Oikos 119: 1136–1148, 2010
doi: 10.1111/j.1600-0706.2009.18104.x
© 2009 The Authors. Journal compilation © 2010 Oikos
Subject Editor: Tim Benton. Accepted 1 September 2009
Rapid development of indirect effects in ecological networks
Stuart R. Borrett, Stuart J. Whipple and Bernard C. Patten
S. R. Borrett ([email protected]), Dept of Biology and Marine Biology, Univ. of North Carolina Wilmington, 601. S. College Rd., Wilmington,
NC 28403, USA, and Center for Marine Science, Univ. of North Carolina Wilmington, Wilmington, NC 28403, USA. – S. J. Whipple,
Skidaway Inst. of Oceanography, 10 Ocean Science Circle, Savannah, GA 31411, USA. – SJW and B. C. Patten, Odum School of Ecology,
Univ. of Georgia, 140 E. Green St., Athens, GA 30602-2202, USA.
Indirect effects are important components of ecological and evolutionary interactions that may maintain biodiversity,
enable or inhibit invasive species, and challenge ecosystem assessment and management. A central hypothesis of Network
Environ Analysis (NEA), one type of ecological network analysis, is that indirect flows tend to dominate direct flows in
ecosystem networks of conservative substance exchanges. However, current NEA methods assume that these ecosystems are
stationary (i.e. time invariant exchange rates), which is unlikely to be true for many ecosystems for interesting time and
space scales. For the work reported here, we investigated the sensitivity of the dominance of indirect effects hypothesis to
the stationary modeling assumption by determining the development rate of indirect effects and flow intensity, as expressed
as the number of transfer steps, in thirty-one ecosystem models. We hypothesized that indirect effects develop rapidly in
ecological networks, but that they would develop faster in biogeochemically based models than in trophically based models.
In contrast, our results show that indirect effects develop rapidly in all thirty-one models examined. In 94% of the models,
indirect flows exceeded direct flows by a pathway length of 3. This indicates that ecological systems do not need to maintain
a particular configuration for long for indirect effects to dominate. Thus, the dominance of indirect effects hypothesis
remains plausible. We also found that biogeochemical models tended to require more of the extended path network than
the trophic models to account for 50% and 95% of the total system activity, but that both types of models required more
of the power series than is typically considered in engineered systems. These results succinctly illustrate the complexity of
ecological systems and help explain why they are challenging to assess and manage.
“Speed is the form of ecstasy the technical revolution has
bestowed upon man” Czech novelist Milan Kundera
(as quoted in Gleick 1999).
Organisms are linked through an intricate network of energy,
matter and informational exchanges. This network allows
one species to influence the distribution and abundance of
others without direct contact. For example, in the Ross Sea,
Antarctica, Adelie penguins Pygoscelis adeliae consume krill
which in turn eat algae like Fragilariopsis sp. (Ainley et al.
2006). This is a classic food chain (a simple network) in
which biomass is transferred by the feeding process. These
direct energy–matter transactions establish a set of direct and
indirect relationships. The predators benefit directly from
eating their prey, but there is also an indirect mutualistic
relationship between the penguins and Fragilariopsis sp. that
passes over a pathway of length m 2. This type of indirect
effect is what enables trophic cascades (Carpenter et al. 1985,
Ainley et al. 2006). In the work reported here, we investigate
the development of indirect relationships in ecosystem
network models.
Patten (1983, 1984, 1985a) introduced to theoretical
ecology the concept that indirect effects are the dominant
form of causation in ecological networks. This implies that
ecosystems have a holistic organization (Patten and Odum
1136
1981, Patten et al. 2002). Today, evidence for the broad
ecological and evolutionary importance of indirect effects
is building in the literature (Strauss 1991, Wootton 1994,
Laland et al. 1999, Diekotter et al. 2007, Hall et al. 2007,
McCormick 2009). For example, Wootton (2002) suggests
that they are an important source of biocomplexity, and
Bever (1999, 2002) shows how they may be a mechanism
that maintains biodiversity and facilitates the success of
invasive species. Treweek (1996) and Mandelik et al. (2005)
discuss how indirect effects present serious challenges for
ecological assessment and management. Thus, understanding the development and propagation of indirect effects is
critical for many aspects of ecology and evolution.
The network of direct and indirect interactions has the
power to transform the effective relationship between organisms (Ulanowicz and Puccia 1990, Patten 1991). For example, Bondavalli and Ulanowicz (1999) used ecological network
analysis to show that, on average, the American alligator of
Big Cypress National Park, Florida, benefits a number of its
prey populations more than it hurts them. Specifically, the
alligator preys on frogs in the swamp, which certainly has a
direct and negative impact on the frog population, but when
the whole network is considered there is a net positive relationship between alligators and frogs. This is in part because
the alligator also preys upon snakes that eat the frogs. Without the alligators, the frogs would be worse off. This follows
the political principle that the enemy of my enemy is my
friend. Patten (1991) and Fath and Patten (1998) observed
that when shifting consideration from only direct transactions to the integral relationships, there tended to be a change
from more negative relationships (competition, predation) to
more positive relationships (mutualism) in ecosystem networks. This transformative power is what Patten calls “the
network variable in ecology” (Patten and Fath 2000).
Network Environ Analysis (NEA) is one type of ecological
network analysis to study the network variable in ecology
(Patten et al. 1976, Patten 1978, Fath and Patten 1999). A
central hypothesis from NEA is that ‘indirect effects are the
dominant components of ecological interactions’ (Patten
1983, 1984, 1985a, Higashi and Patten 1986, 1989). Higashi
and Patten (1989) and Patten et al. (1990) showed algebraically why the ratio of indirect-to-direct flows (I/D) should
tend to increase as network size (number of nodes, n), connectivity (proportion of possible direct links L connected,
C L/n2), and the strength of direct flows and recycled flow
increases. These predictions held true in Fath’s (2004) largescale simulated ecosystem models. In addition, the limited
application of NEA to more empirically-based ecosystem models (as opposed to Fath’s (2004) cyber models) has confirmed
that indirect flows tend to exceed direct, I/D 1 (Higashi and
Patten 1989, Patten 1991, Borrett et al. 2006, Borrett and
Osidele 2007). If true, this hypothesis suggests that the dominant causal forces are non-local and non-obvious.
This hypothesis arose from a new methodology, NEA,
whose critical assumptions are that the system’s model is stationary (i.e. with time invariant coefficients) and usually at
steady state (balanced inputs and outputs, but see Finn 1980,
Shevtsov et al. 2009, for possible ways to relax these assumptions). While ecologists often make these modeling assumptions explicitly or implicitly, they are unlikely to be true of
many ecosystems at many time scales. The static assumption is
required because it lets us partition the energy–matter flux
across the extended network of pathways that can be infinite in
length and number (Patten et al. 1982, Borrett and Patten
2003, Borrett et al. 2007). This issue undermines our confidence in the indirect effects dominance hypothesis because it
is plausible that the ecosystem will change configuration
(structure or functional relationships) before much of the
extended network is utilized. Thus, to better understand the
significance of indirect effects, it is essential to know how
quickly they develop in the extended pathways of ecological
networks. If they develop rapidly – that is only shorter pathways of the extended network are required for indirect flows to
exceed direct flows – then the hypothesized significance of
indirect effects remains plausible. If dominance of indirect
effects requires longer pathways, traveling further into the
extended network, then the hypothesis rests on shaky ground.
In the work presented here we tested two specific hypotheses. First, we examined the hypothesis that indirect effects
develop rapidly in terms of path length in network models of
energy–matter flux in ecosystems. This hypothesis has been
suggested by both Patten (1985b) and Ulanowicz and Puccia
(1990), but to our knowledge it has never been tested.
Empirically, Menge (1997) found that indirect effects
appeared rapidly in experimental communities, so our
measure of indirect effects should also develop rapidly for the
theory to be consistent with such empirical observations.
Second, we hypothesized that indirect effects develop more
rapidly in ecosystem models that focus on biogeochemical
processes rather than trophic processes. Christian et al.
(1996) noted that ecosystem models that focus on biogeochemical processes tend to be less dissipative and have more
aggregated biological nodes than trophically focused models.
Thus, they tend to have more cycling and higher indirect
effects, which we expect to (1) decrease the pathway length
at which indirect flows first exceed direct flows and (2)
increase the pathway length at which we have accounted for
50% or 95% of the total system throughflow. We conclude
the paper by estimating the rate of energy–matter decay in
the extended pathway network, which we use to explain the
observed variation in the development of indirect effects.
Material and methods
To test our hypotheses we used NEA to determine the energy–
matter flux in the extended network of empirically-based
ecosystem models drawn from the literature. In this section,
we first introduce the network models we selected for this
study, and then describe the relevant components of NEA.
Next, we describe how we characterized the development of
indirect effects in the extended network and flux decay. We
conclude with an example analysis to clarify the concepts and
analysis.
Ecological networks
We selected 31 network models of energy–matter flux in
ecosystems from the literature. These network models were
originally constructed for different purposes, but generally
follow the network construction guidelines described by
Fath et al. (2007). We specifically selected models that are
empirically-based in that the original authors quantified
many of the fluxes by either collecting empirical data (in the
field or from the literature) or by building a simulation
model guided by empirical data. We contrast these more
empirically-based models with Fath’s (2004) cyber models or
Webster et al.’s (1975) hypothetical ecosystem models, which
do not use empirical measurements. Table 1 shows that the
models vary in size (number of nodes, n), number of links
(L), connectance (C L/n2), total boundary inputs
G
( z =¤ ni=1 z i ) and total system throughflow (TST). We
use the L1-matrix norm throughout the paper to simplify
our notation (Seber 2008). The L1-matrix norm is the sum
of the absolute values of the elements in a vector or matrix
n
n
M, and is notated as M =¤ i=1¤ j=1 m ij . Equations introduced in this paper may not hold if another norm is substituted, as more generally x y b x y . Table 1 also
shows a few other commonly reported network statistics. In
addition, preliminary analyses determined that in these systems indirect flows (I) were dominant over direct flows (D)
such that I/D 1. These networks are of two types: biogeochemical and trophic.
1137
Biogeochemical networks
Biogeochemical network models are typically based on
a nutrient currency such as nitrogen or phosphorus. As
mentioned in the introduction (Christian et al. 1996, Baird
et al. 2008), when compared to trophic models,
biogeochemical ecosystem models tend to be less dissipative,
have more aggregated biological nodes and higher average
connectance, include inorganic entities and flows derived
from biogeochemical processes such as denitrification, and
have more cycling and larger indirect effects. Table 1 shows
that these tendencies hold true for the models we used in this
study. These models include: (1) 16 seasonal network models
for the Neuse River Estuary first published by Christian and
Thomas (2003) and recently analyzed by multiple investigators (Borrett et al. 2006, Gattie et al. 2006, Schramski et al.
2006, Whipple et al. 2007), (2) an 11-node model of phosphorus flux in Lake Sidney Lanier, Georgia (USA) (Borrett
and Osidele 2007), (3) a long-term averaged 16-node nitrogen flux network for the Baltic Sea that Hinrichsen and
Wulff (1998) constructed by sampling a simulation model
built by Stigebrand and Wulff (1987), (4) two network
models of nitrogen flux and one of phosphorus flux in the
Sylt–Rømø Bight ecosystem (Baird et al. 2008), and (5) two
biogeochemical models of nitrogen (Baird et al. 1995) and
phosphorus (Ulanowicz and Baird 1999) flux in the mesohaline region of the Chesapeake Bay.
Trophic networks
These models generally trace the flux of energy or carbon (an
energy surrogate) through species or groups of species. These
are essentially food web models in which all ecological process that transfer energy or carbon are considered (e.g. eating, respiration, egestion). A critical component of these
models is that they include at least one compartment for
detritus. In our study, we analyzed seven network models
originally developed for parts of Everglades National Park as
part of the Across Trophic Level System Simulation (ATLSS)
project (http://atlss.org/). These included dry and wet
season models for Florida Bay (Ulanowicz et al. 1998),
graminoid marshes (Ulanowicz et al. 2000, Heymans et al.
Table 1. Network properties of biogeochemical and trophic ecosystem models†
Model
G
z
L1(A)
%SCC
0.45
0.45
3.27
3.27
100
100
133
119
9120
20 182
0.91
0.96
68.94
169.42
7
7
7
7
7
7
7
7
7
7
7
7
7
7
11
59
59
36
36
16
0.45
0.43
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.21
0.09
0.09
0.12
0.12
0.15
3.27
3.02
3.27
3.27
3.27
3.27
3.27
3.27
3.27
3.27
3.27
3.27
3.27
3.27
2.75
7.20
7.23
4.10
4.01
2.78
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
44
44
72
72
100
181
187
128
165
100
691
334
90
85
171
176
132
128
291
97
99,613
2,510
73,431
9,402
2,349
8780
6880
12 915
11 980
9863
7907
11 533
15 621
7325
8680
6898
16 814
5732
5739
747
363 692
57 666
484 326
101 092
44 511
0.88
0.85
0.94
0.91
0.94
0.62
0.84
0.96
0.93
0.89
0.85
0.95
0.87
0.75
0.4
0.23
0.66
0.33
0.51
0.67
47.24
36.95
101.42
71.73
99.47
9.89
34.64
174.01
86.58
51.58
38.37
127.34
44.14
18.52
7.13
2.95
20.30
5.44
9.90
14.58
6
125
125
68
66
94
94
59
36
0.33
0.13
0.12
0.12
0.18
0.15
0.15
0.08
0.09
2.15
11.01
10.97
6.85
11.06
14.17
14.16
6.72
2.85
83
82
82
78
91
91
91
83
16
41
548
739
1,419
3,473
1,531
1,532
505,107
888,792
84
1779
2722
2572
7520
3272
3266
1 353406
3 227 456
0.11
0.08
0.14
0.04
0.04
0.1
0.1
0.09
0.19
1.53
1.45
1.91
1.71
1.00
1.74
1.69
1.13
3.07
Currency
n
C
Neuse Estuary 1985 Sp
S
mmol N m2 season1
mmol N m2 season1
7
7
F
1986 W
Sp
S
F
1987 W
Sp
S
F
1988 W
Sp
S
F
1989 W
Lanier
Sylt–Rømø
Sylt–Rømø
Chesapeake Bay
Chesapeake Bay
Baltic Sea
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mmol N m2 season1
mg P m2 d1
mg N m2 year 1
mg P m2 year 1
mg N m2 year 1
mg P m2 year1
tons N d1
Oyster
Florida Bay (dry)
Florida Bay (wet)
Cypress (wet)
Graminoids (dry)
Mangrove (dry)
Mangrove (wet)
Sylt–Rømø
Chesapeake Bay
kcal m2 d1
mg C m2 d1
mg C m2 d1
mg C m2 d1
mg C m2 d1
mg C m2 d1
mg C m2 d1
mg C m2 year 1
mg C m2 year 1
TST
FCI
I/D
Biogeochemical
Trophic
n is the number of nodes or compartments in the ecosystem model, C L/n2 is the connectance of the model; L1(A) is the rate of pathway
proliferation; %SCC is the percent of nodes contained in a large (n 1) strongly connected component in the ecosystem model;
G
G is the sum of the boundary inputs; TST T
is the total system throughflow; FCI is the Finn cycling index; and I/D is the ratio
z
†
of indirect-to-direct flow intensities calculated as (||N||−||I||−||G||)/||G|| using the L1-norm.
1138
2002), and mangrove ecosystems (Ulanowicz et al. 1999,
Heymans et al. 2002), as well as a wet season model for a
cypress wetland (Ulanowicz et al. 1997). These models track
carbon and are some of the largest, most disaggregated networks that have been created to date. We contrast these
models with Dame and Patten’s (1981) oyster reef community ecosystem model because, although its nodes are highly
aggregated (n 6), it is historically important in the development of NEA. We also examined the annual carbon-based
ecosystem models for the mesohaline region of the Chesapeake Bay (Baird and Ulanowicz 1989) and Sylt–Rømø
Bight (Baird et al. 2008).
In total, we included 22 biogeochemical models that represent five distinct systems and nine trophic models representing seven distinct ecosystems in this study. Two of these
systems, the Sylt–Rømø and the Chesapeake Bay, have paired
trophic and biogeochemical models. For simplicity, we chose
to report the results of the Neuse River Estuary models as a
summary mean and standard deviation.
Network environ analysis and indirect effects
Network Environ Analysis (NEA) is a family of input–output
methods that mathematically trace energy–matter through
systems of interest (Patten et al. 1976, Fath and Patten 1999).
It is applied to ecosystem models to investigate the organization of their internal environs (Patten 1978). The methodology includes analyses of structure, throughflow, storage,
utility and control within systems. As the techniques are well
described in the literature (Patten et al. 1976, Fath and
Patten 1999, Fath and Borrett 2006, Gattie et al. 2006), we
only briefly review the relevant components of throughflow
analysis.
In an n node network, let Fnn (f ij ) represent the
observed flow from ecosystem compartment j to compartment
i (e.g. jm i). Note that this orientation is reversed from many
other forms of network analysis, which may be a source of
confusion. However, this orientation is consistent with the
previous notation used in NEA, and a simple transpose can
be applied to re-orient the matrix to match other network
G
analyses when needed. Further, let z n1 be the observed
G
inputs to the compartments and y1n be the observed
boundary losses from the compartments. Formally, our network model is a weighted digraph.
The first step of the NEA throughflow analysis is to calculate the amount of energy–matter flowing into or out of each
node as:
G
n
T in ¤f ij z i and
(1)
G
n
T out ¤f ij y j .
(2)
j1
i1
G
G
G
At steady state, T in (T out )T Tn1 (Tj ). Next, we
calculate the output oriented direct flow intensities as:
Gnn (g ij ) f ij /Tj .
(3)
With this information we can determine the integral flow
intensities, which are the flow intensities over all pathways of
all lengths – the extended path network – between each
model node (i.e. boundary + direct + indirect). We do this by
implicitly using the following equation:
d
N ¤ Gm m=0
I
N
Boundary
1
2
G
!
Gm
!
N G
Direct
(4)
Indirect
Here I (i ij ) G0 is the matrix multiplicative identity and
the elements of Gm are the flow intensities from j to i over all
pathways of length m.
By definition, ecosystems are thermodynamically open
(Jørgensen et al. 1999), which implies that the power series
in Eq. 4 should be convergent. Thus, we can explicitly calculate the exact value of N in Eq. 4 as m m d using the following identity:
N (I G)1
(5)
A historical critique of NEA held that Eq. 4 double counts
pathways (Wiegert and Kozlowski 1984, Pilette 1989).
However, this cannot be true because we recover the exact
G
node throughflow T using the integral flow matrix as:
G
G
(6)
T Nz
Instead of double counting, Eq. 4 represents a true mathematical partition of the observed flows across the possible
pathways.
From these network elements we derive a number of indicators of whole ecosystem organization (Fath and Patten
1999, Fath and Borrett 2006, Borrett and Osidele 2007).
Three of these indicators central to our work here are total
system throughflow (TST), the ratio of indirect-to-direct
flow intensities (I/D), and the Finn cycling index (FCI). We
calculate the indicators as follows:
G
G
G
TST T ¤ ni1 | Tj | F z F y , which is
an indicator of the magnitude or size of the system activity
much like gross domestic product in economies (In Ascendent
ecological network analysis total system throughput is calculated
G
G
as F z y (Ulanowicz 1986). The key difference
is that both boundary inputs and outputs are added here,
instead of just one or the other.); I / D N I G ,
G
which indicates the role of indirect effects in the system. If
I/D 1 then we conclude that indirect effects are dominant;
and FCI (n jj 1) / n jj Tj / TST , which is the ratio
of energy–matter cycled divided by TST (Finn 1980).
In addition, we will also use the concepts of total flow
intensity ( TFI N ), direct flow intensity ( DFI G ),
indirect flow intensity ( IFI N I G ), and boundaryflow intensity ( BFI I n ). When we use the L1-norm,
TFI BFI DFI IFI . TFI is similar to TST as it is the
sum of the flow intensities before they are scaled by the
boundary inputs (outputs).
Development thresholds
We calculated the flow intensities and cumulative flow
intensity in the extended pathway network as
pathway length m increased using Eq. 4. We describe the
1139
development of flow in the network models using three
threshold values:
mID: the pathway at which the magnitude of IFI
exceeds DFI;
m50: pathway at which 50% of TFI is recovered; and
m95: pathway at which 95% of TFI is achieved.
We illustrate these thresholds in Fig. 1 for the Lake
Lanier model. Together, these thresholds indicate how
much of the extended path network is required to support
the hypotheses about ecosystem organization. This approach
is similar in spirit to Lenzen’s (2000) method for calculating the truncation error of system completeness for economic input–output analysis as used in industrial life cycle
analyses.
Flow decay rate
Thresholds provide an intuitive way to describe the accumulation of total flow intensity in the extended path network,
and a direct answer to our initial hypotheses. However, we
can describe the decay rate of energy–matter exiting the system as pathway length increases more precisely using matrix
algebra.
Equation 4 is a generating function (Godsil 1993) that
describes the flow decay as pathway length increases. Thus,
the input energy–matter declines approximately geometrically (Fig. 5a). If G is an irreducible non-negative matrix that
represents a strongly connected component (SCC)(A graph
is strongly connected if it is possible to reach every node
from every other node over a pathway of some length.), then
the dominant eigenvalue of G, L1(G), is the asymptotic decay
rate (see Caswell 2001 or Borrett et al. 2007 for a fuller
explanation of the mathematics). Here, L1(G) is analogous
to the population growth rate, L, recovered from matrix
population models like the Leslie matrix (Caswell 2001). If
L1(G) 1, the system would be gaining more energy–matter
than it is dissipating, but if 0 L1(G) 1 energy–matter is
dissipating from the system. As mentioned previously, ecosystems are open thermodynamic systems, so we expect the
latter condition to hold. As this is a geometric decline, as
L1(G) approaches zero, the decay rate increases and the
system becomes more dissipative.
If only a subgraph of the initial graph is a large (n 1)
SCC (a subgraph of size n 1 is technically a strongly connected component, but this type of SCC is uninteresting for
our purposes), then the dominant eigenvalue describes the
flow decay rate in the subgraph. It is also possible that ecosystem models may be comprised of several linked large
SCCs. In this case, each SCC will have its own asymptotic
flow decay rate which will be the dominant eigenvalue for
the G for the subdigraph associated with each SCC. For
more details on how to do this see Berman and Plemmons
(1979), Allesina et al. (2005) and Borrett et al. (2007).
We used the scc.m MATLAB function provided by
Kevin Murphy (www.cs.ubc.ca/murphyk/Software/index.
html) to identify the strongly connected models in the
ecosystem models. We then found L1(G) for all large SCCs.
Table 1 also reports the %SCC, which refers to the percent
of network nodes participating in the large (n 1) SCCs of
the network.
1140
Figure 1. Illustration of the three indicators (m1D, m50 and m95)
used to characterize the development of flow intensity in the
extended path network. The top panel shows the saturation curve of
cumulative flow intensity for the Lake Lanier phosphorus model.
The bottom panel marks the three development thresholds (m1D 3, m50 5 and m95 2). As the path series is discrete, we conservatively chose to mark the thresholds length immediately beyond when
the threshold is passed (observe m50). Notice that in this model 13%
of the total flow intensity is generated by the boundary input when
m 0 (boundary flow intensity), and a further 10% is generated by
the direct paths when m 1 (direct flow intensity). Thus the indirect flow intensity generated by all paths larger than 1 is 77%.
Example of NEA throughflow analysis
For clarity, we will illustrate the NEA calculations using the
hypothetical ecosystem shown in Fig. 2. This ecosystem
model is comprised of four compartments or nodes (n 4)
that represent (1) primary producers, (2) detritus, (3) detritivores, and (4) a general consumer. The arrows or links in
the model represent the transfer of carbon by any ecological
processes (e.g. photosynthesis, death, consumption, respiration). There are six internal transfers, one boundary input
(z1) into the primary producer compartment, and four
boundary losses (yj), one from each node. Notice that in this
model the detritus, detritivore, and consumer nodes form an
SCC because it is possible to start in any of these nodes and
travel to any other node over a pathway of some length. The
primary producers form a trivial second SCC that lies
upstream from the larger SCC.
As required for NEA, we will assume that this model is at
a static, steady-state. Figure 2b shows the flow magnitudes
for the model. These fluxes are recorded in the F matrix for
G
G
internal transfers and z and y vectors for boundary inputs
and outputs, respectively.
Given these ecosystem data, we can begin our NEA
throughflow analysis. First we calculate the vector of node
throughflows as:
¨ 100 ·
G
n
n
©13.5065¸
T ¤f ij z i ¤f ij y j ©
6.7532 ¸
i1
j1
© 3.7013 ¸
ª
¹
(7)
G
The sum of the elements of T , which is the total system
throughflow (TST), is 123.961. This shows that when we
put 100 units of carbon into the system, we obtain 123.961
m
units of activity. The fact that TST is greater than ²²Z²² is an
example of the multiplier effect of the system that is partmof
the power of the network organization. The ratio TST/²²Z²²
has been alternatively called the average path length (Finn
1980) and network aggradation (Jørgensen et al. 2000).
Next, we use Eq. 3 to determine the direct flow intensity
matrix G to be
0
0
0 ·
¨ 0
© 0.1 0 0.3 0.4 ¸
G ©
0
0.5 0
0 ¸
©0.01 0 0.4 0 ¸
ª
¹
(8)
We use G and Eq. 4 to calculate the integral flow intensity matrix as to be:
0
0
0 ·
¨ 1
© 0.1351 1.2987 0.5974 0.5195¸
N©
0.0675 0.6494 1.2987 0.2597 ¸
©0.0370 0.2597 0.5195 1.1039 ¸
ª
¹
(9)
There are several items to notice when comparing G and
N. First, the elements of the direct flow intensity matrix G are
in the same positions as the elements of F. In contrast, nearly
all of the positions in the integral flow intensity matrix N
show some positive value. This is because the integral matrix
considers the flow intensity over all path lengths, effectively
including indirect interactions. The key insight here is that
what we perceive as direct flow is actually a composite of molecules, each with its own history in the network (Patten et al.
1990). NEA lets us trace their paths mathematically in a way
that is difficult or impossible to do empirically.
A key addition to the NEA throughflow analysis reported
in this paper is the consideration of discrete thresholds to
characterize the development of flow intensity in the extended
network (e.g. over longer path lengths). To do this, we modified the power series of G shown in Eq. 4 as follows:
N I G G2 G3 G4 G5 ––– Gm –––
N
N
TFI
BFI
DFI
IFI
(10)
7.7461 4 1.71 0.914 0.517 0.2739 0.2739 0.1507 {
Notice that Eq. 10 only works when we assume the
L1-norm.
We then used Eq. 10 to determine that mID5 because
²²G2²²²²G3²²²²G4²²²²G5²² 1.8556 DFI 1.71.
Likewise, we find that m50 0 because ²²I²²²²N²² 100% ²I²
51.64%, and m953 because (²
² ²²G²² ²²G2²² 3
²²G ²²)/²²N²² 100% 96.89%. Notice that m952
because ²²I²²²²G²²²²G2²²)/²²N²² 100% f 93.9%.
The final step of our analysis was to determine the asymptotic flow decay rate, L1(G), in the path length power series
shown in Eq. 4. As noted, this is the dominant eigenvalue(s)
of the non-trivial SCCs. In this example, there is only one
non-trivial SCC in which L1(G) 0.5448. This decay rate
suggests that carbon entering the system is dissipated at a
moderate rate.
(
)
Results
Figure 2. Hypothetical ecosystem model used to illustrate network
environ analysis calculations. The notations indicate the following:
fij represents a flow of energy–matter from j to i, zi designates
boundary flow into node i, and yj is a flow from node j across the
system boundary. Corresponding magnitudes of the fluxes are provided below the diagram. In this model, the detritus, consumer,
and detritivore nodes form a strongly connected component
(SCC) because it is possible to start from any of these nodes and
get to any of the others over a pathway of some length. The primary producer node is a trivial SCC that is disconnected from the
larger SCC.
Our results reveal that dominant indirect effects require
short path lengths to achieve, and thus develop rapidly in the
extended path network in all the ecosystem models examined, regardless of temporal changes or model type.
Development of indirect effects
Figure 3 shows that in 29 of the 31 models the cumulative
indirect flow exceeded the direct flow by a pathway length of
three (mID 3). Only indirect pathways of length two and
three were required to exceed the direct flows regardless of
the temporal variability in the Neuse River Estuary models
or the model differences from biogeochemical or trophic
1141
assumptions. The two exceptions to the general pattern are
both trophic models: the dry season Graminoid model whose
I/D is just barely over 1 (I/D 1.000216) and the Sylt–
Rømø whose I/D 1.13. However, even in these cases indirect flows exceeded direct flows by a path length of eleven
and five, respectively. These results suggest that relatively few
of the longer pathways in ecosystem network models are
required for indirect flows to exceed direct flows.
Flow decay rates
Figure 4 shows that the pathway lengths at which 50% and
95% of TFI are recovered are more variable than that for
mID and clearly differs by model type. The median pathway lengths at which 50% and 95% of TFI are recovered in
the biogeochemical models is 29 and 126, respectively,
while in the trophic models the median values were much
lower at 1 and 5, respectively. This difference supports the
hypothesized differences between the model types; it takes
more of the extended network in biogeochemical models to
recover the stated fractions of TFI than it does in the
trophic models.
The Summer 1987 Neuse River Estuary model has the
longest m95 at 524. While this is a large number, it is far
shorter than the infinitely long paths that are included in the
full network analysis. This implies that while paths of all
lengths are required to account for all of the system activity,
in many cases we can substantially truncate our power series
and achieve a close approximation.
Figure 5a illustrates the decay of energy–matter as pathway
length increases in selected models, and panels (b) and (c)
show the asymptotic decay rates. As with m50 and m95, there
is an obvious difference in the decay rates between the biogeochemical and trophic models. The median decay rates in
the two groups is 0.98 and 0.59 respectively. Material is generally lost more rapidly from the trophically based models.
The network with the slowest decay rate was the Neuse
River Estuary model for Spring 1987 (L1(G) 0.99). In this
case, recycling is also the highest (FCI 0.96) such that
little of the nitrogen is lost from the model each time it
passed through one of the nodes. Thus, the system is highly
retentive of the input nitrogen, and in this sense is very efficient. In contrast, the network with the fastest decay rate was
the Graminoids dry season model where L1(G) 0.41. In
this model recycling was the lowest of our sample at FCI 0.04. Each time a quantity of carbon passed through a node,
a large fraction was lost (respired or exported). Activity in
this system depends heavily on the external inputs that drive
it, rather than the internal dynamics.
Thirty of the models contained only one large SCC.
However, the Chesapeake Bay trophic model has two large
SCCs as identified in Table 2. In this case, each SCC has its
own flux decay rate. The smaller SCC (A) has six nodes made
up primarily of pelagic species and a flux decay rate of 0.32.
The larger SCC (B) has 16 species groups that are primarily
benthic in nature and a slower flux decay rate of 0.74.
Figure 3. Pathway length at which indirect flow intensities exceed
direct flow intensities, mID, in 22 biogeochemically (top) and 9
trophically (bottom) based ecosystem models. The m1D for each of
the 16 seaonal models of the Neuse River Estuary was identical.
Figure 4. Development of total flow intensity (TFI) in 31 ecosystem models. Panel (a) and (b) show the pathway lengths at which
50% and 95% of TFI are achieved in biogeochemically and trophically based ecosystem models, respectively.
Development of total system throughflow
1142
The %SCC ranged from a maximum of 100% to a minimum of 44%.
Discussion
The results reported here show that indirect effects tend to
develop rapidly in the extended path network of ecosystem
models. This implies that the hypothesis of dominant indirect effects in ecosystems remains plausible. In this section,
we consider how these results contribute to our understanding of ecological systems and network analysis. We first discuss the significance of this rapid development of indirect
effects. We then explain how the importance of higher-order
pathways in ecological systems might be further evidence for
ecological complexity and suggest that the flow decay rate,
L1(G), is an indicator of system growth. We conclude by
discussing related and future work for this research line.
Figure 5. Decay rate of energy–matter flow in the extended pathway network. Panel (a) shows the decay in flow intensity as pathways become longer in three selected models. Flow intensity is the
total amount of energy–matter flux over pathways of length m.
Here we normalized the total flows by total boundary input to
enable cross model comparison. Panels (b) and (c) show the asymptotic exponential rate of flux decay, L1(G), in biogeochemically and
trophically based ecosystem models. Notice that the closer L1(G) is
to unity, the slower the decay rate. Conversely, the closer L1(G) is to
zero, the faster the decay rate.
Rapid dominance of indirect effects
Dame and Christian (2006) asserted that the fundamental
assumptions of ecological network analysis (ENA) are an
important source of uncertainty that complicates the use of
ENA for ecosystem assessment. These assumptions also
endanger the theoretical inferences about ecosystem organization that have been made using this technique (Ulanowicz
1986, Higashi and Patten 1989, Patten 1998, Jørgensen
2002). However, the rapid development of indirect effects in
the ecosystem networks we analyzed (Fig. 3) bolsters our
confidence in the dominance of indirect effects hypothesis
(Patten 1983, 1984, Higashi and Patten 1986, 1989). This
result suggests that ecosystems do not need to sustain a specific configuration over long periods for indirect effects to
develop. Thus, the phenomenon should not be dependent
upon the static assumptions of the analysis; we expect indirect effects to be dominant in time-varying systems as well,
though this has yet to be systematically tested (Shevtsov et al.
2009).
We want to stress that by ‘rapid development’ we mean
with respect to increasing path length rather than time.
While there must be a relationship between path length and
time, Wiegert and Kozlowski (1984) pointed out that it is
conceptually challenging to specify this connection. This is
because in a well connected system an infinite number of
pathways are realized within the time step of the model. For
example, in the Neuse Estuary model, the full complement
of pathways is achieved in the span of a single season, while
in the Sylt–Rømø Bight models all of the pathways occur in
the span of a year. A further conceptual challenge is that the
physical time required for energy–matter to travel through a
specific compartment may differ by orders of magnitude, for
example if we were to compare bacteria to macroheterotrophs. In the context
of ENA/NEA this ecological difference is
m
captured in T and subsequently G and N. However, the time
step across path lengths as m increases is independent of the
turnover times and is assumed to be constant. Thus, longer
paths must require more time than shorter paths, but what
the specific time unit is remains unclear.
The observed rapid development is consistent with previous research, both empirical and theoretical. For example,
Menge (1997) empirically examined the speed with which
indirect effects developed in marine intertidal interaction
webs. He found that the indirect effects were evident either
in concert with the direct effects or shortly thereafter. This
suggests that the theoretical view of ecosystems that is developing through NEA and the Holoecology Research Program
(Borrett and Osidele 2007, Patten unpubl.) is consistent
with more empirically based discoveries. From the theoretical perspective, Borrett et al. (2006) examined the temporal
variability of indirect effects in a sequence of sixteen seasonal
models of nitrogen flux in the Neuse River Estuary (North
Carolina, USA). They found that although total system
activity varied seasonally as we would expect from the system
biology, the proportion of TST derived from indirect flows
remained surprisingly constant across seasons and years analyzed. Indirect flows always dominated direct flows in the
Neuse Estuary. The dominance of indirect effects seems to be
established rapidly in ecosystems and once established it is
persistent. Further, Borrett and Osidele (2007) showed the
1143
Table 2. Species groups in the two strongly connected components
with more than one node (A and B) of the Chesapeake Bay trophic
(carbon) ecosystem model.
SCC
A (pelagic)
B (benthic)
Node
2
7
8
9
10
35
3
14
15
16
17
18
19
25
26
27
28
29
30
32
33
36
Species1
Bacteria attached to suspended particles
Microzooplankton (Ciliates)
Zooplankton
Ctenophores
Sea Nettles (e.g. Chrysaora fuscescens)
Suspended particulate organic carbon
Sediment Bacteria
Other Polychaetes
Nereis sp. (Polycheate)
Macoma sp.
Meiofauna
Crustacean deposit feeders
Blue crabs
Atlantic croaker
Hog choker
Spot
White perch
Catfish
Bluefish
Summer flounder
Striped bass
Sediment particulate organic carbon
Node numbers as in Baird and Ulanowicz (1989).
1
ecological systems. This complexity contrasts with the apparent simplicity of many engineered systems and mathematical
functions as suggested by their common approximation
using only first or second order terms.
The Taylor series approximation is often a useful way “to
approximate a complicated function near a point we care
about” (Bolker 2008, p. 84). A relatively good approximation usually only requires the first and second order terms
of the Taylor series. This technique is similar in spirit to
Paynter’s matrix exponential method for approximating
solutions to differential equations, which does estimate
the dynamics of a system of equations (Patten 1971). For
engineering applications, Bay (1999) introduced the truncated (two terms) Taylor series expansion to approximate
non-linear physical systems as linear systems. He notes that
this linear approximation is often very good for engineering
purposes.
When we consider the importance of the higher-order
pathways in conjunction with the rapid development of
indirect effects a clearer picture of these ecological systems
emerges. Relatively short path length approximations will
usually reveal the qualitative importance of indirect flows in
the network, but to accurately estimate the total system
activity requires considerably more information about the
extended path network. This succinctly illustrates the complexity of ecological systems.
Ecological significance of the flow decay rate L1(G)
dominance of indirect effects was robust to parameter uncertainty in a network model of phosphorus flux in Lake Sidney
Lanier (Georgia, USA).
The slower development of indirect effects in the trophically based networks for the dry season Graminoid marshes
and the Sylt–Rømø Bight suggests an interesting alternative
interpretation for the development speed. To determine the
relative development speed we could consider the path
lengths required for indirect effects to be dominant relative
to the path length required for 95% of TFI (mID/m95)
rather than for 100% of TFI. As (mID/m95) is 11/5 2.2
and 5/5 1 for the Graminoid and Sylt–Rømø networks,
respectively, we might conclude that these are examples of
slower development of indirect effects. This is apparent when
we compare the values to the 3/9 z 0.33 value for the trophically based model of the Chesapeake Bay and the 3/524 z
0.01 value for the Summer 1987 nitrogen model for the
Neuse Estuary. This perspective provides interesting comparisons and may prove useful in the future, but we argue
that it is inadequate for assessing the sensitivity of the dominance of indirect effects hypothesis to that stationary system
assumption. For this purpose, we must consider the full
complement of pathways used in traditional NEA (i.e. infinite). In this sense, even the Graminoid model’s mID of 11
seems relatively fast.
Evidence for ecological complexity
The clear importance of higher order terms of the path generating function for recovering TFI (Fig. 4), and subsequently TST (not explicitly shown), suggests that Patten’s
“network variable” is a significant source of complexity in
1144
Assuming that there exists one large strongly connected
component in the network, the dominant eigenvalue or
spectral radius of the direct flow intensity matrix, L1(G), is
the asymptotic geometric decay rate of energy–matter as
pathway length increases. Therefore, L1(G) indicates how
dissipative or conversely how retentive the system is. The column sums of G must scale between zero and unity because
ecosystems are open thermodynamics systems (Jørgensen
et al. 1999, Straškraba et al. 1999). This in turn ensures that
0 L1(G) 1(Berman and Plemmons 1979). The more
retentive a system is of the energy–matter introduced into
the system, the larger L1(G) will be. If a system retained all of
the energy–matter input into it L1(G) would equal unity,
and if it did not recycle any of the energy–matter then L1(G)
would be zero. Thus, we might expect L1(G) to correlate
with the Finn cycling index (FCI).
As expected, Fig. 6 shows a strong non-linear relationship
between L1(G) and FCI.
b
We fit the function FCI= L1 (G) to this relationship
2-L1 (G)b
because FCI is 0 when L1(G) is zero and FCI 1 when
L1(G) 1. Using non-linear least squares, we found b 3.6 (p 0.001, with a 95% CI of [3.374, 3.849]). This
strong relationship suggests that L1(G) provides us with
much of the same information as FCI. As L1(G) is the largest root of the characteristic polynomial of the system
matrix, however, it is a fundamental property of the system. The relationship between these two indicators is complicated by the role of the remaining eigenvalues in
determining the flow decay.
We can further explain this relationship between FCI and
L1(G) by realizing that at any time step, energy–matter exiting a node can either flow to another node in the system or
dissipate across the system boundary. If the network contains
a structural cycle, which implies that there is a SCC, then the
fraction of energy–matter that passes to another node in the
system has a chance of being recycled, while the fraction dissipated does not. Thus, decreasing the fraction dissipated at
any node, which will increase L1(G), will likely also
increase FCI.
Since L1(G) indicates the energy–matter retention of the
system, we can link this measure to previous notions of ecosystem growth and stress. Baird et al. (1991) identified a
characteristic of well organized systems to be a tendency for
them to internalize their activity, becoming relatively indifferent to external supplies and demands of energy–matter.
Jørgensen et al. (2000) and Fath et al. (2004) refer to this
type of ecosystem growth as a Form II or growth-to-throughflow. It is based on changes to the internal organization of
the system like increasing connectivity and recycling. It
appears that L1(G) is a fundamental indicator of this type of
growth.
Ulanowicz (1984, 1986) suggested that instead of growth,
increased recycling may reflect increased stress on the system.
We suspect that, as with many of the network indicators of
system condition, there may be an optimal L1(G). The value
of this optimum, however, remains unknown and will likely
differ between models of trophic dynamics and biogeochemical cycles. This expected difference is reflected in the results
shown in Fig. 5 and is consistent with the differences
observed by Baird et al. (2008).
Related work
Our work is similar to several recent contributions to ecological network analysis and input–output analysis. Here, we
discuss the similarities and distinctions of this work.
Lenzen (2007) recently applied structural path analysis
(SPA) from industrial ecology and economic systems to
sixteen ecological or coupled ecology–economic models
drawn from the literature. This interesting technique ranks
the specific paths for each of the n compartments based on
their contribution to the system activity. Lenzen (2007)
reported the number of paths (of various lengths) required
to account for 90%, 95%, 99% and 99.9% of the TST.
This is not the path length, as we report, but the number
of paths of any length required to account for the proportion
of TST. These SPA results are interesting because they suggest that while longer path lengths may be needed, only a
few of the specific pathways may account for much of the
system activity.
Lenzen (2007, p. 338) also identifies the dominant eigenvalue of the direct flow intensity matrix with the asymptotic
flux decay rate, though his terminology is different. Unfortunately, he does not report the number of strongly connected
components in the networks he analyzes. Dame and Patten’s
(1981) oyster reef model is the only overlap in our data
sets, for which we both find L1(G) 0.59. Notice that
Lenzen (2007) analyzes the input oriented matrices while we
focus on the output oriented matrices. Interestingly, the
eigenvalues of the input and output oriented direct flow
intensity matrices are identical in this case. Also, Lenzen
(2007) includes a version of the Florida Bay ecosystem
model, but it appears to be only the strict food web portion
of the model as the number of model nodes is smaller than
the model we are using.
Allesina et al. (2005) found multiple strongly connected
components (SCC) in 17 ecological networks, but only four
had more than one large SCC. One of these networks was a
carbon based trophic model for the mesohaline region of the
Chesapeake Bay. Our analysis of this model also uncovered
two SCCs. The smaller is comprised of mostly pelagic species
and should have a rapid turnover rate, while the larger one is
composed of mostly benthic species and fish. While the species composition of the larger SCC is identical to that
reported by Allesina et al. (2005), the size and composition
of the smaller SCC that they identify differs from what we
report in Table 2. Allesina et al. (2005) suggest that particulate organic carbon and dissolved organic carbon should be
part of this SCC. We are not sure what caused this difference, but we are not confident that we are analyzing the exact
same version of the Chesapeake Bay model, and we used
different algorithms to find the SCC.
Future work
Figure 6. Non-linear relationship between the flow decay rate and
the Finn cycling index for the 31 models included in this study. The
grey line is the solution to the empirically fit function
FCI=
L1 (G)b
. Using a non-linear least-squared method we
2-L1 (G)b
found b 3.60 (p 0.001) with a 95% CI of [3.374, 3.849].
This research is an important step in the development of a
coherent theoretical understanding of ecosystems, but there
remains much to do just in the context of network environ
analysis. Five specific issues are logical next steps.
First, Borrett et al. (2006) noted that there have been
relatively few applications of network environ analysis to
empirically-based ecosystem models. Our results provide
1145
additional support for the dominance of indirect effects
hypothesis, but an important next step is to systematically
test this hypothesis using the published models. A robust
analysis will require collecting models built by a wide variety
of researchers that may represent variation in modeling
assumptions. This is a limitation of our current work. The
models we used were built by a relatively small group of
authors. This could bias our results; however, many of the
models vary in structure and time of construction such that
we expect our results to remain valid.
Second, modeling decisions made by the original authors
regarding the aggregation or disaggregation of species into
functional groups may influence the results of our analyses.
Model aggregation has been studied extensively, but not
with respect to the NEA results. Thus, future work to
determine how model aggregation or dissaggregation affects
NEA properties and the rapid development of indirect effects
would be useful. The work that comes closest to evaluating
this is Fath’s (2004) research, which showed that many of
the NEA properties tended to increase as he increased the
number of nodes (and links) in his cyber–ecosystem models.
This confirmed the algebraic predictions of Higashi and
Patten (1989) and Patten et al. (1990). Thus, we would
expect our results to hold even in finer resolution models
with more nodes and links, even if the direct flow magnitudes declined. This is again a testable hypothesis that
remains to be evaluated.
A third line of research concerns L1(G). We suspect that
there exists a critical threshold of L1(G) for ecological networks
which insures that indirect flows are larger than direct flows.
Likewise, we expect there to be a lower critical threshold of
L1(G) below which direct flows are guaranteed to be larger
than indirect flows. Between the two thresholds we anticipate
there to be a region in which we are unable to predict the
relationship between direct and indirect flows using only the
information in L1(G). Discovering these thresholds might
provide a quick test for the dominance of indirect effects.
Alternatively, we might find that ecological networks will have
L1(G) values that fall between these critical thresholds.
Fourth, we suspect that there is additional information
about ecological systems to be gleaned from the further
application of eigen spectrum analysis to the fundamental
matrices of ENA (Caswell 2001). For example, what is the
ecological interpretation of the right and left eigen vectors of
the adjacency matrix A and the direct flow intensity matrix
G? Like the interpretation of L1(A) (Borrett et al. 2007) and
L1(G), we expect these mathematical components to be ecologically relevant.
Finally, additional work is required to connect our work
with the recent developments toward a time-varying ecological network analysis (Shevtsov et al. 2009). We expect
this development to make ENA/NEA more useful for ecosystem assessment and management, though it will still
require a relatively large and diverse data set.
Summary and conclusions
Our work shows that contrary to the opening quote by
Kundera, speed is not just the domain of humans. Rather,
we have shown that indirect flows rapidly exceed direct flows
in the extended path network of ecosystems. This result held
1146
for models with both trophic and biogeochemical assumptions. It also held true across seasonal variation in the Neuse
River Estuary and across the wet and dry seasons represented
in the ATLSS models. This evidence supports our first
hypothesis that indirect effects develop rapidly in ecological
networks. Thus, we conclude that the dominance of indirect
effects hypothesis remains plausible. Although there was no
consistent difference in mID between biogeochemically and
trophically based ecosystem models, we did find that the
biogeochemical models were generally more retentive of
energy–matter inputs, requiring more of the extended path
network to account for 50% and 95% of the total system
throughflow. While m50 and m95 were less in the trophic
models, they were still larger than the typical truncation of
the related Taylor power series as used in mathematics and
engineering. We argue that this is further evidence for
the complexity of ecological systems that we expect will continue to challenge our ability to properly assess and manage
ecosystems.
Acknowledgements – We would like to thank R. R. Christian, M. A.
Freeze, A. Stapleton and A. K. Salas for their constructive critiques
of manuscript drafts as well as K. R. Arrigo and members of the
UGA Systems Ecology Group and the UNCW Systems Ecology
and Ecoinformatics Laboratory for discussions and feedback on
ideas. This work was supported financially in part the Univ. of
North Carolina Wilmington.
References
Ainley, D. G. et al. 2006. Competition among penguins and cetaceans reveals trophic cascades in the western Ross Sea,
Antarctica. – Ecology 87: 2080–2093.
Allesina, S. et al. 2005. Ecological subsystems via graph theory: the
role of strongly connected components. – Oikos 110: 164–176.
Baird, D. and Ulanowicz, R. E. 1989. The seasonal dynamics
of the Chesapeake Bay ecosystem. – Ecol. Monogr. 59:
329–364.
Baird, D. et al. 1991. The comparative ecology of six marine
ecosystems. – Philos. Trans. R. Soc. Lond. B 333: 15–29.
Baird, D. et al. 1995. Seasonal nitrogen dynamics in Chesapeake
Bay – a network approach. – Estuar. Coast. Shelf Sci. 41:
137–162.
Baird, D. et al. 2008. Nutrient dynamics in the Sylt–Rømø Bight
ecosystem, German Wadden Sea: an ecological network analysis approach. – Estuar. Coast. Shelf Sci. 80: 339–356.
Bay, J. S. 1999. Fundamentals of linear state space systems. –
McGraw–Hill.
Berman, A. and Plemmons, R. J. 1979. Nonnegative matrices in
the mathematical sciences. – Academic Press.
Bever, J. D. 1999. Dynamics within mutualism and the maintenance
of diversity: inference from a model of interguild frequency
dependence. – Ecol. Lett. 2: 52–62.
Bever, J. D. 2002. Negative feedback within a mutualism: hostspecific growth of mycorrhizal fungi reduces plant benefit. –
Proc. R. Soc. Lond. B 269: 2595–2601.
Bolker, B. M. 2008. Ecological models and data in R. – Princeton
Univ. Press.
Bondavalli, C. and Ulanowicz, R. E. 1999. Unexpected effects of
predators upon their prey: the case of the American alligator. –
Ecosystems 2: 49–63.
Borrett, S. R. and Osidele, O. O. 2007. Environ indicator sensitivity to flux uncertainty in a phosphorus model of Lake Sidney
Lanier, USA. – Ecol. Modell. 200: 371–383.
Borrett, S. R. and Patten, B. C. 2003. Structure of pathways in
ecological networks: relationships between length and number.
– Ecol. Modell. 170: 173–184.
Borrett, S. R. et al. 2006. Indirect effects and distributed control in
ecosystems 3. Temporal variability of indirect effects in a sevencompartment model of nitrogen flow in the Neuse River Estuary (USA)—time series analysis. – Ecol. Modell. 194: 178–188.
Borrett, S. R. et al. 2007. Functional integration of ecological networks
through pathway proliferation. – J. Theor. Biol. 245: 98–111.
Carpenter, S. R. et al. 1985. Cascading trophic interactions and
lake productivity: fish predation and herbivory can regulate
lake ecosystems. – Bioscience 35: 634–639.
Caswell, H. 2001. Matrix population models: construction, analysis and interpretation. – Sinauer.
Christian, R. R. and Thomas, C. R. 2003. Network analysis of nitrogen inputs and cycling in the Neuse River Estuary, North
Carolina, USA. – Estuaries 26: 815–828.
Christian, R. R. et al. 1996. Nitrogen cycling networks of coastal
ecosystems: influence of trophic status and primary producer
form. – Ecol. Modell. 87: 111–129.
Dame, J. K. and Christian, R. R. 2006. Uncertainty and the use of
network analysis for ecosystem-based fishery management. –
Fisheries 31: 331–341.
Dame, R. F. and Patten, B. C. 1981. Analysis of energy flows in an
intertidal oyster reef. – Mar. Ecol. Prog. Ser. 5: 115–124.
Diekotter, T. et al. 2007. Direct and indirect effects of habitat area
and matrix composition on species interactions among flowervisiting insects. – Oikos 116: 1588–1598.
Fath, B. D. 2004. Network analysis applied to large-scale cyberecosystems. – Ecol. Modell. 171: 329–337.
Fath, B. D. and Borrett, S. R. 2006. A MATLAB© function for network environ analysis. – Environ. Modell. Soft. 21: 375–405.
Fath, B. D. and Patten, B. C. 1998. Network synergism: emergence
of positive relations in ecological systems. – Ecol. Modell. 107:
127–143.
Fath, B. D. and Patten, B. C. 1999. Review of the foundations of
network environ analysis. – Ecosystems 2: 167–179.
Fath, B. D. et al. 2004. Ecosystem growth and development. –
Biosystems 77: 213–228.
Fath, B. D. et al. 2007. Ecological network analysis: network
construction. – Ecol. Modell. 208: 49–55.
Finn, J. T. 1980. Flow analysis of models of the Hubbard Brook
ecosystem. – Ecology 61: 562–571.
Gattie, D. K. et al. 2006. Indirect effects and distributed control in
ecosystems: network environ analysis of a seven-compartment
model of nitrogen flow in the Neuse River Estuary, USA –
steady-state analysis. – Ecol. Modell. 194: 162–177.
Gleick, J. 1999. Faster: the acceleration of just about everything. –
Pantheon.
Godsil, C. D. 1993. Algebraic combinatorics. – Chapman and Hall.
Hall, S. R. et al. 2007. Food quality, nutrient limitation of secondary production, and the strength of trophic cascades. – Oikos
116: 1128–1143.
Heymans, J. J. et al. 2002. Network analysis of the south Florida
everglades graminoid marshes and comparison with nearby
cypress ecosystems. – Ecol. Modell. 149: 5–23.
Higashi, M. and Patten, B. C. 1986. Further aspects of the analysis
of indirect effects in ecosystems. – Ecol. Modell. 31: 69–77.
Higashi, M. and Patten, B. C. 1989. Dominance of indirect causality in ecosystems. – Am. Nat. 133: 288–302.
Hinrichsen, U. and Wulff, F. 1998. Biogeochemical and physical
controls of nitrogen fluxes in a highly dynamic marine
ecosystem – model and network flow analysis of the Baltic Sea.
– Ecol. Modell. 109: 165–191.
Jørgensen, S. E. 2002. Integration of ecosystem theories: a pattern.
– Kluwer.
Jørgensen, S. E. et al. 1999. Ecosystems emerging: 3. Openness. –
Ecol. Modell. 117: 41–64.
Jørgensen, S. E. et al. 2000. Ecosystems emerging: 4. Growth. –
Ecol. Modell. 126: 249–284.
Laland, K. N. et al. 1999. Evolutionary consequences of niche
construction and their implications for ecology. – Proc. Natl
Acad. Sci. USA 96: 10242–10247.
Lenzen, M. 2000. Errors in conventional and input–output-based
life-cycle inventories. – J. Ind. Ecol. 4: 127–148.
Lenzen, M. 2007. Structural path analysis of ecosystem networks.
– Ecol. Modell. 200: 334–342.
Mandelik, Y. et al. 2005. Issues and dilemmas in ecological scoping:
scientific procedural and economic perspectives. – Impact
Assess. Proj. Appr. 22: 55–63.
McCormick, M. I. 2009. Indirect effects of heterospecific interactions
on progeny size through maternal stress. – Oikos 118: 744–752.
Menge, B. A. 1997. Detection of direct versus indirect effects: were
experiments long enough? – Am. Nat. 149: 801–823.
Patten, B. C. 1971. A primer for ecological modeling and simulation with analog and digital computers. – In: Patten, B. C.
(ed.), System analysis and simulation in ecology. Vol 1. Academic Press, pp. 3–121.
Patten, B. C. 1978. Systems approach to the concept of environment. – Ohio J. Sci. 78: 206–222.
Patten, B. C. 1983. On the quantitative dominance of indirect
effects in ecosystems. – In: Lauenroth, W. K. et al. (eds),
Analysis of ecological systems: state-of-the-art in ecological
modelling. Elsevier, pp. 27–37.
Patten, B. C. 1984. Further developments toward a theory of the
quantitative importance of indirect effects in ecosystems. –
Verh. Ges. Oekol. 13: 271–284.
Patten, B. C. 1985a. Energy cycling, length of food chains, and
direct versus indirect effects in ecosystems. – Can. Bull. Fish.
Aquat. Sci. 213: 119–138.
Patten, B. C. 1985b. Energy cycling in the ecosystem. – Ecol.
Modell. 28: 1–71.
Patten, B. C. 1991. Network ecology: indirect determination of the
life–environment relationship in ecosystems. – In: Higashi, M.
and Burns, T. (eds), Theoretical studies of ecosystems: the network perspective. Cambridge Univ. Press, pp. 288–351.
Patten, B. C. 1998. Network orientors: steps toward a cosmo
graphy of ecosystems: orientors for directional development,
self-organization, and autoevolution. – In: Müler, F. and
Leupelt, M. (eds), Eco targets, goal functions and orientors.
Springer, pp. 137–160.
Patten, B. C. and Odum, E. P. 1981. The cybernetic nature of
ecosystems. – Am. Nat. 118: 886–895.
Patten, B. C. and Fath, B. D. 2000. The network variable in ecology: a partial account of Georgia systems ecology, with research sketches from the Okefenokee. – In: Barrett, G. W. et al.
(eds), Holistic science: the evolution of the Georgia Inst. of
Ecology (1940–2000). Gordon and Breach.
Patten, B. C. et al. 1976. Propagation of cause in ecosystems. – In:
Patten, B. C. (ed.), Systems analysis and simulation in ecology,
Vol. IV. Academic Press, pp. 457–579.
Patten, B. C. et al. 1982. Path analysis of a reservoir ecosystem
model. – Can. Water Res. J. 7: 252–282.
Patten, B. C. et al. 1990. Trophic dynamics in ecosystem networks:
significance of cycles and storage. – Ecol. Modell. 51: 1–28.
Patten, B. C. et al. 2002. Complex adaptive hierarchical systems. –
In: Costanza, R. and Jørgensen, S. E. (eds), Understanding and
solving environmental problems in the 21st century: toward a
new, integrated hard problem science. Elsevier, pp. 41–87.
Pilette, R. 1989. Evaluating direct and indirect effects in ecosystems.
– Am. Nat. 133: 303–307.
Schramski, J. R. et al. 2006. Indirect effects and distributed control in
ecosystems: distributed control in the environ networks of a sevencompartment model of nitrogen flow in the Neuse River Estuary,
USA – steady-state analysis. – Ecol. Modell. 194: 189–201.
Seber, G. A. F. 2008. A matrix handbook for statisticians. – Wiley.
1147
Shevtsov, J. et al. 2009. Dynamic environ analysis of compartmental
systems: a computational approach. – Ecol. Modell. 220:
3219–3224.
Stigebrand, A. and Wulff, F. 1987. A model for the dynamics of
nutrients and oxygen in the Baltic proper. – J. Mar. Res. 45:
729–759.
Strauss, S. Y. 1991. Indirect effects in community ecology – their
definition, study and importance. – Trends Ecol. Evol. 6:
206–210.
Straškraba, M. 1996. Ecology and environmental impact assessment. – J. Appl. Ecol. 33: 191–199.
Ulanowicz, R. E. 1984. Community measures of marine food networks and their possible applications. – In: Fasham, M. J. R.
(ed.), Flows of energy and materials in marine ecosystems: theory and practice. Plenum Press, pp. 23–47.
Ulanowicz, R. E. 1986. Growth and development: ecosystems
phenomenology. – Springer.
Ulanowicz, R. E. and Baird, D. 1999. Nutrient controls on ecosystem dynamics: the Chesapeake mesohaline community. – J.
Mar. Systems 19: 159–172.
Ulanowicz, R. E. and Puccia, C. J. 1990. Mixed trophic impacts in
ecosystems. – Coenoses 5: 7–16.
Ulanowicz, R. E. et al. 1997. Network analysis of trophic dynamics in
south Florida ecosystem, fy 96: the cypress wetland ecosystem. –
Annu. Rep. US Geol. Serv. Biol. Resour. Div. Ref. No. [UMCES]
CBL 97-075, Chesapeake Biol. Lab., Univ. of Maryland.
Ulanowicz, R. E. et al. 1998. Network analysis of trophic dynamics
in south Florida ecosystem, fy 97: the Florida Bay ecosystem.
1148
Annu. Rep. US Geol. Serv. Biol. Resour. Div. Ref. No.
[UMCES]CBL 98–123, Chesapeake Biol. Lab., Univ. of
Maryland.
Ulanowicz, R. E. et al. 1999. Network analysis of trophic dynamics
in south Florida ecosystem, fy 98: the mangrove ecosystem.
Annu. Rep. US Geol. Serv. Biol. Resour. Div. Ref. No.
[UMCES] CBL 99-0073; Tech. Rep. Ser. No. TS-191-99,
Chesapeake Biol. Lab., Univ. of Maryland.
Ulanowicz, R. E. et al. 2000. Network analysis of trophic dynamics
in south Florida ecosystem, fy 99: the graminoid ecosystem.
Annu. Rep. US Geol. Serv. Biol. Resour. Div. Ref. No.
[UMCES] CBL 00-0176, Chesapeake Biol. Lab., Univ. of
Maryland.
Webster, J. R. et al. 1975. Nutrient recycling and the stability of
ecosystems. – In: Howell, F. G. et al. (eds), Mineral cycling in
southeastern ecosystems. US ERDA, pp. 1–27.
Whipple, S. J. et al. 2007. Indirect effects and distributed control
in ecosystems: comparative network environ analysis of a
seven-compartment model of nitrogen flow in the Neuse
River Estuary, USA – time series analysis. – Ecol. Modell.
206: 1–17.
Wiegert, R. G. and Kozlowski, J. 1984. Indirect causality in ecosystems. – Am. Nat. 124: 293–298.
Wootton, J. T. 1994. The nature and consequences of indirect
effects in ecological communities. – Annu. Rev. Ecol. Syst. 25:
443–466.
Wootton, J. T. 2002. Indirect effects in complex ecosystems: recent
progress and future challenges. – J. Sea Res. 48: 157–172.