EPA/600/A-97/086
PATCH: A spatially explicit life history
simulator for terrestrial vertebrates
Nathan H. Schumaker
US EPA Environmental Research Laboratory
200SW351IlSt.
Corvallis, Oregon 97333
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The information in this document has been funded in
part by the U.S. Environmental Protection Agency
under contract CR824682 to Oregon State University.
It has been subjected to the Agency review and
approved for publication.
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to directly examine the cost, in terms of species' viability, of rates and patterns of habitat
alteration.
This chapter is devoted to a description of the PATCH model, but it includes a case study
reminiscent of the analysis of habitat alteration conducted by Wallin et. al. (1994). The model is
presented in considerable detail in hopes of introducing the reader to some of the complications
involved in merging realistic spatial detail with an otherwise simple demographic model. The case
study is intended to serve as an example of the types of questions that can only be addressed using
a model that conducts viability analysis within the confines imposed by landscape pattern.
MODEL DESCRIPTION
History
I first started working on the predecessor to the PATCH model while attending a summer
school on Patch Dynamics organized by S. A. Levin, T. M. Powell, and J. H. Steele, and held at
Cornell University in 1991. The first study to emerge from this modeling effort examined pattern
formation generated from a spatially explicit version of a simple Nicholson-Bailey predator-prey
model (Deutschman et. al. 1993). I continued the development of the model over the period from
1991 to 1995 at the University of Washington. During this period, I used the model to explore
issues of habitat connectivity (Schumaker 1996), and I modified it to create a life history
simulator for the Northern Spotted Owl (Schumaker 1995). An additional two years of work,
from 1995 to 1997, saw the transformation of the spotted owl simulator into the present PATCH
model. The PATCH model will be available, free of charge, from the US EPA beginning either in
late 1997, or early 1998.
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INTRODUCTION
The consequences of habitat alteration for wildlife species include the direct effects of habitat
loss plus a host of indirect effects such as reduced inter-patch dispersal (Thomas et al. 1990,
McKelvey et al. 1993, Schumaker 1995), increased edge effects (Chen et al. 1992), and
conversion of source habitats to sinks (Pulliam et al. 1992, Dunning et al. 1992). Such indirect
effects are difficult to detect, but they can strongly influence a landscape's ability to support the
species that inhabit it (Reid and Miller 1989, Jensen et al. 1993, Lawton 1993, Schumaker 1995,
Schumaker 1996). Complications such as these dictate that management efforts aimed at
preserving wildlife diversity must consider how different species' habitat requirements and
behaviors couple with landscape pattern, and to what extent landscape pattern limits species'
viability.
Unfortunately, few meaningful generalizations exist with which to estimate the consequences
of habitat alteration for wildlife species (Fahrig 1991, Doak and Mills 1994, Schumaker 1996).
The research described here attempts to overcome such shortcomings by making use of a new
spatially explicit life history simulator called PATCH. This model has been recently completed,
but was derived from an existing spotted owl simulator (Schumaker 1995) that has undergone
extensive peer review. PATCH reads Geographical Information System (GIS) imagery directly,
and it uses these data to link every attribute of a species' life cycle to the quality and distribution
of habitat throughout a landscape. The model tracks an entire population of organisms comprised
of individuals that each are born, disperse, breed, and then die. PATCH is designed specifically to
work with a complex landscape composed of habitats of various shapes, sizes, and qualities.
Further, these landscapes can change continuously through time, and in this way the model is able
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Overview
PATCH (figure 1) is a Spatially Explicit Population Model, or SEPM (Dunning et al. 1995).
PATCH stands for a Program to Assist Tracking Critical Habitat (its focus on critical habitat will
become apparent later). The model is distinguished by the attention it pays to landscape pattern,
and by its ability to work with an entire spectrum of terrestrial vertebrates. PATCH directly
imports GIS habitat coverages, and is parameterized with habitat utility indices, territory size,
survival and fecundity information in the form of a population projection matrix (Leslie 1945,
Lefkovitch 1965, Caswell 1989, Gotelli 1995), and estimates of movement ability and behavior.
PATCH is females-only model, is highly parsimonious, and is designed to accommodate a range
of data availability and quality. The outputs generated by the PATCH model include population
size as a function of time, effective survival and fecundity rates (rates that reflect the effect of
habitat quality on the population), and estimates of the importance of each territory-sized parcel
of habitat for the modeled population. These features permit the user to quantify the consequences
of landscape change for population viability, to estimate changes in vital rates corresponding to
habitat loss or fragmentation, and to identify source and sink habitats within a landscape,
PATCH was designed specifically to address the contribution of spatial pattern to the viability
of a wildlife species. A typical use of the model would include establishing a baseline viability
analysis under current landscape conditions. The investigation might stop here, or the model
landscape could be modified, and the consequences of this change for the viability of the
organism would then be assessed by repeating the demographic analysis.
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Patch identification
It is often desirable to identify aggregate features of landscape pattern that correlate with
measures of ecological quality such as population viability, or habitat connectivity (Schumaker
1996). PATCH facilitates this type of analysis because it includes a module devoted specifically to
quantifying landscane pattern. Many different metrics can be developed from measures of
landscape pattern. PATCH'S approach to providing such information is to break a landscape up
into a collection of individual fragments of habitat, and then to provide a limited amount of
information about each one. Indices of landscape pattern, can subsequently be constructed from
this information.
For the purposes of patch identification, PATCH allows each habitat type to be assigned a
weighting value (i.e. species' habitat preferences), which takes the form of an integer between 0
and 99. Any pixel that has been assigned a non-zero weight is treated as habitat, whereas the rest
are considered non-habitat. PATCH then locates individual patches in the imagery using one of
two rules for defining connectivity. One rule specifies that each pixel has only four neighbors that
touch it, while the other implies that each pixel has a total of eight touching neighbors. Based on
the rule that is applied, PATCH then assigns every habitat pixel to one, and only one, patch. The
area, weighted area, interior area, and perimeter are then computed for each patch. Area is
measured as the number of pixels of habitat present. Weighted area is measured as the sum, taken
over every pixel in a patch, of the weighting values assigned to each pixel. Interior area is
computed based on a user defined edge width, which is specified as a number of pixels. A patch's
interior area is defined as the number of pixels that are at separated from the patch's perimeter by
a distance equal to at least one edge width.
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Many well known indices of habitat pattern can be built up from the measures described
above. Examples include perimeter-area ratio, shape index (Patton 1975, Forrnan and Godron
1986), estimates of fractal dimension (Milne 1988, Milne 1991), and patch cohesion (Schumaker
1996). More importantly, this analysis provides the raw data necessary to construct yet
undescribed measures of landscape pattern that might serve as effective indicators of ecological
quality.
Territory allocation
Before PATCH'S demographic analysis can be conducted, it is necessary to break the
landscape being used into an array of territory-sized units. This process, termed territory
allocation, is accomplished by intersecting the GIS image with an array of hexagonal cells.
PATCH was designed with the intent that each hexagon's area would equal the size of a typical
territory for an individual of the species being modeled. In addition to setting the hexagon size,
the user also provides a minimum and a maximum territory size. The minimum size corresponds
to the size of a territory in optimal habitat, while the maximum size would be assumed to occur in
the most marginal habitats. Each individual hexagon within the territory map has two attributes:
its score, and its breeding status. A hexagon's score is computed as the arithmetic average of the
weighting values assigned to each of the data pixels contained within it. Thus the scores are real
numbers between zero and the maximum weighting value assigned to any of the habitat categories
present in the GIS imagery. A hexagon's breeding status is a binary attribute that determines
whether or not breeding is allowed at the site. Breeding status is determined based on the
minimum and maximum territory sizes, as described below.
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The territory minimum and maximum sizes do not affect the hexagon areas. Instead, these
parameters govern the degree to which habitat can be shared across hexagon boundaries in an
attempt to allocate the maximum number of breeding sites throughout the landscape. The territory
allocation algorithm proceeds in several steps. Initially, PATCH computes a threshold score using
the equation
, , ., . , , . , minimum territory size
threshold score = maximum weighting value x r-^ .
hexagon size
This relationship defines the threshold score to be that which would be assigned to a hexagon
containing only the minimum territory size worth of optimal habitat. Any hexagon with a score of
at least this threshold value is automatically labeled suitable for breeding. Hexagons that do not
meet this threshold value still have a chance to be classified as breeding sites, and this depends on
the maximum territory size parameter.
PATCH determines the extent to which habitat can be shared across the hexagon boundaries
using the expression
maximum territory size ,
expansion = • r-* 1.
hexagon size
The expansion parameter defines the maximum amount of habitat, expressed in fractions of a
hexagon, that one cell can borrow from its six immediate neighbors. The maximum territory size
is never allowed to exceed seven times the size of a single hexagon, and thus the expansion
parameter can never exceed the area of a hexagon's six neighbors. After identifying every
hexagon that contains enough high quality habitat to qualify as a breeding site, PATCH builds a
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list of all of the remaining sites that have any habitat at all. These hexagons are sorted by score, in
decreasing order. The hexagons are then allowed, in turn, to borrow habitat from their neighbors
up to the limit set by the expansion parameter. Borrowing continues until a hexagon either meet
the suitability threshold, or exhaust its license to infringe on its neighbors. Habitat can only be
lent once, but hexagons that are unable to meet the suitability threshold return any habitat they
have borrowed in the process.
The amount of habitat that can be borrowed depends on whether the lending hexagon is suitable
for breeding. Suitable hexagons are allowed to lend only what they hold in excess of the threshold
score, while unsuitable hexagons can lend all of their habitat. Borrowing begins with the neighbor
having the largest amount of habitat to lend, and concludes with the neighbor having the least.
This process, coupled with the initial sorting of the borrowing hexagons by score, approximately
maximizes the allocation of suitable breeding sites across the landscape. What borrowing really
entails is one hexagon laying claim to a fraction of the total quality of some (or all) of its
neighbors habitat. If the expansion parameter has a value of 2.5, that implies that portions of each
neighbor can be borrowed until a total of 1.5 hexagons worth of the neighboring habitat has been
claimed. The borrowing process is conducted under the assumption that each lending hexagons
habitat is distributed uniformly throughout its area. It is important to note that the process of
borrowing habitat does not change any features of the territory map other than the determination
of which hexagonal sites are deemed suitable for breeding. The additional energetic costs of
defending a larger territory are approximated through the borrowing process since hexagons that
are labeled suitable for breeding by virtue of having borrowed habitat will have lower scores than
those that had sufficient habitat on their own. These lower scores can then translate into higher
mortality rates and lower reproductive output later in the demographic analysis.
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The construction of a territory map from the inn:..; GIS imagery adds a number of desirable
features to the PATCH model. The structure of a territory map is simnie compared to the GIS
imagery from which it was derived. For the purpose, i demographic modeling, PATCH is only
concerned only with each hexagon's score, whether -: not it is suitable for breeding, and who its
neighbors are. The details of habitat patterning within _-uch hexagon arc not important at this level
of analysis, and they are therefore ignored. The hex;._on map may sometimes constitute the final
product of the analysis, or it may serve as an intermeuiae product that can be peer reviewed. The
PATCH model also contains an editor that allows tr.o :ser to alter the territory map. Using this
tool, alternative future landscapes can quickly be dever.mcd, or "what if questions addressing the
consequences of specific habitat modifications can e easily pursued. PATCH also makes it
possible to randomize the placement of hexagons \v.-.;n a territory map, and this feature can be
used to test hypotheses about the importance of articular orientation of habitat across a
landscape.
Demographic xii; ran (ions
PATCH conducts demographic simulations within ;.c territory maps described in the previous
section. The life cycle is modeled as a series of discrete events that take place on a yearly basis.
The model year begins in the summer with a breedin- ulse, which is followed in the autumn by
the mandatory dispersal of the young-of-the-year (hence referred to as "juveniles"). Next comes
over-winter survival, which is followed by the option.; movements of adult animals in the spring,
and finally a census it taken. The process then begins ,.^am with the summer of the following year.
For simplicity, all mortality is collapsed into the sin_-.c evaluation that takes place in the winter.
There is no additional mortality associated with the ir.uvement process. The model also allows the
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landscape to change in time. The user activates this feature by instructing the model to load new
territory maps at different points during a simulation. A new territory map can be installed at the
start of any given year, and this function can be used even if multiple replicate simulations are
being conducted.
Survival and reproductive information is entered into the PATCH model in the form of a
population projection matrix, and its demographic simulations can be thought of as an extension
of the analysis that would be performed using a matrix model. If an entire landscape consists of
optimal habitat, and if breeding sites are unlimited, then PATCH will generate results essentially
identical to those that would be obtained using a projection matrix. However, to the extent that
high quality habitat is limiting, the model results will differ from those of a projection matrix.
PATCH also differs from a matrix model in that it is individual based, and because its survival,
reproduction, and movement modules all incorporate an element of stochasticity. Decisions
regarding survival probability, reproductive output, and movement behavior, are all made on an
individual basis, and they can be significantly influenced by the quality of the habitat contained
within the hexagonal cells that the organisms occupy.
PATCH does not automatically make the assumption that survival and reproductive output
scale linearly with habitat quality. Instead, the user is provided with a set of six generic
interpolation functions that can be used to describe the manner in which habitat quality affects
these vital rates. These functions are linear, logistic, concave, convex, constant, and piecewise
constant (see figure 1). The user is required to provide survivals and fecundities (number of
females per female that survive to the following year) in the form of a projection matrix. It is also
necessary to specify what quality of habitat these vital rates should be associated with. That is,
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PATCH needs to know if these survivals and fecundities that are supplied to the model will be
realized in the best habitat, or the worst, etc. The user then specifies an interpolation function for
survival, and one for fecundity, and then this portion of the model parameterization is complete.
Other than specifying the model organism's movement behavior (discussed below), there are
only a few remaining parameters for the user to define. It is necessary to specify the duration in
years of each simulation, and the number of replicate simulations that will be conducted. PATCH
must also be told where within the territory map to locate the initial population of organisms, and
what age or stage class they are to he assigned to. It is also necessary to specify whether any of the
hexagons (other than at the edges of the GIS imagery) should be treated as reflecting boundaries.
Lastly, a transient period can be specified before which information about the emigration and
immigration into breeding sites will not be tallied (see below).
The movement module
Three different movement routines are available within the PATCH model. In addition, three
distinct rules can be used to specify a level of site fidelity (the likelihood of remaining on an
occupied breeding site from one year to the next). The options for simulating movement include a
directed random walk, selection of the best available site within a search radius, and selection of
the closest available site within a search radius. These movement routines require that the user
specify a minimum and a maximum movement ability in terms of the total number of steps that
can be taken from a hexagon to one of its six neighbors. It is also necessary to place bounds on
how random vs. directed the movement will be when a random walk is taken. This is done
through the specification of a minimum and a maximum turning rate. The options for site fidelity
are termed high, medium, and low. High site fidelity implies that organisms possessing a territory
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will never relinquish it. Low site fidelity implies that every territorial individual will search yearly
for a new site. If site fidelity is set to medium, then the decision regarding whether to remain on a
territory, or to leave it in search for a superior site, is made based upon the quality of the habitat at
the current location. All of these features of the movement module are discussed in detail below.
The movement routine is called twice per year. It is used to drive the movements of adult
organisms prior to the breeding pulse, and is used later to control the dispersal of the year's new
juveniles (the young-of-the-year). The implementation of the movement routine is slightly
different depending on whether juveniles or adults are moving. Every juvenile is obliged to
disperse away from its natal site, whereas adults may or may not elect to move. Decisions
regarding adult movement are based upon the individual's status (territorial vs. floater), the
quality of habitat currently being occupied (for territorial individuals), and the site fidelity
parameter. Juveniles are not allowed to settle until they have moved at least the minimum distance
specified by the user. In addition, until this minimum distance has been traversed, juveniles travel
with the minimum turning rate (i.e. these movements are made as linear as possible, forcing the
juveniles to move away from the natal site). Adults, on the other hand, are not subject to a
minimum movement distance or the restrictions on turning rate imposed on the juveniles. As a
whole, this scheme provides the user with the flexibility necessary to apply the model to a variety
of organisms using only a small number of parameters.
When a random walk is used for the movement routine, individual organisms take a series of
steps from a the hexagon currently occupied to one of its six neighbors. The direction of the walk
is influenced by the quality of the habitat within which the movement is taking place, the
minimum and maximum turning rate, and the direction previously moved. Individuals taking a
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random walk first look to see if any of their neighboring hexagons arc suitable for breeding and
available. If so, then they elect whether or not to settle in that site. This determination is based
upon the site's quality, and upon the site fidelity parameter. The better the quality of the site, the
more likely an individual is to settle in it. Given this, individuals are more selective when site
fidelity is high (they will not get another chance to move) and are less selective when site fidelity
is low (they will always get another chance to move). In addition, individuals become less
selective as their ability to continue searching diminishes. If a suitable, available, neighbor does
not exist, then individuals will select a neighbor to move into with a ucgree of randomness that is
governed by a general tendency to move towards (but not necessarily to remain in) higher quality
habitats, and by the influence of the turning rate parameter.
The turning rate parameter takes on values between 0 and 100%. When the turning rate is 100
percent, the choice of which of a hexagon's six neighbors to move into will be made randomly (in
the absence of decisions to move specifically to a higher quality site). When the turning rate is
zero, an individual will always move in the direction of the previous step, thus producing linear
motion. The user sets the bounds on the turning rate parameter by setting its minimum and
maximum values. However, at any given location in the landscape, the turning rate that is actually
used is derived from the relationship
hexagon score , . ....
turning rate = r2 x (max turning rate - mm turning rate) + nun turning rate.
maximum score & to
This ensures that the turning rate used in any movement decision falls in the range spanned by the
minimum and maximum turning rates, and that this value increases linearly with the quality of the
hexagon presently occupied. The value obtained for the turning rate is then used in the process of
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selecting the next hexagon to move to. The result of this scheme is that movements are generally
more linear in poorer habitats, and they become more random in higher quality habitats. Thus, the
organisms tend to travel quickly through sparse regions of the landscape, and they perform a more
exhaustive search for available breeding sites when they arrive at clusters of suitable habitat. This
behavior is accentuated by the individuals tendency to move into, but not necessarily to remain in,
high quality habitats.
When individuals searching for breeding sites are instructed to select the best, or closest,
available site within a search radius, the movement process takes on a very different set of
characteristics. For juveniles, the search radius becomes the annulus, centered on the current site,
defined by the minimum and maximum movement abilities. For adults, the search radius becomes
the disk with a radius equal to the maximum movement ability. The quality and availability of
every hexagon within the search radius is examined if the best site is to be selected. If the closest
available site is to be selected, then the search radius is expanded iteratively from the minimum to
the maximum until a suitable hexagon is located. In either case, if the searching individual is
unable to locate a suitable site, a random walk is taken.
The behavior of PATCH'S movement algorithm is also controlled by the site fidelity
parameter. Site fidelity governs the probability that a territorial adult will elect to abandon its
territory in search of one of higher quality. In the spring, just before floaters begin searching for
breeding sites, the territorial adults decide whether or not to venture out in search of a better site.
If the site fidelity parameter is low, then every adult will abandon its territory (if one is held) and
search for another site. If site fidelity is high, individuals holding territories remain on them
indefinitely. When the site fidelity parameter is set to medium, individuals will elect to move if the
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site is expected to behave as a demographic sink. The analysis inv< ed in making this decision is
explained in greater detail in the section on model outputs, below.
Individuals in motion will reflect off of the edges of the GIS ; :age. In addition, the user can
force any hexagon to behave like a reflecting boundary as long a, is not suitable rbr breeding.
This feature is designed to allow the user to prohibit movement ^yond a coastline, or over a
mountain range, etc.
Model outputs
The PATCH model produces an array of output information. I 7CH provides the user with a
histogram showing the number of data pixels of each habitat type esent in the GIS imagery. In
addition, the patch counting algorithm produces a table that di: ::ays the area, weighted area,
interior area, and perimeter of each patch in the landscape. The erritory allocation algorithm
produces a table that displays the score, area, weighted area, and b:-,-jding status of each hexagon.
This table also provides the interior area and perimeter, computed n a patch by patch basis, that
happen to fall within each hexagon. These outputs allow the user o build up a broad range of
pattern-based indices of landscape quality, from the simplest r :asures of habitat area up to
sophisticated estimates of the range of breeding site qualities pres-, :n a landscape.
The demographic model produces five additional outputs. T;,J ^rincipal output file contains
the input parameters used, and an array of information for every r;-vacate run and year. The array
of information includes the mean dispersal distance (juveniles i iv) reported as both the total
distance moved, and as the net displacement from the starting pour.. Also included are the sizes of
the floater and breeder populations, and the number of individual LS in each age/stage class. A
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second demographic output file contains the means and standard deviations, taken over each
replicate run, of the total population size, the number of floaters, breeders, and the number of
individuals of each age/stage class. PATCH also tracks the effective aggregate survival rates and
fecundities, on an age/stage class basis, as mean values taken over all of the replicate runs. This
output data can be thought of as consisting of a unique population projection matrix for each year
of a model simulation. The mean population size on an age/stage class basis, taken over the all of
the replicate runs, can be reproduced exactly from this time series of projection matrices using
matrix multiplication. This output data lets the user track the changes in the survival rates and
fecundities, for each age/stage class, taking place through time.
Lastly, the PATCH model has features that both estimate, and then actually track, whether
portions of the landscape function as demographic sources or sinks. This analysis is performed on
a hexagon by hexagon basis, but it is only done for suitable breeding sites. The potential of a
hexagon to function as a source or sink is evaluated by computing the value of the dominant
eigenvalue (k) of the projection matrix associated with the site. The computation of X incorporates
information about the site quality, the survival and fecundity information supplied to the model,
and the interpolation functions that are being used to assess survival and fecundity in hexagons of
arbitrary quality. This information then provides the user with an initial estimate of the
importance of portions of the landscape for the model species.
The PATCH model also tracks the immigration into, and emigration from, each breeding site,
and uses this information to identify effective demographic sources and sinks throughout the
landscape. Specifically, what PATCH does is to increment a counter each time an individual
leaves a breeding site, and decrement the same counter each time an individual enters the site.
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These counters are referred to as "utility values", and a separate utility value is compiled for each
breeding site. The user specifies at what time the utility data should start being collected, and then
the immigration and emigration data is gathered for even,' subsequent year. Individuals in motion
that simply pass through a site produce no net change in its utility value. On the other hand, when
individuals are born in, and subsequently disperse from a breeding site, this produces an
incremental gain in utility. When individuals move into a breeding site, and then die, this produces
an incremental loss in utility.
PATCH'S source-sink analysis is then presented to the user in three files. The first is a table
that displays the score, lambda-value, and utility value for each breeding site. The second and
third files are raster images of the lambda-values and utility scores, respectively, which can be
directly compared to the territory map.
A CASE STUDY
Methods
I developed a case study that exhibits some of the features of the PATCH model described in
the preceding text, but that also addresses the theme of the 1996 AMIGO workshops. The focus of
these workshops was cross-biome comparisons of the consequences of habitat fragmentation.
This case study examined the response of two wildlife species to two types of habitat
fragmentation in two contrasting landscapes. The model species included a "large" organism
characterized by low reproductive output, high survival, and a large territory size. Also modeled
was a "small" organism characterized by high reproductive output, low survival, and a small
territory size. Both of these species exhibited identical habitat affinities. The two types of habitat
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fragmentation mimicked "aggregated" vs. "dispersed" clear-cutting. Aggregated clear-cutting was
approximated by removing large (100 x 100 pixel) squares of habitat. Dispersed clear-cutting was
approximated by removing small (10 x 10 pixel) squares of habitat. Landscapes were either
subjected to the aggregated or dispersed cutting, not both, and the frequency of dispersed cuts was
always 100 times that of the aggregated cuts. A sequence of landscapes exhibiting increasing
degrees of habitat removal was generated by imposing a specific number of cuts, saving the
resultant image, and then proceeding on with additional cuts. The cuts were placed randomly
across the landscapes, and no attempt was made to prevent their overlapping one-another. Areas
that were clear-cut remained in this state for the duration of a model run.
The landscapes used in this study (figure 2) were simply fabrications intended to illustrate
very different types of underlying habitat pattern. Landscape 1 (figure 2) might typify habitats
distributed along an topographic gradient, whereas landscape 2 could characterize a patchy array
of vegetative communities or habitat patterns resulting from intensive management. Habitat utility
indices (HUI), which are relative measures that designate each habitat's suitability for the model
species, were specified for the different categories present in the two landscapes. The category
depicted in black in figure 2 was assigned a HUI of six, the darkest gray color was given a HUI
value of five, and so on down to the habitat colored white, which was assigned an HUI of one.
Clear-cuts were assigned an HUI of zero. Prior to fragmentation, landscape 1 held identical
numbers of pixels of each habitat type. Landscape 2 held similar, but not quite identical, areas of
each of the habitat types.
The larger model species had a territory size that was ten times that of the smaller species, and
each landscape could hold 6960 of the large territories and 68,854 of the smaller territories. The
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minimum and maximum territory sizes were set to 1/2 and 3/2 the r-uxagon size, respectively. The
population projection matrices, and associated lambda values (C..v,veil 1989), used for the two
species were
small organism
large organism
0.74 1.20
0.20 0.50
0.00 0.50
0.50 0.90
larnncia = 1.124
lambuu = 1.123
For both organisms, the interpolation functions for survival and fecundity were set to linear, and
the movement routine used was a random walk. Individuals searcr.inq; for available breeding sites
were allowed to take a maximum of 25 steps from hexagon to hexnnun. Dispersing juveniles were
obligated to move at least 5 steps before settling. The range o: irning rates was set at the
maximum possible (0 - 100%) and the landscape was initialized '. ith every breeding site filled
with an adult. Simulations were conducted for each combination r,f landscape, ore: sm, and
cutting regime (aggregated vs. dispersed) at each of 26 different leveis of habitat fragmentation. In
every case, five replicate simulations were performed, and the results were averaged across
replicates. Utility data, used to identify demographic sources and sinks, were collected only after
the first 100 years so that transient effects of the model parameter!?,aiion could die down.
Because the two model species had identical habitat preferences, and nearly identical values
for lambda, they could be expected to perform equally well in me absence of complications
arising from spatial pattern. Observed differences in the performance of the two species should be
tied to interactions between the landscape patterns, life history atimuites, and the cutting regimes.
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The goal of this analysis was a qualitative examination of the relative importance of each of these
contributions to the overall success of the model species.
Results
Habitat loss was tracked as the percent degradation of a landscape's quality. The equation
used to obtain percent degradation was
pre-fragmentation pixel weights - Y post-fragmentation pixel weights
lUU X ' '"•"• _,
-fragmentation pixel weights
and thus degradation measured the loss of habitat, weighted by the quality of that habitat. Defined
this way, percent degradation served as a unitless metric for making comparisons between
landscapes and disturbance regimes. The two study landscapes were subjected to 25 different
levels of habitat fragmentation, for each cutting regime, plus the original pre-fragmentation
images. Habitat degradation resulting from this fragmentation spanned a range from zero to 64
percent.
Because the sample landscapes could support larger numbers of the smaller organisms than
the larger ones, comparisons of the model results between species were conducted using a relative
measure of population size. The measure that was used for the relative population size was
mean population size in the pre-fragmentation landscape
mean population size in the post-fragmentation landscape'
where the mean values were computed from the last 50 model years of five replicate simulations.
(five replicate simulations were performed for each combination of model parameters reported
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here). 'T/.o standard deviations derived from the five replicate model simulations were small
compared ro the population size. This measure of relative population size provided a unitless
estimate < i the amount by which the populations declined under the fragmentation pressure. The
relative ronulation size parameter took on values between zero and one.
Ik? principal results of the case study are displayed in figure 3. Relative population size was
tracked :.s a function of percent degradation for both landscapes, disturbance regimes, and
species, "nderlying landscape pattern appeared less important than disturbance regime or life
historv rategy in determining the population response to habitat loss. Not surprisingly,
interactions between body size and disturbance regime nlayed a large role in species persistence.
as evidenced by the differential responses of the two species under the dispersed cutting regime.
These results are encouraging because they suggest that cross-biome comparisons of wildlife
response.*. >;o habitat fragmentation may be useful in spite of inherent differences in landscape
pattern.
Estimates of population size, while critical to viability analyses, provide little or no insight
into me : ~atial patterns of habitat use by model organisms. Summary data exhibiting patterns of
habitat i:,-e. if collected at all, are typically presented as rates of habitat occupancy. PATCH is
designed instead to track immigration and emigration rates into breeding habitat, and from this
information it compiles data on demographic sources and sinks (see Model Description, above).
Source/sinK uat;; are arguably superior to occupancy rate information because they better indicate
the imDortance of different localities for the population under study. Source/sink data were
compiieu for each of the model simulations conducted in this study, and these data provide a
visual analogue to the results displayed in figure 3. For the sake of brevity, I examine here only the
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source/sink data acquired from the pre-fragmentation landscapes, and from landscapes that were
degraded by approximately 42%.
The top panels of each of set of three images in figure 4 display the pre-fragmentation
source/sink data associated with the different landscapes and species. The best sources are colored
dark black, while poorer sources and then sinks are displayed in increasingly lighter shades of
gray. Habitats that were not suitable for breeding, or for which immigration and emigration were
exactly balanced (including those that were never occupied) are displayed in the lightest shade of
gray. Immigration and emigration were compiled on a hexagon by hexagon basis, but for the
purposes of constructing figure 4, these hexagons have been collapsed into small squares (this is
done to minimize the disk space necessary to store the images). Hexagons (shown as the little
squares) that, as a result of fragmentation, contained no habitat whatsoever are colored white in
figure 4. Hexagons that experienced some fragmentation, but that still contained some habitat, are
shown in one of the shades of gray (note in particular that the source/sink maps with the large
species and small clear-cuts contain no white areas).
The source/sink images corresponding to the pre-fragmentation landscapes provide baselines
for the evaluation of the source/sink data in the post-fragmentation landscapes. The center panels
in figure 4 display the source/sink data resulting from simulations conducted with aggregated
clear-cutting. Remnants of the pre-fragmentation source/sink patterns can be clearly observed in
the post-fragmentation images exhibiting these large disturbances. This is less true of the
source/sink data resulting from the dispersed clear-cutting (bottom panels in figure 4). There, the
patterns become blurred, and in the case of the large species plus the dispersed cutting, evidence
of the underlying landscape patterns is lost altogether.
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Figure 4 reinforces the notion, derived from the data of I: jure 5, thai underlying landscape
pattern a less critical determinant of the response to fragmentation than are relationships
between ,.ie spatial scales associated with the model species ar.ci the disturbance, figure 4 shows
that the ,;:fect of the large disturbances is to dramatically lower'::: uualhv of some large pieces of
the lanu .i-aoe. while leaving other areas intact. Both the large I..IQ small organisms were still able
to construct many high quality territories in these landscapes and these areas buoyed up the
simulate-j populations even'at high levels of habitat degradation, """he effect of the small
disturbances, however, was to dramatically reduce the likelihood that a territory of high quality
could re constructed anywhere in the landscape. And this effec; Deeame more pronounced as the
territorv ;ze increased. Consequently, the small model species ;'ared more poorly in the midst of
the dispersed ciear-cutting, and the large organisms did the worst ovenul. These results suggest
that, ail ;ncr things being equal, the consequences of habita; ss that is aggregated across a
landscL;:.- nay be less severe than losses that are more unifonr.iy distributed in space. However,
no attc:r.pt was made here to realistically mimic any tyv: or amaropogenic disturbance,
Moreover, in nature, ail other things are never equal and compi; ;;. ns associated with large-scale
disturbances might negate any advantages suggested by this anaivsis.
CONCLUSIONS
The :-ATCH model was designed to help investigators examine the importance of landscape
pattern ;or an array of terrestrial vertebrate species. The model is oased on a population projection
matrix. ..nd it requires the user to specify a minimum of parameters. While PATCH'S life history
module :s simole, its coupling to spatial pattern through CIS imagery adds a great deal of
compie.nuy to • ::a overall model behavior. Landscape categoric- can be assigned different habitat
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#
utility indices, and these in turn affect the survival, reproduction, and movements of the model
organisms. Individuals compete for high quality breeding sites, and this introduces density
dependence and, at times, metapopulation-like dynamics. PATCH'S many outputs allow an
investigator to conduct viability analyses, to examine the differential effects of habitat pattern (or
loss) on individual age or stage classes, and to both predict and track the importance to the
population of specific habitat units through an analysis of demographic sources and sinks.
A case study was conducted that illustrated some of the workings of the PATCH model, and
that examined an issue central to the 1996 AMIGO workshops. The importance of underlying
landscape pattern, types of anthropogenic disturbance, and species life history characteristics,
were examined in the context of a population viability analysis. The results of the case study
suggest that inherent differences in landscape pattern will not preclude cross-biome comparisons
of the effects of habitat fragmentation on certain wildlife species. The results suggest that the
severity of the impacts to wildlife will instead be determined largely by interactions between the
spatial scales of disturbance and the spatial scales important to the organisms responding to the
disturbance. While this analysis is simple and esoteric, it may be useful to investigators designing
better theoretical and empirical studies of biotic responses to landscape change.
ACKNOWLEDGMENTS
The initial development of the PATCH model was supported by USDA/USFS Grant PNW
90-340 to R. J. Naiman, U.S. State Department Grant 1753-000574 to R. J. Naiman, M. G.
Turner, and R. G. Lee, and NSF Grant BIR9256532 to G. Odell, T. Daniel, and P. Kareiva. More
recent work on the PATCH model has been supported entirely by the U.S. Environmental
Protection Agency. I would like to thank Gay Bradshaw and Pablo Marquet for organizing the
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1996 AMIGO workshops, and for developing and editing this book. Discussions with Gay
Bradshaw, Scott Bergen, and Robin Bjork greatly improved the case study. Barbara Marks
developed the two fabricated landscapes that appear in the text. I would also like to thank Paul
Ringold for reviewing the manuscript.
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LITERATURE CITED
Caswell, H. 1989. Matrix Population Models. Sinauer Associates, Sunderland, Mass. USA.
Chen, JM Franklin, J. E, and T. A. Spies. 1992. Vegetation responses to edge environments in
old-growth Douglas-fir forests. Ecological Applications 2:387-396.
Deutschman, D. H., G. A. Bradshaw, W. M. Childress, K. L. Daly, D. Griinbaum, M. Pascual, N.
H. Schumaker, and J. Wu. 1993. Mechanisms of patch formation. Pages 184-209 in S. A. Levin,
T. M. Powell, and J. H. Steele editors. Patch Dynamics. Lecture Notes in Biomathematics 96.
Springer-Verlag, New York, NY,
Doak, D. E and L. S. Mills. 1994. A useful role for theory in conservation. Ecology 75:615-626.
Dunning, J. B., B. J. Danielson, and H. R. Pulliam. 1992. Ecological processes that affect
populations in complex landscapes. OIKOS 65:169-175.
Dunning, J. B., D. J. Stewart, B. J. Danielson, B. R. Noon, T. L. Root, R. H. Lamberson, and E. E.
Stevens. 1995. Spatially explicit population models: Current forms and future uses. Ecological
Applications 5:3-11.
Fahrig, L. 1991. Simulation methods for developing general landscape-level hypotheses of
single-species dynamics. Pages 417-442 in M. G. Turner, and R. H. Gardner, editors. Quantitative
methods in landscape ecology. Springer-Verlag, New York, New York, USA.
Forman, R. T. T., and M. Godron. 1986. Landscape ecology. John Wiley & Sons, New York, New
York, USA.
Gotelli, N. J. 1995. A Primer of Ecology. Sinauer Associates, Sunderland, Mass. USA.
Jensen, D. B., M. S. torn, and J. Harte. 1993. In our own hands: a strategy for conserving
California's biological diversity. University of California Press, Berkeley, California.
Lawton, J. H. 1993. Range, population abundance, and conservation. Trends in Ecology and
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Evolution 8:409-413.
Lefkovitch, L. P. 1965. The study of population growth in organisms grouped by stages.
Biometrics 21:1-18.
Leslie, P. H. 1945. On the use of matrices in certain population mathematics. Biometrika
35:183-212.
McKelvey, K., B. R. Noon, and R. H. Lamberson. 1993. Conservation planning for species
occupying fragmented landscapes: the case of the northern spotted owl Pages 424-450 in P. M,
Kareiva, J. G, Kinssolver, and R. B. Huey, editors. Biotic interactions and global change. Sinauer.
Sunderland, Massachusetts, USA.
Milne, B. T. 1988. Measuring the fractal geometry of landscapes. Applied Mathematics anu
Computation 27:67-79.
Milne, B. T. 1991, Lessons from applying fractal models to landscape patterns. Pages 199-235 //;
M. G. Turner and R. H. Gardner, editors. Quantitative methods in landscape ecology.
Springer-Verlag, New York, New York, USA.
Patton, D. R. 1975. A diversity index for quantifying habitat "edge". Wildlife Society Bulletin
3:171-173.
Pulliam, H. R., J. B. Dunning, Jr., and J. Liu. 1992. Population dynamics in complex landscapes:
a case study. Ecological Applications 2:165-177.
Reid, W. V., and K. R. Miller. 1989. Keeping options alive: the scientific basis for conserving
biodiversity. World Resources Institute, Washington, DC.
Schumaker, N. H. 1995. Habitat connectivity and spotted owl population dynamics. Ph.D.
Dissertation. University of Washington, College of Forest Resources.
Schumaker, N. H. 1996. Using landscape indices to predict habitat connectivity. Ecology
77:1210-1225.
Thomas, J. W., E. D. Foreman, J. B. Lint, E. C. Meslow, B. R. Noon, and J. Verner. 1990. ..
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conservation strategy for the Northern Spotted Owl: report to the interagency scientific committee
to address the conservation of the Northern Spotted Owl. United States Government Printing
Office, Washington, D.C., USA.
Wallin, D. O., F. J. Swanson, and F. Marks. 1994. Landscape pattern response to changes in
pattern generation rules: Land-use legacies in forestry. Ecological Applications 4:569-580.
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FIGURE CAPTIONS
Figure 1. The PATCH model's control windows. The windows used for displaying imagery are
not shown. The separate panel at the bottom is a second view of the panel directly above it, as
indicated by the arrow. Population projection matrices are entered into the array of numeric fields
immediately above the arrow.
Figure 2. The two sample landscapes used in the study. Each image was 2384 pixels wide and
1031 pixels tall. The proportions of landscape 1 in each of the six habitat types were identical,
while they were only roughly equal in landscape 2.
Figure 3. Results from the model simulations showing the population response to habitat loss.
See the text for the definitions of habitat degradation and relative population size. The upper
figure displays the responses of the small model species to habitat loss in the two landscapes,
while the lower figure is for the large model species. The types of anthropogenic disturbance that
were used in the simulations are indicated next to the curves for which they apply.
Figure 4. The source/sink maps derived from the PATCH model. The best sources are colored
dark black, while poorer sources and then sinks are displayed in increasingly lighter shades of
gray. Habitats not suitable for breeding, or for which immigration and emigration were balanced,
are displayed in the lightest shade of gray. Squares containing no habitat are colored white. In
each case, the upper panel is the pre-fragmentation source/sink map, and the bottom two panels
show the results obtained from approximately 42% habitat degradation. The small model species
appear in images A and C, and the large model species are in images B and D. Landscape 1 is
shown in images A and B. while landscape 2 is displayed in images C and D.
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Figure 1
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Figure 4
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• •. • <*•• •
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NHEERL-COR-2188A
TECHNICAL REPORT DATA
(Please read instructions on the reverse before comp'"i:
1. REPORT NO.
EPA/600/A-97/086
2.
4. TITLE AND SUBTITLE
Computer model helps connect landscape structure to population viability
7. AUTHOR(S) Nathan H, Schumaker
9. PERFORMING ORGANIZATION NAME AND ADDRESS
US EPA NHEERL
200 SW 35th Street
Corvallis, OR 97333
12. SPONSORING AGENCY NAME AND ADDRESS
US EPA ENVIRONMENTAL RESEARCH LABORATORY
200 SW 35th Street
Corvallis, OR 97333
5. REPORT DATE
6, PERFORMING ORGANIZATION
CODE
8. PERFORMING ORGANIZATION REPORT
NO.
10. PROGRAM ELEMENT NO.
1 1 . CONTRACT/GRANT NO.
13. TYPE OF REPORT AND PERIOD
COVERED
14. SPONSORING AGENCY CODE
EPA/600/02
15. SUPPLEMENTARY NOTES:
16. Abstract:
A significant need exists for tools that can help resource managers project the consequences of land management for
biodiversity. In an attempt to provide one such tool, a new spatially explicit life history simulator has been developed to aid
researchers exploring the possible influences of habitat pattern on the viability of populations of terrestrial vertebrates. The
model is especially useful for evaluating the consequences for wildlife species of habitat change through time. The model is
discussed in the manuscript, and a case study is presented that illustrates its use. Results from the case study suggest that
relationships between population viability and anthropogenic stressors developed for one region may, in certain cases, be
extrapolated to other very different regions. This research will be useful for investigators involved in the development of theory
linking population viability analysis to landscape structure and anthropogenic disturbance.
17.
a. DESCRIPTORS
Landscape ecology, conservation biology,
viability analysis, simulation model, projection
matrix, anthropogenic disturbance.
18, DISTRIBUTION STATEMENT
KEY WORDS AND DOCUMENT ANALYSIS
b. IDENTIFIERS/OPEN ENDED
TERMS
1 9. SECURITY CLASS (This Report)
20. SECURITY CLASS (This page)
c. COSATI Field/Group
21. NO. OF PAGES: 32*
22. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION IS OBSOLETE
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