks are reliable
enough for safe emigration from disturbed habitats. For example, a ‘necklace’ arrangement of
landscape elements is very unreliable for migration,
but a ‘loop’ is better (Fig. 5(a, c)). A ‘spider’
arrangement (Fig. 5(b)) is advantageous only for
the central species (Pickett and Cadenasso, 1995).
In general, a network containing patches arranged
Fig. 5. The most typical arrangements of habitats and corridors according to aerial photographs: necklace (a), spider (b),
and loop (c, cited in Pickett and Cadenasso, 1995).
in series cannot be very safe for migration, if
corridors can be destroyed. The reliability of the
migration increases if the level of parallelism is
enhanced partly (e.g. in a bridge system) or totally
along a subnetwork (e.g. graph ‘G’ in Fig. 2). The
best metapopulation network topology is designed
when an engineered corridor connects the critical
habitat directly to any of the suitable habitats. We
have presented the application of a reliability-theory method for analyzing landscape composition
from a topological viewpoint.
Physiognomic analyses may be useful in managing concrete conservation problems and may help
the understanding of metapopulation processes.
Under economical constraints, the optimal design
of an engineered landscape could decrease costs. If
a habitat in a well known landscape structure is
considered critical (for any reason), the presented
simple method is suitable for predicting the corridor worth to be engineered. The possible application of this approach requires knowledge about
landscape structure (e.g. aerial photos) and some
reason to consider a habitat to be critical. Independently of perturbation (i.e. landscape with no
F. Jordán / Ecological Modelling 128 (2000) 211–220
critical habitat), local populations can be characterized topologically and the risk of extinction can
be predicted from this point of view. For migration is a frequent event also in undisturbed landscapes, these considerations may be useful also for
studying the source-sink dynamics of metapopulations and problems of population genetics.
We have analyzed simple hypothetical landscapes with corridors of equal quality (length,
width and cost) and have not taken into account
any characteristics of species. More useful applications could follow only a more detailed method,
including, for example, corridor permeability. We
hope that this study will stimulate further theoretical and field works. For example, an analysis of
the combinations of various reliability-enhancing
factors (e.g. network design with corridor repair),
as well as the comparison of hypothetical networks and real landscape data would be very
interesting. We hope that our viewpoint is worth
consideration in field studies of conservation biology. Because conservation problems become increasingly urgent, reliability engineering also
should be applied in biology.
Acknowledgements
I thank Professor Gábor Vida, István Molnár,
András Báldi, András Takács-Sánta and two
anonymous referees for helpful comments on the
manuscript. I am indebted to A. Báldi for providing a manuscript of his own. This work was
supported by the grant OTKA F 029800.
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