Ecological Modelling 179 (2004) 235–245
Distributed control in ecological networks
Brian D. Fath∗
Biology Department, Towson University, Towson, MD 21252, USA
Abstract
Understanding how “control” is exercised in ecological systems, even giving a more appropriate definition or meaning to
the word “control” in this context, is an important theoretical issue and would increase our ability to manage ecosystems.
Conventionally, in food web ecology, the distinction is drawn between bottom-up and top-down control. In that literature, the
bottom-up hypothesis asserts that the primary producers are the source of system regulation and the top-down hypothesis states
that keystone species at a higher trophic level can regulate the system. However, we know that in reality control of system
behavior is much more complex and distributed than this dichotomy would suggest. Indeed, there is an urgent need for a
succinct, yet more complete and comprehensive conceptual framework for thinking about control and for deriving insights into
what governs ecosystem organization. In an ecosystem, each element contributes to the overall flow-storage pattern observed in
the system through its interactions with the other elements; in this sense, control is distributed among the system elements. Those
pair-wise system interactions can be identified and quantified using network analysis. Since the network analysis methodology
accounts for both the input (recipient-oriented) and output (donor-oriented) influences from each element, it is possible to use this
methodology to move beyond the simple top-down and bottom-up perspective of control. Here, I connect the network analysis
methodology to traditional control theory, reintroduce a network-based control parameter using flow analysis and extend the
methodology to network storage analysis. Model ecosystems are constructed and used to investigate these properties.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Control theory; Distributed control; Ecosystem ecology; Network analysis
1. Introduction
Ecosystems exist as open, thermodynamic, far-from
equilibrium systems. As such, they depend on the
continual input of high quality, low-entropy energy.
This observation has led to the perspective that plants
control the energy gateway for ecosystems and therefore, bottom-up processes have primacy (White, 1978;
Power, 1992). Other studies have shown that higher
trophic level species can significantly affect community populations and thus they control the ecosystem
(Hairston et al., 1960; Paine, 1966, 1974). However,
∗ Fax: +1-410-704-2405.
E-mail address: [email protected] (B.D. Fath).
control of ecosystem behavior is distributed and therefore more complex than this dichotomy would suggest
(Hunter and Price, 1992). An approach recently developed by Getz et al. (2003) uses sensitivity analysis
to analyze control of trophic food chains. This begins
to address some of the complexities involved with
assigning ecosystem control but is limited to trophic
chains and does not address whole ecosystem interactions. Clearly, an ecosystem cannot exist without
energy driving it, but given the fact that ecosystems
are open thermodynamic systems, the question remains: After the initial energy input what controls
the distribution of the energy flow or storage within
the ecosystem? This is a different question regarding
control than would traditionally be addressed, but in a
0304-3800/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2004.06.007
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B.D. Fath / Ecological Modelling 179 (2004) 235–245
natural ecosystem there is no explicit controller, nor is
there an objective function. Instead, an ecosystem is
made up of many different elements each influencing
and being influenced by each other. Based on these
systemic interactions, the elements co-adapt to each
other and to their environment such that they actively
construct or engineer their particular niches (Jones
et al., 1997; Odling-Smee et al., 2003). The observed
energy flow or storage configuration in the ecosystem
is determined in part by each of the system elements
interacting together such that control is distributed
among the elements. Therefore, one way to consider
internal control in an ecosystem is the extent or degree
to which elements influence each other and contribute
to the system’s overall flow-storage pattern.
Having defined internal ecosystem control as being the role each element plays in contributing to the
overall flow-storage pattern, the challenge is to find a
method, preferably quantitative, that assigns a value
commensurate with that element’s network interactions. This value will be taken as a measure of its level
of control in the ecosystem. Bottom-up and top-down
perspectives attempt to measure these interactions, but
a shortcoming is that they often blur the fact that these
activities occur simultaneously as each element contributes to both the generation and reception of flow in
the ecosystem. A more general characterization is to
consider donor-originated (bottom-up-oriented) and
recipient-originated (top-down-oriented) influences,
respectively. In other words, the current bottom-up
and top-down perspectives are insufficiently holistic
to account for distributed control within the system.
A new measure is needed.
In ecological network analysis, an environmental
application (Hannon, 1973) of economic input–output
analysis (Leontief, 1966), it is common to consider
an ecosystem as a network of interacting elements
linked together primarily by the flow of energy or
material through the system. Each element acts as
both a receiver of input from other elements in
the network and as a generator of output to the
other network elements. The elements are connected to each other in the network through these
input and output environments. These unique within
system-boundary “environments” comprise the input environ and output environ, respectively (Patten,
1978b). Using this methodology, it is possible to
quantify the static flow dependencies of each indi-
vidual element in the network through its relative
input–output interactions. The three commonly applied network analyses consider the flow, storage, and
net flow dependencies (Fath and Patten, 1999). The
flow analysis was previously used to develop a control
parameter (Patten, 1978a; Patten and Auble, 1981).
The purpose of this paper is to reintroduce the holistic,
network analysis-based, flow environ control parameter and extend the methodology to storage network
analyses as well. The net flow analysis (utility analysis) may also have application as a control parameter
and is an area to consider further in future research.
2. Background to traditional control theory
Two key concepts of traditional control theory are
input and output reachability (Siljak, 1991). These
concepts relate to how a system can be both manipulated and detected. If the output of a particular compartment is received by another compartment, then the
first compartment is said to be observable. Likewise,
if the output of a system compartment falls within the
input environ of another compartment then the second
compartment is said to be controllable. Input reachability occurs when a path exists to every compartment
from at least one input, and output reachability occurs
when each compartment reaches at least one output.
Kalman (1963) developed the fundamental properties
of complete controllability and complete observability, summarizing that if, within any finite period of
time, any initial state of a given dynamical system can
be forced by inputs into the zero state as output, then
the system is completely controllable; and if, within a
finite period of time, the value of any prior state of the
system taken as input can be determined from outputs,
then the system is completely observable. Consider
the basic state transition and state response equations
for a linear dynamical system (Eq. (1)):
S : ẋ = Ax + Bz,
y = Cx
Rn
(1)
Here, S is the system, x(t) ∈
its state, z(t) ∈ Rm
k
its input, and y(t) ∈ R the output at time t ∈ R. A, B,
and C are constant matrices (Siljak, 1991). Note, that
the first equation in (1) corresponds to the state transition function and the second equation with the state
response function. Also note that u(t) is generally used
in control theory literature for controllable inputs, but
B.D. Fath / Ecological Modelling 179 (2004) 235–245
z(t) is consistent with the notation in network environ
analysis. The conditions for complete controllability
and observability are
rank[B AB . . . An−1 B] = n
(2)
rank[C CA . . . CAn−1 ] = n
(3)
Computing complete controllability for large systems
can be difficult; a more robust and simpler test exists
to determine structural controllability (Siljak, 1991).
When considering structural controllability, input and
output reachability are based only on the system
connections and not the parameter values. Parameter
values may alter conclusions from structural controllability, but the test does allow one to decide if a
system structure can admit to controllability (Siljak,
1991). Regarding input and output environs, a component is at least structurally controllable if part of the
input environ of the controlled component emanated
from the output environ of another component, in this
case one acting like the “controller.” For structural
control to occur, a controller must reach the input
environ of the component it is controlling. Similarly,
a component is structurally observable if its output
environ reaches the input environ of another component acting as a controller (Fig. 1). Note though that
since the “controller” is another system component,
each component ultimately plays the role of observer
and controller for those components within its input
and output environs, respectively. Therefore, each
object exhibits some degree of control over the others. The dual input–output orientation of environs fits
nicely with the duality between controllability and
observability.
237
Another important concept of traditional control
theory is vulnerability. A control system is vulnerable
if the removal of a connection from the corresponding digraph destroys input reachability (Siljak, 1991).
This has important implications for structural changes
within a system because a structural change is one in
which the connections of the system are altered. In an
ecosystem, structural changes can be simple seasonal
variations or can be the result of longer term perturbations such as a change in feeding preference or even
loss of biodiversity. An important question arises: Is
the system still input and output reachable given certain structural changes? A theory has been developed
to identify the minimal set of connections between
components which are essential for preserving input
reachability and structural controllability of a system
(Pichai et al., 1981).
The standard approach to control theory is based
on the cybernetic processes of error generation, error
detection, and error correction. Without errors, there
can be no correction process, and hence, no control.
The correction process requires that the desired state
be reachable. This implies there is a way to detect
errors, therefore the state must be observable. In
network terminology, for control to be effective, the
system must be connected to the output environ of
the controller in such a way as to achieve reachability,
and to the input environ of the controller in such a
way as to achieve observability. When a component
can both observe and control another component a
feedback control structure exists (Fig. 2).
In traditional control theory it is also necessary to
know the “desired” state of the system because error
implies that there is a preferred or reference state, goal,
or direction from which the system has deviated. It is
Fig. 1. Environ representation of a controller demonstrating observability and reachability.
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B.D. Fath / Ecological Modelling 179 (2004) 235–245
Fig. 2. Two objects connected observing and reaching each other through the input and output environs. This is the essence of distributed
control in ecosystems since the object and controller are conceptually and functionally interchangeable.
also possible that there are multiple pathways to reach
a desired goal and that a seemingly local deviation
may in fact still globally lead to the same preferred
outcome. In this case, the concern may be to reach
the long-term goal optimally by minimizing whatever
objective function is pertinent to the control manager.
The goal function is clear in human engineered systems such as automatic pilots or thermostats. As long
as the system is operating within an accepted range, no
corrective action is taken by the controller. But, when
the system deviates (the nose of the plane dips or a
room becomes too cold) a detector is alerted which, if
operating properly, proceeds to compensate for the deviation. The three properties: input reachability, output
reachability, and goal function optimization criterion
are all necessary ingredients for a control system.
3. Ecosystem controller
In ecosystems, the concept of control is much more
tenuous that that described above for several reasons.
First, if there is a preferred state or goal, then it is usually unclear what it is. At best we could observe how
an ecosystem changes over time, such as seasonally or
during succession, but that does not mean there is an
explicit goal. At the macroscale, we promote sustainability as a desired goal, but that is an anthropocentric
concern wrought with ambiguity. This is not to deride
its importance, in fact, it is the encompassing challenge we face, however, there is no easy bridge across
the many levels of organization from organisms to
ecosystems to eco-human complexes. Focusing again
on ecosystems, it is impossible to know for sure what
is the goal of ecosystem development. There are several theories on the subject, which can be divided into
two broad categories: individual or energetic oriented
goals. Odum (1969) introduced a set of properties that
fall into both of these categories. The individual oriented hypotheses place the emphasis on the survival
and competitive nature of the organisms as the overriding process in ecosystem development. The primary,
and perhaps only, goal of the individual is to differentially out procreate its competitors as defined by its
fitness. The energetics hypotheses focus instead on the
energy gradient, which is established within an ecosystem and the processes that occur to sustain or degrade
it. Often the response to the gradient can be observed
in certain emergent ecosystem properties such as maximizing energy throughflow (Lotka, 1922; Odum and
B.D. Fath / Ecological Modelling 179 (2004) 235–245
Pinketon, 1955), maximizing production over respiration ratio (Odum, 1969), maximizing energy degradation (Schneider and Kay, 1994), maximizing energy
storage (useful energy) (Mejer and Jørgensen, 1979),
maximizing ascendancy (Ulanowicz, 1986), or minimizing specific dissipation (Prigogine, 1980). These
energetic-based goal functions are discussed in more
detail in Fath et al. (2001). These two concepts of
individual- and thermodynamic-based goal functions
are not necessarily mutually exclusive. It may be possible someday to link these two hypotheses into a unified theory encompassing the thermodynamics of the
individual because it is the existence of the thermodynamic gradient which gives rise to the individual’s behavior. The individual, which has arisen as a long-term
response to the gradient, could be explained in terms
of energetics, but this has not been accomplished. Ultimately, however, a methodology to explain ecosystem control would have to be connected with the energetics driving the system.
In addition to the uncertainty of the ecosystem goal,
the second problem with applying traditional control
techniques to ecological systems is that it is not obvious which elements, if any, act as the “ecosystem
managers” guiding the direction of the system. Rather,
as stated earlier, the system develops and unfolds as
a result of all the transactions between interacting
elements. Do the primary producers act as controllers
by their limitation in photosynthetically capturing solar radiation? Do the top predators control the system
by regulating energy flow through the system with
their own feeding preferences? Whatever the specific
mode of operation in ecosystems, we see here that
system control is derived from within the system and
as a result of component interactions. From a control
point of view, there is no separate objective controller,
therefore there is no objective method for error detection. Control is decentralized. Components, through
their input and output environs, both observe and
reach other elements. Each component is influenced
by components in its input environ, and influences
components with its output environ as part of the
network. Each component in the system has a role
to play in the overall control of the system behavior.
Here, the degree to which a compartment contributes
to system control is determined by considering the
direct and indirect influence each compartment has
on the transactions between compartments. Network
239
analysis can use these transactions to investigate
control within energy-storage models.
There have been earlier attempts to consider ecosystem control. For example, Straškraba and Gnauck
(1985) blended traditional control theory with ecosystems and proposed three types of ecosystem control:
(1) internal natural control, (2) external natural control, and (3) external control by management. They
are discussed here in reverse order. The third type of
control, external control by management, requires an
anthropogenic controller and goal function, both of
which we identified as lacking in a natural system.
This control type is appropriate for quasi-natural systems. Wastewater treatment plants and agricultural
fields are good examples of quasi-natural, human
controlled biological systems in which external control by management could be applied. Since we are
interested primarily in the internal interactions of
natural systems, external control by management is
not relevant here and is not considered. An argument
could be made that there are no “natural” systems
and that all ecosystems are at least indirectly human
controlled because of our heavy footprint, but this
position is not taken here.
The second type of control, external natural control,
occurs when the system is dominantly influenced by
events that, by definition, originate outside the system
boundaries. For example, the work of Polis (Polis and
Hurd, 1996; Polis et al., 1997) on the desert islands
off the Baja Peninsula has shown that allochthonous
inputs are the main driving force in the system. In
terms of energetics, the energy gradient on these islands is largely established by natural external inputs.
They found that the smaller islands, which have greater
coastline exposure, have greater productivity. This is
an example of external natural control on the island
energetics because the local island production is controlled by the oceanic inputs. Other examples include
aquatic systems that receive significant detrital matter
such that the decomposer food webs are pronounced.
External natural controls are usually seasonal or climatic events such as fire, El Niño, hurricanes, volcanoes or other unpredictable activity. This concept of
external controls is boundary dependent because the
system could be enlarged to explicitly include the important “external” factors.
The third type of ecosystem control, internal natural
control, refers to the inherent interaction and influence
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B.D. Fath / Ecological Modelling 179 (2004) 235–245
of all the connections (structure) and processes (function) that occur naturally between the components
within any ecosystem. In network analysis, system
structure refers to the pathways over which transactions take place while function is the quantitative exchange along these pathways. System transactions are
necessary and sufficient to determine the internal natural control as herein defined. Internal natural control
is most relevant to the distributed control aspects we
are interested in answering. Therefore, the rest of this
paper deals exclusively with internal natural control.
tems may try to maximize energy storage (Mejer and
Jørgensen, 1979) because they would be less likely to
be controlled by other components if they had built up
greater stores of energy. There are circumstances such
as in inverted food pyramids where a high turnover
rate allows a lower standing stock to control a higher
standing stock. The flow and storage control methods
are illustrated first in a simple three-component chain
model and then later with the model of an oyster reef
community (Dame and Patten, 1981).
5. Food chain model
4. Control methodology using network analysis
The initial step in any type of network analysis is
to conceptualize and quantify a model that represents
the transactions in the system. Once this transactive
or transfer model is created, then the various network
analyses can be implemented. Flow and storage analyses each give different information regarding the information. In flow analysis, the direct transfers between
components are normalized by the total throughflow
at each component. In storage analysis, the within system transfers are normalized by the storage at each
component. Although we work primarily with energy
flow models, this methodology is adaptable to any
steady-state network flow model that has consistent
units throughout. Because network analysis is applied
to steady-state systems, we are looking at a snapshot
of the cumulative behavior of the system, which shows
the degree to which each component is responsible for
the system reaching a particular configuration. These
static flow dependencies can be used to measure how
control is distributed among the network components.
Two network analyses, flow and storage, are each
used to calculate the internal dependence of one component on another in a flow-storage compartmental
model. The flow methodology is based on the transfers through the system and it uses each component’s
flow-based input environ and output environ (Patten
and Auble, 1981). In the second method, the contribution of control is related to flow derived from storage
within each component. Therefore, assuming constant
turnover rates, components with greater standing stock
are less affected by the “controlling” flows (influences) of the other components with which they interact. This approach reinforces the hypothesis that sys-
The first system used to demonstrate the control
measures is a three-component food chain. The flows,
fij , from j to i are given in Eq. (4):


0
0
0 100
 30 0
0
0 


F=
(4)

 0 10 0
0 
70
20
10
0
The fourth column is the input into the system from
the environment and the fourth row is the transfer from
the system component’s to the external environment.
The dimensions of flow are mass per area per time
(ML−2 T−1 ). The storage values for this system are
taken to be: x = [200 100 50] with dimensions mass
per area (ML−2 ). In a food chain model, all the flow
up to the next higher trophic level comes directly from
the previous component, and we can see from the input
vector that the only input into this model is into the
first component. All three components have an output
to the external environment.
5.1. Integral component dependence based on
flow analysis
We want to determine a way in which meaningful
information can be elicited about the dependence relationship between pair-wise components in a connected
system. This can be achieved by looking at the contribution of each component to the other component’s
input and output environs, respectively (Patten, 1978a;
Patten and Auble, 1981). Patten and colleagues determined that if component x1 is less important (lower
valued) in the input environ of x2 than it is in the output
B.D. Fath / Ecological Modelling 179 (2004) 235–245
environ of x2 , then it can be said that x1 is dependent
on x2 . In other words, x1 ’s output has less influence
on x2 than x1 receives as input from x2 . Therefore,
x1 is said to be dependent on or controlled by x2 .
Although a connection could be made to top-down
and bottom-up by interpreting the influence the receiving component has on the output environ of the
donor as top-down, and the influencing the generating
component has on the input environ of the receiver
as bottom-up, the top-down and bottom-up labels
give a simplistic view of the element interactions in
reticulated systems. It is through this influence and influencing that each element acts to control the overall
flow distribution in the network. The network-based
control measure is demonstrated in an example below.
The calculation of the network input and output
environs is explained in detail elsewhere (Matis and
Patten, 1981; Patten, 1981, 1982; Fath and Patten,
1999). The brief summary provided here follows from
these earlier developments. For the output environ, the
first step is to calculate the generating or flow-forward
transfer efficiencies, G = gij = fij /Tj , and the receiving or flow-backward transfer efficiencies, G = g ij
= fij /Ti . G and G give the direct flow-forward and
flow-backward efficiencies in the system and are used
to calculate the integral flow matrices. The integral, or
transitive closure, flow matrices, N and N , are given
in Eq. (5), where I is the identity matrix of similar size.
241
For the three-component chain example in Eq. (4), the
transitive closure matrices are


1
0
0


1
0
N =  0.30
(6)
0.10 0.33 1


1 0 0


N =  1 1 0 
(7)
1 1 1
important. The diagonal elements represent the total
storage that occurs in that particular environ. Since the
diagonal elements of the environs are taken directly
from the transitive closure matrices, it is not necessary
to calculate the individual environs as a preliminary
step to the control matrix, but the environ concept is
introduced here because it is critical to understanding
the control measure. As stated above, Patten (Patten,
1978a; Patten and Auble, 1981) used the input and
output environ concept to develop a control matrix as
the ratio of the elements of the output and input integral matrices. Specifically, they defined a control matrix, CX, where cxij = nij /nji (note, they used C in the
original paper, but I use CX so as not to confuse it
with the C matrix used in the storage analysis below).
Values in CX range from 0 to infinity. In this formulation, values on the interval [0,1] represent situations
in which compartment xi controls xj (with 1 corresponding to no control, 0 to total control), whereas on
the interval [1, ∞), xj controls xi in direct proportion
to the value of cxij (Patten and Auble, 1981). Patten
and Auble (1981) determined that expressing control
on the interval [0,1] was preferable, and took values of
1−cxij (for cxij ≤1, without this condition, which was
not stated in the original 1981 paper, one could obtain
negative control values) to denote strength of control
with 0 being no control and 1 being total control. In
most, but not all cases in the their model, control was
donor-controlled, so the resultant matrix typically had
the control values expressed above the major diagonal or can be represented graphically the direction in
which the control was exerted. Here, all results are
presented using the rescaled matrix, cnij = 1−cxij (for
cxij ≤1). Using this methodology, the rescaled control
matrix for the food chain example is given in Eq. (8):


0 1 1


CN =  − 0 1 
(8)
− − 0
The procedure to calculate the ith output environ, Ei ,
is to multiply the transfer efficiency matrix, G, by a
square matrix with the ith column of N on the diagonal. The input environ, Ei , is similarly calculated,
but now the diagonalized matrix of the ith row of N
is multiplied by G with the diagonal elements of Ei
taken from the ith column of N . Matrix multiplication is not commutative, so the multiplication order is
Here, the dependence of x2 on x1 (cn12 = 1) is
100%. Similarly, the dependence of x3 on x1 (cn13
= 1) and x2 (cn23 = 1) is 100%. If either x1 or x2 were
removed from the system then x3 would not receive
any flow. The first component is a gate that flow must
pass through in order for it to reach x2 and x3 . If x1 is
removed, i.e., the gate is closed, then the other components could not receive flow and ultimately could
N = (I − G)−1 ,
N = (I − G )−1
(5)
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B.D. Fath / Ecological Modelling 179 (2004) 235–245
not survive. Clearly, if the primary producers were removed from a system then all other parts dependent
on them could no longer survive. This control measure
biases the donor-oriented or bottom-up forces. In the
food chain model, there are no cycles or feedback, and
so the dependence is entirely bottom-up. The originator of the flow has dominance over all higher trophic
level components. The zeros along the diagonal of CN
indicate that each component is equally important in
its output environ as it is in its input environ (nii = nii ).
5.2. Integral component dependence based on
storage analysis
A second control measure is developed using the
storage environs. As stated above, network storage
analysis uses a ratio of the inter-compartmental flows
and the storage values. Matis and Patten (1981) developed output and input storage environs using the same
techniques as those used above for the flow environs.
Patten (Patten, 1978a; Patten and Auble, 1981) did
not investigate the use of storage environs as a control
parameter. That extension is presented here using the
same basic approach introduced above with the flow
environ control parameter.
The combination of output and input storage environs are used to develop another control parameter.
The output environ is based on the fraction of flow out
from the total standing stock of the donating component, C = cij = fij /xj , and the input storage environ
is based on the fraction of flow out from the standing
stock of the receiving component, C = cij = fij /xi .
The storage dimensions are different than the flow dimensions, so the elements of C and C are not dimensionless. An intermediate step is needed before the
transitive closure matrices can be calculated. The elements are non-dimensionalized by multiplying C and
C by a characteristic time step “dt” and adding the
identity matrix. The time step is specifically chosen to
scale all the elements of C and C between 0 and 1
(Barber, 1978). Selection of an appropriate time step
also ensures that the storage power series converges,
and that the integral storage values can be found. The
diagonals of these new matrices are set equal to zero
and the integral storage or transitive closure matrices,
Q and Q , are calculated. The matrices Q and Q are
used to calculate the output and input storage environs
in an analogous procedure to the flow environs. Now,
the ith output storage environ, MEi , shows the distribution of one unit of storage into the ith component
and the ith input storage environ; and, MEi , shows
the distribution of storage needed to create one unit
of output at the ith component. The control measure
for the storage environs is given by cqij = 1−qij /qji .
For the food chain model, using dt = 1, the storage
transitive closure matrices and control matrix are as
follows:


2
0
0


Q =  1 3.33 0 
(9)
0.5 1.67 5

2

Q =  2
2

0

CQ =  −
−
0
0

3.33
3.33

0
5
1
0
−

1

1
0
(10)
(11)
The storage control matrix looks the same as the
flow control measure. This will in fact always be the
case because the flow and storage mappings are related
through turnover rates (Higashi et al., 1993; Patten and
Fath, 1998). Therefore, the flow and storage ratio control measure will give identical results. In this sequential food chain model, control in the flow and storage
analyses is entirely donor controlled. In other words,
all the storage (biomass) in the higher trophic levels
was once storage (and then flow) from the preceding
components ultimately originating with the primary
producers. In other words, downstream components
are completely dependent on the upstream components for both flow and storage in the food chain
example. In systems with feedback and cycling, such
as the one below, the downstream dependency is not
absolute.
6. Oyster reef model
A second, ecologically derived, example is given
which is based on a model of an oyster reef community (Dame and Patten, 1981). Only the flow matrix
(Eq. (12)) with input and output on the last column
B.D. Fath / Ecological Modelling 179 (2004) 235–245
243
and row, storage values (Eq. (13)), and control matrices (flow and storage) are presented (Eqs. (14)–(15)).


0
0
0
0
0
0 41.47


 15.79 0
0 4.24 1.91 0.33 0 




0
0
0
0 
 0 8.17 0



0
0
0 
F =  0 7.27 1.21 0



 0 0.64 1.21 0.66 0
0
0 




 0.51 0
0
0 0.17 0
0 


25.16 6.18 5.76 3.58 0.43 0.36
0
(12)
x = 2000 1000 2.4 24.1

0 1.00 1.00 1.00

0.80 0.44
− 0

− −
0
−

CN = 
− −
0.15 0


−
−
− −

− 0.47
0.36 0.44
0 1.00
1.00 1.00

− 0


− −
CQ = 
− −


− −

− 0.47
0.80 0.44
0
−
0.15
0
−
−
0.36 0.44
16.3
69.2

1.00 1.00

0.21
− 

0.35
− 


0.25
− 


0
0.74 
−
(13)
(14)
Fig. 3. Flow or storage derived control diagram (arrows are in
direction of control such that arrow from x1 to x6 indicates that
x1 controls x6 with strength 1.00).
0
1.00 1.00


− 


0.35 − 

0.25 − 


0 0.74 

−
0
0.21
(15)
As stated above, the flow and storage control matrices are identical. The flow and storage control
relationships for the oyster reef model are shown in
Fig. 3. In this model, the first component has complete
control over the rest of the components cn1i = cq1i
= 1 for i = 2, 3, 4, 5, 6. All input enters this component and, this is unreachable by flow from the other
components, so the influence of the other components
are not contained in x1 ’s input environ. However, all
of the other components are in its output environ.
Since no flow cycles back from the other components, it cannot be controlled by the other components
based on this uni-directional, flow-only metric. In
most cases the control is exerted as donor-oriented,
although note that the predator compartment (x6 ),
exerts recipient control on x2 -deposited detritus (cn62
= cq62 = 0.47), x3 -microbiota (cn63 = cq63 = 0.36),
and x4 -meiofauna (cn64 = cq64 = 0.44). The only
other example of recipient control in this model is the
control the x4 -meiofauna exerts on the x3 -microbiota
(cn43 = cq43 = 0.15). These recipient-oriented control relationships arise out of the feedback loops that
occur in the model. Also note, all components are
completely observable because the environment acts
as an absorbing state (Kemeny and Snell, 1960).
7. Maximum donor and recipient control
Finally, it is possible to use network control concepts to identify the system configurations that exhibit
the greatest donor and recipient control. Maximum
donor control is found in the sequential food chain
because all the flow to the next component originates from the preceding component (Fig. 4). In other
words, the lowest trophic level generates a pervasive
output environ in which all other components are
dependent. Maximum recipient control occurs in a
244
B.D. Fath / Ecological Modelling 179 (2004) 235–245
Fig. 4. A linear food chain has maximum bottom-up control since
each component is dependent on the previous for its flow source.
Fig. 5. A sink with many inputs has maximum top-down control
since each source has little impact on the sink.
system with one component acting as a recipient of
flow from all the other components (Fig. 5). Here,
the recipient spans a pervasive input environ, such
that all other components contribute to it. Since there
are many inputs to the receiver, no one component’s
input has strong control over it. While these network
structures are unrealistic and idealized, they do provide theoretical endpoints to which other networks
can be compared for their relative strength of donor
and recipient control. In addition, the linear food
chain is used ubiquitously in the ecological literature
and it is useful because any complex network can be
unfolded into a set a food chains of various lengths
(Higashi et al., 1989, 1992).
8. Conclusions
Each ecosystem component has a dual role as a
receiver and generator of transactions and as such is
affected through its input environ and affects its surroundings through its output environ (Patten, 1981,
1982). It is the combination of these environs that
gives the system its characteristic flow-storage configuration. Therefore, the interacting network of all
its separate components controls the ecosystem. Control in ecological systems is, in this sense, distributed.
Energy and matter within an ecological system are
pushed “up” by the energy gradient and pulled to the
“top” by the feeding activity of the predators. Static
flow dependencies that capture these donor and recipient processes can be quantified using network analysis. These values represent the control each component
exerts in the overall system configuration. Network
flow and storage analysis were used to investigate
ecological concepts of donor and recipient control.
Although incorporating both the input and output
environs, the flow and storage methodologies are biased toward donor-oriented control. Therefore, the
higher trophic levels are typically dependent on or
controlled by the lower ones in this methodology. This
makes intuitive sense because the energy capturing
processes create the gradients and allow the network
to exist as a far-from equilibrium, self-organizing
system. But, when feedback and cycling are present,
as was observed in the oyster reef model example, the
donor control is not absolute because higher trophic
level recipient-activated processes can control lower
trophic levels in the system. In conclusion, the control within an ecosystem is decentralized among the
various components. Network analysis is an effective
way of quantifying the static flow dependencies of
both donor and recipient-oriented processes to quantify the control each component has in maintaining
the ecosystem’s overall flow-storage configuration.
Acknowledgements
This research was supported in part by a grant
from Towson University Faculty Development Office
and was prepared while the author was visiting research scientist with the Dynamic Systems Project at
the International Institute of Applied Systems Analysis. The author is indebted to Bernard Patten for
his comments and feedback on multiple versions of
the manuscript and also to Christopher Pawlowski,
Stuart Whipple, and Stuart Borrett for their thorough
reviews and suggestions.
B.D. Fath / Ecological Modelling 179 (2004) 235–245
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