Metapopulation dynamics in a complex ecological landscape

Abstract

We propose a general model to study the interplay between spatial dispersal
and environment spatiotemporal fluctuations in metapopulation dynamics. An
ecological landscape of favorable patches is generated like a L\'{e}vy dust,
which allows to build a range of patterns, from dispersed to clustered ones.
Locally, the dynamics is driven by a canonical model for the evolution of the
population density, consisting of a logistic expression plus multiplicative
noises. Spatial coupling is introduced by means of two spreading mechanisms:
diffusive dispersion and selective migration driven by patch suitability. We
focus on the long-time population size as a function of habitat configurations,
environment fluctuations and coupling schemes. We obtain the conditions, that
the spatial distribution of favorable patches and the coupling mechanisms must
fulfill, to grant population survival. The fundamental phenomenon that we
observe is the positive feedback between environment fluctuations and spatial
spread preventing extinction.

Full text

Metapopulation dynamics in a complex ecological landscape
E. H. Colombo1, ∗and C. Anteneodo1, 2 , †
1Departament of Physics, PUC-Rio, Rio de Janeiro, Brazil
2Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil
We propose a general model to study the interplay between spatial dispersal and environment spa-
tiotemporal fluctuations in metapopulation dynamics. An ecological landscape of favorable patches
is generated like a L´evy dust, which allows to build a range of patterns, from dispersed to clustered
ones. Locally, the dynamics is driven by a canonical model for the evolution of the population
density, consisting of a logistic expression plus multiplicative noises. Spatial coupling is introduced
by means of two spreading mechanisms: diffusion and selective dispersal driven by patch suitability.
We focus on the long-time population size as a function of habitat configurations, environment fluc-
tuations and coupling schemes. We obtain the conditions, that the spatial distribution of favorable
patches and the coupling mechanisms must fulfill, to grant population survival. The fundamental
phenomenon that we observe is the positive feedback between environment fluctuations and spatial
spread preventing extinction.
PACS numbers: 87.23.Cc, 89.75.Fb, 05.40.-a
I. INTRODUCTION
Habitat fragmentation is commonly observed in na-
ture associated with heterogeneity in the distribution of
resources, e.g., water, food, shelter sites, physical fac-
tors such as light, temperature, moisture, and any fea-
ture able to affect the growth rate of the population of a
given species [1]. A fragmented population made of sub-
populations receives in the literature the suitable name
of metapopulation [1–3]. These fragments, also known as
patches, are not completely isolated as they are coupled
due to movements of individuals in space. For model-
ing purposes, as a first step one can adopt a single patch
viewpoint, taking into account the impact of the sur-
rounding population in an effective manner [4–7]. As a
further step beyond the single patch level, one can re-
sort to a spatially explicit model. From this perspec-
tive, deterministic and stochastic theoretical models have
been developed to obtain the macroscopic behavior of the
whole population [2, 8–12]. One of the main results is
the detection of critical thresholds that delimit the con-
ditions for the sustainability of the population, which oc-
curs for a suitable combination of diverse factors, related
to quality and spatial structure of the habitat, migration
strategies and extinction rates. Here, we address related
fundamental questions in metapopulation theory propos-
ing a model that includes a general dispersion process,
incorporating random and selective dispersal strategies.
Additionally, we investigate the model dynamics on top
of a complex ecological landscape whose spatial structure
can be tuned, ranging from spread to aggregated patches.
In Sec. II, we will describe in detail each part of the
model. The spreading process and spatial configuration
of the ecological landscape are described in Secs. II A
∗eduardo.colombo@fis.puc-rio.br
†celia.fis@puc-rio.br
and II B, respectively. The results, reported in Secs. III
and IV, focus on the impact of the spatial arrangement
of the habitat on its overall viability, that is, on the long-
time behavior of the population size. Mainly numerically,
and with the aid of analytical considerations, we inves-
tigate the impact of habitat topology, spread range and
stochasticity in the long time behavior of the population
size, compared to the corresponding uncoupled metapop-
ulation.
II. MODEL
Let us start from the local dynamics perspective. We
assume that the rules that govern the dynamics of a single
patch can be primarily modeled by the logistic or Ver-
hulst expression [13], since it mimics reproduction and
intraspecific competition which are the two fundamen-
tal deterministic driving factors. In mathematical terms,
the evolution of the population density (number of indi-
viduals per unit area), ui, in each patch iis described
by
˙ui=aiui−bu2
i,(2.1)
where aiand bare real parameters. Such simple model
allows to predict the relaxation towards a steady state
that can be null (ui= 0) or not (ui=ai/b > 0), de-
pending on the interplay between the population growth
given by the intrinsic growth rate aiand the intraspecific
competition for resources modulated by parameter b.
In real systems, however, the evolution is not deter-
ministic. Stochasticity is introduced mainly by (i) the
inherent complexity of the environment, which produces
fluctuations in the growth rate (external, environmental
noise) and (ii) variations in the birth-death process (in-
ternal, demographic noise) [4]. These effects have been
previously incorporated to Eq. (2.1) as [4, 7, 14, 15]:
˙ui= [ai+ση•ηi(t)]ui−bu2
i+σξ√ui◦ξi(t),(2.2)

2
where σηand σξare positive parameters, and ηiand
ξiare assumed to be mutually independent zero mean
and unit variance Gaussian white noises. The environ-
mental noise term which introduces fluctuations in the
growth rate, modulated by ση, is expected to have exter-
nal origins, then, its correlation even if small is non-null,
justifying the use of Stratonovich calculus (•) to treat
its multiplicative nature. The demographic noise, mod-
ulated by σξ, represents fluctuations in the reproduction
process of each independent individual, then its magni-
tude is proportional to the square root of population size,
so that its variance is the sum of the variances of in-
dependently identically distributed individual stochastic
contributions [14, 15]. Moreover, under the assumption
of uncorrelated and non-anticipative noise, Itˆo calculus
(◦) is the adequate choice [4, 14–16].
We assume that Eq. (2.2), which is known as canonical
model [4, 14, 15], defines the local dynamics that takes
place on each site iof a lattice. Moreover, we will say
that a patch iis favorable if it induces positive growth
at low densities (i.e., ai=A+
i>0) and unfavorable if it
is adverse to support life (i.e., ai=A−
i<0).
Spatial coupling is introduced by migrations from one
patch to another. Then, the full model can be expressed
by
˙ui=aiui−bu2
i+DΓi[u] + σηui•ηi(t) + σξ√ui◦ξi(t),
(2.3)
where the additional term DΓi[u], with D > 0, is given
by the net flux Γi[u] towards patch i. It is the nonlocal
term that couples the set of stochastic differential equa-
tions (2.2). We model the populational exchange between
patches based on two behavioral strategies: one where
the individuals spread in space diffusively, driven by den-
0
20
40
60
80
100
0 20 40 60 80 100
0
0.5
1
1.5
2
2.5
Figure 1. Ecological landscape and population distribution in
a square lattice of linear size L= 100. Each lattice cell repre-
sents a patch. The landscape is defined by the configuration
of favorable (positive growth rate) and unfavorable (negative
growth rate) patches. A favorable patch is denoted by a black
open square. The population density (number of individuals
per unit area) is represented by shades of green.
sity differences, and another where individuals transit
selectively, mainly driven by patch-quality differences.
The precise form of these exchanges will be presented
in Sec. II A.
Finally, we construct a complex arrangement of favor-
able and unfavorable patches, that will be defined in
Sec. II B. For the sake of simplicity, we consider a bi-
nary landscape, where sites can be in any of two states,
A+
i=−A−
i=A > 0, as assumed in previous stud-
ies [17, 18]. A typical configuration of the model system
in a square lattice is illustrated in Fig. 1.
A. Nonlocal coupling
In order to define the coupling scheme let us state some
considerations. First, let us assume that spatial spread is
conservative, preserving the number of individuals during
travels, and also that it is nonlocal, in the sense that
individuals can travel long distances over the landscape,
for example like butterflies and birds [8, 19].
Furthermore, it is reasonable to assume that active in-
dividuals like butterflies, birds and also terrestrial ani-
mals use their perception and memory to increase the
efficiency in the search for viable habitats. Spatial knowl-
edge can be acquired, for instance, by a direct visualiza-
tion, previous visit or by the perception of the collective
dynamics. The spatial information stored by the indi-
viduals can yield optimized routes between favorable re-
gions. In fact, there is a relation between spatial memory
and migration strategy [20]. We introduce this trait by
allowing individuals to have access to information about
the spatial distribution of patch quality. This will origi-
nate selective routes towards favorable patches and some
directions will be preferred. Otherwise, if individuals do
not have any information about the ecological landscape,
or if they do not have memory, uncorrelated trajectories
(random movements) can emerge. In fact, this has been
the focus of works on animal foraging, where optimal effi-
ciency in resource search occurs without previous knowl-
edge of food distribution [21]. This type of behavior has
isotropy as a main trait, indicating directional indiffer-
ence.
We contemplate both scenarios by modeling spread
through a diffusive component together with a contribu-
tion of direct routes connecting favorable patches, gov-
erned by quality differences. The relative contribution
of both mechanisms is regulated by parameter δ, with
0≤δ≤1 tuning from the ecologically driven (δ= 0) to
the purely diffusive (δ= 1) cases. Moreover, we assume
that coupling is weighted by a factor γ(dij ) that decays
with the distance dij [22] between patches iand j, as will
be defined below. Then, the flux Jij from patch ito jis
given by
Jij = [δ+ (1 −δ)αij ]γ(dij )ui≥0,(2.4)
where αij ≡(aj−ai)/(4A) + 1/2. Hence, the total flux

3
is
Γi[u] = X
j6=i
(Jji −Jij )
=X
j
γ(dij ) [δ(uj−ui) + (1 −δ)(αji uj−αij ui)] .
(2.5)
The total density is conserved by the exchanges described
by Eq. (2.5), as can be seen by summing over i. It indi-
cates that individuals tend to move towards patches with
fewer individuals and better quality. For δ= 1, Eq. (2.5)
represents a generalization of the Fick’s law for nonlocal
dispersal driven by density gradients. For δ= 0, with
our definition of αij , and binary patch growth rate, the
possible values of αjiuj−αij uiare
j
iA−A
A(uj−ui)/2uj
−A−ui(uj−ui)/2
This means that, when the quality of two patches is dif-
ferent, the flux occurs in the direction of the higher qual-
ity, weighted by the out-flowing population density (low-
est quality patch). Only when the quality is the same,
diffusive exchange can occur, to allow a network of favor-
able patches.
Concerning the factor that takes into account the dis-
tance between patches, there is empirical evidence [8, 23]
that the frequency of occurrence of flights between
patches decays with the distance, which is reasonable due
to the increase of energetic cost. Although diverse decay
laws are possible, we will assume exponential decay of
the weight γwith the traveled distance `, as observed for
some kinds of butterflies [8, 23, 24], that is
γ(`) = N−1exp(−`/`c),(2.6)
where `cis a characteristic length (the average traveled
distance) and the normalization constant Nis such that
the sum of the contributions of all patches equals one.
Operationally, we will truncate the exponential at `'
8`c<< L, where Lis the linear characteristic size of the
landscape.
B. Ecological landscape
In nature, the arrangement of the ecological landscape
is built by many distinct processes, occurring in many
time scales, creating complex spatiotemporal structures.
Then, beyond the inclusion of the environmental noise
η, it is also important to take into account the spatial
organization of patches [8, 9, 25, 26].
Heterogeneity and patchiness are adequate to cap-
ture the complexity of diverse ecological systems [27–31].
Here we propose to use as complex ecological landscape
aL´evy dust [21] distribution of favorable patches on a
square domain of size L×Lpatches, with periodic bound-
ary conditions. Over a background of adverse patches
(ai=−A), we construct a L´evy dust of favorable patches
(ai=A) given by the sites visited by a L´evy random walk
with step lengths `drawn from the probability density
function
p(`)∝1/`µ,(2.7)
with 1 ≤`≤L. This protocol has been used in the
study of different problems [21, 28, 29], but we apply it
here in the study of metapopulation dynamics. It allows
to mimic a general class of realistic conditions [27, 29–
31] and to tune different habitat landscapes through pa-
rameter µ, from widely spread (for µ= 0) to compactly
aggregated in a few clusters separated by large empty
spaces (for µ= 3), as illustrated in Fig. 2.
µ= 0.5µ= 1.5
µ= 2.0µ= 3.0
Figure 2. Ecological landscape for different values of the ex-
ponent µthat characterizes the distribution of L´evy jumps
given by Eq. (2.7), used to build the configuration of favor-
able patches in a square domain with L= 100. Black cells
indicate positive growth rate A(favorable patches) and white
cells negative growth −A. In all cases the density of favorable
patches is ρ= 0.1.
We quantify the change in the spatial structure by
computing the probability distribution of the distance
dbetween favorable patches Pµ(d) (see Fig. 3). For
the density ρ= 0.1 used in the figure, when µ.1,
patches are typically far from each other. For high val-
ues (µ&3), the generating walk approaches the standard
random walk, creating a much more clustered structure,
evidenced by the peak at short distances. However the
shape of Pµ(d) changes with ρ. When the patch density
ρis high, the shape of Pµ(d) resembles that of the uni-

4
Pµ(d)
d
µ= 0
µ= 1.0
µ= 2.0
µ= 3.0
0
0.01
0.02
0.03
0.04
0 10 20 30 40 50 60 70
Figure 3. Probability distribution of the distance between fa-
vorable patches for different values of exponent µin Eq. (2.7),
for density ρ= 0.1 and lattice size L= 100 (200 configura-
tions were used). Fluctuations are due to the discrete nature
of the possible distances in the lattice. The dotted lines are
a guide to the eye. The solid line represents the probability
distribution for the distance between uniformly distributed
random points in continuous space, drawn for comparison:
P(d) = 2πρd if d < L/2, P(d) = 2πρd(1 −4arcos(L/[2d]),
otherwise.
form arrangement even for large µ, while at low densities
Pµ(d) presents a peak at small dsince the resulting con-
figuration of patches is very localized even for small µ,
as will be discussed in Sec. IV C. Furthermore, Pµ(d) is
also sensitive to L, but we kept Lfixed (L= 100), even
if some properties may have not attained the large size
limit, as far as µand ρallow to scan many qualitatively
different possibilities of landscape structure.
C. General considerations about the model
The set of parameters {D, δ, `c}regulate the nonlocal
dynamics. While Dis the strength of the nonlocal cou-
pling, δcontrols the balance between diffusion and di-
rected migration, and `cdefines the coupling range. The
ecological landscape is characterized by ρand µthat set
the density and the degree of clusterization, respectively.
In the results presented in the following sections, we
will restrict the analysis to a region of parameter space
relevant to discuss the main phenomenology of the model.
Thus, we will set A=b= 1 in all cases. We will also
consider L= 100 and typically ρ= 0.1. Concerning
the noise parameters, we set ση=σξ= 0 to analyze the
deterministic case in Sec. III and turn noise on by setting
ση=σξ= 1 in Sec. IV. This choice is based on previous
works [4, 14]. Indeed, population size can be sub ject
to large fluctuations as demonstrated by experimental
data [19].
We performed numerical simulations of Eq. (2.3) on
top of different landscapes, by preparing the system in
the stationary state of the deterministic and uncoupled
case, i.e., ui(0) = max{ai/b, 0}for all i, plus a small
noise. Integration of Eq. (2.3) was carried out with
Euler-Maruyama scheme with a time step ∆t= 10−3.
Stratonovich noise was implemented by performing a
shift in the drift to obtain the corresponding equivalent
Itˆo version [32].
III. DETERMINISTIC CASE (ση=σξ= 0)
Before proceeding to study the full model, we consider
the deterministic case. Locally, when stochastic contri-
butions are neglected, the asymptotic value of the popu-
lation size for each patch is ui=ai/b. Introducing non-
local effects, the population size might change. If pop-
ulation exchanges between patches are guided solely by
their quality (δ= 0), then, the favorable-patch network
will conserve the initial population size, so no interest-
ing phenomena occur from the viewpoint of extinction.
However, when δ > 0, the diffusive behavior induces ex-
ploration of the neighborhood independently of habitat
quality, which leads to the occupation of unfavorable re-
gions making likely the death of individuals.
10−12
10−10
10−8
10−6
10−4
10−2
100
102
0 200 400 600 800 1000
n
t
10−2
100
102
104
100101102103
Figure 4. Deterministic (ση=σξ= 0) time evolution of the
total population density nin different initial landscapes, for
δ= 1, ρ= 0.1, D= 10, `c= 0.5 and µ= 1.7. This particular
set of values of the parameters results in about half of 50
realizations leading to extinction. We use a dotted line to flag
the ones that tend to extinction exponentially fast and a solid
line for those that lead to population survival. Alternatively,
the same data are represented as a log-log plot in the inset.
By numerical integration of Eq. (2.3) we obtain the
time evolution of the total population density n(t) =
PL2
i=1 ui(t). In Fig. 4 we show the outcomes for fixed
values of the model parameters and different initial con-
ditions (different landscapes). While some of the realiza-
tions lead to exponential decay of the population other
ones attain finite values at long times. Several different
non null steady states can be attained. Notice however,

5
that the steady values of different realizations are all be-
low that of the uncoupled case, ρL2A/b = 1000 for the
parameters of the figure. Hence, diffusion favors the de-
crease of the total population density and the occurrence
of extinctions, as expected.
In order to investigate how the fraction of survivals
changes with the topology, we plot in Fig. 5 the number of
survivals per realization, fs, as function of the landscape
parameter µ, for several values of the coupling coefficient
D. Besides the initial condition used throughout this
paper (see Sec. II C), we observed that a perturbation
of the null state also leads to the same results of Fig. 5.
For given µ, increasing Dfavors the occurrence of extinc-
tions as already commented above. For given D, below a
threshold value of µthe population gets extincted in all
the realizations, while above a second threshold it always
survives (for the finite number of realizations done), be-
tween thresholds both states, the null and non null ones,
are accessible. The number of non null stable states in-
creases with µ.
Summing over all ithe deterministic form of Eq. (2.3),
one finds that the steady solution ˙n= 0 must satisfy
Piaiui=bPu2
i, which has infinite solutions between
the fundamental null state and the uncoupled case solu-
tion (the only stable one for D= 0). The condition for
stationarity of the total density depends only on the local
parameters, since fluxes are only internal, however, the
coupling and landscape can stabilize configurations other
than the trivial ones. Furthermore, in the Appendix, we
performed an approximate calculation to show that, for
small D(recalling that ai=±A), the null state is stable
if
A−D(1 −γµ)>0,(3.1)
0
0.2
0.4
0.6
0.8
1
0.5 1 1.5 2 2.5 3
fs
µ
D= 5
D= 10
D= 20
Figure 5. Fraction of surviving metapopulations fs(over 100
realizations) in the deterministic case (ση=σξ= 0) as a
function of exponent µin Eq. (2.7) that gives the degree of
clusterization. The other parameters are δ= 1, ρ= 0.1,
`c= 0.5, for the values of the coupling coefficient Dindicated
on the figure. In this and following figures, dotted lines are a
guide to the eye.
where 0 ≤γµ≤1 is a factor that mirrors the topology, as
defined in Eq. (A3), varying from γµ=ρfor the uniform
case µ= 0 to γµ= 1 in the limits of large µor large
ρ. Despite this approximate expression fails in providing
accurate threshold values, it predicts that survival is fa-
cilitated by larger Aand spoiled by increasing D. It also
qualitatively predicts the impact of the topology, as far
as it indicates that the destructive role of diffusion can be
compensated by a large enough degree of clusterization
of the resources given by large γµ.
IV. STOCHASTIC CASE
First let us review some known results about the local
(one site) dynamics, which is obtained in the limit D→0
of Eq. 2.3 (canonical model). In the deterministic case,
the two-state habitat [17, 18] leads to local extinction
(if ai=−A) or finite population (if ai= +A). The
presence of stochastic contributions changes the stability
of the patches. When ai=−A < 0, the local extinction
event predicted deterministically is reinforced by noise.
For ai= +A > 0, the demographic (Itˆo) noise ξ(in the
presence of the Stratonovich noise η) leads to extinction
in a finite time. But this time diverges as σξ→0[4, 6].
This divergence is due to the fact that when only the
Stratonovich noise ηis present, the null state is strictly
not accessible, for any noise intensity, in the continuous
model. That is, the external noise ηreduces the most
probable value of the population size, that becomes very
close to zero, but non null, when ση>p2A/b [33].
The population stability can be quantified by the mean
time to extinction Taveraged over realizations starting
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
T
ση,σξ
ση
σξ
Figure 6. Single patch dynamics. Mean extinction Ttime vs
noise strengths ση(fixing σξ= 1, triangles), that modulates
the fluctuations in the growth rate, and σξ(fixing ση= 1,
diamonds), that modulates the demographic noise. Symbols
correspond to numerical simulations averaged over 500 sam-
ples and the full lines to the theoretical prediction given by
Eq. (4.1). The curve for variable σξdiverges in the limit
σξ→0.

6
at u(0). For Eq. (2.3) with D= 0, Tis given by [14],
T=Zu0
0Z∞
z
exp Rv
zΨ(u)du
V(v)dvdz , (4.1)
where Ψ(u) = 2M(u)/V (u), with M(u) = au −bu2+
σ2
ηu/2 and V(u) = σ2
ηu2+σ2
ξu. The results of Eq. (4.1)
are in good accord with those from numerical simula-
tions, as illustrated in Fig. 6. When the noise intensity
decreases, the time to extinction always increases, being
divergent in the limit σξ→0.
A. From local to global behavior
In this section we investigate the effects introduced by
patch coupling, i.e., when D6= 0. Nonlocal contributions
redistribute the individuals in space, driven by density
and quality gradients. In Fig. 7 we show that D6= 0 pre-
vents the extinction events that occur when D= 0 (see
Fig. 6). Therefore, in contrast to the deterministic case,
now spatial coupling is constructive. On the other hand,
noise has also a constructive role when D6= 0, differently
to the uncoupled case, not only preventing extinction but
also contributing to the increase of the population (as in
the case D= 10). In a previous work [7], we already ob-
served the constructive role in population growth of lin-
early multiplicative Stratonovich noise in contrast with
the destructive behavior of its Itˆo version. Therefore, en-
vironmental noise and coupling have a positive feedback
effect on population growth, as shown in Fig. 7.
We will compute the long-time total population density
n∞≡limt→∞ n(t), which is useful to be compared with
0
0.5
1
1.5
2
0 20 40 60 80 100
n/n0
t
D= 10−2
D= 10
Figure 7. Temporal evolution of n/n0, the total population
density relative to the initial value n0(set as the uncoupled
deterministic value). For δ= 0.5, ρ= 0.1, `c= 0.5, µ= 2.0,
ση=σξ= 1 and values of the coupling coefficient Dindicated
on the figure. We highlight a single realization (black full line)
for each set of 50 realizations (gray lines). The dashed line at
n=n0is plotted for comparison.
the initial value n0≡n(0) = ρL2u0=ρL2(A/b), that
represents the asymptotic total density in the determin-
istic uncoupled case. Then we will measure the long-time
relative total population density E≡ hn∞i/n0, that rep-
resents a kind of efficiency, where the brackets indicate
average over landscapes and noise realizations.
In the upper panel of Fig. 8 we plot the long-time rela-
tive value Eas a function of D. We see that for very small
values of D, the population is non null, although the fi-
nal relative population density Eis smaller than one.
Moreover, for given D, the long-time relative value Eis
smaller when the diffusive component is absent (δ= 0).
In all cases, Efirst increases with Dand even exceeds
the value E= 1, indicating again that not only the noise
has a constructive role in preventing extinction but also
in promoting the increase of the initial total population.
When the diffusive strategy is present (δ > 0), the in-
0
0.5
1
1.5
2
10−610−510−410−310−210−1100101102
E
D
δ= 0.0
δ= 0.5
δ= 1.0
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
E
δ
D= 0.01
D= 5
D= 10
D= 100
Figure 8. Long-time relative total population density, E≡
hn∞i/n0, as a function of the coupling coefficient D, for dif-
ferent values of δ(upper panel) and Eas a function of δ, for
different values of D(lower panel). Recall that parameter δ
regulates the balance between the diffusive (δ= 1) and se-
lective (δ= 0) strategies. The other parameters are ρ= 0.1,
`c= 0.5, µ= 2.0 and ση=σξ= 1. The symbols represent
the average over 20 samples and the vertical bars the standard
error. The dashed line at E= 1 is plotted for comparison.

7
crease of Eoccurs up to an optimal value of the coupling
D(with E > 1), above which the Edecays. Hence,
there is a nonlinear effect that does not reflect the linear
combination in Eq. (2.4), as shown in the lower panel
of Fig. 8. The diffusive component, despite being much
less efficient, like in placing individuals in unfavorable
regions, acts with greater connectivity. Then, for small
D, the nonlocal contribution of the diffusive coupling is
much higher than in the δ= 0 case, leading to a higher
population size. In fact, the abrupt transition in the con-
nectivity of the spatial coupling is mirrored in the abrupt
change suffered by Eas δbecomes non null. Contrarily,
for high values of D,δ= 0 is more efficient due to high
damage caused by an intense dispersal towards unfavor-
able regions, which in the case of Fig. 8 are the majority
of the sites. All these observations highlight the impor-
tance of the diffusive strategy, that can become more ef-
ficient than the ecological pressure driven by the quality
gradient.
B. Habitat topology and coupling range
The nonlocal contribution results from the combina-
tion of the spread strategies, interaction range and topol-
ogy, characterized by δ,`cand µ, respectively. Fig. 9
shows the long-time relative population density Eas
function of µwith different values of `cfor δ= 1 and
δ= 0.
E > 1 means that the combination of habitat topol-
ogy and spatial coupling range leads the population to
profit from the environment fluctuations, increasing its
size. The region E > 1 is bigger when individuals are
selective with respect to their destinations (δ= 0) and
increases with `c. For the diffusive strategy (δ= 1),
E > 1 is attained only in a clustered habitat (large µ)
together with short-range dispersal (small `c). We have
already seen that in a sparse habitat, diffusion represents
a waste, specially if the dispersal is long-range. Instead,
when δ= 0, the habitat does not need to be so clustered
or the range so short for population growth. In this in-
stance, the optimal combination occurs in a clustered
habitat but with long-range coupling. Finally note that,
as `cincreases, Ebecomes independent of the topology.
C. Density of favorable patches
Another important issue is the influence of the density
ρof favorable patches in the dynamics. Until now, we
have kept it constant to highlight the effects of the hetero-
geneity of the habitat and of the coupling schemes in the
longtime behavior of the total population size. In terms
of the protocol used to generate the ecological landscape,
ρnot only changes the proportion of favorable patches
but also reshapes the distribution of distances between
favorable patches. In Fig. 10 we show three different
outcomes of the spatial structure and the corresponding
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.5 1 1.5 2 2.5 3
E
µ
δ= 0
ℓc= 0.2
ℓc= 0.5
ℓc= 1.0
ℓc= 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0.5 1 1.5 2 2.5 3
E
µ
δ= 1
ℓc= 0.2
ℓc= 0.5
ℓc= 1.0
ℓc= 1.5
Figure 9. Long-time relative total population density, E≡
hn∞i/n0, as a function of exponent µin Eq. (2.7) for different
values of the coupling range `c, when δ= 0 (selective strategy,
upper panel) and δ= 1 (diffusiv strategy, lower panel), with
ρ= 0.1, D= 20 and ση=σξ= 1. The symbols represent the
average over 20 samples and the vertical bars the standard
error. The dashed line at E= 1 is plotted for comparison.
distance distribution for a fixed value of µ= 2. For low ρ,
patches organize in a kind of archipelago structure, that
is much smaller than the system size, and the distance
resembles that obtained for large µwhen ρ= 0.1. For
high ρ, many points of the domain are visited creating a
distance distribution that approaches the homogeneous
form. For µhigher than the value of the figure, pro-
files very similar to those shown in Fig. 10 are obtained.
Meanwhile, for small values of µ, the distribution is al-
most invariant with ρ, being very close to that of the
uniform case. This is due to frequent flights with lengths
of the order of system size. Concerning the factor γµ
that reflects the topology, as defined in Eq. (A3), it can
be affected by ρmore through the amount of favorable
patches nvthan by its indirect consequences on the spa-
tial distribution Pµ.
In Fig. 11, we show Eas a function of ρfor the case
µ= 2. By comparing the outcomes for different values of
δ, we see the impact of distinct connectivities. In order

8
ρ= 0.01 ρ= 0.1ρ= 0.8
0
0.02
0.04
0.06
0.08
0 10 20 30 40 50 60 70
Pµ(d)
d
0
0.02
0.04
0.06
0.08
0 10 20 30 40 50 60 70
Pµ(d)
d
ρ= 0.01
ρ= 0.1
ρ= 0.8
Figure 10. Ecological landscape (upper panels), with favor-
able patches in black, probability distribution of the dis-
tance between favorable patches, averaged over 100 land-
scapes (lower panel). Three different values of ρ, indicated
on the figures, were considered. In all cases, L´evy exponent
µ= 2.
to interpret this figure, recall that the initial population
density n0is proportional to the number of favorable
patches nv, namely n0=nvA/b =ρL2.
For δ= 1, Epresents a minimum value for ρ'0.15.
Beyond this value, Egrows with ρattaining the value of
the full favorable lattice. In the opposite limit of vanish-
ing ρ(no favorable patches), Ediverges as far as, accord-
ing to the model, (intrinsically) favorable patches are not
necessary to promote growth, due to the noisy growth
rate. However, if noise is reduced, then the stochastic
dynamics approaches the deterministic one, where the
population will certainly go extincted.
Now, turning our attention to the δ= 0 case, Eis
monotonically increasing with ρ, also attaining a limiting
value when ρ→1. Differently from the diffusive case,
there exists a critical value ρc= 4 ×10−4(nv= 4) for
population survival.
For small ρ, it is curious that the role played by the
connectivity, according to the model, makes the diffusive
behavior more efficient, while selective moves are impor-
tant at high values of ρ. In this case, when the system
is approaching a fully favorable landscape, the long-time
relative population density Etends to be the same for
different values of δ. For intermediate values of ρ, we see
that the selective strategy overcomes the diffusive one
(but never overcomes the combined scheme).
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
E
ρ
δ= 0.0
δ= 0.5
δ= 1.0
Figure 11. Long-time relative total population density, E≡
hn∞i/n0as a function of the favorable-patch density ρ. Dif-
ferent values of the strategy balance parameter δwere also
considered as indicated on the figure. Recall that δallows to
tune from the purely diffusive strategy (δ= 1) to the selec-
tive one (δ= 0) The remaining parameters are A=b= 1,
D= 20, `c= 0.5, µ= 2.0, ση=σξ= 1 and L= 100. The
symbols represent the average over 20 samples and the ver-
tical bars the standard error. The horizontal line represents
E= 1.
V. CONCLUDING REMARKS
We implemented a general model in which the local
dynamics, ruled by the canonical model [14], was cou-
pled through different schemes on top of a complex land-
scape. This setting allowed to study the role of the habi-
tat spatial structure and the stochastic fluctuations on
the long-time state of the metapopulation. We restricted
the analysis to a region of parameter space relevant to
display the main features and the interplay between the
different processes involved. For the deterministic case,
we have shown that, for small coupling coefficient D, the
distribution of favorable patches must be clustered (large
µ) enough for survival, while below a critical value of µ
extinction occurs. For the stochastic case, we have shown
that noise in combination with spatial coupling has a
constructive role, that drives the population to survival,
in contrast to the decoupled case where isolated patches
would be extincted in finite time. We also studied the ef-
fects of the spreading strategy, pointing out that a mixed
strategy (diffusive dispersion plus selective routes) will
result in a larger population size (Fig. 8).
Furthermore, when the population survives in long-
time observations, we analyzed the steady state by means
of the quantity E=hn∞i/n0, which is the quotient be-
tween the average long-time population size and its un-
coupled deterministic value. This allowed us to high-
light the outcome of the combination of spatial coupling
and stochasticity, when compared to the case where both
are neglected (i.e. σξ=ση=D= 0). On the one
hand, the deterministic dynamics shows that diffusion

9
decreases population size. On the other, stochasticity is
the main responsible to lead population to extinction.
Then, both mechanisms when considered separately are
harmful to population conservation. However, our results
show that the combination of both mechanisms is con-
structive. This occurs for clustered habitats (large µ).
The constructive effect can be enhanced when dispersion
is short-range (small `c) under the diffusive strategy, or
when dispersion is long-range under the selective strat-
egy.
Our model could be improved in several directions. For
instance by considering correlated environment fluctua-
tions, exhaustible resources, etc. But, despite simple, the
model shows the impact of spatial coupling, spatiotem-
poral fluctuations and their interplay, allowing to foresee
the qualitative conditions for population survival as well
as the optimal dispersal strategy.
Appendix A: Stability of deterministic steady states
In order to study how steady state stability is affected
by spatial coupling, let us assume that the population is
located at the favorable patches, which is true for small
D(that is, close to the uncoupled case), and that the cou-
pling is purely diffusive (δ= 1). For a favorable patch,
the deterministic form of Eq. (2.3) reads
˙ui=Aui−bu2
i+DX
j6=i
(uj−ui)γ(dij )
= (A−D)ui−bu2
i+DX
j6=i
ujγ(dij ),(A1)
recalling that Pj6=iγ(dij ) = 1. To estimate the last term,
that represents the flow of individuals from the neighbor-
hood towards patch i,Jin
i, we consider that uj≈ui. In
this case
Jin
i=uiX
j6=i
γ(dij ),(A2)
where the sum effectively runs over the nvfavorable
patches. The average over arrangements of a landscape
γµ≡ hPj6=iγ(dij )ican be estimated as
γµ=nvZPµ(`)e−`/`cd` . (A3)
It depends on µand on the density ρ, such that it varies
from ρ(when µ= 0) to 1, in the extreme cases of either
maximal density or very large µ. That is, γµincreases
with µ, with ρand with `ctoo. Then, Eq. (A1) can be
approximated by
˙ui'(A−D[1 −γµ])ui−bu2
i≡Gui−bu2
i.(A4)
If G > 0, the population will grow and assume a finite
value, bounded by the carrying capacity. Meanwhile, D
diminishes the effective growth rate G, that becomes neg-
ative for sufficiently large D, namely for
D > A/(1 −γµ) (A5)
indicating decrease of the population. In fact notice in
Fig. 5 that the smaller Dthe less frequent the extinction
events for a given µ. This effect can be mitigated by
the landscape, through parameter γµ, when the density
of favorable sites or clusterization associated with large
µincreases. Eq. A5 also provides the linear stability
condition for the null state. If G < 0, the population
will decrease and go extincted.
ACKNOWLEDGEMENTS
C.A. acknowledges the financial support of Brazil-
ian Research Agencies CNPq and FAPERJ. E.H.C. ac-
knowledges financial support from Coordena¸c˜ao de Aper-
fei¸coamento de Pessoal de N´ıvel Superior (CAPES).
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