El:OLOfilrm
HODELLIHG
ELSEVIER
Ecological Modelling 95 (1997) 301 310
Residence time of matter and energy in econetworks at steady
state
Bo-Ping Han*
Institute of Hydrobiology, Jinan University, Guangzhou 510632, PR China
Received 25 July 1995; accepted 2 July 1996
Abstract
In consequence of interactions between compartments, the matter or energy residence time in an econetwork is in nature
distinct from that in a compartment. Based on the analysis of econetwork structure, a strategy is developed to calculate the
matter or energy residence time in a general econetwork and the effects of self-, direct- and indirect interaction on
econetwork residence time. Two typical examples are used to illustrate the strategy, the results show that total residence time
equals the ratio of total standing stock to total system outflow or total system inflow instead of the ratio of total standing
stock to total system throughput. O 1997 Elsevier Science B.V. All rights reserved
Keywords: Residence time; Interaction; Econetwork
1. I n t r o d u c t i o n
Networks are very convenient representations of ensembles of transactions, such as that might occur in
economic systems, neural systems or ecosystems. Their basic units are compartments with relatively independent structures and functions. The matter and energy flow networks of ecosystems are termed ecological
networks, simply, econetworks (Han, 1993a; Han, 1993b). There has been substantial theoretical interest in
econetwork analysis for the past two decades. In the early 1970's, Hannon (1973) applied the Leontief
input-output model to analyze the energy flow in ecosystems. The development of flow network analysis might
be ascribed to the research work finished by Patten et al. (Patten et al., 1976; Ulanowicz, 1979; Higashi et al.,
1989). The flow analysis of econetwork is an attempt to understand the comprehensive characteristics of matter
and energy flow in an econetwork. Cycling efficiency and residence time are two significant measures showing
the comprehensive characteristics (Hannon, 1979). There have existed many literatures on cycling efficiency
(Finn, 1976; Patten, 1984). Residence time is the time for particles such as matter and energy to stay at an
econetwork until its leaving out of the econetwork. It depends on econetwork structure and its state. The
residence time in an econetwork is greatly distinct from that in a compartment which the past researches mainly
focused on (Herendeen, 1989). The difference above results from the interactions between compartments in an
econetwork. Some researches have shown that residence time is related to the measure of ecosystem stability
* Fax: + 86-20-85516941.
0304-3800/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved.
PH S 0 3 0 4 - 3 8 0 0 ( 9 6 ) 0 0 0 4 4 - 0
302
B.-P. Han/ EcologicalModelling95 (1997)301-310
(DeAnglis, 1980; Voris and O'Neill, 1980). In the present paper, we intend to explore the strategy to calculate
the residence time of matter or energy in an econetwork at steady state.
2. Structure analysis
To appreciate the effects of interactions in an econetwork, we give some definitions and make the analysis of
econetwork structure at first. In an econetwork, a compartment denotes a functional holon such as a population
or a community. The set of all compartments in an econetwork is defined as compartment space (CS), every
compartment is denoted by a number, CS = (0, 1, 2 . . . . . N ) indicates that an econetwork consists of N + 1
compartments. For an econetwork, there follows N > 2, i.e, there are three compartment at least. Let Pi~ (i,
j ~ CS) be the transition probability from compartment i to j. According to stochastic process theory, some
definitions and propositions yield as follows.
2.1. Classification of compartments
Definition 2.1: Pu(k) is the k step transition probability of particles from i to j, if there exists k > 1 and
Pu(k) > 0, then the particles from compartment i can attain compartment j, expressed as i o j . If Pu(k) = 0
when k > 1, then i ~ j.
Definition 2.2: If i ~ j and j ~ i, then i and j are commutative, expressed as i o j . Element gij is the total
transition probability for particles from compartment i to j over all time. One gets that
gij = E Pij(k) O < P i j ( k ) ~ g i j
(2,1)
k=l
In particular, when j = i, ggi is called the recurrent probability from i to i.
Definition 2.3: If there exist ~'.iecsgij=O and ~,~csgji ~ O, then compartment j is termed a source
compartment (SC), i.e., it only has outflows.
Definition 2.4: If there exist Y"ia cs gu :~ 0 and E i ~ c s gyi ~ 0, then, compartment j is termed a transient
compartment (TC), i.e, it has both inflows and outflows.
Definition 2.5: If there exist Ei s c s gu -~ 0 and Ei ~ cs gj~ = 0, then, compartment j is termed a mergence
compartment (MC), i.e, it has inflows only.
For any econetwork, therefore, its compartment space can fall into three subspaces, source compartment
space (CS s), transient compartment space (CS T) and mergence compartment space (CS M)" For convenience, we
assume that CS = {0, 1, 2 . . . . .
N}, CS T = {1, 2 . . . . .
n}, CS M = {n + 1, n + 2 . . . . .
N}, and all source
compartments are merged into compartment 0, i.e., CS s = {0}.
2.2. Classification of paths
A path is a sequence of compartments and links between compartments, e.g., ~ i ~ j ~ g --* h, it shows the
flow way of particles in an econetwork, if a path contains k compartments, the length of path is equal to k - I.
Definition 2.6: If a path contains no repeating compartments, then it is a simple path, e.g., --->i ~ j --->k ~ ,
otherwise, a compound path, e.g., ~ i ~ k ~ j ~ k --->h --->.
Definition 2.7: If its originating and terminating compartments are the same, then the path is a cycle path.
When a cycle path contains no other repeating compartments except for its originating and terminating
compartments, it is a simple cycle path, otherwise, a compound cycle path.
Proposition 2.1 (the path decomposition law): In an econetwork, all paths can be decomposed into simple
paths and compound paths, and all compound paths can fall into simple paths and simple cycle paths (Hail,
1993a; Han, 1993b).
Definition 2.8: If an econetwork contains no compound paths, then, it is termed a simple econetwork;
otherwise, a compound econetwork.
B.-P. Hart/Ecological Modelling 95 (1997) 301-310
303
3. The first transition matrix Q! and the first structure matrix M I
For an econetwork, its CS = {0, 1, 2 . . . . . N}, and CS s = {0}, C S T = {1, 2 . . . . . n}, CS M = {n + 1, n + 2,
. . . . N}, If i ~ j , let lij(k) be the minimum number of times that particles from i reach j by the kth step, one
reads that
[ij(k) =
{10 P~J(k) 4~O
Pij(k) = 0
(3.1)
Let Pit
be the one step transition probability from i to j, Pij(k) the k step transition probability, we define the
matrix Q, which element (i, j) is Pij(i, j ~ CS T) as the first transition matrix. Let mij be the expect value of
lij(k), there follows
9C
~C
mij= ~-~ Pij(k) "lij(k)=
k=t
E Pij(k)
(3.2)
k=l
The element mij represents the cumulative transition probability for particles from i to j. In particular, the mii
are related to the cycling times. The matrix M I, for which the (i, j)th element is mij, is termed the first
structure matrix. For a steady state econetwork, we find that
Mj = ( I -
(3.3)
QI) -1
where I is a unit matrix. The structure matrix (M I) was first introduced by Hannon (1973). Patten et al. (1976)
called it 'transitive closure matrix'. The structure matrix, as an important concept of econetwork analysis,
presents the total interactions between compartments.
4. The residence time in a compartment
At steady state, single compartment residence time may be defined in two equivalent ways, (1) the average
time that particles of energy and matter placed in a compartment remain there before leaving the compartment,
(2) the average time that particles now in the compartment have been continuously there. The compartment
residence time in a steady state ecosystem is
= xi/TOF i
TOFi= ]~ Fij
(4.1)
RT,
(4.2)
jECS
Where x i is the standing stock of compartment i, TOF i, the total outflow of compartment i, Fij the flow from i
to j. At an unsteady state, the residence time is strongly dependent on the dynamics of econetwork.
Accordingly, we define the unsteady state residence time as
RT i = ( x
where
Xi
i+2 i.At)/TOF i, i ~ C S
T
(4.3)
is the increase rate of stock x v
5. The residence time of matter and energy in an econetwork
The particle residence time in an econetwork is different from that in a single-compartment because of the
interaction between compartments. Generally, the total residence time of particles in an econetwork depends on
three ways: (1) the state characteristics of econetwork, (2) the flow strength of particles, (3) the structure of
econetwork.
B.-P. Han / Ecological Modelling 95 (1997) 301-310
304
5.1. The particle residence time in a simple econetwork
Fig. la is a simple econetwork, i.e., it has no cycle path, there, CS s = {0}, CS T = {1, 2, 3}, CS M = {4}. RT~ is
the particle residence time in compartment i, i E CST, the total residence time (TRT) of a particle in the simple
econetwork follows
TRT = P01 "RTI + P01 ' Pl2 "RT2 + P01 "Pi2 "P23 " RT3
(5.1)
To generalize, the paths shown in Fig. la can be decomposed into two simple paths (Fig. lb and c), the total
residence time in the simple econetwork is the sum of the residence time in the terminating transient
compartments of all available paths (RTSP), and in the originating compartments of all available paths
(RTOCi), therefore Eq. (5.1) can be rewritten as
2
1
TRT = Y'. RTSP, + Y'. RTOC,
i=1
(5.2)
i=1
where
2
E RTSPi = Po, "P,2" RT2 + Po, " P l 2 ' P 2 3 " RT3
(5.3)
i=1
1
Y~ RTOCi = eoj "RTI
(5.4)
i=l
For any simple econetwork, the total number of available simple paths is NSP. the total numbers of available
originating compartments that its input fo~ ~ 0 is NOC, then a generalizing expression immediately results from
NSP
NOC
T R T = Y'~ RTSPi+ Y'~ RTOCi
i=l
(5.5)
i=l
5.2. The particle residence time in a compound econetwork
In the econetwork shown in Fig. 2, the path, ~ 1 ~ 2 ~ 3 ~ 1 ~ , is a cycle path, so the econetwork is a
compound econetwork. Therefore all paths in the econetwork can be decomposed into simple and compound
paths. There are two available simple paths, i.e., --* 1 ~ 2 ~ and --* 1 --* 2 ~ 3 --*. Theoretically, there are
numberless compound paths by virtue of cycling process and interaction. In a compound econetwork, let NSP
be its total number of available simple paths, NCP its total number of available compound paths except for cycle
paths, NOC its total number of available originating compartments, RTSPi is the residence time that a particle
(a)
(b)
(c)
Fig. 1. (a) is a simple econetwork, (b) and (c) indicate two simple paths in (a).
B.-P. Hbn / Ecological Modelling 95 (1997) 301-310
305
Fig. 2. A c o m p o u n d e c o n e t w o r k with a s i m p l e c y c l e path: ~ 1 ~ 2 -~ 3 ~ I 4 .
stays at the terminating transient compartment after it flows along the ith available simple path, RTCPi is that
after it flows along the ith available compound path. Total residence time of particle in a compound econetwork
yields as follows,
NSP
NCP
NOC
TRT = ~., RTSPi + E RTCPi + E RTOCi.m,i
i=1
i=1
(5.6)
i=1
Clearly, Eq. (5.6) can be modified as
NSP
NOC
NCP
NOC
TRT = ][2 RTSPi + ~ Poi" RT, + Y'. RTCPi + Y'~ eoi .RT i . ( m , i - 1)
i~l
i=1
i=l
(5.7)
i=1
In the Eq. (5.7), the first two items on the right side represent the residence time from simple paths, the same as
Eq. (5.5); the last two items on the right side represent the residence time from compound paths, i.e., from the
recycling process.
5.3. The particle residence time in general econetwork
In view of Section 5.1 and Section 5.2. The recognization of path is a key step to calculate residence time.
Patten (1985) gave a method to recognize and calculate simple paths. But for a compound econetwork, we
hardly recognize every compound path. The number of compound path should be infinite by cycling process.
Therefore it is necessary to find a analysis strategy that obviates having to enumerate every compound path.
Eq. (3.3) can be expanded as
MI=I +Q x+Q( + . . . +Q~-I +Q~ + . . .
For a simple econetwork, its maximum of path length (MPL) is not larger than n - 1, n is the size of
transient compartment. Then, Q~ = 0, when k > n - 1
Ml =I +Q l +Q~ + ... +Q~-I
Let P0~ be the transition probability of particle from the source compartment to transient compartment i, and
P0 = {P01, P02 . . . . . P0,}, RT = {RT 1, RT 2 . . . . . RT,,}. By use of Eq. (5.5), there follows
NOC
RTOC = ]~ P 0 i RTi = P0" I- RT T
(5.8)
i=1
NSP
RTSP = ~ RTSP~= P0" (QI + Q~ + - . - + Q~ + Q~*~ + . - . + Q ~ - ~ ) RTr
(5.9)
i=l
A general identity produces by use of Eq. (5.5).
TRT = P0 'Ml" RTT
(5.10)
By induction, Eq. (5.10) holds for a compound econetwork. Since the analysis is based on the equilibrium
state feature, Eq. (5.10) is suitable for the steady state econetworks. For an unsteady state econetwork, there
does not exist Eq. (3.3). But, we may approximately obtain TRT by use of Eq. (5.10) as the transient state value
of TRT, while RT i, is calculated by Eq. (4.3).
306
B.-P. Hart/ Ecological Modelling 95 (1997) 301-310
6. The effects of interaction on residence time
The total residence time of matter or energy in an econetwork indicates the capacity of ecosystem, and it
relies on the properties of compartments and the structure of paths transferring interactions. The residence time
in an ecosystem is traditionally defined as the ratio of the standing stock of a composite compartment to its
throughput. The ratio, in fact, only reflects the characteristics of a composite compartment, not that of
ecosystem. In Eq. (5.10), the interactions between compartments are obviously emphasized. According to the
three types of interaction, i.e., self-, direct- and indirect interaction, the structure matrixes can be rewritten as
that
MI = M I ( S ) + M E ( D ) + M I ( I D )
M I ( I D ) = a~ + Q3 + . . .
MI(S ) = I
M I ( D ) = Qt
+ O ~ + Q~+l + . . .
(6.1)
Where the symbols S, D and ID implies self-, direct- and indirect interaction, respectively. Therefore, the first
structure matrix ( M 0 is crucial to express the network properties of ecosystems quantitatively. Further, the total
residence time (TRT) can be expressed as
T R T = TRT s @ TRT o 4- TRTIo
TRTs = P0" M I ( S ) . RT T
TRTID = P0" M , ( I D ) . RT T
TRTD = eo" M I ( D ) " RTT
(6.2)
In Eq. (6.2), the effects of the three kinds of interactions on the residence time of energy or matter in an
econetwork are presented.
7. T w o examples
The energy econetwork shown as Fig. 3 is a simple econetwork with 8 compartments (Qiu, 1982). The
energy flow F,7 is in kcal m -a y - l, the energy standing stock x i in kcal m -2. The energy standing stocks in 6
transient compartments and the flows between compartments are as follows, x I = 200, x 2 = 8.3, x 3 = 9.9,
x 4 = 85.0, x 5 = 6.2, x 6 = 0.7, F01 = 1096, Fl2 = 422, El4---400, FI7 = 274, F23 = 48, F24 = 160, F26 = 94,
F27 = 120, F36 = 3, F37 = 45, F45 ~-- 33, F46 = 22, F47 = 505, F56 = 2, F57 = 31, F67 = 121.
,0,
Fig. 3. The energy flow econetwork in a marine ecosystem with 8 compartments, Co: solar energy, C~: phytoplankton communities, C2:
zooplankton communities, C~: sea-surface fish communities, C4: benthonic fauna, C5: benthonic fishes, C6: bacteria communities, C7:
dissipative energy.
B.-P. Han/ Ecological Modelling 95 (1997) 301-310
307
X3 (03~ '~/-~ [=35
Fig. 4. The flow network of nitrogen, phosphorus and kalium in an agroforestry ecosystem with 6 compartments (Han and Wu, 1996). Co:
Source compartment, CI: the plant above ground; C2: the litter falls; C3: the soil; C4: the plant under ground. C5: the mergence
compartment.
According to the definitions of Qj and Eq. (3.3), the first flow matrix QI and the first structure matrix Mj
are
0.000
0.000
Q! ~-- 0.000
0.000
0.000
0.000
0.385
0.000
0.000
0.000
0.000
0.000
0.000
0.114
0.000
0.000
0.000
0.000
0.365
0.379
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.059
0.000
0.000
0.000
0.223
0.063
0.039
0.061
0.000
1.000
0.000
0.000
M! =
0.000
0.000
0.000
0.385
1.000
0.000
0.000
0.000
0.000
0.044
0.114
1.000
0.000
0.000
0.000
0.511
0.379
0.000
1.000
0.000
0.000
0.030
0.022
0.000
0.059
1.000
0.000
0.110
0.246
0.063
0.043
0.061
1.000
The cycling index (CI) of energy is equal to zero. The residence time vector of energy in transient
compartments (RT) is {0.1825, 0.0197, 0.2063, 0.15t8, 0.1879, 0.0058). The total residence time of energy in
the econetwork (TRT) is equal to 0.2829 y. TRT s, TRT D and TRTID equal 0.183 y, 0.063 y and 0.037 y,
respectively.
The flow network of nitrogen, phosphorus and kalium in an agro- forest ecosystem of the
Populus/Euramericana-Triticum sativum-Zea mays community shown as Fig. 4 is a compound econetwork
with 6 compartments (Han and Wu, 1996), XI: the standing stock of nutrient element in the plant above ground;
X2: the standing stock of nutrient element in the litter falls; X3: the standing stock of nutrient element in the
soil; X4: the standing stock of nutrient element in the plant under ground, the flow of trophic element F u is in
kg ha- I y- ~, the standing stock of trophic element x i in kg ha- t. The standing stocks and flows of nitrogen,
phosphorus and kalium in the econetwork are as follows, X~ ~ (299.0, 67.8, 205.7), X 2 ~ (32.7, 8.2, 48.7),
X3 e(368.0, 265.0, 629.0), X4~(18.0, 3.1, 28.1), Fj2E(24.4, 3.8, 34.0), F~s ~(277.0, 63.99, 171.3),
F23 e (15.6, 2.1, 24.2), F2s ~ (8.8, 1.7, 9.8), F34 E (297.4, 67.0, 204.2), F35 ~ (2.14, 1.04, 2.5), F4~ ~ (296.3,
66.99, 203.4), F43 e (1.1, 0.01, 0.8), F01 ~ (5.1, 0.8, 1.9), F03 e (282.84, 65.93, 181.7), the three values in
brackets are for nitrogen, phosphorus and kalium, respectively.
B.-P. Han / Ecological Modelling 95 (1997) 301-310
308
Similarly, the first flow matrixes and the first structure matrixes for nitrogen, phosphorus and kalium are as
follows,
0.0000
0.0000
Q,(N) = 0.0000
0.9963
0.0810
0.0000
0.0000
0.0000
0.0000
0.6393
0.0000
0.0037
0.0000
0.0000
0.9929
0.0000
1.0542
0.6691
M~(N) = 1.0466
1.0541
0.0853
1.0542
0.0847
0.0853
0.0548
0.6765
1.0581
0.0585
0.0544
0.6716
1.0505
1.0581
0.0000
0.0000
QI(P) = 0.0000
0.9999
0.0561
0.0000
0.0000
0.0000
0.0000
0.5526
0.0000
0.0002
0.0000
0.0000
0.9847
0.0000
1.0315
0.5613
1.0157
1.0315
0.0579
1.0315
0.0570
0.0579
0.0320
0.5701
1.0317
0.0322
0.0315
0.5614
1.0159
1.0317
0.0000
0.0000
QI(K) = 0.0000
0.9961
0.1656
0.0000
0.0000
0.0000
0.0000
0.7118
0.0000
0.0039
0.0000
0.0000
0.9879
0.0000
1.1318
0.7958
M~(K) = 1.1180
1.1317
0.1874
1.1318
0.1852
0.1874
0.1339
0.8087
1.1362
0.1379
0.13231
0.7989 [
1.1224
1.1362
Mr(P)=
The ~sidence times for three kinds of elements to stay at compartments and econetworks are listed in Table
.
According to the residence time analysis for the two examples above, it is found that econetwork residence
time does not equal the sum of all compartment residence times, the effects of indirect interactions on
econetwork residence times are different in the two ecosystems. In example 1, there exists no cycling path, the
indirect interactions act by the simple indirect paths. But in example 2, the indirect interactions consist mainly
of cycling paths. And the greater the cycling index is, the stronger the effect of indirect interactions (TRTIo) is.
Therefore, the diversity of flow paths in an ecosystem is a significant factor influencing the composition of total
residence time.
Table 1
The cycling indexes and residence times of nitrogen, phosphorus and kalium in the econetwork shown as Fig. 4 (residence time in year)
N
P
K
CI
RT l
RT 2
RT 3
RT4
TRT5
TRT D
TRT m
TRT
0.0536
0.0306
0.1186
0.9920
1.0015
1.0020
1.3402
2.1579
1.4324
1.2286
3.8948
3.0431
0.0605
0.0463
0.1376
1.2244
3.8601
3.0219
0.0609
0.0464
0.1370
1.2072
1.2501
1.8057
2.4925
5.1566
4.9646
CI (cycling index) is referred to the formula of Patten et al. (1976).
309
B.-P. Han / Ecological Modelling 95 (1997) 301-310
Table 2
The ratio of total standing stock to total system throughput and the ratio of total standing stock to total system outflow in the econetwork
shown as Fig. 4
N
P
K
TSS (totalstanding stock)
717.7
344.1
911.5
TST (totalsystem throughput)
922.74
206.63
650.20
TSO (totalsystem outflow)
287.94
66.73
183.60
TSS/TST
0.7778
1.6653
1.4019
TSS/TSO
2.4925
5.1566
4.9646
TSS = Y',~ , x,, TST = ,v~_ ~TOF,,TSO : ~ 4 F,~.
An interesting argument is if the compartmental residence time ( x J T O F s ) can be directly developed to
calculate the total residence time of energy or matter in econetworks. The ratio of total standing stock to total
system throughput has been used as the total residence time, so the comparison of the ration with TRT in Eq.
(5.10) will be useful to understand TRT. In the first example, it is found that TRT equals the ratio of total
standing stock to total system outflow or total system inflow (E~= ~F0~) instead of the ratio of total standing
stock to total system throughput. Table 2 shows the same results for three nutrient elements nitrogen,
phosphorus and kalium in the second example. Therefore total residence time (TRT) is only a function of total
system standing stock and total system outflow, that is, for an ecosystem with constant total standing stock and
total outflow, its total residence time of matter or energy does not change with the structure of ecosystem. The
conclusion is useful to calculate the total residence time of matter or energy in complex ecosystems. Total
residence time defined by Eq. (5.10) serves to understand the residence process of matter or energy in
ecosystems.
8. Summary
On the basis of structure analysis, a group of simple formulae is obtained to computate the total residence
time (TRT) of particles such as energy and matter in an econetwork. The total residence time (TRT) of energy
or matter in an econetwork is naturally different from the compartmental residence time (RTi), because there
exist interactions between compartments. The first structure matrix ( M I) representing the interactions among
compartments may fall into three items, i.e., I (self interaction), QI (direct interaction) and M ~ - I - Q I (indirect
interaction). The effects of three kinds of interactions on the econetwork residence time are defined as TRT s,
TRT D and TRTID which strongly depend on the diversity of interaction paths and compartment properties. In
the two examples, TRT is equal to the ratio of total standing stock to total system outflow instead of the ratio of
total standing stock to total system throughput which has been used as the total residence time. The identity
between TRT and the ratio of total standing stock to total system outflow indicates that total residence time is
not a system index related to the structure of ecosystems, so we have a simple method to calculate the total
residence time for a complex ecosystem. The formulae of total residence time expressed in the present paper
reveal the effect of interaction and structure on the residence process of matter or energy and provide a
quantitative explanation for applying the formula of compartmental residence time to total residence time.
References
DeAngelis, D.L., 1980. Energy flow, nutrient cycling, and ecosystem resilience. Ecology, 61(4): 784-771.
Finn, J,T., 1976. Measure of ecosystem structure and function derived from analysis flows. J. Theor. Biol., 56: 363--380.
Han, B.P., 1993. The analysis of econetwork and its simulation in computer. Dissertation for Ph.D., Amoy University, pp. 40-48.
Han, B.P., 1993. The econetwork and econetwork analysis. Chin. J. Nat., (7-8): 467-50.
Han, B.P. and Wu, G., 1996. Network analysis of nitrogen, phosphorus and Kalium flow in a typical agroforestry ecosystem. Chin. J. Appl.
Ecol.. 7(1): 19-22.
310
B.-P. Han/ Ecological Modelling 95 (1997) 301-310
Harmon, B., 1973. The structure of ecosystems. J. Theor. Biol., 41: 535-556.
Harmon, B., 1979. Total energy costs in ecosystems. J. Theor. Biol., 80: 271-293.
Herendeen, R., 1989. Energy intensity, residence time, energy, and ascendency in dynamical ecosystems. Ecol. Model., 44: 143-151.
Higashi, M., Bums, T.P. and Patten, B.C., 1989. Food network unfolding: an extension of trophic dynamics for application to natural
ecosystems. J. Theor. Biol., 140: 243-261.
Patten, B.C., Bosserman, R.W., Finn, J.T. and Cale, W.G., 1976. Propagation of cause in ecosystems. In: B.C. Patten (Editor), Systems
analysis and simulation in ecology, Vol. IV. Academic Press, New York, pp. 475-579.
Patten, B.C., 1984. Modified cycling index for ecological application. Ecol. Model., 25: 89-83.
Patten, B.C., 1985. Energy cycling in the ecosystems. Ecol. model., 28: 1-71.
Qiu, Z.X., 1982. Plant Ecology. Advance Education Press, Beijing, pp. 320-324.
Ulanowicz, R.E., 1979. Complexity, stability and self-organization in natural communities. Oecologia, 43: 295-298.
Voris, P.V. and O'Neill, R.V., 1980. Functional complexity and ecosystem stability. Ecology, 61(6): 1350-1360.