THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR METAPOPULATIONDYNAMICS AND CONSERVATION
By
MIGUEL A. ACEVEDO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2013
c⃝ 2013 Miguel A. Acevedo
2
To my mother who taught me the values of perseverance and hard work
3
ACKNOWLEDGMENTS
First and foremost, I would like to thank my advisor Dr. Robert J. Fletcher, Jr, for
believing in my ideas and sharing with me his wealth of knowledge and for providing
invaluable guidance. It was a great honor to be his student. I am grateful to Dr.
Raymond Tremblay and Dr. Elvia Melendez-Ackerman for providing the data analyzed in
Chapter 1. I would also like to thank Dr. J. Cole Smith and Jorge Sefair for their patience
and for their invaluable help with the mathematical formulations in Chapter 3. I would
also like to thank my committee, Dr. Madan Oli, Dr. Katherine Sieving and Dr. Robert
Holt for their help at multiple stages of my degree. I would like to thank Andre Revell,
Conor Egan and Kaan Kerman for their help in the field. This work greatly benefited from
discussions with Dr. Benjamin Bolker, Dr. Severine Vuilleumier, Dr. Christine Miller and
members of the QSE3 IGERT. I am grateful to Dr. Jeff Hostetler, Aida Miro, Raya Pruner,
Nathan Marcy, Dr. Jason Evans, Dr. Kyle McCarthy, Varun Goswami, Divya Vasudev,
Rajeev Pillay, Chris Rota, Kristen Aaltonen, Eva Kneip, Binab Karmacharya, Noah
Burrell, Katie Haase, Luke McEachron, Madelon Van de Kerk, Irina Skinner, Mauricio
Nunez-Regueiro, Brian Reichert, Ellen Robertson, Irina Skinner, Rebecca Wilcox, and
Dr. Jennifer Seavey . Funding was provided by the NSF Quantitative Spatial Ecology,
Evolution and Environment (QSE3) Integrative Graduate Education and Research
Traineeship Program (IGERT) grant 0801544 at the University of Florida and an NSF
Doctoral Dissertation Improvement Grant (DEB–1110441). Funding was also provided
by the School of Natural Resources and Environment and the Department of Wildlife
Ecology and Conservation at the University of Florida.
I would also like to thank my family, specially Melba Torres and Carlos Milan, and
my good friends Dr. Rey Rivera and Dr. Melanie Mediavilla for their support. I am also
particularly grateful to Christine Gutierrez for her invaluable support.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
CHAPTER
1 THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR CONNECTIVITYAND PATCH OCCUPANCY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 Study System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Site Occupancy Model and Parameter Estimation . . . . . . . . . . 15
1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.2 Colonization and Extinction . . . . . . . . . . . . . . . . . . . . . . 241.4.3 Asymmetric Dispersal and Metapopulation Modeling . . . . . . . . 26
2 THE PROXIMATE CAUSES OF ASYMMETRIC MOVEMENT . . . . . . . . . . 29
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1 Study Area and Focal Species . . . . . . . . . . . . . . . . . . . . 312.2.2 Study Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2.1 Mark-recapture . . . . . . . . . . . . . . . . . . . . . . . . 312.2.2.2 Field experiment . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.3 Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3.1 Mark-recapture . . . . . . . . . . . . . . . . . . . . . . . . 372.2.3.2 Field experiment . . . . . . . . . . . . . . . . . . . . . . . 38
2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 OPTIMAL SITE SELECTION STRATEGIES: PROTECTING AGAINST WORST-CASEDISTURBANCE SCENARIOS . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Network Flow Interdiction Models . . . . . . . . . . . . . . . . . . . 493.2.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2.1 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . 52
5
3.2.2.2 Connectivity assessment . . . . . . . . . . . . . . . . . . 543.2.2.3 Disturbance: simulating a worst-case scenario . . . . . . 553.2.2.4 Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.2.5 General model assumptions . . . . . . . . . . . . . . . . 57
3.2.3 Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.1 Roseate Terns: Step-by-Step . . . . . . . . . . . . . . . . . . . . . 593.3.2 Skipjack Tuna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.1 Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.2 Conservation Planning for a Worst-case Scenario for Connectivity 653.4.3 Uncertainty and Worst-case Scenario Planning . . . . . . . . . . . 67
APPENDIX
A MODEL SUMMARIES AND ADDITIONAL FIGURES . . . . . . . . . . . . . . . 68
B CONNECTIVITY ASSESSMENT: MATHEMATICAL DETAILS . . . . . . . . . . 73
C DISTURBANCE OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . 74
D PROTECTION OPTIMIZATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
E MODEL ITERATIONS FOR WEIGHTED LIFE EXPECTANCY OPTIMIZATION 80
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6
LIST OF TABLES
Table page
1-1 Model selection of occupancy models predicting colonization and extinctionof Lepanthes rupestris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2-1 Model selection for each experimental treatment . . . . . . . . . . . . . . . . . 42
A-1 Parameter estimates and standard errors (SE) for the multi-season occupancymodel that resulted in the best fit . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A-2 Parameter estimates and standard errors (SE) for the model that included asymmetric dispersal kernel (Ssym) as a site covariate . . . . . . . . . . . . . . . 69
E-1 Model iterations showing the interaction between disturbance and protectionstages when pT = [3773, 2467, 350, 172] . . . . . . . . . . . . . . . . . . . . . . 80
7
LIST OF FIGURES
Figure page
1-1 Various approaches to model patch connectivity . . . . . . . . . . . . . . . . . 13
1-2 Calculation of the δ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1-3 Distribution of δij in the Lepanthes system . . . . . . . . . . . . . . . . . . . . . 21
1-4 Partial relationships between extinction, colonization and connectivity measures(Ssym and Sasym) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2-1 Observed inter-patch transitions of C. vittiger in a 30× 30 m plot . . . . . . . . 32
2-2 Diagram showing the study design . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-3 Linear regression analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2-4 Partial relationships of the best fit model explaining the rate of movement inthe area treatments of the field experiment . . . . . . . . . . . . . . . . . . . . 41
2-5 Movement rate (± SE) at the patch level in the experimental treatments. . . . . 45
3-1 Schematic diagram of the three stages of the modeling process: assessment,disturbance and protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3-2 Spatial network representation of Roseate Tern colonies in northeast USAand Skipjack Tuna in the Western Pacific . . . . . . . . . . . . . . . . . . . . . 54
3-3 Example of the iterative process between the disturbance and protection modelsfor a protection budget of u = 2 and allowing b = 2 patches to be disturbed . . 60
3-4 Sensitivity of the optimal life expectancy in the protection stage given changes(ϵ) in probability of staying in the same patch (�qii ). . . . . . . . . . . . . . . . . . 61
3-5 Optimal life expectancy (z) given all possible protection budgets (u) and numberof patches allowed to be disturbed (b) . . . . . . . . . . . . . . . . . . . . . . . 62
A-1 Spatial representation of patches (boulders and tree trunks) sampled for Lep-anthes rupestris occurrence in the Luquillo Experimental Forest. . . . . . . . . 69
A-2 Theoretical distributions of the parameter δij . . . . . . . . . . . . . . . . . . . . 70
A-3 Predicted partial relationships between colonization, extinction and patch areafor both phorophytes for the best-supported model . . . . . . . . . . . . . . . . 71
A-4 Partial relationship between patch area and detectability . . . . . . . . . . . . . 72
8
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR METAPOPULATIONDYNAMICS AND CONSERVATION
By
Miguel A. Acevedo
May 2013
Chair: Robert J. Fletcher, JrMajor: Interdisciplinary Ecology
Variations in movement have broad implications for many areas of ecology and
evolution. For instance, in metapopulation theory, movement plays an important role
promoting (meta)population persistence over time. Most metapopulation modeling
approaches describe patch connectivity using pair-wise Euclidean distances resulting in
the simplifying assumption of a symmetric connectivity pattern. Symmetric connectivity
may be rarely observed in nature where organisms move responding to environmental
cues or advection sources through heterogeneous landscapes. Assuming symmetric
dispersal when movement is asymmetric may result in biased estimates of colonization,
extinction and persistence, which has important implications for management and
conservation. Here I studied the implications of asymmetric connectivity for patch
colonization and extinction. I leveraged a long-term, time-series on colonization-extinction
dynamics in the wind-dispersed orchid Lepanthes rupestris and found that models
that accounted for these asymmetries had better fit. The implications of asymmetric
connectivity to metapopulation dynamics may be contingent on the driving mechanism
behind them. I used a combination of an observational study of individual movements
and translocation experiments to test for possible mechanisms of directed movement
in the cactus-feeding insect, Chelinidea vittiger and found that variations in patch size
were the most important driver of directed movements in this species. I also developed
9
a novel site selection model that optimally selects patches that will best preserve
connectivity given a worst-case disturbance scenario.
10
CHAPTER 1THE IMPLICATIONS OF ASYMMETRIC DISPERSAL FOR CONNECTIVITY AND
PATCH OCCUPANCY
1.1 Introduction
Variation in movement has broad implications for evolutionary biology (Kawecki
and Holt, 2002), community ecology (e.g., Salomon et al., 2010; Tilman et al., 1994),
and population dynamics (Armsworth and Roughgarden, 2005; Revilla et al., 2004;
Wiegand et al., 1999). In metapopulation ecology, movement is a fundamental process
for metapopulation dynamics and local population persistence over time (Hanski,
1998). Often, the focus on movement emphasizes dispersal from natal environments
or previous breeding locations (Colbert et al., 2001). Here, we broadly use the term
dispersal to reflect these movements and related movements that generate variation in
emigration, immigration, and colonization and extinction rates (sensu Vuilleumier et al.,
2010).
Most metapopulation approaches describe colonization-extinction dynamics
following the area-isolation paradigm (Dias, 1996; Hanski, 1998; Pellet et al., 2007).
Under this paradigm, extinction is negatively related to patch area assuming that
population size increases with patch area; (Hanski, 1998), and colonization is negatively
related to isolation from other patches. Patch isolation is often quantified using patch
distance-dependent connectivity measures (i.e., the inverse of isolation) that assume
the probability of colonization declines with distance to surrounding occupied patches,
which act as sources of propagules. To do so, these measures typically use an assumed
dispersal kernel weighted by the occupancy state and area of surrounding patches
(Hanski, 1994).
While these connectivity metrics have proven useful, these metrics make assumptions
regarding the movement behavior of the species. For instance, a negative exponential
function of Euclidean distance is commonly assumed in many patch connectivity
measures (Hanski, 1998; Moilanen and Nieminen, 2002). Assuming a kernel based
11
on Euclidean distance results in the simplifying assumption of symmetric dispersal in
which the likelihood of an individual dispersing from patch i to patch j is the same as the
likelihood of an individual dispersing from patch j to i . Nevertheless, several factors in
nature often cause an asymmetric pattern of dispersal, such as spatial heterogeneity
(e.g., Ferreras, 2001; Prevedello and Vieira, 2010) and advection sources (e.g., wind,
ocean and marine currents; (Keddy, 1981; Treml et al., 2008).
An asymmetric pattern of dispersal may also arise due to variations in patch or
inter-patch attributes (Figure 1-1). At the patch level, variation in patch area may result
in a greater likelihood of dispersal towards large patches because active dispersers
may better detect or prefer large patches. For passive dispersers, large patches may
simply act as large targets or because these are large targets for passive dispersers (i.e.
target effects; (Gilpin and Diamond, 1976; Lomolino, 1990); Figure 1-1a). Quantitatively,
these asymmetries due to patch level variations have been traditionally incorporated
in connectivity measures by weighting a dispersal kernel by patch area. Inter-patch
attributes, such as advection sources (e.g., wind, ocean or river currents), may result
in a greater likelihood of dispersal in the direction of the advection source than in the
opposite direction (Figure 1-1d). That is, the effective distance in the direction of the
advection source may be less than the effective distance in the opposite direction
(Figure 1-1d).
Connectivity measures currently used in metapopulation modeling generally lack
a formal way to incorporate this kind of asymmetry in effective distance. Moilanen and
Hanski (1998) provide ways to account for habitat type-specific effective distances (h);
however, the effective distance measure is still symmetric (i.e. dhij = dh
ji ; Figure 1-1c).
Other alternate parameterizations employ least-cost paths (LCP) between patches
(Chardon et al., 2003). Even though they have been shown to be better descriptors of
effective distance than Euclidian distance (Sawyer et al., 2011), in most applications
these LCP are also symmetric (LCPij = LCPji ; Figure 1-1b).
12
Figure 1-1. Various approaches to model patch connectivity. Patch connectivity inIncidence Function Models (a) takes into consideration variations in patcharea. Least cost paths (b) replace Euclidean distance by the least cost pathbetween two patches. Other proposed measures (c) scale Euclideandistance depending on habitat type (or land-cover/use type). Even thoughthese modifications, dij = dji in all these modeling approaches, which makesthem inapplicable to model organisms that disperse asymmetrically drivenby advection sources (d).
Recent metapopulation theory hypothesizes that failing to acknowledge dispersal
asymmetries leads to an overestimation of patch connectivity, resulting in biased
estimates of colonization and extinction (Bode et al., 2008; Vuilleumier et al., 2010;
Vuilleumier and Possingham, 2006). There are very few empirical tests on these
hypotheses, in part because we lack an adequate way to quantitatively incorporate
asymmetric dispersal due to inter-patch attributes in metapopulation models.
Here, we leverage a long-term, time-series on colonization-extinction dynamics
in the wind-dispersed orchid Lepanthes rupestris to test for asymmetric dispersal.
We assess for the role of asymmetries for colonization and extinction dynamics by
extending the connectivity function of Hanski (1994) to account for dispersal asymmetry
due to wind advection. We expected asymmetric dispersal would be prevalent in this
13
metapopulation and that such asymmetries would alter colonization and extinction
estimates relative to models assuming dispersal symmetries. Given the wind-dispersed
nature of the species, we also expected that target and rescue effects will play a crucial
role in the estimation of colonization and extinction probabilities.
1.2 Methods
1.2.1 Study System
Orchids, such as Lepanthes rupestris, that grow on rock (lithophytic habitat) and/or
trees (epiphytic habitat) are appropriate to study metapopulations with asymmetric
dispersal because they often live in spatially discrete ephemeral habitats and are
passively dispersed by directed sources such as wind (e.g., Snall et al., 2005;
Tremblay et al., 2006). Moreover, many epiphytic and lithophytic orchids are subject to
colonization and extinction dynamics due to their small population sizes and stochastic
reproductive success driven, in part, by dispersal and pollinator limitation (Ackerman,
1995; Olaya-Arenas et al., 2011; Tremblay, 1997).
Lepanthes rupestris is a small, wind-dispersed orchid (leafs 1.3 4.3 cm, shoots
15cm in height and flowers of < 6 mm) commonly found along the riverbeds of the
Luquillo Mountains in Puerto Rico (Ackerman, 1995). This patchily distributed orchid
anchors its roots to the substrate and roots are often covered by moss living on the
surfaces of trees or rocky boulders. The seeds are microscopic with a mean dispersal
distance of 4.8 m (Tremblay, 1997). Given the small size of its seeds, little is known
about the fate of Lepanthes seeds after dispersal and the presence of seed banks in this
species has been debated. Whigham et al. (2006) found that temperate orchid seeds
in experimental conditions were viable even after 7 years; however, Lepanthes seeds
are expected to die if they fall in the river. Given the flood dynamics of tropical rivers
(Johnson et al., 1998), the effect of seed banks in the metapopulation dynamics of this
orchid is likely negligible.
14
A permanent plot for the study of the metapopulation dynamics of L. rupestris
was established in Quebrada Sonadora in the Luquillo Experimental Forest (LEF; 18◦
18 N, 65◦ 47 W) in 1999. This permanent plot is composed of 1000 occupied and
unoccupied boulders and tree trunks (patches hereafter; Tremblay et al., 2006). Most
patches (920) were mapped to within approximately 10cm (x, y, z position) relative
to the center of each patch using metal rulers, a sighting compass and a clinometer
(Figure A-1). The presence/absence of L. rupestris was censused twice a year from
1999–2008 (17 censuses). Additional descriptions of the site and sampling procedure
can be found in Tremblay et al. (2006). Patch size (i.e., total moss area), was estimated
as the perimeter occupied by the individuals in a patch using a 150 cm2 mesh. The
average wind direction was calculated from daily measures from the nearest weather
station located (∼1km from study site) in El Verde Biological Station (Ramırez and
Melendez-Colom, 2003).
1.2.2 Site Occupancy Model and Parameter Estimation
Incidence function models (IFM; Hanski, 1994; Moilanen and Nieminen, 2002)
are probably the most common occupancy modeling approaches used to estimate
colonization and extinction dynamics in a metapopulation context. Although IFMs have
been applied successfully to a wide range of species, they make two assumptions
that are difficult to meet in most applications. First, they assume that the species
is always detected where it is present (i.e. perfect detection). This is rarely the
case, where an observed absence may be either a true absence (the population is
locally extinct) or simply that the species was not detected, which may result in an
overestimation of extinction (Moilanen and Nieminen, 2002). Second, IFMs assume that
the metapopulations are at a Markovian pseudo-equilibrium in which the occupancy
status of each patch at time t is given only by the patch status at time t − 1. This
assumption is difficult to test and some studies argue that it is appropriate to assume
nothing about the equilibrium state of the metapopulation (Erwin et al., 1998; Pellet
15
et al., 2007). In this study, we applied an alternate method, a dynamic occupancy model,
in which these two assumptions are relaxed (MacKenzie et al., 2003).
The dynamic (i.e., multiple-season) occupancy modeling approach resembles
Pollocks robust design (Pollock, 1982) in that there are two types of sampling periods.
Primary periods are used to estimate colonization and extinction parameters. The
population is assumed to be open between these primary periods. Within the primary
periods, sites are surveyed multiple times (secondary sampling). The population is
assumed to be closed (no immigration, emigration, births, or deaths; see Rota et al.
(2009) for an assessment of this assumption) between these secondary sampling
periods, which are used to estimate detection probabilities. For each secondary
sampling period, there are three occupancy possibilities: presence, absence (which may
correspond to imperfect detection or an actual absence), or missing data (MacKenzie
et al., 2003).
Our primary sampling periods are years (1999-2008, n = 10). The secondary
sampling periods consisted of two censuses that were performed each year. One
census was conducted at the beginning of the year (January-February) and the second
in the summer (July-August). This model formulation allows the system to be open
to colonizations and extinctions during the wet season (August-December). Tropical
storms are common throughout the wet season causing flash floods, which may be
responsible for most local extinctions and anomalous strong winds, which may increase
the magnitude of dispersal events potentially resulting in more local colonizations.
Even though some colonizations and extinctions may happen in the dry season, our
conclusions are based on relative comparisons (symmetric vs. asymmetric models; see
below) and the potential violation of the closure assumption will affect both instances
equally, with minimal effect in the final relative comparisons. Given the small size of
the orchid, we expected variation in detectability especially in patches with small areas
16
of moss where it may be difficult to discern between moss and a small L. rupestris
individual.
The effects of area (Aj ) and patch connectivity (Si ) were added as covariates
to model colonization, extinction and/or detection probabilities. We calculated patch
connectivity in two ways, one of which treats the distance effect in the dispersal kernel
as symmetric and a second that adjusts effective distance to allow for asymmetric
movement. Patch connectivity with a symmetric dispersal kernel was calculated using
the general approach applied in many Incidence Function Models (Moilanen and
Nieminen, 2002):
S sym
i =
N∑j =i
exp(−αdij)Aj (1–1)
where N is the total number of patches in the landscape, 1/α is the average
dispersal distance of the species, dij is the Euclidean distance between patches i and
j and Aj is the area of the target patch j (Hanski, 1994, 1998). Note that the effective
distance between patches in this model is symmetric (i.e. dij = dji ). This formulation
can result in asymmetric patch connectivity due to variation in patch area, but not due
to variations in effective distance among patches (i.e. inter-patch attributes). Also note
that the connectivity measures we employed do not incorporate occupancy data as is
traditionally done in some IFMs measures of isolation where the occupancy state of
patch j is incorporated as a binary indicator variable to restrict potential re-colonization
only from previously occupied sites. A binary incorporation of occupancy state assumes
perfect detection, which we found is not the case in our system (Figure A-4). Given that
occupancy state was not incorporated in either symmetric or asymmetric measures of
connectivity, this modification did not affect the general conclusions.
We modified Equation 1–1 to incorporate the effect of average wind direction in the
estimation of connectivity. Patch connectivity with an asymmetric dispersal kernel was
calculated as:
17
Sasym
i =
N∑j =i
exp(−α dij
1− δij
)Aj (1–2)
where δij describes the difference between the angle of wind direction and the angle
between patches i and j in radians with respect to the horizontal axis (Figure 1-2). This
modeling approach resembles the strongly asymmetric distance-dependent theoretical
model from (Vuilleumier et al., 2010) in which δ is incorporated as a modifier inside the
exponential decay function. For simplicity, δij is scaled by π to constrain δij ∈ [0, 1],
δij =|θij − θwind |
π. (1–3)
Figure 1-2. Calculation of the δ parameter. The δij parameter is calculated as thedifference of the angles between patches i and j, and the angle of winddirection with respect to the horizontal axis
For both measures of patch connectivity (Ssym and Sasym), connectivity decreases
with increasing distance between patches. In Sasym when δij = 0, the effective distance
between patches is the same as Euclidean distance, and hence Sasym = Ssym; however,
18
as δij increases, the effective distance between patches increases resulting in decreased
connectivity when compared to Ssym. This modification allows an asymmetric dispersal
kernel because the effective distance between patches in the direction of wind is less
than the effective distance between the same patches in the opposite direction (against
wind direction; Figure 1-1d).
The distribution of the δ parameter will depend on the spatial arrangement of the
patches with respect to the angle of the advection source. For instance, if patches are
randomly arranged in space, we will expect a uniform distribution of δs (Figure A-2a,
A-2b). In contrast, if patches are spatially aligned parallel to the angle of the advection
source (e.g., along a river when wind advection is going downstream), we will expect
a bimodal distribution (Figure A-2c, A-2d). The bimodal nature of the distribution is
expected because the difference in the angles was scaled by π, and thus δij + δji = 1.
As the difference between δij and δji increases, effective distance becomes more
asymmetric, and the distribution will become more skewed towards the limits of the
distribution.
We fitted six kinds of occupancy models. Each represented a hypothesis of the
underlying mechanism governing colonization and extinctions in the L. rupestris
metapopulation. Traditionally, patch occupancy models in a metapopulation context
originate from the area-isolation paradigm. Hence, what we named the (1) “area/isolation
model”, includes patch connectivity as a site covariate for colonization and patch area
as a site covariate for extinction (i.e. γ(S), ϵ(A); where S represents a measure of
connectivity either with a symmetric or asymmetric dispersal kernel, and A describes
patch area). Connectivity may also decrease extinction probability by the rescue
effect of nearby patches (e.g., Hanski, 1998). The (2) “rescue effects” model includes
connectivity as a site covariate for colonization and both connectivity and patch area are
incorporated as interacting site covariates for extinction i.e. γ(S), ϵ(A ∗ S). Given the
wind-dispersed nature of Lepanthes rupestris, we expect that larger patches may have
19
a greater likelihood of receiving seeds because they are larger targets. Hence, we also
fitted a (3) “target effects” model in which patch connectivity together with patch area are
incorporated as a covariate for colonization and patch area as a covariate for extinction
i.e. γ(S ∗ A), ϵ(A). We fitted a (4) “target / rescue effects” model, which included both
target and rescue effects i.e. γ(S ∗ A), ϵ(S ∗ A). For each of these models (1–4) we fitted
a model with Ssym and another with Sasym as a site covariate. Because connectivity may
not necessarily be an important factor predicting colonization and extinction dynamics
(Pellet et al., 2007), we also fitted a (5) model that had moss area as site covariates for
colonization and extinction, but no connectivity metric were included. Previous analyses
for this species have found that colonization and extinction dynamics may be different
depending on phorophyte type (tree or rocky boulder; Tremblay et al., 2006). Moreover,
a preliminary analysis showed that models that included phorophyte type as a site
covariate for colonization, extinction and detectability had a better fit than models that
did not incorporated it. Hence, for all of these models (1–5) we include phorophyte
type as an interacting site covariate for colonization, extinction and detectability. We
also included patch area as a site covariate for detectability in all models, because
this preliminary analysis also showed that model fit increased. Finally, we also fitted
a (6) null (intercept-only) model with no covariates to compare with more complex
models. We fitted each model using maximum likelihood with covariates scaled and
centered, and ranked each occupancy model based on Akaike Information Criterion
(AIC; Burnham and Anderson, 2002). Models with the lowest AIC were considered most
parsimonious. Occupancy models were fitted using the package unmarked (Fiske and
Chandler, 2011) in R (Team et al., 2011).
1.3 Results
The calculated values for δ ranged between 1.11 × 10−6 and 9.9 × 10−1. The
distribution of the δij parameter showed a bimodal distribution, skewed towards the limits
20
of the distribution. This distribution was calculated based on an average wind direction
of 2.02 rad (Figure 1-3).
Figure 1-3. Distribution of δij in the Lepanthes system
The “target/rescue” and “target” effects models had better fit than the “no connectivity”
model suggesting that patch connectivity is an important predictor of patch dynamics
in this system. The most parsimonious model was the “target/rescue effects” model
that included patch area and patch connectivity with an asymmetric dispersal kernel
(Sasym) as site covariates for both colonization and extinction (“asymmetric model”
hereafter). The asymmetrical model had considerably more support than the similar
model (“target/rescue effects”) that included the patch connectivity measure with a
symmetric dispersal kernel as a site covariate (Ssym; “symmetric model” hereafter; Table
21
1-1). In all five model-types, the “asymmetric” form of the model had a relative better
than the corresponding “symmetric” model.
Table 1-1. Model selection of occupancy models predicting colonization and extinction ofLepanthes rupestris. Notation: ψ, occupancy;γ, colonization; ϵ, extinction; p,detectability. Covariates: Ssym, connectivity with symmetric kernel; Sasym,connectivity with asymmetric kernel; A, moss area; Ph, phorophyte type
Model �AIC AIC weight1 ψ(.) γ(Sasym ∗ A ∗ Ph) ϵ(Sasym ∗ A ∗ Ph) p(A ∗ Ph) 0 9.70E-012 ψ(.) γ(Sasym ∗ A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 7.57 2.20E-023 ψ(.) γ(Ssym ∗ A ∗ Ph) ϵ(Ssym ∗ A ∗ Ph) p(A ∗ Ph) 10.86 4.30E-034 ψ(.) γ(Ssym ∗ A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 20.64 3.20E-055 ψ(.) γ(A ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 24.46 4.80E-066 ψ(.) γ(Sasym ∗ Ph) ϵ(Sasym ∗ A ∗ Ph) p(A ∗ Ph) 30.02 2.90E-077 ψ(.) γ(Ssym ∗ Ph) ϵ(Ssym ∗ A ∗ Ph) p(A ∗ Ph) 31.32 1.50E-078 ψ(.) γ(Sasym ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 38.93 3.40E-099 ψ(.) γ(Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 41.89 7.80E-1010 ψ(.) γ(S ∗ Ph) ϵ(A ∗ Ph) p(A ∗ Ph) 42.63 5.40E-1011 ψ(.) γ(.) ϵ(.) p(.) 106.57 7.00E-24
Both models (asymmetric and symmetric) predicted similar partial relationships
between colonization, extinction and connectivity (Figure 1-4). Both models predicted
a positive relationship between increasing connectivity and colonization probability
for both phorophytes. Also, both models predict that extinction probability decreases
with increasing connectivity for the rock phorophyte. However, models differed in the
prediction of extinction probabilities for tree phorophyte. The asymmetric model predicts
that extinction probability decreased with increasing connectivity, while the symmetric
model predicts a fixed relationship (Figure 1-4).
In both models, moss area was positively related to colonization and detection
(Figure A-2, A-3), but it was negatively related to extinction. Detectability was high
(p > 0.80) and positively related with patch area in both phorophytes (Figure A-3).
1.4 Discussion
Asymmetric dispersal may be the rule more than the exception in nature (Gustafson
and Gardner, 1996). Nevertheless, most parameterizations of metapopulation models
22
Figure 1-4. Partial relationships between extinction, colonization and connectivitymeasures (Ssym and Sasym). Panels (a) and (b) show the partial relationshipsfor the rock phorophyte, while (c) and (d) show partial relationships for treephorophytes. Shade areas represent 95% confidence interval of theestimates. Note that the range of the connectivity axis represent the valuesof Ssym and Sasym scaled and centered.
23
assume symmetric effective distances between patches. Here we developed a novel
modification of the connectivity formulation traditionally used in incidence function
models that allows for asymmetric effective distances between patches when dispersal
is directed by an advection source. Our results on the wind-dispersed orchid Lepanthes
rupestris suggests that asymmetric dispersal was prevalent in this system and that its
colonization-extinction dynamics were better described by a metapopulation model that
incorporate this modified asymmetric connectivity formulation.
1.4.1 Detectability
Even though Sasym was a relative better descriptor of connectivity than Ssym (models
that incorporated Sasym had always better fit than models that incorporated Ssym),
this model potentially over estimates connectivity. The connectivity measures used
(Equations 1–1 and 1–2) do not incorporate the occupancy state of patch j , which is
commonly used to restrict potential (re)colonization only from occupied patches. This
incorporation implicitly assumes perfect detection, which we found is not the case in this
system.
Imperfect detection is often regarded as an important factor in the study and
management of animal populations (e.g., MacKenzie et al., 2003). Nevertheless, there
is increasing evidence that the probability of detection in plant occupancy studies is
high but often p < 1 (Kery and Gregg, 2003; Kery et al., 2006). Our study showed that
detectability increased with increasing moss area. L. rupestris is a particularly small
orchid; hence the chances of detecting a single individual (particularly if it is missing
leaves) are expected to increase with increasing moss area. Also, generally in epiphytic
species, larger phorophytes will hold more individuals, which will also increase the
probability of detection (Fedrowitz et al., 2012; Snall et al., 2005; Tremblay et al., 2006).
1.4.2 Colonization and Extinction
Our results suggest the potential for both target and rescue effects as important
mechanisms predicting colonization and extinctions (Brown and Kodric-Brown, 1977;
24
Gilpin and Diamond, 1976; Lomolino, 1990). As patches become larger and more
connected they have an increased likelihood of being colonized because they become
bigger targets. Uncertainty also increased with increasing connectivity and area (Figure
1-3 and Figure A-3). In this system, patches are spatially clustered in space with very
few really isolated patches, and there are also very few really large patches, which
may account for the lack of variation at the end of the connectivity and area spectrum.
Even though wind dispersed plant species are characterized for having high gene flow
estimates (Hamrick et al., 1995), L. rupestris has a relatively short mean dispersal
distance and most of the seeds may fall below the maternal plant (Ackerman et al.,
1996). This may be due to the relatively low wind speed in the forest understory and
the relative low frequency of high winds. The importance of the interaction between
connectivity and patch size for colonization has been also reported for other epiphytic
species. For instance, tree diameter and connectivity have been positively related
to colonization of the epiphytic lichen Lobaria pulmonaria (Gustafsson et al., 1992;
Snall et al., 2005) and the epiphytic bryophytes Nyholmiella obtusifolia, Orthortichum
speciosum, Pylaisia polyantha, and Radula complanata (Hazell et al., 1998). The
importance of target effects to predict colonization has also been described in active
dispersers, such as the butterfly Maculinea nausithous (Hovestadt et al., 2011).
The partial relationships between connectivity and extinction for the symmetric
model showed a relatively constant relationship while the best-ranked model (asymmetric)
showed a negative relationship for tree phorophyte. Tree phorophytes are located
relatively higher in elevation than rock phorophytes (Figure A-1) and their occupancy
status may depend more on wind-dispersed seeds.
Metapopulation models often make the plausible assumption that population size
increases with increasing patch area (Hanski and Hanski, 1999). Lower extinction rates
in large patch areas may be the result of positive density-dependent regulation, which
have been previously reported for L. rupestris (Rivera Gomez et al., 2006). Previous
25
research on other epiphytic plants identifies loss of substrate (mostly tree mortality) as
one of the most important driver of local extinction (Snall et al., 2003; Tremblay, 2008).
In our system flooding is the most important driver of extinction. Even though there has
been substrate loss by flooding which always resulted in local orchid extinction, these
were relatively infrequent (15 trees and 2 boulders; (Tremblay et al., 2006). Trees had,
in general, higher extinction rates than boulders. This is consistent with a previous study
on L. rupestris that found that trees had almost double extinction rates as boulders
(Tremblay et al., 2006). Substrate loss due to flash flooding was higher for trees, which
may suggest that flooding affects extinction in tree phorophytes greater than boulders.
There may be other factors (covariates) that may affect the colonization and
extinction dynamics of L. rupestris, in addition to connectivity, area and phorophyte
type. For instance, seed establishment in orchids is limited by their association with
mycorrhizae (Dressler, 1993; Rasmussen and Whigham, 1993). Also, the amount
of moss moisture may also affect seed establishment (Tremblay et al., 2006). These
variables were not included in this model due to the difficulty of incorporating these into
a long-term metapopulation monitoring program. Nevertheless, their potential interaction
with asymmetric connectivity to predict colonization and extinction dynamics remains
unexplored.
1.4.3 Asymmetric Dispersal and Metapopulation Modeling
There are several examples of organisms, including both active and passive
dispersers that occur as metapopulations with asymmetric connectivity. In the closely
related species L. eltoroensis, individuals more frequently colonized boulders and
trees between 270◦ and 90◦ (Tremblay and Castro, 2009). In aquatic systems, ocean
and river currents are important drivers of asymmetric dispersal for both vertebrates
and invertebrates (Lutscher et al., 2007; Treml et al., 2008; Watson et al., 2011a).
Asymmetric dispersal pattern have also been found in active dispersers such as
Everglades Snail Kites (Rostrhamus sociabilis plumbeus), cactus bugs (Chelinidea
26
vittiger Fletcher et al., 2011) and the endangered Iberan lynx (Lynx pardinus; Ferreras,
2001). Moreover, an individual based model developed by Gustafson and Gardner
(1996) showed that altering landscape heterogeneity resulted in asymmetric rates of
immigration and emigration among resource patches. Hence, asymmetric connectivity
may be the rule more than the exception, given that symmetric connectivity may only be
applicable for organisms which dispersal is not affected by advection sources, spatial
variation in resources or that live in homogeneous landscapes. Such examples are
uncommon in nature.
We found that connectivity was important to predict colonizations and extinctions,
and that asymmetric dispersal is prevalent in the system. This is shown by the skewed
distribution of the δ parameter towards the limits of the distribution and because
the best-ranked model included Sasym. A previous analysis on 10 species of birds,
amphibians and butterflies showed that adding connectivity as a site covariate did not
improved model fit compared to constant colonization parameters (Pellet et al., 2007).
This study argues that using the symmetric connectivity measure traditionally used
in incidence function models may not be adequate to model most species because it
does not account for density-independent movements among patches (e.g., response
to an advection source or taxis) which may significantly affect dispersal rates. The
Sasym connectivity measure accounts for some of these density-idependent movements.
It significantly improved model fit, even when it only took into consideration average
wind direction, which is a broad way to describe advection effects. We expect that if
we incorporate patch-specific measures of wind direction, model fit would improve
even more. In this system which is composed of 920 patches, it is unfeasible to collect
patch-specific wind direction (or other advection); however, in other applications in
smaller metapopulations, this may be feasible.
The lack of recognition of asymmetric connectivity in empirical metapopulation
studies may be due in part to the absence of an estimation framework that can
27
incorporate asymmetric dispersal. Here we provide a general framework using
multi-season occupancy modeling using connectivity as a covariate. The asymmetric
connectivity measure that we applied (Equation 1–2) can be generally applied to any
other simple advection sources (e.g., water current direction) or it can be modified
to adapt it to other systems. For example, wind direction can be replaced by the
angle of riverine or marine currents. The incorporation of asymmetric connectivity
into metapopulation modeling will potentially result in an increase in the accuracy of
conservation and management decisions (Beger et al., 2010).
28
CHAPTER 2THE PROXIMATE CAUSES OF ASYMMETRIC MOVEMENT
2.1 Introduction
Variation in movement has broad implications for many areas of ecology and
evolution, including altering local adaptation via gene flow (Kawecki and Holt, 2002),
community structure (Liebhold et al., 2004; Salomon et al., 2010; Tilman et al., 1994),
and population dynamics (Armsworth and Roughgarden, 2005; Revilla et al., 2004;
Wiegand et al., 1999). In metapopulation theory, movement is particularly critical
because provides the means for (meta)population persistence over time (Hanski, 1998).
Most metapopulation modeling approaches describe inter-site connectivity using
pair-wise Euclidean distance between patches (Dias, 1996; Hanski and Hanski, 1999).
Using these distances typically results in the simplifying assumption of a symmetric
movement pattern in which the probability of moving, p, from patch i to patch j is
the same as moving in the opposite direction (i.e. pij = pji ). A symmetric pattern of
movement may be expected when all the factors that affect dispersal are identical in all
directions (Bode et al., 2008; Gustafson and Gardner, 1996); however, such symmetry
might be rarely observed in nature. Instead, an asymmetric pattern of movement
(e.g. pij = pji ) may more likely emerge from organisms moving across spatially
heterogeneous environments (Ferreras, 2001; Prevedello and Vieira, 2010). Recent
models that incorporate this complexity suggest that movement asymmetries can have
substantial effects on metapopulation dynamics (Armsworth and Roughgarden, 2005;
Vuilleumier et al., 2010; Vuilleumier and Possingham, 2006).
A variety of mechanisms may give rise to asymmetrical pattern of movement. For
instance, asymmetric movement may be the result of environmental heterogeneity such
as variation in habitat quality or patch size (Ferreras, 2001; Gustafson and Gardner,
1996; Holt, 1996; Pulliam, 1988). Individuals may have adapted to seek better habitats
by actively dispersing from low- to high-quality patches (often larger in size; e.g.,
29
Ferreras, 2001; Gustafson and Gardner, 1996), or high quality patches may be net
producers of emigrants to low-quality patches due to density dependence (e.g., in
source-sink dynamics; Holt, 1985; Pulliam, 1988).
Other mechanisms may likely differ between passive and active dispersers. In
passive dispersers, asymmetric movement may be the result of advection through a
directional medium (Armsworth and Bode, 1999; Keddy, 1981; Schick and Lindley,
2007; Treml et al., 2008). In active dispersers, asymmetries may be the result of
changes in the organisms movement behavior due to environmental cues (e.g.
taxis; Compton, 2002). For instance, many species of insects are attracted to visual
or chemical cues that direct their movement (Prokopy and Owens, 1983). Some
Homoptera, Hymenoptera and Hemiptera direct their movement upwind while tracking
volatile chemical signals from vegetation (anemotaxis; e.g. Compton, 2002; Moser et al.,
2009; Williams et al., 2007). The likelihood of an individual to move to a particular patch
may also depend on the patch occupancy state or the abundance conspecifics (Serrano
and Tella, 2003; Serrano et al., 2001; Smith and Peacock, 1990).
Here we use a combination of an observational and experimental approach to
test for mechanisms of asymmetric movement in the cactus-feeding insect, Chelinidea
vittiger. This species is entirely dependent on patchy prickly pear cactus (Opuntia
spp.), where it feeds, breeds and aggregates (Miller et al., 2012). Recently, Fletcher
et al. (2011) found that movement in this species across a patch network was highly
directional, which altered assessments of landscape connectivity. We hypothesized that
three primary mechanisms may cause directed movements: (1) positive anemotaxis
(movement upwind), (2) movement toward larger patches, and (3) conspecific attraction.
Wind may help C. vittiger detect volatile olfactory cues of Opuntia, potentially resulting
in individuals directing their movements towards patches that are located downwind
(Schooley and Wiens, 2004). Similarly, C. vittiger may actively search for larger patches
of Opuntia, because of enhanced resource availability (see Schooley and Wiens, 2005).
30
Because of the aggregative behaviors of this species (Fletcher and Miller, 2008; Miller
et al., 2012), conspecific attraction may occur and also cause directed movements.
To address these mechanisms, we first re-analyzed data from a mark-recapture study
across a 56 patch network (Fletcher et al., 2011) to test for the magnitude of movement
asymmetry and if wind direction or patch area may explain observed variation in
movement across the patch network. Second, we conducted a translocation experiment
in which we manipulated patch area, wind advection and the presence of conspecifics to
determine if, and the extent to which, C. vittiger biases movements in response to these
factors.
2.2 Methods
2.2.1 Study Area and Focal Species
The study was conducted at the Ordway-Swisher Biological Station (29.4◦N,
82.0◦W), in Putnam County, Florida, USA. In this area, C. vittiger uses Opuntia hu-
mifusa, primarily occurring in old-field habitats, which are common at the station. O.
humifusa is native to the eastern U.S. and typically grows in dry, sandy soils in sandhills,
old fields, prairies, or scrub.
We focused our research on adult movements of C. vittiger. Adults of C. vittiger are
winged, but rarely fly; instead, adults typically walk between cactus patches through
an unsuitable matrix (De Vol and Goeden, 1973; Schooley and Wiens, 2004). Median
dispersal ranges from 1–2.5 m/day (Schooley and Wiens, 2004, 2005), which makes
movement tractable and an ideal system to study short-term dispersal (Figure 2-1b).
2.2.2 Study Design
2.2.2.1 Mark-recapture
We re-analyzed data published by Fletcher et al. (2011) in the context of connectivity
modeling. The data consisted of individual movements of marked C. vittiger in all
patches (n = 56) inside a 30 × 30 m plot (Figure 2-1a). Patches were defined following
Schooley and Wiens (2004), where cladodes (i.e., cactus pads) < 25 cm apart were
31
Figure 2-1. Observed inter-patch transitions of C. vittiger in a 30× 30 m plot. (a) Greencircles represent patches that included movement while black circlespatches that did not. Note how most of the connections go in only onedirection showing a pattern of high asymmetric movement. (b) Distribution ofmovement distances divided by the number of patches available formovement (availability). Circle size is proportional to patch size.
considered the same patch. Censuses were conducted from September 2008 to
November 2009 every 2–3 weeks (except during the winter) for a total of 21 censuses.
During each census, all individuals on each patch were counted (both nymphs and
adults), and all adults were sexed and individually marked on the pronotum with nontoxic
permanent marker. Even though this plot was not a closed system, additional recapture
rates in the surrounding area were low.
To address the mechanisms for asymmetric movement, we estimated patch area
using the equation for an ellipse (area = major radius × minor radius × π; Schooley
and Wiens, 2005). The major and minor radii were measured using a ruler. We used
data from the nearest weather station to the site (< 1km), to estimate wind direction. We
averaged daily wind direction from January 1, 2009 to November 27, 2009 to produce
a single estimate of average wind direction relative to each pair-wise combination of
patches in the network.
32
2.2.2.2 Field experiment
We used a random block design in which each block contained four plots, one for
each treatment (wind, patch area, and conspecifics) and a control (Figure 2-2). Each
plot included four patches of Opuntia humifusa cladodes located in the four cardinal
directions. The release patch was located in the center of the plot and consisted of
a single cladode (Figure 2-2). We used a small release patch to catalyze movement
(Castellon and Sieving, 2006) because a single cladode is below the minimum patch
area requirements for C. vittiger (Schooley and Wiens, 2005). All experimental patches
were located 1-m away from the release patch, which is within the perceptual range of
this species (Fletcher et al., 2013).
All plots consisted of 32 cladodes that were distributed among four patches,
arranged in different proportions between the patches depending on the treatment. In
the wind and conspecific treatments, and in the control plot, each experimental patch
was composed of eight cladodes per patch (Figure 2-2a,b,c). In the area treatment,
one patch consisted of 16 cladodes (large patch); another consisted of eight cladodes
(medium patch) and two of four cladodes (small patches; Figure 2-2d). These cladodes
were fresh, with no visible feeding marks and were collected in nearby old-fields (< 100
m).
The wind treatment consisted of placing a battery operated Caframo Kona black
color (7cm diameter) fan located 5–10cm away from a randomly selected experimental
patch and 10cm from the soil with wind directed towards the release patch (Figure
2-2b). The fan was operated by a 12V battery and continuously blowing during the
sampling period (48 hrs). Wind speed at the wind treatment experimental patch was
14.8 km/h and 4.7 km/h at the release patch. These wind variation included speeds
previously reported in the literature in a study that found a strong relationship between
wind direction and C. vittiger movement (0.5–7.75 km/h; Schooley and Wiens, 2005).
33
Figure 2-2. Diagram showing the study design. Experimental blocks were composed offour plots, one for each treatment (a) control, (b) wind, (c) conspecifics, and(d) variations in patch area. The study included 3 blocks of this design andall treatments were assigned randomly.
34
The conspecifics treatment consisted of placing one individual male in one patch
and one individual female in another patch in the same plot (Figure 2-2c). We included
males and females separately because previous studies have found differences
in aggregative behavior for males and females (Miller et al., 2012). Treatments
were assigned to the patches randomly. The individual cladodes that contained the
conspecifics were enclosed with fine mesh to prevent the individual from leaving the
patch (Miller et al., 2012). The mesh allowed for visual and/or pheromonal cues to be
potentially detected by the released individuals.
We considered control patches to be included in the treatment plots that were not
subject to the treatment. For example, in the wind treatment plots, only one out of the
four patches had a fan, such that the other three patches in the plot were considered
control plots. Similarly, in the conspecific treatments the two patches that did not have
conspecifics were considered control patches. In the area treatment, patches with eight
cladodes were considered control also. In addition to the control patches, we included
in the block design a control plot in which each patch was composed of 8 cladodes and
releases and censuses were done in the same way as in the treatment plots.
We also performed two types of procedural controls, one for the conspecifics
and another for the wind treatment, to ensure the observed rates of immigration for
each treatment were driven by the treatments and not an artifact of incorporating a
fan or mesh in the plot. We conducted the procedural control experiment for the wind
treatment in the same plot plots (i.e. same location and same cactus patches in each
block) where the wind experiment took place; the only difference was that the fan was
placed blowing in the direction away from the release patch. By doing this, we keep the
potential effects of having a non-natural object in the landscape (i.e. black fan producing
noise), but removed the effect of wind. Similarly, we conducted the procedural control
experiment for the conspecifics treatment in the same plots where the conspecific
treatment experiments took place (i.e. different plots than the wind procedural control
35
experiments). We enclosed the same two patches at used for the conspecific treatments
with mesh, with the only difference being that a conspecific individual was not included.
Sampling was done in the same was as in the other experimental treatments (see
below).
At the beginning of the study, plots were depleted of any other cactus and
individuals of C. vittiger within 10 m of the release patch. Plots were located at least
35 m apart from the nearest experimental plot. Locations of the treatments at both the
plot and patch level were assigned at random.
Individuals were captured in nearby cactus patches (< 100 m). These were
bred and reared in deli containers in a screen house. Nymphs were raised in groups
with fresh and healthy O. humifasa cladodes. As soon as they became adults, they
were marked on their pronotum using non-toxic permanent marker and transferred to
individual containers until they were released in the experimental plots. A greenhouse
experiment found that marking individuals in this way does not alter survival rates
(P > 0.5; Fletcher, unpublished).
In each trial, two males and two females were released at the released patch. The
location of each of these individuals in the plot was censused after 24 and then after 48
hours of being released. Each observed individual was removed from the plot as soon
as its location was noted. We conducted 7 (5 for experimental treatments and 2 for the
procedural control) trials in each of the three blocks between July and November 2012.
Previous studies have found that matrix resistance may be an important determinant
in the movement of C. vittiger (Schooley and Wiens, 2005). To control for this effect, we
measured the structure of the matrix from the release patch to each experimental
patch. We measured matrix structure at 15 points within a 100 × 50cm strip transect
(5 measures in each edge of the transect and 5 in the center spaced every 20 cm)
between the release patch and the experimental patch. For each point, we quantified the
number of times vegetation touched a graded pole in each 10 cm category (vertically)
36
to a maximum of 90 cm. We averaged the number of layers and the maximum height of
each set of 15 measures to characterize the matrix between the release patch and each
experimental patch. Both measures (number of layers and maximum height) were highly
correlated (rp = 0.85); therefore, here we present only the data for maximum height to be
consistent with other investigations on this species (Schooley and Wiens, 2005).
2.2.3 Analyses
2.2.3.1 Mark-recapture
We represented the movements of C. vittiger as a matrix W were wij represents
the total number of movements from patch i to patch j . We calculated the proportion
of symmetric links as the number of links in W in which wij − wji = 0 (i.e. the number
of movements was the same in both directions) divided by the total number of links
that had movement. While Fletcher et al. (2011) noted that movement appeared highly
directional, they did not assess if this putative pattern was greater than what would be
expected based on chance. To test if this proportion was different from the proportion
of symmetric links expected by chance, we compared the proportion of symmetric links
in this matrix of movements W, to the proportion of symmetric links in 1000 randomly
generated movement matrices. These random matrices had the same number of
movements as W, but the positions of the links were randomly assigned among the
links that had movement (i.e. shuffling the weights of the matrix while keeping the same
topology).
We used a linear regression model to test for the relationship between a pattern of
asymmetric movement and variations in patch size and the angle of wind direction. The
response variables for both of these models was the difference in movement between
i and j , �wij = |wij − wji |. Similarly, we calculated the difference in patch area as
�Aij = |Ai − Aj | where Ai is the area of patch i . We also calculated the difference (δij ),
between the angle of wind direction and the angle between patches i and j , in radians
37
with respect to the horizontal axis. For simplicity, δij is scaled by π to constrain δij ∈ [0, 1].
See Chapter 1.
2.2.3.2 Field experiment
To test for the effect of the treatments (variations in patch area, wind and conspecific
presence) driving the directed movements of C. vittiger, we modeled the rate of
movement from the release to each of the experimental patches. This rate of movement
described the proportion of the released individuals that moved from the release patch
to an experimental patch in a period of 48 hrs. This is an appropriate response given
that experimental patches were located 1-m away from the target patch and the average
individual movement rate is < 1 m/day (Figure 2-1b).
Each treatment was analyzed independently because variation within plots for
different treatments made pooling data across treatments impractical. In the conspecific
treatment, we fitted separate models to describe the rate of movement of males, females
and another for both sexes together, because previous research has shown different
aggregative behaviors between males and females C. vittiger (Miller et al., 2012). For
each treatment consideration (area, wind, and conspecifics male, females and both),
we fitted five types of models that represented different hypotheses explaining the
rate of movement. An (1) intercept-only model representing the null hypothesis that
the experimental treatments are not driving the movements of C. vittiger. We, also
fitted a model representing the (2) treatment. Because each treatment was model
independently, the variable used was different for each treatment. In the area treatment,
variation in patch area was represented by the number of cladodes in each patch (4,
8 or 16 cladodes). In the wind treatment, the presence of wind was represented by a
binary variable indicating the presence or not of the fan in the experimental patch. In
the conspecific treatment (response of males, females and both), the presence of a
conspecific was treated as categorical variable with three factors: male, female and
control. The effect of the matrix, represented by the maximum average vegetation height
38
between the release patch and the experimental patch, was incorporated into models
both as a (3) single main effect, and also as an (4) additive and (5) interacting covariate
with the treatment.
We used a negative binomial regression to model the rate of movement toward
experimental patches with an offset representing the total number of individuals
released (4 individuals, except for the response of males and female conspecifics
in which two individuals were released). The negative binomial distribution was an
appropriate choice because the count data showed over-dispersion. In addition, we
compared model fit to a Poisson distribution and the negative binomial resulted in better
fit (all negative binomial models resulted in AIC values < 2 units than the corresponding
Poisson model). In a preliminary analysis, we incorporated the date of the census,
block and plot as random effects, but their variance resulted < 10−8 and thus their
incorporation did not significantly increase model fit.
Negative binomial regressions were fitted by maximum likelihood using the MASS
package in R. We used Akaikes Information Criterion (AIC) to compare model fit and
choose the most parsimonious model.
2.3 Results
We observed a total of 70 movements in the 30 × 30m patch network (see Fletcher
et al., 2011, for more details). The proportion of symmetric connections in this network
(0.04) was lower than the average in the randomly generated movement networks
(0.10 ± 0.04; z = −1.77,P = 0.04). Patch area ranged from 1.25–252 cm2. Average
wind direction was 131 ± 99◦, with average prevailing wind speeds of 4.07 ± 3.03 km/hr.
We found a weakly significant, positive relationship between the difference in movement
(�wij ) and difference in patch area (�Aij ) between patches (R2 = 0.09,P = 0.05;
Figure 2-3). In contrast, there was no significant relationship between the difference in
movement and increasing deviation from the angle of wind direction (R2 = 0.03,P =
0.25; Figure 2-3).
39
Figure 2-3. Linear regression analysis. Figure shows the relationship between thedifference in movement (�wij ), and (a) the difference in patch area (�Aij ) and(b) deviation from average wind direction (δij ) in the mark-recapture study
In the field experiment, a total of 240 individuals were marked and released of which
86 were recaptured. We recaptured 19 individuals in the area treatment, 16 individuals
in the wind treatment, 25 (14 females and 11 males) in the conspecifics treatment and
26 in the control plot. The most parsimonious model explaining the rate of movement
in the area treatment included the number of cladodes (patch size) interacting with the
maximum height of the vegetation in the matrix as covariates (Table 2-1). This model
predicted a relative small and constant rate of movement to all size patches when the
matrix was low, but a higher rate of movement towards bigger patches when the matrix
vegetation was higher than the median (Figure 2-4). The null model (intercept only) was
the most parsimonious model explaining the rate of movement in the wind treatment
and control plots. A null model was also the most parsimonious explaining the rate of
movement in the conspecific treatments when modeling the response of both males and
females together and when modeling the movement of females. When modeling the
40
rate of movement of males in the conspecifics treatment, the most parsimonious model
incorporated the presence of conspecifics; however, this model had no better fit than the
null (intercept-only) model (�AIC < 2; Table 2-1). The null (intercept-only) model was the
most parsimonious model explaining the rate of movement in both procedural controls
(Table 2-1).
Figure 2-4. Partial relationships of the best fit model explaining the rate of movement inthe area treatments of the field experiment. This model included patch area(number of cladodes) interacting with vegetation height. Lines represent 1stquartile (low matrix height), median (moderate matrix height) and 3rdquartile (high matrix height) of maximum vegetation height. Shaded areasrepresent the 95% confidence intervals
2.4 Discussion
Asymmetric dispersal may be the rule more than the exception in nature. While
recent metapopulation models suggest that movement asymmetries can have
substantial effects on metapopulation dynamics, empirical tests of dispersal asymmetries
across landscapes remain rare with even fewer tests of the mechanisms driving such
patterns. We found that movements of C. vittiger were generally asymmetric and better
explained by movements from small-to-large patches.
41
Table 2-1. Model selection for each experimental treatment. The number of cladodes isrepresented by n.cladodes, matrix refers to the maximum height of vegetationin the matrix
Treatment Sex Response Model k �AIC DevianceArea Both N.cladodes*Matrix 4 0 36.1
N.cladodes+Matrix 3 1.4 38.8N.cladodes 2 3.5 37.4Matrix 2 12 36Intercept only 1 12.1 37.3
Wind Both Intercept only 1 0 33.5Wind 2 1.2 33.7Matrix 2 1.9 33.7Wind+Matrix 3 3.4 33.8Wind*Matrix 4 5.6 33.6
Conspecifics Both Intercept only 1 0 38.3Matrix 2 1.4 38.6Conspecifics 3 2.7 38.7Conspecifics+Matrix 2 4.6 38.3Conspecifics*Matrix 3 9.2 38.1
Males Conspecifics 3 0 35Intercept only 1 1.7 36.1Conspecifics+Matrix 4 2.2 35Matrix 2 3.8 36.1Conspecifics*Matrix 6 4.7 33.9
Females Intercept only 1 0 60.2Conspecifics 3 1.6 57.5Matrix 2 2 60.1Conspecifics+Matrix 4 3.6 57.3Conspecifics*Matrix 6 5.6 54.9
Control Both Intercept only 1 0 54.59Matrix 2 2.1 54.6
Procedural Cont. Both Intercept only 1 0 20.8Fan 2 0.68 21.78
Intercept only 1 0 20.35Mesh 2 1.5 20.35
42
Both the observational and experimental results showed no support for prevailing
or artificial wind driving movement decisions. Even though previous studies have
shown that C. vittiger directs its movements towards prevailing winds (Schooley and
Wiens, 2003) in open-short grass prairies in Colorado, Fletcher et al. (2013) also
found little support for this wind hypothesis while studying the perceptual range of C.
vittiger in our study area. They suggested that wind speeds at old-field habitats in our
study area are relatively lower than in the open short-grass prairie habitat than where
(Schooley and Wiens, 2003) conducted their study. Nevertheless, in our experiments,
artificial wind speeds at ∼ 10cm from the ground were within the range of those in
the open short-grass prairie (Schooley and Wiens, 2003). This discrepancy may be
explained by at least two alternative hypotheses related to habitat differences between
open short-grass prairie in Colorado and old-field habitats in Florida. First, C. vittiger
populations at OSBS may have developed alternative ways to detect patches that
do not include relying on wind because wind is not a prevalent factor in this system.
Alternatively, the type or amount of olfactory cues may differ between O. humifusa and
O. polycantha (the Opuntia species in Schooley and Wiens, 2003).
We found no strong support for the presence of adult conspecifics driving
the directed movements of C. vittiger. Nevertheless, multiple studies have found
that this species uses social information to guide its behavior (e.g., Fletcher and
Miller, 2008; Miller et al., 2012), suggesting that there may still potential for other
(non-tested) conspecific interactions driving directed movements. Given that conspecific
interactions have been documented for both aggregative and non-aggregative species
(e.g., Fletcher, 2009; Serrano and Tella, 2003), and can affect colonization rates
(Hahn and Silverman, 2006), functional connectivity estimates (Zeigler et al., 2011)
and metapopulation dynamics (Alonso et al., 2004; Ray et al., 1991), its potential
implications as a driver of asymmetric dispersal remains an alternative hypothesis to be
further tested.
43
Patch size has been found to be an important determinant of patch immigration
rates (Bowler and Benton, 2005). Both observational and experimental results showed
support for the hypothesis that variation in patch area is a driving mechanism of directed
movements that result in an asymmetric pattern of movement. In the field experiments,
the rate of immigration increased with increasing patch size. Two potential mechanisms
may account for this pattern. First, organisms may be dispersing by a passive-diffusive
process (Kindvall and Petersson, 2000; Travis and French, 2000), which makes large
patches (i.e. patches with greater circumference) more prone to be detected by a
randomly moving individual. On the other hand, there is ample evidence showing that
animals tend to move toward larger patches (e.g., Diffendorfer et al., 1995), suggesting
that other factors may be covarying with patch size and individuals may have strong
preference for larger patches (Bowler and Benton, 2005). Our results suggest that C.
vittiger prefer large patches. If individuals of C. vittiger were randomly finding patches
in the area treatment, we would have expected the immigration rate to be proportional
to the linear dimension of the patch, because all patches were located at the same
distance from the release patch (Bowman et al., 2002). Given that in our experimental
design the linear dimension of the patches was proportional to the number of cladodes,
we would have expected the average movement rate towards the 16 cladode patches to
be 4 times that of the 4-cladode patches. Instead, we found that the rate of movement
towards the 16 cladodes patch was 11 times that of the 4 cladodes patches (Figure
2-5c). Moreover, the average movement rate towards the 16 cladode patches was 2.5
times that of the 8 cladode patches suggesting a non-linear relationship between patch
size and immigration rate (Figure 2-5c). C. vittiger may direct their movements towards
large cactus patches simply because larger patches provide a greater amount of food
resources. Alternatively, large patches may provide more spines for females to lay eggs
or may include complex vertical structures that may serve as refugia from predators,
thereby decreasing predation risk.
44
Figure 2-5. Movement rate (± SE) at the patch level in the experimental treatments. Thesolid gray line represents the mean immigration rate for patches in thecontrol treatment and the dashed line the associated standard errors
A combination of matrix characteristics and other landscape features such as patch
size and isolation are the most important drivers of inter-patch movements (Bender and
Fahrig, 2005). Moreover, variations in matrix permeability and landscape characteristics
may result in an asymmetric pattern of dispersal
The cost of dispersal is an important determinant of whether dispersal leads to
fitness benefits (Bowler and Benton, 2005). Our study system could be an example in
which, even though increased matrix structure may provide resistance to movement
(Schooley and Wiens, 2005), it may increase survival by decreasing predation risk.
Predation estimates for C. vittiger are scarce, but spiders have been shown to predate
them and other cactus-feeding insects (De Vol and Goeden, 1973; Miller, 2008).
Regardless of treatment, most plots in which we observed no recaptures had little or no
matrix structure. These individuals either were predated or permanently emigrated from
the experimental plot, suggesting that the risk of longer distance dispersal may be less
than the risk of being exposed moving through the matrix without cover. On the contrary,
45
moving through a more complex matrix may provide cover in which individuals could
move more safely and assess their surroundings for a patch to settle. A similar pattern
has been found in other taxa. For instance, predation risk for forest birds is higher while
trying to cross an agricultural matrix than secondary forests with relatively more vertical
structure
Inter-patch matrix heterogeneity and movement behavior has strong implications
for population dynamics and biodiversity patterns (Armsworth and Roughgarden, 2005;
Salomon et al., 2010). We found that patch size and matrix structure were the most
important predictors of inter-patch movements and immigration rate in C. vittiger, leading
to directed movements that resulted in an asymmetric pattern of dispersal. Given that a
relationship between patch size and matrix heterogeneity is common in many taxa, our
results support the idea that asymmetric dispersal may be common in nature (Gustafson
and Gardner, 1996). Variation in landscape features such as matrix structure and
patch size may be important drivers of asymmetric dispersal and hence have broad
implications for (meta)population dynamics.
46
CHAPTER 3OPTIMAL SITE SELECTION STRATEGIES: PROTECTING AGAINST WORST-CASE
DISTURBANCE SCENARIOS
3.1 Introduction
Human-induced habitat loss and global climate change have been identified as
some of the most important large-scale drivers of biodiversity declines (Fahrig, 2003;
Lindenmayer and Fischer, 2006; Mawdsley et al., 2009; Sala et al., 2000); however,
there is much uncertainty about their future ecological consequences (Elith et al., 2002;
Kujala et al., 2012; Moilanen et al., 2006a; Thomas et al., 2004). There is an increasing
interest in identifying conservation strategies that can ameliorate some of the expected
future impacts of these drivers, while at the same time accounting for uncertainties
(Moilanen et al., 2006b), because failing to account for these uncertainties may result in
inadequate conservation and management actions (Regan et al., 2005).
Managing networks of protected areas is one of the most common biodiversity
protection strategies (Heller and Zavaleta, 2009; Kukkala and Moilanen, 2012; Mawdsley
et al., 2009). Often, these area networks are designed under a systematic conservation
planning framework that employs quantitative tools for strategically selecting reserves.
These quantitative tools identify areas for the representation and/or persistence of
biodiversity given socioeconomic or biological constraints. These problems can be
described as linear or integer optimization programs (that maximize or minimize a linear
function over a set of linear constraints) and benefit from a wide array of models and
algorithms that have been developed in the operations research field (Cabeza and
Moilanen, 2001; Kukkala and Moilanen, 2012; Margules and Pressey, 2000; Moilanen,
2008; Moilanen et al., 2009). The end goal of these models is to determine the optimal
set of resource patches to protect, given a conservation objective. A wide array of
objective functions have been employed in these optimization models, such as the
maximization of species representation, the maximization of species persistence or the
minimization of costs (Cabeza and Moilanen, 2001). Recently, the focus of these models
47
have shifted from these pattern-based objectives to process-based objectives that drive
ecosystem functioning and species persistence (Possingham et al., 2005).
Connectivity is an ecological process that promotes species persistence in
fragmented landscapes, the maintenance of biodiversity, and other ecosystem
processes (Crooks and Sanjayan, 2006; Heller and Zavaleta, 2009; Lindenmayer
et al., 2008; Possingham et al., 2005; Taylor et al., 1993; Tischendorf and Fahrig,
2000). Improving connectivity is one of the most frequent recommendations when
managing landscapes for future environmental changes because highly connected
sites are expected to have greater persistence and population densities (Gillson et al.,
2013; Hanski, 1998; Heller and Zavaleta, 2009). The rationale behind this idea is that
increasing connectivity will allow individuals to respond to future changes by moving into
new regions (Krosby et al., 2010). Even though connectivity is commonly considered
in a variety of management actions, recently its relevance for systematic conservation
planning has been acknowledged (Fagan and Lutscher, 2006; Moilanen et al., 2007,
2009; Wilson et al., 2009) by incorporating a penalizing factor in the objective function to
promote the selection of sites that are closer to each other (e.g., Ball et al., 2009; Beger
et al., 2010).
Decisions based on information from reserve selection tools are, however, subject
to different types of uncertainties (Pressey, 2004). These uncertainties can be generally
classified into epistemic (related to lack of knowledge or data) or linguistic (due to
misunderstanding of concepts or terms) (Regan et al., 2002). Statistical epistemic
uncertainties include those arising from insufficient data. Non-statistical epistemic
uncertainties arise from plausible assumptions made in the planning process such
as static environmental or biological conditions, such as those related to static
species distributions or static species interactions (Moilanen et al., 2006a). Yet
future anthropogenic actions, climate change or natural disasters may change both
environmental and biological conditions in uncertain ways (Elith et al., 2002; Thomas
48
et al., 2004). Uncertanity analysis is rarely incorporated into systematic conservation
planning. Some uncertainty analysis methods such as sensitivity analysis, scenario
building and information gap approaches have been proposed as feasible ways to
address statistical uncertainties (Moilanen et al., 2006b; Regan et al., 2005), but we
currently lack a formal approach to account for non-statistical epistemic uncertainty.
Here we describe a novel application of network flow interdiction models, that
deals with non-statistical epistemic uncertainty by optimally determining which patches
to protect based on a worst-case disturbance scenario for connectivity. We describe
the modeling framework and apply it to two empirical systems to show its application,
assumptions, sensitivity, and potential modifications.
3.2 Methods
3.2.1 Network Flow Interdiction Models
Ecological spatial networks consists of nodes that represent spatial locations such
as forest patches, grasslands, ponds, lakes, or animal colonies and links (“arcs” from
now on) that describe some flow between node pairs. In ecological spatial networks,
arcs typically describe the flow of individuals (proportion or the probability of movement)
between nodes.
Arcs and nodes may be vulnerable to disturbances that impair flow in the network.
The purpose of network flow interdiction models is to assess vulnerabilities in networks
and plan protective measures (Brown et al., 2006; Smith, 2010). These models analyze
the worst-case scenario for network infrastructure by assuming an intelligent adversary
that has limited resources (i.e. a budget on how many arcs or nodes can be disturbed).
The adversary’s objective is to maximize damage to the network. The end goal of
network flow interdiction models is to determine the value of protecting a given set
nodes or arcs that will best maintain network flow after this worst-case disturbance.
Even though interdiction models are widely applied in military and anti-terrorism
scenarios, they can be generally applied to any kind of scenario when a decision maker
49
is concerned with designing a robust system that withstands any possible combination
of disruptions. This viewpoint is often taken when analyzing critical systems (e.g., power
systems and national defense infrastructure), where resilience to worst-case disruptions,
accidental or intentional, is vital.
In ecological spatial networks, large-scale disturbances, such as land use or natural
disasters, may decrease connectivity. This decrease may have important ecological
consequences. For instance, the loss of an important patch in a metapopulation may
cause local extinctions or a decrease in metapopulation persistence (Hanski, 1998).
Given the uncertainty releated to future disturbances in natural systems, network flow
interdiciton models are an appropriate modeling tool to assess the vulnerability of an
ecological spatial network to a potential worst-case scenario.
Here we applied a defender-attacker model (i.e., Stackelberg game Von Stackelberg,
1952), which involves two players: a leader and a follower. The leader picks a subset
of patches to protect, which become immune to disruption. The follower then acts
to optimally disrupt a subset of unprotected patches with the goal of minimizing
connectivity. We emphasize that the follower need not be a malicious or sentient agent,
but rather simply a representation of a worst-case scenario given the leaders protection
strategy.
3.2.2 Model Description
In the context of connectivity conservation, a common goal is to identify a land
protection strategy given a budget, under the assumption that connectivity may be
reduced in the future. Disturbances may reduce connectivity by limiting the ability of
individuals to move within the landscape as a result of decreased matrix permeability or
patch quality (Kindlmann and Burel, 2008). Here we consider the case where protection
and disturbance occurs at the patch (node) level; however, the modeling approach can
be easily modified to incorporate disturbances to the matrix (arcs), see Section 3.4.1.
50
Decreasing patch-quality reduces connectivity by increasing patch level mortality and
decreasing movement.
The modeling approach consists of three interrelated stages: assessment,
disturbance, and protection (Figure 3-1). In the assessment stage we use patch
transition probabilities (including probabilities of staying in a given patch) and individual
survival associated with each patch (Figure 3-2) to quantify connectivity. These
transition probabilities and survival are often estimated from mark-recapture data in
a multi-state modeling framework (Nichols and Kendall, 1995; White et al., 2006). We
describe these transitions as a reducible Markov chain (see Section 3.2.2.2) to quantify
total individual life expectancy. This connectivity measure describes the amount of time
an individual remains alive and moving through the network.
The usefulness of various connectivity measures for management and conservation
has been debated in the literature (Calabrese and Fagan, 2004; Kindlmann and
Burel, 2008). Generally, measures that incorporate realistic estimates of movement
in combination with integrative fitness currencies (e.g., survival) are preferred over
measures that make plausible assumptions about movement (Belisle, 2005). Hence,
quantifying connectivity as total individual life expectancy is an appropriate measure
because it incorporates precise between-patch movement probability estimates (from
mark-recapture), site fidelity, and patch-specific survival.
A protection strategy is attained by optimizing disturbance and protection actions.
First, a worst-case scenario for connectivity is assessed by determining an optimal
disturbance action (a set of patches to disturb) that will minimize connectivity (total
individual life expectancy). These disturbance actions may include factors that decrease
patch quality such as deforestation, fire, hurricanes, flooding or changes in abiotic
factors. Second, we simulate an optimal protection action (an optimal set of patches
to protect) that will mitigate the worst-case scenario by maximizing the amount of
connectivity loss prevented (i.e. maxmin bilevel problem). These protection measures
51
include actions that prevent patch deterioration such as protecting against future
development, reduce human use (e.g., reduce fishing or hunting), managing for fire
or control predator populations. In this context, we are uncertain about how future
disturbances will affect connectivity, hence, we protect against the worst-possible
scenario (Figure 3-1).
Figure 3-1. Schematic diagram of the three stages of the modeling process:assessment, disturbance and protection. The protection and disturbancemodels interact recursively to assess the optimal set of patches to protectgiven a worst-case scenario for connectivity.
3.2.2.1 Case studies
We applied the network flow interdiction modeling framework to two published
datasets: adult Roseate Terns (Sterna dougallii) living in northeastern USA (Spendelow
et al., 1995) and Skipjack Tuna (Euthynnus pelamis) living the western Pacific (Hilborn,
1990). These datasets included survival, site fidelity and between-patch movement
estimates.
52
Roseate Tern colonies were located in Bird Island (BI) in Marion Massachusetts,
Great Gull Island in New York (GG), Falkner Island in Connecticut (FI) and Cedar Beach
(CB) in Long Island, New York (Figure 3-2a). Individual adults were captured yearly from
1988–1990 in these sites using treadle traps and individually marked with color bands.
See Spendelow et al. (1995) for more details on potential bias due to colorband loss and
fading, and how they controlled for it. We used the arithmetic mean for 1988–1990 of
their survival, transition and colony size estimates (see below).
Skipjack Tuna were tagged and recovered (>150,000 individuals tagged), monthly
from 1977–1981, in seven fisheries sites that included waters near Palau (PAL), Yap
(YAP), Truk (TRK), Ponape (PON), Papua New Guinea (PNG), Solomon Islands (SOL)
and the international waters between YAP, TRK, PON and Indonesia (Figure 3-2b).
The data analyzed were collected as part of the Skipjack Tuna survey and assessment
program of the South Pacific Commission. Fish tagged and/or recovered from Fiji were
excluded from their analyses. For more details on the data or the estimates see Hilborn
(1990).
Both datasets varied in the number of patches, scale of movement, survival, and
level of phylopatry. Survival was more variable for the Skipjack Tuna (0.664–0.99) than
the Roseate Terns (0.74–84). Phylopatry was generally lower in the Skipjack Tuna
(0.60–0.87) than in the Roseate Terns (0.88–0.98). Both species had similar ranges of
between-patch movement rates (Skipjack Tuna: 0–0.11; Roseate Terns: 0–0.09).
Survival and transition probabilities for both species were estimated using variations
of the Arnason-Schwarz multi-state model. In this multi-state modeling framework,
movement is modeled as a first-order Markov chain, which models the stochastic
movement patterns of individuals among patches, accounting for the possibility that an
individual dies at each time step. This multi-state approach allows the decomposition
of transition probabilities into survival (Si ) and movement (�qij ). Note that in this case
mortality happens before an individual moves. See Spendelow et al. (1995) for
53
comparison with alternate modeling approaches to estimate survival and transition
probabilities.
Figure 3-2. Spatial network representation of Roseate Tern colonies in northeast USAand Skipjack Tuna in the Western Pacific. The color of the arcs correspondto the probability of movement from patch i to j . Note that self arcs had thelargest probabilities showing high site fidelity.
3.2.2.2 Connectivity assessment
We calculate functional connectivity as the total individual life expectancy, which
describes the number of time-steps an individual remains alive in the network. To
add more biological reality to this measure, we weight total life expectancy by patch
abundance to account for the number of individuals subject to conditions in patch i . At
each time-step an individual has some probability of moving to another patch, staying in
the same patch, or dying. The units of this measure are the same as the input data. For
example, if the estimated transition probabilities correspond to a period of two weeks,
then the life expectancy describe how many periods of two weeks an individual will
survive.
We calculate individual life expectancy by modeling transition probabilities using a
reducible Markov chain. Accordingly, we consider a landscape composed of discrete
patches denoted by P = {1, ..., |P|}. The Markov chain contains |P| + 1 states, where
the additional state is an absorbing state d that corresponds to mortality (Figure 3-2).
54
The Markov chain states are given by M = P∪{d}. We initially represent this Markov
chain as a matrix �Q, comprised of elements �qij for i ∈ P and j ∈ P, and �qdi = 0,
�qid = 1 − Si for each i ∈ P, and �qdd = 1. Each column of �Q represents a patch in the
network, except for the last column, which represents mortality. By assumption we have
that �qij ≥ 0 for all i , j ∈ M and∑
j∈M �qij = 1 fro all i ∈ M.
In the absence of any disturbances, the total individual life expectancy (z) can be
calculated as follows. Let p be a |P|-dimensional column vector whose i th element, pi ,
describes the abundance in patch i ∈ P, and let �A be the submatrix of �Q obtained by
removing the last row and column of �Q (Ross, 2006). See Appendix B for additional
mathematical details of the connectivity assessment.
z = pT (I− �A)−11, (3–1)
where I is the |P| × |P| identitiy matrix and 1 is the |P|-dimensional column vector of
all ones. In the unweighted case p = 1, and hence life expectancy is based solely on the
transition probabilities.
3.2.2.3 Disturbance: simulating a worst-case scenario
The disturbance model identifies a worst-case scenario for connectivity by
selecting a limited number of unprotected patches to disturb that will minimize total
life expectancy. Here we define disturbance as any action that may decrease patch
quality. To describe the disturbance decision we use the binary variable yi , which equals
1 if patch i ∈ P is disturbed, and equals 0, otherwise. Disturbance to any given patch will
affect the individual life expectancy by changing the transition probabilities in the Markov
chain. We define qij as the updated transition probability of going from state i ∈ M to
state j ∈ M. The magnitude of this change depends on the disturbance regime, which
is described by the parameters δij , γij , and ηij . Biologically, these parameters describe
how a disturbance action in a patch (decreasing habitat quality) affects patch-specific
55
survival and the between-patch movements associated with the disturbed patch. We
assume that disturbance actions will decrease connectivity; hence, δij and γij are both
nonnegative. For instance, if patch j is disturbed (yj = 1), then the probability of moving
from patch i to patch j (qij ) decreases by δij . Similarly, if patch i is disturbed (yi = 1),
qij decreases by γij . The parameter ηij is used as a correction term when both yi = 1,
and yj = 1. Also we set δii = γii = 0, meaning that if patch i is disturbed the updated
probability of staying in the same patch (�qii ) is affected only by ηii (see Appendix C for
more details).
qij =
�qij − δijyj − γijyi + ηijyiyj if i , j ∈ P
�qid +∑k∈P
(δikyk + γikyi − ηikyiyk) if i ∈ P and j = d
�qij if i = d and j ∈ M
(3–2)
where P describes the set of all patches in the network and M the set of all states
in the Markov chain. We estimate δij , γij , and ηij as a percentage of the initial transition
probability �qij . To this end, we define δij = αij�qij , γij = βij�qij , and ηij = ωij�qij , where
αij , βij ∈ [0, 1] and ωij ∈ [αij + βij − 1,αij + βij ], ∀i , j ∈ P.
We select the optimal set of patches (H) to disturb by means of the mixed-integer
linear program described in Appendix C.
3.2.2.4 Protection
The end goal of the protection model is to determine the optimal subset of patches
to protect. We define binary variables wi , ∀i ∈ P , which equal 1 if and only if patch i is
protected. The w variables affect the y -variables in the disturbance problem because
yi = 0 whenever wi = 1.
The protection model maximizes the potential loss in the total life expectancy that
can be prevented by protecting a particular set of patches. We use the integer linear
56
program described in Appendix D to determine an optimal set of patches to protect,
given a set of possible disturbance actions.
3.2.2.5 General model assumptions
We assume that individuals follow a random walk between patches. That is, an
individual attempt to move from one patch to another is independent of the sequence of
patches previously visited. This assumption is common to other connectivity modeling
approaches (McRae et al., 2008). We also assume that �qid > 0, for all i = 1, ... , |P|,
which means that no matter the patch where the individual is currently living, there is
always a positive probability of death. We consider the survival rates (Si ) estimated from
mark-recapture as true survival; however, these survival estimates may be negatively
biased due to the difficulty of discerning between mortality and permanent emigration
(Gilroy et al., 2012). We consider the network to be a closed system, hence, the only
available patches to move are included in P. We assume individual homogeneity in the
factors that affect movement and survival. We also assume that movement decisions
and mortality are density independent.
3.2.3 Scenarios
We illustrate the modeling framework by building different protection scenarios for
each species. We developed scenarios in which the protection budget (u) was roughly
half the landscape (Roseate Tern: u = 2, b = 2; Skipjack Tuna: u = 3, b = 4). We
optimized both the weighted and unweighted total life expectancy for the Roseate Tern,
because the data included patch abundances. We optimized only the unweighted total
life expectancy for the Skipjack Tuna, because patch abundances were not included in
the dataset. We also show how total individual life expectancy varies for all combinations
of protection budgets and number of patches allowed to be disturbed.
Because there was no data available to estimate δij and γij , we generated δij =
αij�qij and γij = βij�qij as described in Section 3.2.2.3. Parameters αij and βij were
randomly generated on the continuous uniform distributions with bounds 0 and 0.35,
57
which represents disturbance actions of low to moderate magnitude. Furthermore, we
generate parameters ωij on the continuous uniform interval [αij + βij − 1,αij + βij ]. All the
code was written in C++, using CPLEX 12.3 optimization software.
3.2.4 Sensitivity Analysis
We performed two types of sensitivity analysis. First, we assess the robustness
of the solution (set of optimal patches to protect) given changes in the input data (�Q).
Second, we study how the optimal individual life expectancy changes with increasing
protection budget (amount of sites allowed to be protected) or number of patches
allowed to be disturbed. We observed high site fidelity in both species (high probability
of individuals staying in the same patch). We wanted to know how much change in
these probabilities will take to change the optimal solution. Recall that ωii describes the
proportional decrease in the probability of staying in patch i given that the patch was
disturbed. We vary ωii(1+ ϵ), where ϵ ∈ [−1, 0], to assess how much variation is required
to change the optimal solution.
From Section 3.2.2.3 we have that qii = �qii(1+ωiiyi) ∀i ∈ P, showing that a source of
change in qii is ωii . We define ϵ as the rate of change in ωii , so that qii = �qii(1+(1+ϵ)ωiiyi)
∀i ∈ P. Since qii ∈ [0, �qii ], we know that ϵ ∈ [−1, 0]. In addition, we have that qii = �qii
∀i ∈ T , so we only allow ϵ to affect qii ∀i /∈ T . We define z(ϵ) as the optimal life
expectancy obtained after solving the protection problem using (1 + ϵ)ωii (instead of ωii )
∀i ∈ T among the input data.
Define z(ϵ) as the life expectancy calculated by assuming that set of patches T
is protected, as input data. The ratio ρ(ϵ) = z(ϵ)/z(ϵ) measures the change in the
optimal total life expectancy as a function of ϵ. If ρ(ϵ) = 1, then T is still optimal after
the perturbation given by ϵ, whereas if ρ(ϵ) < 1 then T provides a suboptimal solution
with a gap from the optimal life expectancy given by 1− ρ(ϵ). Given that abundance data
was not available for the Skipjack Tuna, for consistency we perform this analysis on the
unweighted individual total life expectancy.
58
3.3 Results
3.3.1 Roseate Terns: Step-by-Step
Total individual life expectancy (unweighted), in the case that no patch is disturbed
was z = 20.43. If we allowed at least half of the network to be disturbed (b = 2 patches),
with a protection budget of two patches (u = 2). The worst-case for connectivity
happens when the set of patches H = {2, 4} are disturbed, and a corresponding total
life expectancy of z = 15.59 (Figure 3-3b). The corresponding optimal set of patches
to protect that will prevent the most loss of total life expectancy is T = {1, 4} with an
associated life expectancy of z = 15.97 (Figure 3-3j).
When the weighted total life expectancy is optimized using patch abundances
pT = [3773, 2467, 350, 172] (Spendelow et al., 1995), the total life expectancy when
no patch is disturbed was z = 32368.9. The worst-case scenario for connectivity
happens when patches H = {1, 2} are disturbed with an associated life expectancy of
z = 20545.7. The corresponding optimal set of patches to protect is T = {1, 2} with an
associated life expectancy of z = 30638.9 (Appendix E).
3.3.2 Skipjack Tuna
The total individual life expectancy in the case that no patch is disturbed was
z = 46.07. If we allowed at most four patches to be disturbed (i.e. b = 4), and have
a protection budget of at most three patches (i.e. u = 3), the optimal strategy is to
protect the set of patches T = {2, 3, 7}, with an associated optimal total individual life
expectancy of z = 35.99.
3.3.3 Sensitivity Analysis
We found that, in both species, as the protection budget (u) decreases, total life
expectancy decreases and as the number of patches allowed to be disturbed increases,
total life expectancy decreases asymptotically (Figure 3-4). When optimizing the
unweighted total individual life expectancy, the optimal solution for the Skipjack Tuna
was more robust than the optimal solution for Roseate Terns (Figure 3-5). The optimal
59
Figure 3-3. Example of the iterative process between the disturbance and protectionmodels for a protection budget of u = 2 and allowing b = 2 patches to bedisturbed. The figure shows the (a) total individual life expectancy withoutany protection or disturbance, the (b) worst-case scenario for connectivityand (j) the optimal protection against this worst-case scenario.
60
solution for the Skipjack Tuna remained optimal for ϵ ∈ [−0.7, 0], which implies that site
fidelity estimates can decreased by at most 70% and the solution will remain optimal.
On the contrary, the optimal solution for the Roseate Tern network remained optimal
for ϵ ∈ [−0.2, 0], which implies that side fidelity estimates can decrease at most 20%
for the solution to remain optimal. When considering the weighted total individual life
expectancy, the solution remained optimal regardless of the amount of variation (Figure
3-5).
Figure 3-4. Sensitivity of the optimal life expectancy in the protection stage givenchanges (ϵ) in probability of staying in the same patch (�qii ).
3.4 Discussion
Systematic conservation planning tools provide guidance for the proper management
of protected area networks. Uncertainty is present in many stages of this planning
process, and often times not acknowledged, which may result in inadequate management
actions. Recently, methods have been developed that test the robustness of site
selection models given uncertainty in the input data (Moilanen et al., 2006a,b); however,
even when uncertainty in the data is accounted for, there is still uncertainty about
future changes in the system due to human disturbances, or natural disasters. Here
we presented a novel application of network interdiction models that deals with these
61
Figure 3-5. Optimal life expectancy (z) given all possible protection budgets (u) andnumber of patches allowed to be disturbed (b). Life expectancy decreaseswith decreasing protection budget and with increasing number of patchesallowed to be disturbed
unaccounted uncertainties by proposing a protection plan given a worst-case scenario
for connectivity.
We group the data requirements to parameterized the model into two: movement
data and scenario building variables. Movement data includes transition estimates
(ψ), patch survival (S) and patch abundances. It also includes the estimates on how
disturbance in a patch will affect the transition probabilities (γ, δ and η). Scenario
building variables include the maximum number of patches allowed to be disturbed (b)
62
and the protection budget (the maximum number of patches to protect u). The data
requirements to parameterize the interdiction model may be an important limitation of
this approach. Even though between-patch transition and survival estimates are one
of the best data available to quantify landscape connectivity (Belisle, 2005; Kindlmann
and Burel, 2008), these datasets are scarce in the literature because it requires much
field effort and they are complex to fit in a multi-state mark-recapture context (Calabrese
and Fagan, 2004). Nevertheless, these datasets are becoming more common in
conservation efforts. Parameter estimates that describe how movement changes when
patches are disturbed can be taken from the literature when available (e.g., Bechet
et al., 2003; Bowne et al., 2006). If these estimates are not readily available for a
particular species, the model can be parameterized assuming different scenarios for
these parameters.
The sensitivity of the optimal solution to variations in the data will depend on the
structure of the network. We found the model to be very robust to changes in site
fidelity estimates for the Skipjack Tuna, but less robust for the Roseate Terns. The
Roseate Tern network consisted of only 4 nodes with little variation in site fidelity and
mortality probabilities (Figure 3-2), which means that even small changes in some of
the movement estimates may significantly change the optimal solution. In contrast,
the robustness of the solution for the Skipjack Tuna network may be due to increase
variation in site fidelity and mortality probabilities. The optimal protection for Roseate
Tern network, when considering the weighted total individual life expectancy, was very
robust. This is because the abundances in the optimal patches to be protected was an
order of magnitude greater than the other patches.
Scenario building variables are useful to understand various scenarios for
connectivity, depending on how many patches are expected to be disturbed and the
protection budget. Generally, life expectancy is expected to decreases with increasing
number of patches allowed to be disturbed and increase with increasing number of
63
patches allowed to be protected (Figure 3-5). The shape of these relationships will be
specific for each application and will be a useful tool for managers making prioritizing
decisions. In some instances, increasing the protection budget by one patch may result
in a significant increase in life expectancy, suggesting that increasing the protection
budget may be useful. In other instances, decreasing the protection budget may not
necessarily result in a significant life expectancy decrease suggesting that a reduced
protection budget may be appropriate.
3.4.1 Modifications
The network interdiction model presented here is a flexible framework that
can be applied to a wide array of instances in conservation and management. This
approach can be applied to organisms, both active and passive dispersers, for which
between-patch movement and survival data are available. Even though we describe
a model in which the goal is to asses vulnerabilities to patch disturbances, the model
can be modified to assess vulnerabilities in the matrix. In this case the disturbance
and protection models will optimize life expectancy given the protection or disturbance
of arcs in the network instead of nodes. This modification will allow the identification
of places in the matrix to perform restoration activities or where to build corridors to
increase connectivity.
The cost of acquiring or protecting a patch is commonly incorporated in most
prioritization models (Cabeza and Moilanen, 2001; Kukkala and Moilanen, 2012;
Margules and Pressey, 2000; Moilanen, 2008; Sarkar et al., 2006). In this modeling
approach the protection model is given a budget or maximum number of patches to
protect. This inherently assumes that the cost of protecting each patch will be the same.
Variations in cost can be incorporated as constraints in both disturbance and protection
models.
Prioritizing disturbance actions to minimize connectivity for invasive species is a
potential alternate application of network interdiction models. In this case the problem
64
may be simpler because it only requires the disturbance model, in which we identify the
worst-case scenario for connectivity. Hence, the optimal set of patches that minimizes
connectivity will be the best patches to disturbed to control the invasive species.
3.4.2 Conservation Planning for a Worst-case Scenario for Connectivity
The few prioritization models that account for connectivity follow the area/isolation
paradigm from metapopulation theory that states that the probability of recolonization
increases with decreasing isolation (Hanski, 1998) and attempt to increase structural
connectivity by either minimizing the perimeter length of the network or the distance
between patches (Ball et al., 2009; Beger et al., 2010; Nalle et al., 2002; Williams
et al., 2004). Here, when prioritizing areas for the Roseate Terns, patches 1 and 4 were
selected as the optimal patches to be protected, even when the distance between these
two patches is the longest when compared to other between-patch distances in this
network (Figure 3-2a). Similarly, the optimal solution for the Skipjack Tuna includes the
node that represents international waters, which is the most isolated patch in terms of
distance (Figure 3-2b). This patch was part of the optimal protection solution because
it had high site fidelity and very low mortality. These results suggests that minimizing
between-patch distance may not be always an appropriate criteria for prioritization.
The practicality of increasing structural connectivity between patches has been
debated. Decreasing distance between patches may decrease the probability of
local extinction and reduce potential inbreeding, but it may increase the spread
of diseases or the persistence of exotic species (Williams et al., 2005). Hence, it
has been argued that increasing structural connectivity (i.e. distances between
sites without information on how organisms actually move among sites) may not
be an optimal management strategy, when prioritizing areas for an uncertain future
(Hodgson et al., 2011, 2009). Measures of structural connectivity do not incorporate
information about the behavior of the species, or how moving through landscape affects
individual fitness. Here, the optimal protection is assessed optimizing a connectivity
65
measure based on precise movement data (i.e. movement probabilities are quantified
through rigorous mark-recapture methods) and patch-specific survival estimates,
which ameliorates significantly some of the concerns that have been brought about
connectivity conservation for future changes (Doerr et al., 2011).
The current paradigm of metapopulation ecology also determines patch importance
by considering patches individually (Hanski, 1998). Highly connected patches are
considered more important to maintain ecological process in a spatial network (Bunn
et al., 2000; Minor and Urban, 2008). Moreover, the resilience of a network is often
measured by quantifying how many individual nodes can be removed without altering
network connectivity (Bodin and Saura, 2010; Bunn et al., 2000; Pascual-Hortal and
Saura, 2006; Urban and Keitt, 2001; Watson et al., 2011b). Here we presented an
alternative method by considering the importance of a particular subset of nodes instead
nodes individually. Considering a combination of patches instead of the sum contribution
of individual patches, often results in non-intuitive solutions. This is a common result in
the operations research literature (Brown et al., 2006).
Solutions from the network flow interdiction models, like any other optimization
model, are simply a tool for decision support. Its purpose is to guide the management
process and its solutions should not be necessarily considered in isolation (Williams
et al., 2004). Instead, it should be considered in conjunction with other models with
different objectives as part of the conservation planning process. Pressey and Bottrill
(2009) summarized the conservation planning process in 11 stages. These stages,
include the identification of (1) the scope, (2) the stakeholders, (3) the spatial extent
and the (4) goals of the conservation program, (5) the collection of socio-economic
variables and threats, (6) collect data on biodiversity, (7) set conservation targets,
(8) evaluation of the existing area network, (9) identification of important areas, (10)
applying the conservation actions and (11) monitoring. The modeling framework
presented here encompasses stages 6 to 9 of this process. The data collection (stage 6)
66
requires between-patch transition probabilities and patch specific survivals. The network
interdiction framework also allows to set the specific conservation objectives (stage 7)
in terms of the number of patches to be protected (and the number of patches to be
allowed to be disturbed). The connectivity assessment stage (Section 3.2.2.2; stage 8)
which calculates the expected life expectancy given the current conditions. The result of
the model is the set of patches that will minimize future decreases to the life expectancy
of individuals given the scenarios built in in the model (stage 9).
3.4.3 Uncertainty and Worst-case Scenario Planning
Uncertainty may be present in multiple stages of the conservation planning
process, but rarely acknowledged. Information gap decision theory has been proposed
as an efficient framework to incorporate statistical uncertainties in conservation
planning (Moilanen et al., 2006a,b). The aim of information gap decision models is
to identify solutions that are robust to the presence of uncertainty. Even when statistical
uncertainties are incorporated in the planning process by the use of information gap, or
other sensitivity analysis, other non-statistical uncertainties may still affect the system in
the future and result in inadequate management actions. The network flow interdiction
model acknowledges that there are additional uncertainties which are difficult to quantify,
but uses the available data to protect against the worst possible disturbance. Hence,
even when climate change or regional-scale changes in environmental conditions may
affect the network in an unpredictable manner, the system has already been protected
against the worst possible scenario.
67
APPENDIX AMODEL SUMMARIES AND ADDITIONAL FIGURES
Table A-1. Parameter estimates and standard errors (SE) for the multi-seasonoccupancy model that resulted in the best fit; ψ(.), γ(Sasym ∗ A ∗ Ph),ϵ(Sasym ∗ A ∗ Ph), P(A ∗ Ph).
Colonization Extinction DetectionEstimate SE Estimate SE Estimate SE
INTERCEPT -4.897 0.198 -2.724 0.285 2.142 0.113Sasym 0.330 0.174 0.565 0.200A 0.008 0.001 -0.016 0.005 0.004 0.001Ph(TREES) 0.800 0.383 0.575 0.415 -0.410 0.184Sasym : A 0.002 0.001 -0.016 0.005Sasym : Ph(TREES) 0.410 0.255 -0.414 0.394A : Ph(TREES) -0.003 0.006 -0.005 0.010 0.001 0.002Sasym : A : Ph(TREES) -0.015 0.005 0.004 0.012
68
Table A-2. Parameter estimates and standard errors (SE) for the model that included asymmetric dispersal kernel (Ssym) as a site covariate; ψ(.), γ(Ssym ∗ A ∗ Ph),ϵ(Ssym ∗ A ∗ Ph), P(A ∗ Ph).
Colonization Extinction DetectionEstimate SE Estimate SE Estimate SE
INTERCEPT -4.867 0.195 -2.60 0.267 2.134 0.112Ssym 0.231 0.192 0.558 0.219A 0.007 0.001 -0.021 0.006 0.004 0.001Ph(TREES) 0.800 0.409 0.400 0.400 -0.413 0.184Ssym : A 0.001 0.001 -0.018 0.005Ssym : Ph(TREES) 0.289 0.284 -0.667 0.380A : Ph(TREES) -0.002 0.007 -0.001 0.010 0.001 0.002Ssym : A : Ph(TREES) -0.008 0.005 0.020 0.009
Figure A-1. Spatial representation of patches (boulders and tree trunks) sampled forLepanthes rupestris occurrence in the Luquillo Experimental Forest.
69
Figure A-2. Theoretical distributions of the parameter δij . A random spatial arrangement(a) results in a relatively uniform distribution (b). If patches are alignedalmost perfectly in the same angle of wind direction (c), it results in abimodal distribution highly skewed to the limits of the distribution (d).
70
Figure A-3. Predicted partial relationships between colonization, extinction and patcharea for both phorophytes for the best-supported model
71
Figure A-4. Partial relationship between patch area and detectability for (a) rock and (b)tree phorophyte for the best-supported model
72
APPENDIX BCONNECTIVITY ASSESSMENT: MATHEMATICAL DETAILS
We calculate total individual life expectancy (�gi ) as the number of transitions that
an individual living in patch i survives before dying (going to state d) (Ross, 2006). This
calculation is derived based on the following linear equation. The expected lifetime, �gi ,
starting from patch i ∈ P, is given by one plus
• zero if the individual dies at the next period (with probability 1− Si )
• �gi if the individual successfully reaches patch j ∈ P (with probability �qij ).
The relationship can be written as
�gi = 1 +∑j∈P
�qij�gij , (B–1)
or in matrix from:
I�g = �A�g+ 1. (B–2)
If (I− �A) is invertible (which we assume to be the case, by a a slight perturbation of
�A if necessary), then
�g = (I− �A)−11. (B–3)
The weighted total life expectancy, z , is calculated as:
z = pT (I− �A)−11 = pT�g. (B–4)
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APPENDIX CDISTURBANCE OPTIMIZATION
The disturbance optimization model includes the initial transition probabilities given
by �qij (∀i , j ∈ M) and the estimated impacts on the transition probabilities caused by
the interdictor’s disturbances, which are given by δij , γij , and ηij (∀i , j ∈ M). A formal
statement that describes the calculation of qij , ∀i , j ∈ M is given by Equation 3–2.
To illustrate the use of ηij , consider the case in which disturbing patches i and j
together reduces qij by less than the sum of the individual disturbances δij + γij . In this
case, ηij is a positive correction term. On the other hand, if there is a synergistic effect
attributable to jointly disturbing patches i and j , such that disturbing i and j together
reduces qij by more than the individual disturbances, then ηij is as a negative correction
term. We assume that ηij ∈ [δij + γij − �qij , δij + γij ], so that in case of disturbing both
patches i and j , we have qij ∈ [0, �qij ]. In the case that only one patch is disturbed, we
require the condition �qij − max{δij , γij} ≥ 0 so that qij ∈ [0, �qij ]. Also, we set δii = γii = 0
without loss of generality, meaning that if patch i is disturbed, then qii is updated only
according to ηii ∈ [−�qii , 0].
We define the parameter b ∈ {1, · · · , |P| − 1} as the maximum number of patches
that can be disturbed. Our model also uses the parameter �gi (∀i ∈ P), given by Equation
B–3. We know that �gi is an upper bound on the number of transitions spent in transient
states, because the transition probabilities between transient states are either reduced
when some patches are disturbed or remain unchanged when no patch is disturbed.
The model identifies those patches that should be disturbed to minimize the
individual total life expectancy. The disturbance optimization model uses variables yi ,
∀i ∈ P, and updated transition probability variables qij , ∀i , j ∈ P as defined by the first
case of Equation 3–2.
The disturbance problem objective is given by
min z = pTg, (C–1)
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and g is again calculated as
g = (I− A)−11, (C–2)
where A is obtained by deleting the last row and column from Q. By substituting
qij using Equation 3–2, and writing Equation C–2 in functional form, we can rewrite
Equation C–2 as
gi −∑j∈P
�qijgj +∑j∈P
δijyjgj +∑j∈P
γijyigj −∑j∈P
ηijyiyjgj = 1, ∀i ∈ P. (C–3)
Note that each equation in Equation C–3 is nonlinear in the y and g variables. To
linearize these equations in Equation C–3, we introduce the additional variables wij
(= yigj ) and xij (= yiyjgj ). The resulting system is given by
gi −∑j∈P
�qijgj +∑j∈P
δijwjj +∑j∈P
γijwij −∑j∈P
ηijxij = 1,∀i ∈ P. (C–4)
The system shown in Equation C–4 is the first set of constraints in our model.
This set describes not only the way to update the transition probabilities given the
interdictor’s actions (q variables), but also the calculation of the expected time spent in
transient states (g variables). To guarantee that wij = yigj , we introduce the set of linear
constraints shown in Equations C–5–C–8. Note that if yi = 1, then wij = gij . If yi = 0,
then wij = 0, as desired.
wij ≥ 0, ∀i , j ∈ P (C–5)
wij ≤ gj , ∀i , j ∈ P (C–6)
wij ≥ gj − �gj(1− yi), ∀i , j ∈ P (C–7)
wij ≤ �gjyi , ∀i , j ∈ P (C–8)
Similarly, we define the set of constraints shown in Equations C–9–C–13 to
guarantee that xij = yiyjgj . Since the term yiyjgj is also nonlinear, we introduce the new
75
variable vij and the corresponding set of constraints shown in Equations C–13–C–16
to guarantee that vij = yiyj . Note that if yi = 0, then vij = 0 and xij = 0. Moreover, if
yi = yj = 1, then vij = 1 and xij = gj , as desired.
xij ≥ 0, ∀i , j ∈ P (C–9)
xij ≤ gj , ∀i , j ∈ P (C–10)
xij ≥ gj − �gj(1− vij), ∀i , j ∈ P (C–11)
xij ≤ �gjvij , ∀i , j ∈ P (C–12)
vij ≥ 0, ∀i , j ∈ P (C–13)
vij ≤ yi , ∀i , j ∈ P (C–14)
vij ≤ yj , ∀i , j ∈ P (C–15)
vij ≥ yi + yj − 1, ∀i , j ∈ P. (C–16)
The maximum number of patches that can be disturbed is considered in Equation
C–17, while the constraint shown in Equation C–18 defines the binary nature of the
y -variables.
∑i∈P
yi ≤ b (C–17)
yi ∈ {0, 1}, ∀i ∈ P (C–18)
The mixed-integer linear problem consists in minimizing the objective function shown in
Equation C–1 subject to constraints in Equations C–4–C–18.
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APPENDIX DPROTECTION OPTIMIZATION
The protection optimization model includes the parameter u ∈ {1, · · · , |P| − 1} to
denote the protection budget (the maximum number of patches that can be protected).
We let the binary decision variable wi equal 1 if and only if patch i ∈ P is protected.
First observe that all feasible protection solutions obey the constraints:
∑i∈P
wi ≤ u (D–1)
wi ∈ {0, 1}, ∀i ∈ P. (D–2)
Equation D–1 enforces the maximum number of patches that can be protected,
while constraints in Equation D–2 define the binary nature of the decision variables.
The objective for the protection problem is less straightforward to model. Conceptually,
let H be the set of all cardinality b subsets of P, i.e., the set of all possible disturbance
solutions. For every H ∈ H, let zH be the individual life time given disturbance set H.
We can define binary variables δH,∀H ∈ H, which equal 1 if and only if the protection
action “disables” disturbance set H (i.e., if there exists an i ∈ P such that i ∈ H and
wi = 1). Then we can state constraints:
∑i∈H
wi ≥ δH, ∀H ∈ H. (D–3)
The protection problem objective is denoted by
max
{min
H∈H:δH=0
zH
}. (D–4)
However, such a problem is non linear and requires exponentially many constraints
and variables. As an alternative consider the feasibility problem given by F(H):
Constraints in Equations D–1 and D–2,
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∑i∈H
wi ≥ 1 ∀H ∈ �H, (D–5)
where �H ⊆ H is a subset of the set H. Note that F( �H) is feasible if and only if all
possible disturbances in the set �H can be disabled by a protection action. Therefore,
consider the following algorithm, in which an upper bound value is initially set to UB= �z
(computed by Equation B–4), �H is initially empty, and which the protection solution is
given by w∗ = 0.
• Step 1. Solve F( �H), and if feasible solution w exists, go to the second step;otherwise, go the third step.
• Step 2. Solve the optimal disturbance problem augmented with the constraints thatyi ≤ wi ,∀i ∈ P, which enforce the condition that no protected patch is disturbed.Let H be an optimal solution to the disturbance problem, given w, noting thatconstraints yi ≤ 1− wi ,∀i ∈ P, are added to the disturbance problem. Add H to �H.If UB > zH, then set UB = zH and w∗ = w. This step returns to step 1.
• Step 3. Terminate with an optimal solution given by w∗ having an objective UB.
Two notes are necessary regarding this algorithm. The first regards the convergence
and correctness of the algorithm. Each time F( �H) is solved, one more subset is added
to �H. A second issue is that solving F( �H) in early iterations of the algorithm yields
protection strategies that are blind to many possible disturbance solutions that are not
yet in �H, but are likely to be added to �H in future iterations. Accordingly ,we can apply a
lower-boundary objective function to F( �H).
To estimate the lowest possible life expectancy we define ~g as a vector whose
entries ~gi represent the mean number of transitions spent in the transient states, given
that the Markov chain starts at state i . To guarantee that ~g is a lower-bound on life
expectancy, we assume that the transition probabilities between transient states are the
minimum possible, which is equivalent to assume that the transition probabilities from
any transient state to the death state are maximum. To this end we define the transition
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matrix ~Q, whose entries ~qij are given by
~qij = (1−max{δij , γij , δij + γij − ηij})�qij . (D–6)
By our construction of the transition probabilities in Equation D–6, no disturbance can
produce a total life expectancy of less than pT~g, and so pT~g is a valid lower bound on
total life expectancy. The objective function for F( �H) in this case can be written as
f (w) =∑i∈P
pi(�gi − ~gi)wi , (D–7)
where pi(�gi − ~gi) is an estimate of the potential loss in the total life expectancy that can
be prevented if patch i is protected.
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APPENDIX EMODEL ITERATIONS FOR WEIGHTED LIFE EXPECTANCY OPTIMIZATION
Table E-1. Model iterations showing the interaction between disturbance and protectionstages when pT = [3773, 2467, 350, 172]
Iteration Stage Optimal solution z �z
1 Protection wT = [ 0 0 0 0 ] 20545.7 −10093.2Disturbance yT = [ 1 1 0 0 ]
2 Protection wT = [ 1 1 0 0 ] 30638.9 0Disturbance yT = [ 0 0 1 1 ]
3 Protection wT = [ 1 0 1 0 ] 27084.8 −3554.1Disturbance yT = [ 0 1 0 1 ]
4 Protection wT = [ 1 0 0 1 ] 26465.4 −4173.5Disturbance yT = [ 0 1 1 0 ]
5 Protection wT = [ 0 1 1 0 ] 24499.1 −6139.8Disturbance yT = [ 1 0 0 1 ]
6 Protection wT = [ 0 1 0 1 ] 23846.9 −6792Disturbance yT = [ 1 0 1 0 ]
80
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BIOGRAPHICAL SKETCH
Miguel A. Acevedo was born in San Juan, Puerto Rico in 1980. He graduated from
the University of Puerto Rico with a B.S. in biology 2002. He later graduated with a M.S.
in the Tropical Biology Program from the University of Puerto Rico-Rıo Piedras in 2006.
Miguel received his Ph.D. from the University of Florida in the spring of 2013 where,
as part of the Quantitative Spatial Ecology, Evolution and Environment (QSE3 IGERT),
developed an interdisciplinary dissertation research that incorporated spatial ecology,
network theory and concepts from industrial and systems engineering. He hopes to
return someday to Puerto Rico and join their professorate conducting quantitative
ecological research, and teaching.
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