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DISPERSAL CHOICE ANALYSIS
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| Main goals |
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Identify the landscape features that
facilitate or impede individual movements, if any,
based on potentially any dispersal data.
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Quantify the maximum distance of
inhospitable habitat an organism can cross.
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| Introduction |
Despite the fact that dispersal has been
recognised to be a fundamental process to drive the
distribution, the viability and the evolution of species, it is
still poorly understood and over-simplified in most studies. Although dispersal has often been assumed to
be independent of the landscape (Fig. 1, path 1), the features
of the landscape can often facilitate or impede individual
movements. A common problem in dispersal studies is the fact
that records of dispersal paths, obtained for example from
radio-tracking, typically consist in series of locations,
discrete in time and space. We therefore generally do not know
the path followed by an individual between them. Straight lines
were generally used to represent such path, but nowadays, least-cost path modelling
allows a better
representation of individual movements. Indeed, the path inferred
between two observed locations can be calculated in order to
maximise the crossing of facilitating features and to minimise
the crossing of barriers or inhospitable habitats (Fig. 1, path
2). See how cost functions work on the
online ArcGIS page. However:
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How do we know whether individuals use
corridors for their dispersal or not? Or more generally,
do all substrates provide the same connectivity and if not, how can we identify the features that may facilitate or impede
individual movements?
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Figure 1.
Whereas dispersal has often been modelled as a straight movement
between patches regardless of the features of the landscape
(path 1), using cost distance modelling to represent the
preference of individuals to move through specific substrates
and avoid others (path 2) may be more realistic in many species
and landscapes.
Cost distance modelling
relies on the assignment of a resistance value (also called cost
or friction value) to each cell of the map that denotes the
difficulty of the organism or its reluctance to cross the map
cell. However, the resistance values are unfortunately often
guessed based on expert judgment or on proxy data. Moreover,
their assignment generally assumes that individual movements are
decided on a cell-by-cell basis, and decisions made on a larger
scale such as gap-crossing ability are ignored.
Ignoring the spatial
distribution of these resistance values implies that a path
crossing several small stretches of inhospitable habitat (Fig.
2, path 1) can have the same cost as a path crossing fewer but
larger stretches (Fig. 2, path 2). However, it is possible that
an organism might be willing to use the first path but not the
second one. This therefore brings another question:
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Figure 2. Using cost distance modelling
while ignoring the gap crossing ability of species often leads
to the fact that paths 1 and 2 are equally probable, which is
often unrealistic.
I propose here an approach I developed to answer these questions.
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Methods |
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To illustrate the approach, let's consider an
organism living in mature forests. We want to test the
hypothesis that this species is reluctant to cross pastures and
prefers using woody vegetation to disperse. After fitting some
individuals with radio-transmitters, we recorded their
location every day during their dispersal (daily dispersal
steps). Our dataset at this point only
consists of a table containing the radio-tagged individuals, the
date and time of each resighting and the corresponding locations
where the individuals were found.
If we want to test whether
pasture has a low degree of connectivity compared to woody
vegetation, we can assign each cell of the map
representing pasture with a high resistance value, i.e. 10,
compared to those representing woody vegetation with a value of
1, meaning that an individual is 10 times more likely to cross a
cell of woody vegetation than a cell of pasture. Most GIS
software can then calculate the cost distance between two
consecutive recorded locations, which indicates the likelihood of
the movement between the two points, assuming that our
assignment of resistance values is right. This assignment of the
values to each pixel of the map represents our
hypothesis about landscape connectivity.
Our
connectivity hypothesis can be considered as
satisfying only if we can demonstrate that individuals seek to
minimise the cost distance of the path they choose, i.e. they
prefer moving through low resistance substrates and avoid high
resistance ones. For this purpose, we can match each chosen
dispersal step (each pair of consecutive locations) to a sample
of non-chosen available alternatives of similar length, and
calculate the cost distance for these alternative paths also
(Fig. 3).
The least-cost paths (and
the cost distances that can be calculated
in most GIS software package) typically look like this:
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Figure 3. This figure shows
a dispersal step (in yellow, starting point in the centre) matched
to 10 random alternatives (in red) of similar length,
chosen among a wider set of random locations situated in woody
vegetation (in black). The least-cost paths
are calculated assuming that movements occur most likely in
native forest (resistance value of 1), less in pine forest
(value of 2), even less in shrubland (3), and impeded by pasture (10).
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Conditional logit models can then be used to test whether the chosen
steps have a lower cost distance on average than their matched
alternatives. Such model can be fitted using a Cox proportional hazards model
with Breslow ties as their likelihood functions are similar (Chen & Kuo 2001).
The data only need to be prepared to be equivalent to survival data, where the
individual "dies" at time 1 for the chosen alternative,
or "survives" and becomes censored at time 2 for
the non-chosen alternatives (Kuhfeld 2001). The theory behind the conditional
logit model in an ecological context has been described
in Fortin et al. (2005) and for the closely related multinomial
logit regression model in Cooper & Millspaugh (1999).
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Here is an example of how the model can be fitted using the great
free statistical software
R, using the coxph
function from the survival library:
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And for the old-fashioned people, here is an example of how
to fit the same model using Proc PHREG in SAS®:
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Note that this model only has two hierarchical levels, the alternative points within
the dispersal steps (defined by StartPtID in this case),
i.e. the chosen and non-chosen alternatives,
can only be compared from
each starting point. It is therefore assumed that actual
dispersal steps are independent within and between individuals.
A close look (plots etc.) at the data should be sufficient to detect if this
assumption is erroneous.
From the fitted model, we get the value and
significance of the coefficient β of the utility function U
of path i defined as:
Ui = β.RelativeCosti (equation 1)
A negative value of β that is statistically significant
indicates that individuals prefer to go to locations that are
well connected, i.e. for which paths go through corridors and
avoid barriers most often.
The probability pi that an individual chooses the path i among j
possibilities is:
pi = exp(Ui) / ∑exp(U1...j) (equation 2)
If β is significantly
negative, individuals' movements follow the map of resistance
values we created, which is already satisfying. We can now go
further and try different values of resistance, e.g. increase
the resistance of pasture to movements and assign different
values to different woody vegetation types. If the dataset is kept the
same among models, i.e. the random alternatives matched to the
chosen dispersal steps are identical across models, then we can
use the Akaike Information Criterion (AIC) to
compare the models and find the model that fits the data the
best.
Likewise, we can
incorporate some behavioural traits in the
model such as the gap crossing ability of
species (see above). This can be achieved by assigning a
resistance value to each cell of pasture that changes as a
function of its distance to the nearest woody vegetation. A
linear or Gompertz function can be used, and several models with
various function parameters can be tested as above (Fig. 4).
Figure 4.
To test for the gap crossing limitation of
species, the resistance value of each cell of pasture can change
as a function of its distance to the nearest woody vegetation.
Several functions can be used, tested, and their comparison
based on models' AIC would indicate which one fits the data the
best.
The best model will
indicate which hypothesis about landscape connectivity fits the
data the best. If the best model indicates a strong gap crossing
limitation, we can then calculate the largest gap crossed by the
species by first calculating the least-cost paths between
consecutive dispersal points based on the cost map of the best
model to infer the movement paths, and then by extracting the
largest gap crossed by these paths by splitting these paths
based on the vegetation cover layer in the GIS.
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| Benefits of the method |
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Objective.
Whereas the assignment of resistance values used to
be quite subjective, the comparison of alternative
hypotheses based on AIC allows to test whether a factor is
truly influencing species movements.
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Can be potentially applied
to any available dispersal data, without having
to rely on expensive research design.
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Quantitative.
The approach allows to quantify the resistance values/functions
associated with landscape elements.
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Applicable
to most landscapes and species to test most hypotheses about
dispersal.
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Flexible.
Most factors promoting or limiting species movements can be
represented in a cost map. Avoidance of roads, buildings or
human habitation for example can be modelled by assigning a cost
to each map pixel that decreases with distance from the these
features, easily calculated in the GIS, and gap crossing ability
can be modelled as described above. Furthermore, several factors
can be combined in the calculation of the cost map; the
resistance value of each pixel would simply be a weighted sum of
the different factors.
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Generalizable.
By directly incorporating some behavioural traits in the
hypotheses, the functions used to represent these traits, once
fitted and validated, are likely to be extrapolated to other
landscapes.
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Extendable.
One can also include in the models some characteristics of the
dispersing individuals such as sex, size, age or sibling status
to analyse or control for variations in dispersal behaviour
among individuals. Such models would then be a mixture of
conditional and multinomial logit models, but they can be fitted
the same way as previously described.
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The results are easy to present to the
public, stakeholders, land and wildlife managers. Cost
distances and dispersal
frequencies can be represented
on maps such as the following ones:
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Figure 5. Cost distance from
a given patch (in centre). The height and colour
represent the cost distance from the source. Note how the patch on the right-hand side is unreachable to dispersers because of its isolation.
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Figure 6. Movement potential from
the same given patch (top plateau in red). The height and colour
represent the probability of movement from the source, obtained
by multiplying the cost distance from the source (Fig. 5) and the
probability density function of cost distances achieved by
dispersers between their natal territory and their settlement.
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| "OK, but how can I do all this?" |
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Assuming
that you already have a shapefile containing some
dispersal points, the main task is to
calculate the cost distances of
the chosen and random locations for each dispersal
step for the conditional logit model.
Fortunately, I wrote
a toolbox in Python for ArcGIS that automates
most of the operations. You can download
it for free at the end of this page.
The toolbox contains two scripts:
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Dispersal choice analysis - for ArcGIS 92.py
From a shapefile containing consecutive
dispersal locations, this script selects random
locations to be matched to the chosen
destination for each dispersal step and
calculates the cost distance of these points for
one of several cost rasters. The output is a
table where each row represents a dispersal
step, chosen or not, with the associated cost
distance (one column for each cost raster). This
table can be used directly in R or SAS for
fitting conditional logit models using the codes
provided above.
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Dispersal paths and gaps - for ArcGIS 92.py
From a shapefile containing consecutive
dispersal locations, this script calculates the
least-cost path between them, given one or
several cost rasters, and extracts the maximum
distance crossed over a specified substrate (for
gap crossing ability - see above).
What do I need to run the scripts?
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ArcGIS 9.0-9.2 with an ArcInfo license and spatial
analyst. This is the expensive part of the
process... I hope one day to
be able to adapt the scripts to GRASS.
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A shapefile with dispersal points, ordered by
individual and by date/time.
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A shapefile with relevant random points that will
be used as the alternative points. You can
create such shapefile using the free extension
Hawth's Analysis Tools for ArcGIS.
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One or more cost rasters calculated in ArcGIS to
represent the various hypotheses about landscape
connectivity.
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A lot of patience... Everything in ArcGIS 9.x takes time,
but more particularly the calculation of cost distances
and least-cost paths. For instance, in the case of 100
recorded dispersal steps, each matched to 10 alternatives, and under
10 different cost sets, the script needs to calculate 10
000 least-cost paths, while a single calculation may take
several minutes depending on the length
of the step, the map resolution and the
computer power.
However, the script offers the possibility for
the user to be notified by e-mail upon
completion of the script (whether successfully
or not), and, in case of successful completion,
the output table can also be sent.
What these scripts do NOT do:
For more details, see the help
file provided with the script.
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| Download the script for ArcGIS |
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The script is available both for
ArcGIS versions 9.0 to 9.2:
- Download the toolbox for ArcGIS
9.0 and 9.1 (available soon) -
- Download the toolbox for ArcGIS
9.2 -
LICENSE & COPYRIGHT
This software is copyrighted and
is the intellectual property of the author. You
(users) are granted license to use, install and
freely distribute this software without limit. You
are expressly forbidden to sell this product, or in
any way attempt to make a profit by distributing it
(this includes distributing it on a website that
sells advertising space).
Although you may distribute this software, I ask
that you refer other interested users directly to
this web site to ensure they have acquired the
latest version.
Implicit in the use of this product is the
understanding that:
- No technical support is offered for this product.
- The product is provided AS-IS, without warranty of
any kind.
- YOU are responsible for ensuring that the output
of this tools is accurate, relevant, consistent, and
otherwise error-free.
- The author assumes no responsibility for any
suffering you may experience as a result of the use
(or misuse) of this software.
- The author does not warrant that this software is
bug free.
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References |
CHEN Z. & KUO L.
(2001). A note on the estimation of the multinomial logit model
with random effects. The American Statistician, 55, 89-95.
COOPER A.B. & MILLSPAUGH J.J. (1999). The application of
discrete choice models to wildlife resource selection studies.
Ecology, 80, 566-575.
FORTIN D., BEYER H.L., BOYCE M.S., SMITH D.W., DUCHESNE T. & MAO
J.S. (2005). Wolves influence elk movements: behavior shapes a
trophic cascade in Yellowstone National Park. Ecology, 86,
1320-1330.
KUHFELD W.F. (2001). Multinomial logit, discrete choice
modeling: an introduction to designing choice experiments, and
collecting, processing and analyzing choice data with the SAS
system. Sas Technical Report TS-621, SAS Institute, Cary, North
Carolina, USA.
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