Modeling Habitat Split: Landscape and Life History Traits
Determine Amphibian Extinction Thresholds
Carlos Roberto Fonseca1*, Renato M. Coutinho2, Franciane Azevedo2, Juliana M. Berbert2,
Gilberto Corso3, Roberto A. Kraenkel2
1 Departamento de Botânica, Ecologia e Zoologia, Universidade Federal do Rio Grande do Norte, Natal, Brazil, 2 Instituto de Fı́sica Teórica, Universidade Estadual Paulista,
São Paulo, Brazil, 3 Departamento de Biofı́sica e Farmacologia, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Abstract
Habitat split is a major force behind the worldwide decline of amphibian populations, causing community change in
richness and species composition. In fragmented landscapes, natural remnants, the terrestrial habitat of the adults, are
frequently separated from streams, the aquatic habitat of the larvae. An important question is how this landscape
configuration affects population levels and if it can drive species to extinction locally. Here, we put forward the first
theoretical model on habitat split which is particularly concerned on how split distance – the distance between the two
required habitats – affects population size and persistence in isolated fragments. Our diffusive model shows that habitat
split alone is able to generate extinction thresholds. Fragments occurring between the aquatic habitat and a given critical
split distance are expected to hold viable populations, while fragments located farther away are expected to be unoccupied.
Species with higher reproductive success and higher diffusion rate of post-metamorphic youngs are expected to have
farther critical split distances. Furthermore, the model indicates that negative effects of habitat split are poorly compensated
by positive effects of fragment size. The habitat split model improves our understanding about spatially structured
populations and has relevant implications for landscape design for conservation. It puts on a firm theoretical basis the
relation between habitat split and the decline of amphibian populations.
Citation: Fonseca CR, Coutinho RM, Azevedo F, Berbert JM, Corso G, et al. (2013) Modeling Habitat Split: Landscape and Life History Traits Determine Amphibian
Extinction Thresholds. PLoS ONE 8(6): e66806. doi:10.1371/journal.pone.0066806
Editor: Christopher Joseph Salice, Texas Tech University, United States of America
Received February 4, 2013; Accepted May 8, 2013; Published June 20, 2013
Copyright: ß 2013 Fonseca et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was supported by the Conselho Nacional de Desenvolvimento Cientı́fico e Tecnológico - CNPq (CRF, GC, RAK), Coordenação de
Aperfeiçoamento de Pessoal de Nı́vel Superior - CAPES (FA), Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (RMC,JMB), and Internacional
Center for Theoretical Physics - ICTP-SAIFR (RAK). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the
manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: cfonseca@cb.ufrn.br
the matrix searching for an isolated forest fragment. This
compulsory bi-directional migration causes drastic declines on
amphibian populations [17].
At the landscape scale, habitat split decreases the richness of the
amphibian community due to the extinction of aquatic larvae
species [16]. More importantly, the richness of amphibians with
aquatic larvae has been demonstrated to be more strongly affected
by habitat split than by habitat loss and habitat fragmentation
[16].This process causes bias in communities towards amphibians
with terrestrial development, since these species are able to breed
successfully in forest fragments even in the absence of a water
source [16,18].
The habitat split concept has also contributed to conservation
issues. In a recent complementarity exercise for the identification
of key Neotropical ecoregions for amphibian conservation, the
differentiation between species with aquatic and terrestrial
developmental generated a more comprehensive coverage of
priority ecoregions than when species were pooled together [19].
Also, by analyzing how the incidence of habitat loss and habitat
split varies across a regional landscape, the selection of a minimum
priority set of watersheds for amphibian conservation could be
optimized [20].
Habitat split is a worldwide phenomenon, being particularly
common in biodiversity hotspots where habitat fragmentation is
Introduction
One-third of the world’s amphibian species are threatened,
more than 40% have declining populations, and 168 species
probably went extinct in the last five centuries [1]. In biodiversity
hotspots, 2841 amphibian species are facing an unprecedented
contraction of their geographic area [2], being threatened by
habitat loss and fragmentation [3]. Many theoretical models have
been proposed to capture the complexity of such processes, from
the theory of island biogeography [4,5] to complex spatially
explicit metapopulation models [6,7]. The basic predictions of
these models have been corroborated for different taxa, including
protozoa [8], butterflies [9,10], birds [11], and mammals [12, but
see 13,14,15]. However, since amphibian species exhibit marked
ontogenetic habitat shifts, being strongly affected by habitat split
[16], the predictive power of such models is limited.
Habitat split is defined as human-induced disconnection
between habitats used by different life history stages of a species
[16]. For forest-associated amphibians with aquatic larvae,
deforestation causes spatial disjunction between the habitat of
the larvae, ponds and streams, and the habitat of the adults, the
forest fragments. At the local scale, habitat split compels adults to
traverse the anthropogenic matrix to reach breeding sites and
recently metamorphosed juveniles to walk haphazardly through
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Modeling Habitat Split
will posit that the adults also obey a diffusion equation, assuming
that the forest fragment is spatially homogeneous and that the
adults will haphazardly look for food and escape from their
predators before reproduction. However, individuals walking in
the matrix or in the forest fragment can exhibit different diffusion
rates.
Juveniles that reach the fragment are dynamically equivalent to
adults, so we will assume that there are no juveniles in the
fragment, J(x,t)~0 when L1 vxvL2 . On the other hand, adults
migrate through the matrix back to the river for reproduction.
This however, is a directed movement, much more like advection
rather than diffusion, and is not modeled explicitly. Mathematically, these assumptions translate into two diffusive equations. The
first equation is defined for juveniles in the matrix and the second
for adults in the forest fragment:
intense and human settlements are generally concentrated on the
valleys where water is available for agriculture, industry, and
human consumption [2]. The Brazilian Atlantic Forest, for
instance, is now distributed in 245,173 forest fragments from
which 83.4% are smaller than 50 ha [21]. In a typical Atlantic
Forest landscape, less than 5% of the fragments are connected to
streams [17]. In fact, ‘‘dry fragments’’ are the rule and their
distance to the nearest stream can vary widely from a few meters
to several kilometers. Predicting how the amphibian populations
respond to such landscape alteration is essential for conservation.
We provide, here, the first theoretical model for habitat split. A
minimum diffusion model shows that habitat split generates
critical split distances for population persistence in forest
fragments. The model predicts how life history traits, such as
juvenile dispersal ability and recruitment, determine the extinction
threshold. Furthermore, it predicts how population size is affected
by the quality of the matrix and its distance from the breeding
habitat. The model has relevant implications for amphibian
conservation landscape design.
Materials and Methods
The Model
In this section we develop a model designed to capture the main
consequences of habitat split on populations of amphibians with
aquatic larvae. This means a model that has enough ingredients to
provide a basis for predictions without, however, taking into
account particularities of any specific amphibian species. The
main point the model is set to address is that of population decline
and local extinction. The spatial configuration of the model is
shown in Figure 1. The reproductive site of the amphibians is at
the river at x~0, whereas the forest fragment, where the adults
live, has a size s~L2 {L1 . The shortest distance from the
fragment to the river is L1 , from now on called split distance, a new
metric for habitat fragmentation studies. We have chosen to work
in a one-dimensional context. Extensions to a two-dimensional
space can be implemented, but the main features are already
present in our model.
Habitat split consequences on the amphibian population are
directly connected to the fact that the population is stagestructured. Accordingly, we introduce two variables, J(x,t) and
A(x,t), which represent juveniles and adults densities, respectively.
We will assume that after leaving the reproductive site the juvenile
amphibians move in a haphazard way through the matrix. This
assumption is based on the fact that in the pristine environment
they did not have to search for the terrestrial habitat, therefore
they lack adaptations that allow them to find the forest fragments
in a directional fashion. From the modeling point of view, this
suggests that a diffusion equation is appropriate to describe the
spatial aspects of the juveniles in the matrix. In the fragment, we
LJ
L2 J
~DJ 2 {mJ J
Lt
Lx
ð1Þ
LA
L2 A
~DA 2 {mA A,
Lt
Lx
ð2Þ
Where DJ and DA are the diffusion coefficients for juveniles and
adults in the matrix and the forest, and mJ and mA the respective
mortality rates. At the fragment border L2 , several scenarios are
possible, depending on the landscape beyond L2 : the boundary
may be completely absorbing if there is a very hostile matrix, or
totally reflexive if the environment is as good as in the fragment, or
it can be something in between. This point will be discussed in
detail in the next section and for now we consider a general
formulation [22]:
LA
x~L2 ~bAx~L2 :
Lx
ð3Þ
LA
LJ
Dx~L1 ~DJ
Dx~L1
Lx
Lx
ð4Þ
{DA
If b~0, we have a completely closed patch at L2 and the adults
will turn back towards the fragment interior. This condition is used
when we do not want to take into account size effects of habitat
patch, that is, when the patch is large. The opposite limit, b??,
corresponds to the situation in which all individuals that reach the
border L2 will leave the modeled landscape.
When juveniles reach the fragment, they become adults and,
since adults cannot turn into juveniles, the border x~L1
represents a completely absorbing boundary for juveniles.
Moreover, the rate at which new adults arrive at the fragment
must be the same as the rate of juveniles leaving the matrix. These
conditions are expressed in the following boundary conditions:
DA
J x~L1 ~0:
Figure 1. The spatial configuration for the habitat split model.
The model has three main landscape elements: the river (or any aquatic
breeding habitat), at L0 , the inhospitable matrix and the forest
fragment. Split distance is defined by L1 while fragment size (s) is
defined by L2 {L1 .
doi:10.1371/journal.pone.0066806.g001
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ð5Þ
The fourth and last boundary condition models the reproductive behavior of the amphibians. For simplicity, adults are assumed
to exhibit a constant recruitment rate r, so that the rate at which
new juveniles are generated at the river is proportional to the total
number of adults in the fragment. We also take into account that it
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Where c1 , c2 , f1 and f2 are integration constants. With help of
boundary conditions (5,6) we find explicitly their values. This
result is found in the supplementary material S1. Equation (10)
makes sense only for real positive solutions. We derive such
conditions from JDx~0 w0. If this condition is not satisfied the
population will go extinct as the null solution turns out to be the
only stable one in this case. We prove in supplementary material
S1 that JDx~0 w0 is equivalent to:
takes a certain time, t1 , for the influx of juveniles to respond to a
variation in the number of adults. Population size is controlled by
competition at the river, so we introduce a saturation parameter
K, which can be interpreted as the maximum rate of juveniles that
can be generated. The mathematical expression of this condition is
the following:
LJ
rN
Dx~0 ~
,
Lx
1z Kr N
{DJ
ð6Þ
2
3
r
1zb
41{ pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi 5w1: ð11Þ
mA cosh ( mJ =DJ L1 )
be mA =DA s ze{ mA =DA s
where r is the recruitment rate and N is the total number of adults
in the fragment at a past time. Notice that the negative sign
accounts for the fact that J(x) is always a decreasing function of x.
The total number of adults in the fragment at a past time is given
by:
N~
ðL
2
A(x,t{t1 )dx:
L1
In this expression, b~
ð7Þ
d2A
~mA A,
dx2
Stationary Solutions
The general form of the stationary solution as a function of x,
the distance from the river, is depicted in Fig. 2, where we assume
now b~0. To help the visualization we plot the juveniles and
adults in the same figure; for 0vxvL1 the density in the y-axis
refers to juveniles while for L1 vxvL2 the density of the adults is
plotted. As a first approach we assume that diffusion coefficients of
adults and juveniles are the same: DJ ~DA ~1. On the other hand
we suppose a large difference in mortalities of juveniles and adults,
we use mA ~0:01vv0:01vvmJ . The values of mJ are shown in
the picture. In this and the following plot we use
L1 ~s~K~r~1. The general behavior of this solution points
to populations that decrease in the matrix and tend to stabilize in
the fragment.
We also explore in Fig. 2 the dependence of the population in
the fragment on juvenile mortality. We plot solutions for three
distinct mJ , simulating matrixes of different quality. As expected,
an increase in the mortality leads to smaller populations and, as a
preview of the next subsection, this trend suggests the existence of
a threshold in this model.
ð8Þ
The Existence of a Critical Split Distance
The split distance, L1 , is an important landscape metric which
has great influence on the existence of a non-zero stationary
solution of the model and therefore on the viability of the
population. At this point we explore the most important
conceptual result of this work. The model introduced in this
article predicts an extinction threshold for L1 . This means, there is
a critical split distance L1 such that if L1 wL1 the amphibian
population goes locally extinct. In other words, if the split distance
is larger than a certain value, the population does not persist in this
landscape.
In Fig. 3 we show the population size in the fragment as a
function of the split distance L1 , for three different values of the
juvenile mortality mJ . The presence of a critical value L1 (the point
where the curves intercept the x-axis) is clearly seen. This figure
ð9Þ
where we have changed partial derivatives for ordinary ones as J
and A depend only on x.The couple of linear equations (8,9) has
the solution:
pffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffi
J(x)~c1 ex mJ =DJ zc2 e{x mJ =DJ
pffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
A(x)~f1 ex mA =DA zf2 e{x mA =DA ,
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ð12Þ
Further, notice that b~0 is the same as taking the limit s??,
that is, considering an arbitrarily large patch.
Equations (1,2) do not contain any density dependent terms:
they are linear. As discussed above, the population control term
appears only in the boundary conditions, namely in equation (6).
Moreover, the fact that these conditions include a time delay
makes it impossible to obtain exact solutions in general. However,
when we seek for stationary solutions, that is, solutions such that
LJ=Lt~0 and LA=Lt~0, the time delay plays no role anymore
and we can find the solutions and – more important – the
existence criteria for non-zero solutions.
The stationary solutions are obtained by setting to zero the
time-derivatives in equations (1,2) which leads us to:
DA
mA DA
r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
w1:
mA cosh ( mJ =DJ L1 )
Results
d2J
~mJ J
dx2
b{
interpretation in population dynamics: the recruitment should be
large enough to replace the population, otherwise, the population
disappears. In the special case where we take to be zero, we have:
In this model we suppose that the most important factor limiting
amphibian flow is of the juveniles that start at the river and cross
the matrix to the forest fragment. The return of the adults to the
river is assumed in equation (6) to be advective. These conditions
introduce two phenomenological constants r and t1 . The first of
them, the recruitment, takes into account the fertility of adults, the
survival of tadpoles and the adult mortality in the matrix. The time
t1 is the sum of the time to cross the matrix, mate, reproduce,
mature eggs and develop juveniles capable of crossing the matrix.
Although describing a different system, this set of equations and
boundary conditions is similar to the one presented in [23].
DJ
pffiffiffiffiffiffiffiffiffiffi
mA DA
pffiffiffiffiffiffiffiffiffiffi
. This condition has a standard
bz
ð10Þ
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Modeling Habitat Split
is the minimum to maintain the population (r~1) when the
favorable habitat is connected to the reproduction site (L1 ~0).
The curves of Fig. 4 show an increase of L1 with recruitment
translating the fact that a higher reproductive success allows for
larger split distances. The reason of this behavior is that the
recruitment compensates the mortality in the matrix.
The three curves in Fig. 4 examine the influence of the diffusion
of the juveniles, DJ ; for a given r, a larger diffusion coefficient
allows a larger split distance for the population. In this way DJ also
counterbalances the mortality in the matrix: higher dispersal
ability helps to deplete the effect of habitat split.
Dependence of the Critical Split Distance on Landscape
Metrics
As we have seen in the previous sections, a critical split distance
appears. When we took the border of the fragment as completely
reflexive (b~0), the dependence of the critical split distance on the
size of the fragment disappeared: no matter how large the
fragment, once a critical split distance is attained, the population
goes locally extinct. On the other hand, we can introduce a nonzero value for b, representing a partially absorbing boundary at
L2 . In this case, a flux of adults to the outside of the fragment
exists, making it still more difficult for the population to persist. To
illustrate this point, we plotted in Fig. 5 the adult population in the
fragment as a function of the split distance for three different
fragment sizes in the case where b~1. It is clear that the
population is always smaller the smaller the fragment is,
representing a typical area effect.
Figure 2. Stationary solutions of the model as a function of
space. In the model, all distances are measured according to the
reproductive site that is taken as zero. The fragment starts at L1 and
ends at L2 . Each curve refers to a given juvenile mortality in the matrix
(mJ ). The parameters used were r~K~mJ ~DJ ~DA ~1, b~0 and
mA ~0:01.
doi:10.1371/journal.pone.0066806.g002
also explores the influence of mJ on L1 ; as expected there is an
inverse relation between L1 and mJ . A more inhospitable habitat
(large mJ ) will make the critical split distance smaller.
Dependence of the Critical Split Distance on Life-history
Parameters
Discussion
One of the most relevant life-history traits for our analysis is the
recruitment r, a parameter that measures the reproductive success
of the amphibians. Indeed, in our model r encompasses three
biological processes: fertility of the reproductive adults, the survival
of the tadpoles until they emerge from the aquatic habitat to
become able to cross the matrix and the adult mortality in their
way back to the river to reproduce. In Fig. 4 we show the critical
split distance, L1 , as a function of the recruitment, r, for three
distinct values of diffusion coefficients of the juveniles DJ . The
point (L1 ~0, r~1) is a limit case; for this situation the recruitment
Amphibian populations are declining worldwide [24,25].
Several non-exclusive hypotheses have been proposed to explain
such a widespread pattern, including the emergence of Batrachochytrium dendrobatidis, a highly virulent fungus [26,27], ultraviolet-B
radiation, climate change [28], pollution [29], introduction of
exotic species [30], habitat loss and fragmentation [1,31,32], and,
more recently, habitat split [16]. Here, we explore the theoretical
consequences of habitat split for the conservation of amphibian
species with aquatic larvae. However, the model can be of
relevance for other organisms exhibiting marked ontogenetic
habitat shifts. For instance, insects with indirect development, such
Figure 3. Population size of adults in the fragment as a
function of the split distance, L1 This picture points a critical split
distance L1 above which the population gets extinct. The three curves
indicated in the figure represent different values of the juvenile
mortality mJ that can be caused by differences in matrix quality. The
parameters used were r~K~DJ ~DA ~1, b~0 and mA ~0:01.
doi:10.1371/journal.pone.0066806.g003
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Figure 4. Critical split distance as a function of life history
traits. The model explores two factors that modulate the habitat split
effect: the recruitment rate and the diffusion coefficient of the juveniles.
The parameters used were r~K~mJ ~DA ~1, b~0 and mA ~0:01.
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Figure 5. Curve of adult population size versus split distance. In this figure we point the effect of the size of the fragment on the critical split
distance. This figures show that large forest fragments have only a limited effect to reduce the habitat split local extinction prevision. The parameters
used r~K~mJ ~DJ ~DA ~L1 ~b~1 and mA ~0:01.
doi:10.1371/journal.pone.0066806.g005
exhibit farther extinction thresholds. Therefore, for a given
fragmented landscape submitted to habitat split, species with
lower diffusion ability are expected to be present in a smaller
number of fragments and ultimately be regionally extinct earlier.
Additionally, the probability of extinction in response to habitat
split gradients can be expected to be higher for species with lower
diffusion ability.
Amphibians vary considerably in dispersal ability. Across
species, the frequency distribution of maximum dispersal distance
fits an inverse power law [41].While 56% of the amphibian species
presented maximum dispersal distances lower than one kilometer,
7% could disperse more than 10 km [41]. However, those are data
for adults. For the habitat split model, the main parameter to be
estimated is the diffusion rate of the post-metamorphic juveniles.
Although such data is more difficult to be obtained, one expects
that the mean dispersal ability of juveniles should be lower than of
adults due to their smaller body size, lower energetic reserves and
higher sensitivity to environmental stress [42].
The reproductive success is also a crucial life history parameter
determining how far is the critical split distance for a given species.
In the model, reproductive success is defined as the average
number of post-metamorphic juveniles produced per adult living
in the fragment. Since this parameter varies between species, we
expect that species with higher reproductive success will be able to
keep viable populations in forest fragments that are farther away
from the breeding site in comparison to species with lower
reproductive success. We envisage that in future individual-based
models, recruitment could vary between individuals according to
their conditions.
Reproductive success is positively correlated to clutch size but
also a function of the survival rate of the aquatic larvae before
metamorphosis. Body size is possibly a good inter-specific
predictor of reproductive success. Body size has a strong positive
inter-specific relationship with clutch size, even after controlling
for the phylogeny [43]. Furthermore, for pond-breeding anurans
of three different families (Bufonidae, Hylidae and Ranidae), there
is a positive relationship between body size and egg-diameter [42].
as dragonflies and damselflies, have been recently demonstrated to
suffer from alterations in the physical structure of the riparian
vegetation that disconnect the aquatic habitat of the larvae from
the terrestrial habitat of the adults [33].
Our diffusive model reveals that habitat split alone can generate
extinction thresholds. Fragments located between the breeding site
and a given critical split distance are expected to contain viable
amphibian populations. In contrast, populations inhabiting
fragments farther from such critical distance are expected to be
extinct. The theoretical existence of extinction thresholds has been
also demonstrated for habitat loss and fragmentation [6,34,35]. In
this case, increase in the proportion of habitat loss above a certain
level causes an abrupt non-linear decay in population size.
Simulation models based on percolation theory suggest that this
can be simply attributed to structural properties of the fragmentation process [36]. However, biological mechanisms such as
minimal home range, minimal population size, and the Allee effect
contribute to such extinction thresholds [35].
The model can be also interpreted from a breeding site
perspective. Split distance is expected to have a negative impact on
the occupancy of matrix-inserted ponds. Indeed, a field study with
Rana dalmatina demonstrated that the number of egg-clutches in
ponds declines exponentially with increasing distance from a
deciduous forest. Ponds less than 200 m from the forest edge were
considered the most valuable for the species conservation [37]. For
Rana temporaria, the occupancy of ponds for breeding purposes is
influenced by the distance from suitable summer habitats [38].
Furthermore, some studies have shown that the richness of
amphibians in ponds is negatively related to the distance to forest
patches [39,40].
The habitat split model predicts that amphibian species with
different life history traits will exhibit different extinction
proneness in response to a given landscape setting. One key
feature determining the critical split distance (i.e. the distance
value of L1 at which the predicted abundance is zero) is the
diffusion rate of the post-metamorphic juveniles in the matrix.
Amphibian species with higher diffusion rate are expected to
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Therefore, species with larger body size can be expected to exhibit
larger critical split distances. The survival rate of the aquatic
larvae, however, depends on biotic interactions, such as predation
and competition. Furthermore, riverine systems are nowadays
submitted to multiple anthropogenic generated stressors [44],
agrochemicals [29], and emerging diseases [26] that can
dramatically alter mortality rates.
The model predicts that the extinction threshold can be pushed
away from the breeding site only at the cost of increasing the size
of the fragment, depending on the kind of boundary at the
interface habitat/exterior matrix. However, although enlarging
the size of the fragment can allow for larger splits, even in the case
of a very large fragment the critical split distance remains finite:
local extinction can occur even for infinite fragments when the
split distance is larger than a critical value. The implications of
such results for conservation are straightforward. When habitat
loss is intense and small fragments are the rule, the best landscape
scenario for the conservation of forest-associated amphibians with
aquatic larvae is the preservation of the riparian vegetation.
The quality of the matrix is also a key element defining the
critical split distance. Higher quality matrix generates lower
mortality rates of post-metamorphic juveniles enabling recruited
individuals to successfully reach forest fragments that are farther
away. For empirical studies this parameter is critical since
anthropogenic matrix vary widely in quality, from intensively
used cattle and crop fields to ecologically-managed tree monocultures [45]. Furthermore, roads are also important matrix elements
that can jeopardize the bi-directional migration of amphibians
[46].
In biodiversity hotspots [2], in particular, landscape design is
expected to play a crucial role in the conservation of the aquatic
larvae species. For instance, the Brazilian Atlantic Forest is home
of one the most species rich amphibian fauna of the world [2],
containing at least 300 endemic amphibian species [47]. Nowadays, only 11.7% of its original cover is left, and although the
protection of the riparian vegetation was, until recently, insured by
the Brazilian Forest Code (4771/65), habitat split is a common
feature in the landscape [16]. Not surprisingly, several amphibian
populations have declined recently [48,49,50] and many more are
expected to pay the extinction debt [51]. The habitat split model
reinforces the view that the conservation and restoration of
riparian vegetation should be properly enforced [52].
Metapopulation models assume disjunct breeding patches
containing individual populations that exist in a shifting balance
between extinctions and recolonisations via dispersing individuals
[6]. Realistic models on metapopulations have incorporated patch
area, shape, isolation besides the quality of the intervening matrix
[7]. The metapopulation concept has been applied to amphibians,
showing even structured genetic outcomes [53]. Despite that, the
application of metapopulation models to amphibians has been
questioned in several grounds [14,41]. We envisage that future
metapopulation models, when designed to species exhibiting
marked ontogenetic habitat shifts, will generate more accurate
predictions by the incorporation of habitat split effects.
Supporting Information
File S1 Mathematical formulation of the model, including the derivation of the stationary solution and a study
of its stability.
(PDF)
Acknowledgments
The authors thank André de Roos (Amsterdam) for fruitful discussions.
Author Contributions
Conceived and designed the experiments: CRF GC RAK. Performed the
experiments: CRF RMC FA JMB GC RAK. Analyzed the data: CRF
RMC FA JMB GC RAK. Wrote the paper: CRF RMC FA JMB GC
RAK.
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