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                    Ecological Modelling 132 (2000) 77 – 94

     Modeling landscape functions and effects: a network
    Scott G. Leibowitz a,*, Craig Loehle b, Bai-Lian Li c, Eric M. Preston d
  National Health and En6ironmental Effects Research Laboratory, U.S. En6ironmental Protection Agency, 200 SW 35th Street,
                          Cor6allis, OR 97333, USA
    National Council of the Paper Industry for Air and Stream Impro6ement, 552 S. Washington Street c224, Naper6ille,
                             IL 60540, USA
            Department of Biology, Uni6ersity of New Mexico, Albuquerque, NM 87131-1091, USA
              Department of Geosciences, Oregon State Uni6ersity, Cor6allis, OR 97331, USA


  Landscape functions, including sediment and nutrient trapping, pollutant degradation, and flood control, are often
adversely affected by human activities. Tools are needed for assessing the effects of human activities at the landscape
scale. An approach is presented that addresses this goal. Spatially-explicit ecosystem units and their connections are
used to define a transport network. A linear transport model is a tractable approach to landscape analysis for
assessment purposes. The ability of each unit to provide ecosystem goods and services is considered explicitly in terms
of its place in the network. Based on this simple model, landscape-level effects of impacts to the functioning of a given
ecosystem unit can be calculated. Effects of changes in network structure (due to changes in the flow regime) can also
be assessed. The model allows several useful concepts to be defined, including change in buffer capacity, free capacity,
an ordinal ranking of the relative importance of ecosystem units to overall landscape functioning, and differentiation
of cumulative versus synergistic effects. Utility functions for valuation of landscape function are also defined. The
framework developed here should provide a foundation for the development of analytic tools that can be applied to
assessment and permitting activities. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Cumulative impacts; Wetland assessment; Source/sink ecosystems; Landscape valuation; Network connectivity

                                  CFR 1508.7) recognize indirect and cumulative
1. Introduction
                                  effects, tools for estimating these effects are cur-
                                  rently inadequate. This is particularly so for ef-
  Environmental assessments and permitting have
                                  fects of impacts on wetlands. Wetlands are
traditionally focused on impacts to a single
                                  typically connected by water flow to surrounding
ecosystem unit. Although US regulations (40
                                  watersheds and often to other wetlands or water-
                                  ways. Wetland functions can generally not be
 * Corresponding author. Tel.: +1-541-7544508; fax: +1-
                                  evaluated properly without considering this pat-
                                  tern of connectivity, which is essentially a land-
 E-mail address: (S.G. Leibowitz).

0304-3800/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 0 ) 0 0 3 0 6 - 9
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

scape-level property (Bedford and Preston, 1988;       streams for removal of nonpoint source pollution
Lee and Gosselink, 1988; Johnston, 1994; Bed-         such as sediment (Xiang, 1993). This approach is
ford, 1996).                         fairly easy to use, but the buffer strip concept is
  Landscape functions, including sediment and        narrow in its focus. At the other extreme, fully
nutrient trapping, pollutant degradation, and         detailed, mechanistic ecosystem models (e.g.
flood control (Hemond and Benoit, 1988), are          Jørgensen and Nielsen, 1994; Tim and Jolly, 1994;
often adversely affected by human activities.         Fitz et al., 1996; Feng and Molz, 1997) can be
While the concept of landscape function is simple,      developed. Such models require excessive data
making quantitative statements about impacts to        collection and are essentially research projects
landscape function has been possible only in the       rather than tools for routine analysis of impacts.
context of extremely complex spatial models.         We are not aware of any approaches that fill the
Such models are not suitable for routine applica-       middle ground between excessive complexity and
tion to permitting or impact assessment (Hirsch,       excessive specificity. Our approach is a beginning
1988; Abbruzzese and Leibowitz, 1997; McAllister       at meeting this need.
et al., 2000). We therefore develop a modeling          Because of our interest in ultimately providing
framework for quantifying landscape function,         tools that can be applied to routine management
and for tracing the propagation of impacts over        applications (Abbruzzese and Leibowitz, 1997;
space and time across a landscape. We particu-        Hyman and Leibowitz, 2000; McAllister et al.,
larly seek to make explicit the manner in which        2000), the modeling framework must meet the
changes in ecosystem properties or connectivities       following criteria: (1) It must be simple enough
affect other ecosystem units and overall system        that sophisticated modeling expertise is not re-
outputs.                           quired to use it; (2) it must be possible to estimate
  The model we present describes landscape func-       parameters based on available data rather than
tion as a result of ecosystem interactions and        requiring detailed experiments for calibration; (3)
environmental impacts. Ecosystem units are con-        the model must be analytic to the extent possible,
sidered in terms of their input – output behaviors      to allow calculation of indices without the need
(Lamont, 1995), although our approach also al-        for numeric simulation; (4) output need not be
lows for compartments (e.g. trophic groups)          highly accurate, but only need be approximately
within ecosystem units. The model is general, can       correct, since the model is to be applied in routine
be applied to various landscapes and classes of        (i.e. non-controversial) regulatory applications;
ecosystems, and can include natural as well as        (5) the effect of changes in network configuration
human impacts. Our particular focus is on land-        must be calculable; and (6) the model framework
scape function in terms of the utility of ecosys-       must allow extension of models to more sophisti-
tems (particularly wetlands) as measured by their       cated functionality. The model presented below
effectiveness as buffers, transformers, filters, or      only partially meets these criteria; it still relies on
producers of goods. Thus the model also includes       numerical rather than analytical solution, and its
a component for landscape valuation, in order to       application therefore requires modeling expertise.
focus attention on the societal benefits that         Thus the model is still beyond routine manage-
ecosystems produce. Thus, in this paper we are        ment applications. However, we believe the
not concerned with ecosystem integrity per se,        framework provides a foundation for the develop-
except in terms of the production of ecological        ment of analytic tools at a level of resolution
goods. For example, even polluted and disclimax        appropriate to assessment and permitting activi-
wetlands or riparian forest zones can act as filters      ties. In the following sections, we develop the
for sediment runoff (Peterjohn and Correll, 1984;       model framework, discuss a set of measures that
Cooper et al., 1986; Whigham et al., 1988) and as       can be used to assess the effect of cumulative
buffers for flood waters.                   impacts on landscape function, and provide an
  A GIS-based modeling approach has been used        approach for landscape valuation of these func-
to evaluate the effectiveness of buffer strips along     tions and impacts through the use of utility func-
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94           79

tions. Several examples of how the model           materials through the unit (i.e. exports exceed
can be applied are then presented. We begin first       imports). A sink ecosystem has a positive net
with a series of definitions to formalize our con-       removal (the capacity for removal is greater than
cepts of cumulative impacts and landscape func-        production), thereby reducing the flow of materi-
tion.                             als through the unit (exports are less than im-
                               ports). For a neutral ecosystem, production and
                               removal are equal, or are both zero, so that
2. Cumulative impacts and landscape function         there is no net effect on throughput; corridors
                               and barriers would be examples of neutral ecosys-
  We define a landscape as a compound spatial         tems. Source, sink, and neutral ecosystems need
unit composed of various component ecosystems.        to be defined relative to a particular material,
We consider here a landscape composed of n          since an ecosystem could be a source for one
internal ecosystems. This assemblage is bound by       material and a sink for another. The value
some shared geomorphology (Forman and Go-           of the production or removal of that material
dron, 1986), such as a watershed. Since the land-       then depends on the particular use. For example,
scape need not be a closed system, transfers are       a plant could act as a pest and be harmful from
included to represent ecosystems outside the         an agricultural perspective, but the same plant
boundary via exports and imports. The funda-         could also serve as a food source and thus be
mental ecosystem unit can be considered either a       beneficial from a wildlife perspective. An
cell (unit area) or a polygon. Each ecosystem         ecosystem is a promoter with respect to a particu-
occupies a unique, spatially explicit location        lar material and user if it is either a source of a
within the landscape, and the entire landscape is       material that is beneficial to the user, or a
composed of ecosystems. Each of these can be         sink for a harmful material. Conversely, a demoter
classified into a distinct ecosystem class (e.g.        is either a source of some harmful material
forested, wetland, agricultural, deepwater aquatic,      or a sink for a beneficial one. Thus, sources and
or other), but each cell or polygon of a given class     sinks can be either promoters or demoters, de-
is modeled separately. To quantify landscape         pending on the nature of the material being pro-
function and cumulative effects, we are interested      cessed.
in the spatial configuration of landscape units and        To illustrate these concepts, a landscape con-
their combined interactions and properties.          sists of dry terrestrial ecosystems, wetlands, and
That is, we are most concerned with ecosystem         permanently flooded deepwater aquatic ecosys-
properties that are affected by human             tems such as rivers, lakes, estuaries, or oceans.
alterations of the spatial structure of the land-       Considering their biogeochemical role in the land-
scape as they influence ecosystem processes. A         scape, terrestrial ecosystems generally act as
good example is the interruption of normal hy-        source; wetlands serve as source, sink, or are
drologic function in the Mississippi Delta result-      neutral; and aquatic ecosystems function as sink
ing from levees on the Mississippi River and         or are neutral. These definitions are mathemati-
canals and roads cut into the marsh, which may        cally equivalent to the classification of equilibria
have contributed to massive wetland loss (De-         in systems of autonomous differential equations,
Laune et al., 1983). For proper understanding,        and lead toward our model-building.
this problem must be evaluated in terms of the          In this paper we refer to human changes and
spatial interactions of the many ecosystem units       actions as impacts and the consequences as effects.
involved.                           Three different kinds of ecosystem impacts can be
  Next, we define the following kinds of ecosys-       defined. Con6ersion is the direct loss of an ecosys-
tems, based on the net effect they have on ecosys-      tem through transformation of an area into a
tem throughput. A source ecosystem is a system        different ecosystem class. An example would be
having a positive net production (production ex-       conversion of wetland to agricultural land by
ceeds removal), thereby adding to the flow of         drainage. Degradation of a system does not
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

change the ecosystem class, e.g. the wetland re-       3. A modeling framework for landscape function
mains a wetland; however, ecosystem processes
are affected. For example, the introduction of          A perfectly mechanistic spatial transport model
hazardous materials into a wetland could elimi-        can be extremely difficult to develop. Theoreti-
nate microbes critical for denitrification. Finally,      cally a synergetic approach (Haken, 1983, 1993)
network impacts result from changes in spatial        can provide a general guideline for this problem.
connectivity. For example, a wetland can be hy-        Synergetics deals with systems that are composed
drologically separated from an adjacent wetland        of many subsystems. General properties of the
by constructing a raised road. Conversion of a        subsystems are their nonlinear dynamics as well as
wetland by filling not only transforms the unit        their nonlinear interactions. These systems pro-
into a different ecosystem class but can also         duce spatial, temporal, and functional structures
simultaneously alter network flow patterns.          by self-organization at a macroscopic scale. The
For a given impact to a given ecosystem, the         interdisciplinary approach of synergetic theory
effects include the direct effects to the ecosystem      and the slaving principle (Haken, 1983, 1993)
itself, plus the indirect effects that this change      have demonstrated that the behavior of a complex
causes in other ecosystems. Cumulati6e impacts        system on macroscopic scales is independent of
and cumulati6e effects can then be defined           any details of the microscopic nature of the sub-
as the sum of all these impacts and effects over       systems and their interactions. As a real example
some time and space. Note that impacts and          of this, recent studies have recognized the effects
degradation could be defined more neutrally, so        of catchment size upon the relative roles of hill-
as to include beneficial as well as harmful          slope processes, channel routing, and network
changes. Our purpose here, however, is to provide       geomorphology in the hydrological response of
formal definitions for terms and concepts that         natural catchments (Robinson et al., 1995). For
apply to impact assessment. The model itself is        small catchments, hillslope response is more im-
neutral and can include both harmful or beneficial       portant than network response. With increasing
changes.                           catchment size, response becomes increasingly
  Landscape function can now be defined as the        dominated by the network response. Here we are
net effect of all ecosystems on landscape through-      considering landscape function in terms of the
put of a particular material. Landscape function       coarser spatial scale that is dominated by the
depends on the quantity of sources and sinks,         network response; however, we note that in prac-
their relative strength in removing or producing       tice this macroscopic scale is usually defined sub-
materials, and their pattern of spatial            jectively by the modeler.
connectivity. Landscape-le6el effects are the direct       Given this macroscopic scale, the detailed na-
and indirect effects resulting from some specific       ture of the subsystems becomes unimportant near
impact(s), i.e. the change in the input – output       critical regions of system instability. This result is
configuration of the landscape following an im-        particularly relevant for impact assessment at the
pact. Landscape function and effects may be mea-       landscape scale. In our case, the parameters or
sured relative to different endpoints. For          variables describing the individual parts of a land-
many substances, the effect that interests us is the     scape system are not well known or are not
net output from the landscape unit, such as the        known at all. On the other hand, measurements
sediment escaping a system of wetlands. That is,       on some macroscopic properties of the system can
the output variable is measured at the point where      be performed. Synergetics suggests that directly
it exits the system. In other cases, it is the state     using these measured data is appropriate to char-
level within some units or compartments (e.g.         acterize these macroscopic phenomena in time
toxicant loading in fish). In the context of the        and space. In this context, spatial scale is the
above definitions, we can now develop a frame-         network response-dominated landscape that we
work and models for assessing effects of these        suggested above; the temporal scale is recovery
impacts.                           times for processes controlling particular ecosys-
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94           81

tem functions (Preston and Bedford, 1988), such        precipitation or deposition). Finally, losses by
as water quality improvement, flood control, or         processing can occur. These removal processes, R,
habitat support.                        could include storage, evaporation, harvesting,
  There are still technical problems in directly       and chemical transformation:
using synergetic approaches for impact assess-
                                R= f(Storage, Evaporation, Harvesting,
ment, however, because of the need for abstract
mathematical analysis. We overcome this               Transformation,…)             (3)
difficulty as follows. We are not interested so
                                 Production can occur if an ecosystem produces
much in the self-organizing properties of the sys-
                                an amount of a material. Examples include or-
tem as in the changes resulting from a modest
                                ganic matter and sediment. For illustrative pur-
impact. For example, pond ecosystems may natu-
                                poses, we assume that production, P(i, j ), is at a
rally fill with sediment and eventually become dry
                                constant rate under steady-state conditions, rather
land, but this rate of change is small relative to
                                than being a percentage of the current state S.
potential impacts due to human activity that are
                                Thus for particulate organic matter being input
of concern. Impacts so severe that the system
                                into a wetland, its fate is defined by transport and
must resume self-organization, such as recovery
                                removal processes, as above, but the ecosystem
from a major landscape impact, must be assessed
                                may also produce particulate organic matter for
using traditional, detailed methods. We may
                                export at rate P( j ). The import/export structure
therefore assume that our field description repre-
                                of an ecosystem is shown in Fig. 1. Network
sents a quasi-steady state and thus use a linear
                                transport structure (with removal and production
approximation to the true nonlinear dynamics;
                                processes not shown) is depicted in Fig. 2.
this allows us to use a linear approximation for
                                 This formulation gives the following set of lin-
transport between compartments or ecosystems
                                ear equations for transport and processing:
and transformations within compartments or
                                         n      m
                                dS( j )
ecosystems. For a series of units, not necessarily
                                    = Z( j )+ % I(i, j )− % E( j, k)
on a regular grid, numbered {1,…, n}, we define          dt       i=1     k=1
transfers between ecosystems i and j using the
                                      − R( j )S( j )+P( j )
transfer coefficient h(i, j ). The transfer into unit j
                                             n      m
from unit i per unit time, T(i, j ), is
                                    = Z( j )+ % h(i, j )S(i )− % h( j, k)S( j )
                                           i=1      k=1
T(i, j )=h(i, j )S(i )                (1)
                                      − R( j )S( j )+ P( j )        (4)
and the transfer out of unit j to unit k is
                                where import into the unit, I(i, j ), and export out
T( j, k)= h( j, k)S( j )               (2)
                                of the unit, E( j, k), are equal to the respective
  All transfers are defined as constant percentages      transfer functions (Eqs. (1) and (2)). For a con-
of the state S(i ) at a given time t. If detailed       stant exogenous input Z( j ) for all j, and starting
time-varying data become available, the above         with S( j )= 0 for all j, this linear model
steady-state transfer matrix can be easily adapted       quickly reaches an equilibrium for all S( j ). When
to a time-inhomogeneous one. The transfer matrix        dS( j )/dt =0, we have a steady state solution of
allows for both downhill drainage transport and        S( j ):
cyclic transport, as between tidal wetland units or                 n
                                     Z( j )+ % h(i, j )S(i )+ P( j )
different compartments within a pond. Generally,
h B 1.0 and imports\ exports holds for ecosystem        S( j )=                     (5)
                                         % h( j, k)+R( j )
units that act as sinks (e.g. of sediment) or neutral
ecosystems.                                   k=1

  Exogenous import to the system (forcing) oc-         By working down-gradient, Eq. (5) can provide
curs via an input function Z(i ), which allows         the steady state solution for unidirectional flows.
import to any compartment (as, for example,          When there are bidirectional flows (as in a tidal
                 S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

wetland), steady-state solutions must be obtained         ecosystems at the next level down. We may then
                                  recursively calculate the equilibrium state S for
by running Eq. (4) to equilibrium. Next, consider
                                  each compartment i as
the special case where downslope movement of
                                   Level 1 (farthest up-gradient):
material dominates landscape dynamics. We may
in such cases partition the ecosystems based on
                                        Z1(i ) +P1(i )
their level L as determined by connectivity (Fig.         S1(i ) =            for i= i1,…,im ;
2). Ecosystems within a level do not exchange
                                       %h(i, j )+ R1(i)
                                                   j =j1,…, jn
matter but may each interact with each of the                 j

Fig. 1. Ecosystem unit definition. A unit is connected in space to other units, but can remove or produce substances internally.
Removal can include storage, evaporation, harvesting, chemical transformation, etc.

Fig. 2. Decomposition of a landscape network into a hierarchical transport structure. Levels are gradient defined, e.g. Level 3 is
down-gradient of both Levels 1 and 2. Note that Ti, j = Ei, j =Ii, j.
                                                    :             ;
                S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94                      83

                                                        h(i, j )DZ1(i )
                                 DI2( j )= % DE1(i, j )= %
where the subscript helps identify the level, i is the
                                       i           i
                                                    %h(i, j )+R1(i )
index for the ecological units at level 1 (e.g. i1 =1
and im =4 in Fig. 2), and j is the index for the units                           j
at level 2 that level 1 units connect to. Note that

                                                :              ;
there are only exogenous imports to level 1 ecosys-       For ecosystem k at level 3 this becomes
tems because of their up-gradient position.
                                      Áh( j, k)% h(i, j )DZ1(i ) Â
 Level 2 (for each j ):
                                      à              Ã
      Z2( j )+ %h(i, j )S1(i ) +P2( j )                      %h(i, j ) +R1(i )
                                      à              Ã
            i                                j
                                 DI3(k)= %
S2( j )=                                                               (11)
                                      à              Ã
         %h( j, k) + R2( j )                       %h( j, k)+ R2( j )
                                      Ä               Å
          k                                   k

for i=i1,…,im ; j= j1,…, jn ;                  and so on for each lower level. Exactly the same
                                 approach applies to a change in production sce-
  k =k1,...,ko                    (7)
                                 nario. In cases where levels cannot be determined
etc. for the lower levels, where the indices are the       due to cycles or complex water flow patterns (e.g.
same as in Eq. (6), but k is the index for compart-       a lake that drains out into two different water-
ments at level 3.                        sheds), these same indices can be determined nu-
  Using Eq. (6), we can find changes in any           merically from the changes in steady-state values
ecosystem, DS(i ), and changes in export from any        found by running Eq. (4) to equilibrium.
ecosystem to assess sensitivity of different parts of        For a spatially distributed model such as a
the network. The steady state change in S(i ) (level       watershed, Eq. (4) will yield a qualitatively correct
1) resulting from a change in input Z(i ) is (assum-       (though not precise) storm hydrograph (demon-
ing no change in production):                  strated under examples, below) because the reten-
                                 tion and release of water on each cell creates a lag
         DZ1(i )
DS1(i )=                        (8)    effect. If a contaminant is not processed or re-
      %h(i, j )+ R1(i )                  moved, it will continuously build up and non-equi-
                                 librium will be established. This lack of

The change in export from this unit to unit j at the       equilibrium is a measure of impact because it
next level (level 2), is                     means that the system processing capacity has
                                 been exceeded. For such nonlinear behaviors of
                  h(i, j )DZ1(i )
DE1(i, j )= h(i, j )DS1(i ) =             (9)    the system, we could define the intermediate or
                %h(i, j ) +R1(i )        alternate steady-states (such as a quasi-metastabil-
                                 ity) and threshold of irreversibility. In that case, we

  We may use Eq. (9) to obtain a more general          can still use the above set of equations to approx-
expression for tracing effects of changes in exoge-       imately describe them by defining the threshold or
nous inputs on downstream input loadings. Con-          switching step functions of the parameters of the
sider a change in input DZ1(i ) to all ecosystems i       above equations.
at level 1. From each ecosystem i in level 1,            This same formalism can be extended to the
material is distributed potentially to each ecosys-       compartments within an ecosystem. Compart-
tem at level 2, and so on to the third level. Each        ments within a unit are modeled at steady state
path transfers remnants of the initial change in         with linear transfers. For example, for a pond we
input as a cascade. Assuming that exogenous           can model nitrogen as the dissolved N, phyto-
inputs and production at all lower levels remain         plankton N, and fish N compartments (Fig. 3).
constant, the Z and P terms drop out for these          Transfers between all of these compartments can
levels. After equilibration with the new input load-       occur, including cycles such as nutrient leakage
ings Z1(i ) across level 1, we may find the total         from phytoplankton and excretion from fish. The
change in input loading to ecosystem j at level 2        ecosystem connections to other units can be based
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

        Fig. 3. Internal structure of a pond ecosystem unit, showing imports, exports, and losses.

on just transfer of dissolved N, or may include the        Determining the effects of error propagation on
transfers of portions of all compartments (e.g.         model results is fairly straightforward. If uncer-
phytoplankton and fish washing downstream to           tainty exists in an input loading, this uncertainty
the next pond). Because of the cycles in such a         can be treated as if it were a perturbation to the
                                import, DI. The effect on downstream compart-
system, Eq. (4) must be solved to equilibrium to
find steady-state compartment values, rather than        ments or system output can then be calculated as
Eq. (5) or Eq. (6).                       discussed above. If uncertainty exists on transfer
  Similar to some concepts developed in wetland        rates or processing rates at a particular ecosystem,
hydrology (Mitsch and Gosselink, 1993; Kadlec,         this uncertainty will diffuse down-gradient ac-
1994; Boyd, 1995), we can define the residence          cording to the connections that exist. The magni-
time or retention time (tr) of a substance in          tude of the effect of this uncertainty will depend
ecosystem j based on the parameters defined in
                                on how much of the total change at a given level
Eq. (4):
                                passes through the particular ecosystem (see dis-
          S( j )
tr( j )=                            cussion of centrality below). For uncertainty that
     %h( j, k)S( j ) +R( j )S( j )             exists across the network, one can make multiple
                                random draws from expected parameter distribu-
                                tions and calculate a distribution of expected out-
   =                    (12)
                                puts. If uncertainties in a given parameter in each
     %h( j, k)+R( j )
                                ecosystem are randomly distributed, then positive
where units are mass for nutrients or other quan-
                                and negative deviations will tend to cancel as one
tities, or volume over inflow rate for water.
                                moves downstream, and the system output uncer-
  The above framework for transport and trans-
                                tainty will be less than the individual range of
formation of materials can be usefully applied to
                                values. If, on the other hand, error in a parameter
floodwater retention and release, sediment trans-
                                estimate will be common to all ecosystems (e.g. if
port and filtration, nutrient and organic matter
                                a single estimate for a parameter is used across
cycling, and contaminant transport and transfor-
                                the network), then the extent of output uncer-
mation. It provides a useful basis for calculation
                                tainty will be a multiplicative function of the
of several indices and can set the stage for more
detailed simulations (if necessary).              number of levels.
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94            85

4. Indices of landscape function               or removes much of the input signal. If BB 1,
                               then the system is a source and it amplifies an
  Given the above modeling framework, we next        input signal, such as when the ecosystems are
develop several measures of landscape function.        producers of the item in question. What we may
Included are the definition of buffering capacity, a      particularly wish to know is how the system re-
measure of free capacity, an index of spatial sensi-     sponds to a change in a parameter of a unit or to
tivity, and a re-examination of the concept of        a change in the type or connections of a unit or
cumulative impacts.                      units. We are thus interested in the change in
                               buffering capacity, which may be positive or nega-
4.1. Buffering capacity                    tive, as a consequence of the impact or change in
                               question. This will simply be
  For practical purposes, we often define ecosys-
                               DB = BN − BI                  (14)
tem function in terms of the ability to process
some material, such as removing sediment or nu-        where the ‘N’ and ‘I’ subscripts refer to nominal
trients, degrading a contaminant, or retaining        and impacted conditions, respectively. Buffer ca-
flood waters. In all of these cases, we are inter-       pacity should be maximal for an undisturbed
ested in how the system processes an import to        landscape, at least for factors such as nutrients,
create an export, such as downstream nutrient         water, and sediment, though not necessarily for
levels. We may also be interested in the state of a      the processing of contaminants. We may consider
system, such as when eutrophication is the re-        as an example a river floodplain which evolves by
sponse variable of interest. For a linear model, the     geological processes to retain water and sediment.
system export is a linear function of import. We       This optimal buffer capacity assumption applies
may use ecological buffering capacity (Jørgensen       strictly only to mature landscape units, not to
and Mejer, 1977) to characterize the input/output       those undergoing rapid uplift and erosion, for
or input/ecosystem state relationships. The con-       example. Certain human alterations of this land-
cept of buffer capacity is widely used in chemistry,     scape (e.g. putting in levees, filling wetlands) can
particularly as a measure of the ability of a solu-      reduce buffer capacity. In those cases
tion to meet the addition of base or acid with
                               BN \ BI \ 0    ƒDB \0
only minor changes in pH. Buffer capacity is                                 (15)
given by
                                 For landscapes considered as processors of ma-
  (F DF                           terial, buffering capacity and changes in buffering
B=                         (13)
  (C DC                           capacity are very useful indices.
where B is buffer capacity, F is the input or
loading (forcing function), and C is the output or      4.2. Free capacity and input/capacity ratio
state variable displaced by the forcing function.
The buffer capacity describes the amount of load-        Free capacity is the capacity of the landscape
ing (e.g. input of organic matter, nutrients or        for absorbing (retaining or removing) a substance
toxic compounds) necessary to cause a unit          above the level currently absorbed, and can be
change in a state variable affected by the loading      related to ecological buffer capacity. In the air-
(e.g. the steady-state concentration of nutrients in     plane analogy of Ehrlich and Ehrlich (1981), free
some compartment or in system export). Because        capacity would be the number of rivets that can
the model is linear, B is clearly defined, for any       be lost without compromising wing integrity. If
particular spatial configuration, by the slope of       we look at free capacity strictly as the ability of a
the input versus output response. For B \ 1 the        system to absorb more input without a change in
system dampens input signals for the particular        output, then free capacity exists only when ecosys-
output signal being measured (i.e. the system is a      tem components completely remove some sub-
                               stance. In this case B= and thus buffering
sink). This corresponds to a process that degrades
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

                               4.3. Spatial sensiti6ity/redundancy
capacity and free capacity are each calculable
under different sets of conditions. For example, a
change in sediment load entering a very large          It is often useful to identify the extent to which
wetland will not affect the export of sediment        some points or bottlenecks in the landscape are
from the wetland over a very wide range of          more critical than others for landscape function-
changes in inputs because most of the sediment is       ing. Such spatially critical points require special
deposited near the edge where it enters. We can        examination in a landscape-scale assessment. An
model this using our framework by setting           explicit analysis of network structure like that
R(i ) = 1 for sediment below some threshold. In        presented here is essential for identifying this
some cases, of course, this may appear to apply        property. With respect to materials that can satu-
when we discretize the effects of interest. For        rate the system, such as sediment input, the order
example, we may group lakes into oligotrophic,        of flow critically determines which ecosystem
mesotrophic, and eutrophic classes, in which case       units will become saturated first and are thus the
an ecosystem can be said to have free capacity for      most sensitive. We have several ways to measure
an input (nutrient) as long as it stays within a       spatial sensitivity of a landscape network and to
category. However, in reality the ecosystem does       create a spatial sensitivity map for assessment and
respond to each addition of nutrients by changing       decision making. First, we can define a network
state in a continuous way, so there is only free       sensitivity ratio (NSR) as the ratio between
capacity with respect to our discrete classification      change in input loading to a compartment j at
of condition.                         level L to the change in total input loading at the
  Rather than looking at free capacity in terms of      up-gradient level L− 1:
not changing the performance of the system, as
                                        % DIL (i, j )
above, we may look at it relative to some regula-
tory standard. In this case we may say that free       NSRL ( j )=                   (16)
                                       % DZL − 1(i )
capacity exists as long as we do not approach the
regulatory threshold (measured at some output                 i

point or at points within the landscape). In this       which takes into account both removal processes
case we may have both changes in buffering with        and pathways of distribution between levels.
respect to absolute system state and free capacity      Within a level, we may also compute centrality, C.
with respect to the regulatory standard. We may        For a given change in input loading, centrality is
also compute the input/capacity ratio, which tells      the ordinal rank, within a level L, of the total
us how close we are to an input level that will        change in flow passing through each ecosystem:
exceed the given regulatory threshold (ratio\ 1)
                               CL ( j )= rank[Dh(i, j )SL (i )]        (17)
or cause a change in state. Finally, we may define
free capacity with respect to ecosystem utility (see       The unit with the highest change in flow has the
below).                            highest centrality. Centrality can be computed
                               using only DZ as input. These two indices show
  The concept of free capacity has several uses.
For a network cascade where a substance is se-        spatially and in terms of network level which
quentially degraded or removed, free capacity is       ecosystems are most subject to excessive loadings
governed not just by R, but by the number of         or impacts, and which therefore may need more
levels. We can therefore test the effect of reducing     study or protection. We may also use the change
                               in buffering capacity DBi, defined earlier, as an
or increasing the number of levels on free capac-
ity. For example, damming a river creates a se-        index of spatial sensitivity. We can map these
quential series of lakes, which may increase         values to locate spatial sensitivity at a landscape
the free capacity for sediment removal. As an-        scale. We can also compare current loading of
other example, we may use the free capacity con-       ecosystems with their maximal capacity (e.g. re-
cept to identify the best location for restoring a      tention or dissipation capacity). If the current
wetland.                           loading is greater than their maximal capacity, the
               S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94            87

ecosystem unit becomes a bottleneck in the land-         In contrast, a synergistic effect occurs when
scape. This allows us to locate ecosystems acting as      impacts have disproportionate effects. An example
bottlenecks, whose remediation would most im-         is the increased variability in river water levels as
prove system performance.                   flood plain area is reduced (Lee and Gosselink,
  Redundancy exists if the removal of a unit does       1988), indicated by the nonlinear response of effect
not significantly change the output from the land-       to impact. In this case, buffer capacity is reduced.
scape as a whole. For a simple cascade, in which        Operationally, if the magnitude of effect of a unit
each unit dissipates a constant percentage of its       of impact increases with the level of past impacts,
input, there is no redundancy, and each unit is        then a synergistic effect is indicated.
equally important. If, however, units can remove         We may use this approach to identify effects that
all of an input (e.g. sediment) but can become         are cumulative in time. From a steady state analy-
saturated, then there may be considerable redun-        sis (as Eq. (5)), cumulative effects cannot be ob-
dancy as long as the upper-level units do not         served. But it often happens that it may take a
become saturated or filled. A cascade of ponds         considerable time to reach a steady state condition,
with respect to sediment approximately meets these       such that a change over time may be observed. In
conditions because most of the sediment settles out      the field, the observation of continued directional
in the first pond. We may also view redundancy         change may be combined with the steady-state
with respect to unit conversion. If removal of a unit     estimate from the landscape analysis to estimate
results in redistribution of its inputs to other units,    the eventual condition that might be reached by a
then redundancy exists as long as the maximum         system and how long this might take. System
dissipation rate or retention capacity of these units     behaviors can be indicators of cumulative effects in
receiving increased input is not exceeded. Thus, the      time. For example, after an apparent functioning
sensitivity of a single unit can be a function of the     of a system for removal of a substance, a unit may
level of input, and the degree of redundancy can be      become saturated, and the overall landscape will
a function of both level of input and time.          become more leaky (exports will increase). This
                                process can be modeled using our linear approxi-
4.4. Cumulati6e and synergistic effects            mation (Eq. (4)) by incorporating a threshold for
                                the capacity of an ecosystem compartment (e.g.
  It is now possible to develop an analysis of        phosphorus retained by vegetation) or for a toxic
landscape function in terms of cumulative effects.       substance removal rate.
Not all landscapes exhibit landscape function for         We may further characterize cumulative
all pollutants or resources of interest. For an air      effects as follows. When we have characterized
pollutant like ozone, induced damage occurs at all       the inputs, transfers, and transformations of a
points based on various factors, but damage at one       landscape, it may occur that some substance of
point may not cascade through other ecosystem         interest does not achieve steady state levels in all
units. True landscape function requires connec-        compartments. A contaminant may slowly
tions by which flows of material between units         build up or sediment may begin to fill a reservoir.
affect production, or by which material is pro-        We cannot calculate a change in buffer capacity
cessed (dissipated or retained) as it flows between       because this calculation is based on achieving a
units. Cumulative effects can be either additive        new steady state. Clearly, with time the effect
(linear) or synergistic (nonlinear) (Beanlands et al.,     continues to worsen (not necessarily linearly) as the
1986). For land conversion on an upland from          substance builds up. What we can do in this case
forest to other uses, effects on timber production       is consider some standard or threshold that we
are cumulative but not synergistic, because an area      wish to avoid exceeding (eutrophication, fish kill
removed from production is independent of other        by metals, pond filling by sediment, etc.). We may
areas (though there may be synergistic effects on       then calculate the time to failure tF using the
water quality downstream from the deforested          dynamic version of the model, Eq. (4). For a
areas).                            hydropower dam, we might consider a tF due to
                  S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

sediment filling of 200 years to be acceptable,            substrate for these plants. However, sediment
because the dam will have been a profitable in-            could be harmful to submerged aquatic vegeta-
vestment over this span, but tF =20 years is not           tion, because it could decrease ambient light levels
acceptable. The tF considered acceptable will de-          and therefore reduce productivity. Although
pend on the seriousness of the threshold and the           ‘target use’ can be defined broadly to refer to a
ease and cost of a remedy should the limit be            particular population or even some specific use by
exceeded. Retention time (Eq. (12)) influences our          that population, we generally consider the utility
estimate of the seriousness of surpassing a             of a given material with respect to a particular
threshold for toxic substances that have built up          ecosystem class.
(consider naturally flushing rivers vs. aquifers, for          The utility of a material is also concentration-
example).                              dependent. For example, trace metals are essential
                                   nutrients at low concentrations, but they can be-
                                   come toxic at higher levels. Because our approach
                                   deals with throughput of materials, we specifically
5. Landscape valuation
                                   consider material fluxes. Thus, our formulation
                                   depends on the material, its flow rate, and the
  Any discussion of landscape benefit or harm
                                   target use.
must consider not only the specific material but
                                    The benefit or harm of a specific material for a
also the target use. For example, the addition of
                                   particular use can be described by a utility func-
sediment to a small lake can benefit emergent
                                   tion (Fig. 4). Consider, for example, the utility of
aquatic vegetation, because this material provides
                                   river discharge with respect to humans. The nor-
                                   mal range of channel flow provides benefits such
                                   as water supply, dilution of pollutants, and ade-
                                   quate draft for navigation. During drought years,
                                   low discharge becomes harmful, because pollu-
                                   tants are concentrated, water supplies are limited,
                                   and navigation is hampered. Similarly, discharge
                                   also becomes harmful at flood stage because of
                                   damage to property and life (Fig. 4a).
                                    Similarly, a utility curve for application of ni-
                                   trogen fertilizer might show a linear increase in
                                   benefit to a nitrogen-limited plant until some
                                   other nutrient becomes limiting. At that point,
                                   benefit remains constant in spite of further in-
                                   creases. Finally, nitrogen applications can become
                                   so high that they damage vegetation, and thus the
                                   nutrient becomes harmful (Fig. 4b).
                                    Utility curves for hydrology and many different
                                   nutrients can be derived with existing informa-
                                   tion. For example, the discharge utility curve
                                   could be constructed from economic data such as
                                   navigation and flood damage records, and nutri-
                                   ent curves could be produced from existing nutri-
                                   ent uptake data. Derivation of a curve that
                                   describes the utility of a biological population is
                                   much more difficult. One possible scenario is that
Fig. 4. Hypothetical utility functions for valuation of land-
                                   at intermediate numbers a population provides
scape-level effects of impacts, for hydrologic (a), water quality
                                   moderate benefit, while at larger numbers the
(b), and habitat support (c) functions.
                S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94            89

                                  Since utility is flow-dependent, ‘beneficial’ and
                                 ‘harmful’ are relative terms. However, we can
                                 characterize a material as marginally beneficial or
                                 harmful by referring to the slope of the utility
                                 function, dU/dJ for change in flow J. Thus, in-
                                 creasing the flow of material is beneficial if the
                                 slope of the utility function is positive; if the slope
                                 is negative, the material is marginally harmful. A
                                 slope of zero means that the material has no
                                 marginal utility, because the function neither in-
                                 creases nor decreases.
                                  We now consider the effect of an ecosystem on
                                 the utility of a material. In this case, we are
                                 specifically interested in the effect an ecosystem
                                 has on throughput, i.e.
                                 dJ( j )
Fig. 5. Relationship of promoter and demoter ecosystems to
                                     = E( j, k)− I(i, j )           (20)
their status as sources or sinks.                 dt
                                  From our previous definitions of source and
population becomes a pest and is harmful. As the
                                 sink ecosystems, export is greater than import for
population size is reduced, however, its utility
                                 source ecosystems, and thus dJ/dt \ 0; for a sink
increases because of its rarity (Fig. 4c). The con-
                                 ecosystem, dJ/dt B0 since exports are less than
tribution of biodiversity to ecological function is
                                 imports. Given the effect of an ecosystem on
poorly understood, and thus utility curves for
                                 material throughput and the slope of the utility
specific populations are at this point a matter of
                                 curve, we can now provide the following formal
                                 definitions based on our earlier discussion of pro-
  The utility of a material to a particular ecosys-
                                 moter and demoter ecosystems: An ecosystem is a
tem class is given by the utility function, U[Ji,
                                 promoter with respect to a given material flow J
el (c)], where Ji is the flow of material i through
                                 and with respect to a particular ecosystem if it is
ecosystem l of class c, el (c). Total ecosystem
                                 a source of that material (dJ/dt \ 0) and if the
utility is obtained by summing over all i materials
                                 marginal utility of that material with respect to
for ecosystem l:
                                 that use is beneficial (dU/dJ \ 0), or if the ecosys-
                                 tem is a sink of that material (dJ/dt B 0) and the
U(l) =%U[Ji, el (c)]                 (18)
                                 marginal utility of that material is harmful (dU/
                                 dJB 0). Similarly, an ecosystem is a demoter with
Total landscape utility U(L) is then the summa-
                                 respect to a particular material and use if it is a
tion of the above equation for all n ecosystems:
                                 source of a material with a negative marginal
                                 utility (dJ/dt \ 0 and dU/dJ B 0) or if it is a sink
U(L)= % U(l)                     (19)
                                 of a beneficial material (dJ/dt B 0 and dU/dJ \

 This analysis assumes that we have formulated         0). This classification is illustrated in Fig. 5. A
our utility functions so that they are additive (are       neutral ecosystem is neither a promoter nor a
on a comparable scale of value). Agreement on          demoter with respect to a given material, because
this can be reached within an organization or by a        either E( j, k)= I(i, j ) or both=0.
group, but different groups often disagree on            From this, we can see that free capacity could
valuation of different goods, posing an unsolved         also be defined alternatively as the difference be-
problem in general for comparing the summation          tween the current level (of say sediment or nutri-
of multiple impacts or impacts on multiple re-          ent input) and that level at which the marginal
sources or goods.                        utility switches from positive or zero to negative.
                 S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

6. Examples                            onto an initially dry land area was simulated for
                                 five timesteps. The resulting flow at the juncture
 It is useful to illustrate the approach presented        of the two valleys is a storm hydrograph (Fig. 6).
here with case studies. The purpose of the exam-         Thus dynamic landscape responses governed by
ples is to show that these simple models are           hydrology such as sediment transport (Heede et
capable of modeling serious phenomena and can           al., 1988) are within the scope of this approach.
capture critical aspects of ecosystem impact. It is        This linear formalism can also be extended to
not intended that they necessarily provide high          model changes in hydrologic response using the
accuracy; this may require more detailed mecha-          Unit Hydrograph for total streamflow (Post et al.,
nistic simulation (e.g. Reckhow and Chapra,            1996), which is a linear compartment approach
1983; Mauchamp et al., 1994; Feng and Molz,            except for the evaporation dependence on
1997); the model is meant to serve in cases where         temperature.
such detailed simulation is not desired or feasible,
                                 6.2. Example 2: landscape effects of wetland loss
and where an approximation is acceptable (Ab-
bruzzese and Leibowitz, 1997).
                                   An important problem at the landscape scale is
                                 effects of wetland loss by conversion. Wetlands
6.1. Example 1: storm hydrograph
                                 have been shown to be important filters for sedi-
                                 ment and nutrients (Osborne and Wiley, 1988;
  It is natural to ask if a linear transport model is
                                 Whigham et al., 1988; Phillips, 1989; Johnston et
realistic. Whereas Eqs. (5) – (7) provide equilibria
                                 al., 1990; Detenbeck et al., 1993; Gale et al., 1994;
values for mass and transfers between units, Eq.
                                 Gilliam, 1994). We may apply the methods devel-
(4) can be used for dynamic cases. A watershed
                                 oped here to this problem. We take as an example
was modeled to test for the ability to produce a
                                 denitrification in a landscape. Denitrification is an
storm hydrograph. A Y-shaped stream was mod-
                                 important process for removing excess nitrogen
eled. Each arm of the Y was divided into two
                                 resulting from agricultural activities and urban
reaches. On each, the land along each side was
                                 green area fertilization. Denitrification is more
modeled such that 10% of the water on it ran off
                                 rapid under the anaerobic conditions of a wetland
into the stream at each timestep. Above each land
                                 (Barber, 1984; Hanson et al., 1994). A single
unit, a higher land unit drained down in a similar
                                 watershed was modeled. The water from two up-
manner onto the streamside units. A pulse of rain
                                 land units runs off onto two wetlands which feed
                                 into a further wetland which acts as the headwa-
                                 ter for a stream (Fig. 7). As this stream flows, it
                                 passes through bordering wetlands, with 40% of
                                 stream flow passing through these wetlands. The
                                 upland areas lose 30% of their N per timestep by
                                 washout downslope (rate exaggerated for illustra-
                                 tive purposes) while wetland areas lose only 10%
                                 per timestep. Streams lose 100% of their current
                                 N downstream per timestep. Denitrification oc-
                                 curs at a rate that is higher on the wetland
                                 (0.2/timestep) than on the upland (0.1/timestep).
                                 A nominal atmospheric input of 0.2 N/timestep
                                 per unit area is assumed. For this model, Eq. (5)
                                 or Eqs. (6) and (7) can be used to calculate
                                 steady-state values for units 1–5 and 10 by work-
                                 ing downslope from 1 and 2 (1 and 2 form level 1,
Fig. 6. Simulated storm hydrograph using simple linear model
                                 etc.). However, the downstream units exchange N
for a Y-shaped stream watershed.
                 S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94            91

                                  ing capacity has gone down from 19.25 to 8.56 or
                                  by 10.7. This drastic decrease in buffering capac-
                                  ity measures the impact of wetland conversion in
                                  the context of this particular landscape. While it
                                  should be kept in mind that this analysis is some-
                                  what qualitative, and this particular example is
                                  really only loosely based on the specific processes
                                  of denitrification, the analysis approach can easily
                                  be refined with better parameter values without
                                  losing the simplicity of the analytic approach.
                                  Further, this type of analysis provides a basis for
                                  deciding whether more detailed mechanistic mod-
                                  eling is warranted. As mentioned above, the cal-
                                  culated changes in output can also provide a basis
                                  for calculating utility functions.
                                   It is possible to calculate residence times from
                                  this model using Eq. (12). The residence time, the
                                  mean time a unit of N resides in an ecosystem, is
                                  2.5tr for the uplands, 3.33tr for the lowlands, and
                                  0.71tr for the lower stream reaches, where tr
                                  reflects the time step for the rate constants in the
                                  model (i.e. if parameters were d − 1 then tr = d).
                                  Retention time is a useful parameter for depicting
                                  where in the landscape a unit of a substance
                                  spends more or less time.
Fig. 7. Simple watershed used in illustrating landscape model.

                                  6.3. Example 3: sediment retention by wetlands
back and forth between the stream and the bor-
dering wetlands, so Eq. (4) must be solved by
                                   An example based on sediment retention will
simulation to steady state. Under this nominal
                                  illustrate the concepts of free capacity and input/
scenario, the watershed N output rate (level in
                                  capacity ratio. The same watershed is used for
stream unit 12) is 10.4. We may calculate the
                                  illustration (Fig. 7), with the same parameters for
buffer capacity by adding an N input to ecosys-
                                  transport. Upland watersheds amplify sediment
tem 1, as would occur during farming, of 50 N
                                  input by a factor of 1.3, as producers of sediment.
units per step. The new equilibrium watershed
                                  Wetlands reduce sediment by capturing it and
output rate is 13.0. Thus an input of 50 units of N
                                  consolidating it. If the input is below a threshold,
increases N output by only 2.6 units due to deni-
                                  all sediment input is captured and consolidated.
trification, giving a buffer capacity of 50/2.6 or
                                  Above this level it builds up and the excess is
B= 19.25. This quantifies the extent to which the
                                  released at the same rate as water (0.1/step). The
landscape system processes N.                   result of this model, which is still linear except for
  We may now quantify the effect of wetland            the threshold effect, as sediment loading is in-
conversion. Unit 3 was converted to upland by           creased, is shown in Fig. 8. Up to about 96 input
altering its parameters. The new output N level is         loading, no sediment can be detected in the export
12.22, showing a decrease in N processing (in-           (level in stream unit 12). Above this point, export
creased leakage) over the nominal case. When            increases linearly. At zero input, the free capacity
buffering capacity is now tested by increasing N          is 96. At say 60, the input/capacity ratio is 60/96
input to unit 1 as before, the new output level is         or 63%. Note that at the point where free capacity
18.06, which gives B = 50/5.84 = 8.56. The buffer-         is zero, and in fact all the way past the values
                  S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94

evaluated, the lower reach streamside wetlands           also developed for cumulative impacts and land-
have not exceeded their filtration capacity and are         scape function that is clearly tied to the landscape
still consolidating all sediment inputs they receive.        formalism. This leads to the development of a
We see in this case that a very simple threshold-          number of landscape indices that should be useful
type process can be incorporated into the model           in evaluating impacts to landscape function.
with only a slight change, leading to the ability to          Second, an approach for landscape valuation is
study free capacity and threshold effects.             presented that is based on landscape function.
                                  Rather than relying on economic (Stevens et al.,
                                  1995; Costanza et al., 1997) or energetic
7. Discussion                            (Costanza, 1980; Odum, 1995; Brown and
                                  Herendeen, 1996) considerations to define a mar-
  The work presented here defines landscape             ket or inherent value, this approach incorporates
function specifically in terms of the net effect           the predefined and often subjective values of any
ecosystems have on landscape throughput. Im-            party that is benefited or harmed by landscape
pacts are similarly considered in terms of the net         functions. These parties can be defined broadly to
change they cause in landscape throughput. A            include particular groups of people or agencies,
main feature is that functions and effects depend          specific animal or plant populations, or — as we
not only on the number and magnitude of sources           demonstrated in our formulation — even whole
and sinks, but also on their network connectivity.         ecosystems. We believe this is an appropriate way
Thus our approach allows landscape function and           of assessing landscape values in situations where
impacts to be analyzed in a specifically spatial           different stakeholders can have diametrically op-
manner. The key to doing so is the use of a             posing views of the same resource.
network transport formalism that focuses on              Third, the approach is general enough that it
transformation and processing functions per-            can be applied to any kind of landscape flow.
formed by individual ecosystems.                  Although we used hydrologic and water quality
  Although network-based approaches are not            examples in developing and demonstrating the
new (e.g. Finn, 1976; Ulanowicz, 1980; Aoki,            model, the concepts can also be applied to biolog-
1992; Higashi et al., 1993; Patten and Higashi,           ical flows. In that case, import, production, re-
1995), this particular application makes several          moval, and export would represent immigration,
contributions. First, the framework provides a           birth, death, and emigration, respectively. Source
unified formalism for considering impacts within           and sink ecosystems would then be interpreted in
a landscape context. A standardized vocabulary is          much the same way as elucidated by Pulliam and
                                  colleagues (Pulliam, 1988; Pulliam et al., 1992): a
                                  sink ecosystem has insufficient reproduction to
                                  maintain a viable population, yet the population
                                  can persist because of continual immigration from
                                  source ecosystems (which produce a net
                                    Finally, we have made progress in developing a
                                  ‘middle ground’ approach that provides a founda-
                                  tion for developing tools that can be applied to
                                  permitting and assessment activities. We simplify
                                  our formulation by using a ‘quasi-steady state’
                                  linear approximation and assuming unidirectional
                                  flows between levels. If these assumptions cannot
                                  be met, results can still be simulated. Although
                                  landscapes are certainly dynamic and not at
Fig. 8. Illustration of how free capacity can influence sediment
                                  steady state, this framework would be valid so
                 S.G. Leibowitz et al. / Ecological Modelling 132 (2000) 77–94                 93

                                  Bedford, B.L., 1996. The need to define hydrologic equivalence
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