On the Diversity of Trophic Structures and Processes in Ecosystems

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Ecological Modelling 107 (1998) 51–62

On the diversity of trophic structures and processes in ecosystems

Bo-Ping Han *

Institute of Hydrobiology, Jinan Uni6ersity, Guangzhou 510632, People’s Republic of China

Accepted 19 November 1997

Abstract

The paper analyzes the overall diversity of trophic structures and processes at the organizational level of ecosystems. The overall diversity based on Lindeman’s trophic dynamics is considered as one-dimensional diversity. By unfolding ecosystems, trophic structures and processes of ecosystems are expressed in two-dimensional space along compartment and trophic level axes. By use of the Shannon-Weaver diversity index, the overall diversity of two-dimensional distributions of standing stocks or throughflows, which are significantly different from those defined in one-dimensional space, is determined. When flows between compartments are partitioned across trophic levels we can determine the overall diversity of three-dimensional distribution of throughflows over two compartment axes and a trophic level axis. The relationships between these overall diversity indexes defined in the different dimensional spaces are formulated by use of trophic niches and trophic functions as suggested by Higashi et al. (1992). The three-dimensional diversity of throughflows fall into three parts. The first identifies the overall diversity of two-dimensional distribution along compartment and trophic level axes. The second indicates the average diversity of resources utilized in an ecosystem. The third specifies the transfer efficiency of flows in an ecosystem. The three-dimensional diversity of throughflows may support a new framework to understand trophic structures and processes. Two real ecosystems are examined through the calculation of overall diversity indexes. The results confirm the differences between diversity indexes defined in different dimensions. Out of all the diversity indexes, those related to paths (to the third-dimension) are more powerful to reveal differences in trophic structures and processes between the two ecosystems. © 1998 Elsevier Science B.V. All rights reserved.

Keywords: Diversity; Trophic structure; Network; Dimension

1. Introduction * Present address: Environmental Impact Assessment Cen- tre, Tekniikante 21B, 02150 Espoo, Finland. Tel.: +358 9 Ecosystem diversity, due to its relation to sta- 70018680; fax: +358 9 70018682; e-mail: han@eia.ﬁ bility, drew intensive attention and became a sub-

52 B.-P. Han / Ecological Modelling 107 (1998) 51–62

stitute for ecosystem complexty (MacNaughton, When analyzing trophic behavior, however, Burns 1977; Han and Lin, 1993). MacArthur (1955) (1989, 1994) and Higashi et al. (1992) found Lin- considered diversity to enhance stability, whereas deman’ principle difficult to apply to trophic pro- May (1972) derived an opposite outcome. Many cesses in complex food networks, thus, they indexes have been developed to measure ecosys- generalized Lindeman’s idea by the concept of tem diversity. Most of them link species richness unfolding ecoystems. Unfolding a food network and abundance, e.g. h-diversity and i-diversity. enables an analytical definition of trophic level In fact, these indexes merely manifest community that preserves Lindeman’s orginal meanings. In characteristics. Ecosystem diversity, as cited in the addition, the concept of trophic niches suggested literature, usually has different meanings for dif- by Higashi et al. (1992) is useful in understanding ferent needs. For example, ecologists who go in detailed trophic structures and processes in for resource conservation usually consider ecosys- ecosystems. The present study attempts to take a tem diversity as the diversity of the types of comparative analysis of available ecosystem diver- ecosystems. Therefore, in discussing ecosystem di- sity indexes and to construct a general index that versity, we must confine it to a certain aspect or a measures the diversity of trophic structures and hierarchical level. Pimm (1984) listed four elemen- processes in an ecosystem on the basis of unfold- tary variables comprizing ecosystem diversity ing ecosystems. (complexity): species richness, evenness, connectance and interaction strength. The first two variables are associated with community 2. Paths and flow diversity of ecosystems organization. The other two pertain to ecosystem organization. Thus, h-diversity and i-diversity 2.1. Connectance and path di6ersity can not be termed as ecosystem structural diversity. Zhou et al. (1991) presented an index of Since flow analysis was introduced by Hannon ecosystem diversity which consisted of species (1973), compartment models have been replaced richness and interacting relationships among all by network models to some degree. The replace- ecosystem compartments. However, it should be ment is not only a simple change of concepts but pointed out that compartments and species are also a change of ideas. In compartment models, different structural elements. The compartment as compartments are elementary structural elements. usually used in compartmental or network models In network models, however, paths are an alterna- refers to a trophic holon, which consists of several tive basic structural element in addition to com- species at the same trophic state. Therefore partments. Hence, analysis and recognization of adding species richness directly to relationships paths are critical to flow network analysis. Patten between compartments is, at least logically, unrea- (1985) gave detailed definitions of paths and re- sonable. At the organizational level of an ecosys- vealed the contribution of paths to the complexity tem, diversity mainly relies on the number of of flow behaviour in ecosystems. In the following, compartments and the interactions between them let us look at the role of paths in ecosystem (connectance and interaction strength). Since the structures.

flow of energy or matter is virtually a trophic In an econetwork with n compartments, Fij is " process, flow diversity reflects to a great degree the flow from compartment i to j. When, Fij 0,

trophic structures and processes in ecosystems. assume aij =1, otherwise aij =0. The matrix com- Ulanowicz and Norden (1990), Patten (1995) and posed of aij is called an adjacency matrix (A). An Christensen (1995) considered flow diversity to be adjacency matrix and its powers (Ak, k=1, 2, …) an index that could be used to measure the char- reflect the connectance between compartments by acteristic structure of ecological networks. paths of various lengths. Connectance, defined by The quantitative analysis of trophic structures Pimm (1984), equals the ratio of the number of or processes began with Lindeman’s food chain nonzero elements to the maximum number of all based trophic dynamic model (Lindeman, 1942). possible links, using only links that involve paths 中国科技论文在线 http://www.paper.edu.cn

B.-P. Han / Ecological Modelling 107 (1998) 51–62 53

n n % % with a length of 1. The sum of all powers of an HII =− (Fij/TST) log(Fij/TST) (2.4) adjacency matrix is i=1 j=0 or G A = % Ak (2.1) n n T % % k=1 HII =− (Fij/TST) log(Fij/TST) (2.5) j=1 i=0 where G is maximum path length. The elements of In Eqs. (2.4) and (2.5), a flow variable between A reflect the connectance between compartments T two compartments is treated as an independent by paths of all lengths. A path is a way to tranfer unit. This means that H represents not only matter and energy, which originate through input II direct connection but also interacting relation- from the environment. When the input of an ships among all compartments. By the definition originating compartment in a path is nonzero, the of throughflow (T ), Eq. (2.4) is expanded as path is called a basic path. Otherwise, if the input i follows, of an originating compartment in a path is zero, n the path is called an expanding path. Expanding % HII =− (Ti/TST) log(Ti/TST) paths can be merged into a basic path (Han, i=1 " 1996b). For compartment i, if its input, F0i 0, n % then assume I0i =1, otherwise I0i =0. We get the + (Ti/TST)Hi (2.6) number of all basic paths with length k, i=1 n % B(k)=I ·Ak−1 ·C (2.2) Hi =− (Fji/TST) log(Fji/TST) (2.7) 0 j=0

where I0 =(I01, I02, ···, I0n). C is a volume vector Hi reﬂects the diversity of resources utilized by which all elements equal 1. If cycling paths exist, compartment i, thus the second term in the right

the sum matrix (AT) tends to diverge and G is hand of Eq. (2.6) is the average of all Hi (i= infinite. In this case, the number of paths can no 1, 2, …) or the diversity of conditional probabil- longer be used to measure ecosystem structure. In ity. fact, when G is greater than a given value, the energy or matter transfered by these paths with length G is negligible. The given length is usually 3. Trophic structure diversity of ecosystems defined as the maximum path length, i.e. G. 3.1. Macroscopic trophic le6els and trophic niches of compartments 2.2. Flow di6ersity Lindeman (1942) assigned a species to a single Investigating the development of an ecosystem, trophic level according to the number of times Ulanowicz (1980) defined development capacity as that energy-matter assimilated was previously uti- n lized by other species since the energy-matter en- % HI =− (Ti/TST) log(Ti/TST) (2.3) tered the ecosystem. However, most species i=1 generally utilize energy-matter that has been pre- Sn where TST is the total system throughflow ( i=1 viously assimilated for different numbers of times Ti) and Ti is the throughflow of compartment i.In in the ecosystem (Burns, 1989). The recognization Eq. (2.3), all flows through a compartment are of trophic levels in an ecological network is criti-

assigned to a single feature. HI is usually em- cal in showing the trophic behaviour of compli- ployed to measure the overall diversity of cated ecosystems. Higashi et al. (1989) developed throughflow and its maximum is associated with the idea of partitioning standing stocks and flows the number of compartments. Ulanowicz and in a flow network according to the length of paths Norden (1990) provided another index of flow connecting to external inputs (basic paths) (Han, diversity, 1996b). Han (1996a) quantitatively expressed this 中国科技论文在线 http://www.paper.edu.cn

54 B.-P. Han / Ecological Modelling 107 (1998) 51–62

Table 1 Higashi et al. (1992) represented the trophic Two-dimensional distribution of total standing stock over all niche of a compartment by its distribution over all trophic levels and all compartments trophic levels, and the trophic function of each Level k Compartment i compartment on any trophic level by the diversity of resources utilized on the previous trophic level. i=1 i=2 … i=n Total The measures of Higashi et al. (1992) to empha- size the trophic behaviour of each compartment k=1 x1(1) x2(1) xn(1)… 1 supports a basis for us to explore further the k=2 x1(2) x2(2) … xn(2) 2 …………… … trophic structural diversity of an ecosystem as a k=G x1(G) x2(G) … xn(G) G whole. Total x1 x2 … xn 3.2. O6erall di6ersity of trophic structures and processes idea by means of matrix analysis. The trophic level defined by the length of basic paths is called First, let wki =xi(k)/ , the fraction of total the macroscopic trophic level. It should be argued standing stock () on compartment i and trophic that there is a tacit assumption made in the level k, then wk· = k/ and w·i =xi/ . Accord- definition of macroscopic trophic levels—namely ing to the principle of two-dimensional distribu- that input to a compartment is distributed evenly tion, the overall diversity of total standing stock among the outputs from that compartment in over all compartments (C) and all trophic levels proportion to the magnitudes of various outputs (L) can be obtained by the Shannon-Weaver di- (Burns, 1989). Nevertheless, unfolding and aggre- versity index, i.e. gation of flow networks make us clearly under- n G stand the trophic processes and efficiencies % % DX(CL)=− wki log wki (3.1) involved in the transfer of energy-matter input in i=1 k=1 ecosystems. In an ecosystem, we assume xj(k)to n be the kth trophic level partion of the standing % DX(C)=− w·i log w·i (3.2) stock (xj) of compartment j and Ti(k)tobethe i=1

kth trophic level partion of throughﬂow of com- G partment i, thus we have the two-dimensional % DX(L)=− wk· log wk· (3.3) distribution of total standing stocks along all k=1

compartments and trophic levels as shown in G n % % Table 1. For the total system throughﬂow (TST) DXL(C)=− wk· (wki/wk·) log(wki/wk·) of an ecosystem, there is a similar two-dimen- k=1 i=1 (3.4) sional distribution shown as in Table 2. n G % % Table 2 DXC(L)=− w·i (wki/w·i) log(wki/w·i) Two-dimensional distribution of total system throughﬂow over i=1 k=1 all trophic levels and compartments (3.5)

Level k Compartment i where DX(C) and DX(L) are the distributional diversity indexes of total standing stock along i=1 i=2…i=n Total compartments and along trophic levels respec- u tively. DXC(L) and DXL(C) are the two diversities k=1…T1(1) T2(1) Tn(1) 1 u of conditional probability, they can be interpreted k=2 T1(2) T2(2) … Tn(2) 2 ……… … …… as the average distribution diversity of total u k=G T1(G) T2(G) … Tn(G) G standing stock over all trophic levels or over all compartments. The following identity always Total T1 T2 … Tn TST holds: 中国科技论文在线 http://www.paper.edu.cn

B.-P. Han / Ecological Modelling 107 (1998) 51–62 55 DX(CL) =DX(C)+DX (L) C . (3.6) Higashi et al. (1992) deﬁned D for compart- DX(L) DX (C) i = + L ment i as

Similarly, we can formulate the overall diversity G % of a two-dimensional distribution of throughﬂow Di =− (Ti(k)/Ti) log(Ti(k)/Ti) (3.17) from Table 2. k=1

n G or % % DT(CL)=− qki log qki (3.7) G i=1 k=1 % Di =− (qki/q·i) log(qki/q·i). (3.18) n k=1 % DT(C)=− q·i log q·i (3.8) i=1 Here Di characterizes the distribution diversity

G of throughﬂow through compartment i on the % DT(L)=− qk· log qk· (3.9) trophic level axis. Similarly, DTC(L) can be k=1 evaulated as G n % % n DTL(C)=− qk· (qki/qk·) log(qki/qk·) DT (L)= % q D . (3.19) k=1 i=1 C ·i i (3.10) i=1

n G For standing stocks, DXC(L) can be re-ex- % % DTC(L)=− q·i (qki/q·i) log(qki/q·i) pressed as follows, i=1 k=1 n (3.11) % DXC(L)= w·iDi. (3.20) i=1 DT(CL)=DT(C)+DTC(L)=DT(L)+DTL(C). (3.12) Since the trophic niche of each compartment is represented by the distribution of its throughflow u Where qki =Ti(k)/TST, qk· = k/TST, q·i =Ti/ over all trophic levels, DTC(L) measures the aver- TST. Since DT(C) is identical to HI described by age trophic niche of all comarptments. Eq. (2.3), we have 6 DT(CL)=HI +DTC(L). (3.13) 3.3. Flow di ersity under unfolding networks In comparison with the trophic niches of com- By unfolding a flow variable, Fij can be parti- partments as suggested by Higashi et al. (1992), tioned along trophic levels. Let Fij(k) be the frac- DT (C) can be rewritten as L tion of Fij on the kth trophic level. Corresponding G to the flow diversity in Eq. (2.4) we get the flow % % DTL(C)= qk·D (k) (3.14) diversity of three-dimensional distribution as fol- k=1 lows n % % u u G n n D (k)=− (Ti(k)/ k) log(Ti(k)/ k). (3.15) % % % i=1 HIII =− Pijk logPijk (3.21) k=1 i=1 j=0 Here D%(k) denotes the diversity of resources where P =F (k)/TST. For simplicity, assume utilized by the kth trophic level as a whole (Hi- ijk ij P =SG F (k)/TST. Then Eq. (3.21) falls into gashi et al., 1992). DT (C) is the average of D%(k) ij· k=1 ij L two parts, over all trophic levels. According to the linear partition assumption of n n % % standing stocks and throughflows, there exists HIII =− Pij· log Pij· i=1 j=0 relationships as follows (Higashi et al., 1992; Han, n n G 1996b) % % % − Pij· (Pijk/Pij·) log(Pikj/Pij·). i=1 j=0 k=1 xj(k)/xj =Tj(k)/Tj =Fij(k)/Fij (3.16) (3.22) 中国科技论文在线 http://www.paper.edu.cn

56 B.-P. Han / Ecological Modelling 107 (1998) 51–62

G n n % % % The ﬁrst right hand term of Eq. (3.22) equals − P·jk (Fij(k)/Tj(k)) log(Fij(k)/Tj(k)) k=1 j=1 i=0 the ﬂow diversity (HII) expressed by Eq. (2.4) G n under two-dimensional distribution. By use of Eq. % % = (Tj(k+1)/TST)Cj(k+1) (3.16), we have k=1 j=1 G n % % G + (Tj(k)/TST)Ej(k). (3.27) % k=1 j=1 D j =− (Pijk/Pij·) log(Pijk/Pij·) k=1 Here, G % n =− (Fij(k)/Fij) log(Fij(k)/Fij). (3.23) % k=1 Cj(k+1)=− (Fij(k)/Tj(k+1)) G i=1 % (T (k) T ) log(T (k) T ) =− j / j j / j log(Fij(k)/Tj(k+1)) (3.28) k=1

Ej(k)=−log(Tj(k+1)/Tj(k)). (3.29)

Hence, Dj indicates the distribution diversity of Higashi et al. (1992) interpreted Cj(k+1) as the throughﬂow through compartment j over all diversity of resources utilized by the (k+1)th trophic levels. Eq. (3.22) can be further simpliﬁed trophic level component of compartment j feeding as on the kth trophic level component. That is, the

n first right hand term in Eq. (3.27) is the average % HIII =HII + P·j·Dj (3.24) diversity of resources over all trophic levels in an j=0 ecosystem. Since the ratio Tj(k+1)/Tj(k) can be Evidently, the difference between HIII and HII is considered as the progressive (transferring) effi- determined by the average of Dj over all compart- cency of throughflow of compartment j at the ments. trophic level k, Ej(k) can be thought of as an n % index of progressive efficiency of compartment j. Let P·ij = Fij(k)/TST, then Eq. (3.22) can be i=0 Eq. (3.25) can be further rewitten as rewitten as HIII =DT(CL) n G % % G n HIII =− P·jk log P·jk % % j=1 k=1 + (Tj(k+1)/TST)Cj(k+1) k=1 j=1 G n n % % % G n − P·jk (Pijk/P·jk) log(Pijk/P·jk) % % k=1 j=1 i=0 + (Tj(k)/TST)Ej(k). (3.30) (3.25) k=1 j=1 In Eq. (3.30), the second right hand term char- or acterizes the average diversity of resources utilized n G % % in an ecosystem. The third right hand term repre- HIII =− Pi·k log Pi·k i=1 k=1 sents the contribution of the transferring effi- G n n ciency of flows to the overall diversity. % % % − Pi·k (Pijk/Pi·k) log(Pijk/Pi·k). k=1 i=1 j=0 (3.26) 4. Analysis of two typical ecosystems In Eqs. (3.25) and (3.26), the first right hand term equals the overall diversity of two-dimen- In order to demonstrate trophic structure and sional distribution along a compartment axis and process diversity of ecosystems under different a trophic level axis (DT(CL)). In order to identify dimensions, we analyze two typical ecosystems. the role of each compartment in three-dimen- The first is an intertidal oyster reef ecosystem sional diversity, the second right hand term of Eq. (Dame and Patten, 1981), and the second is a (3.25) can be rearranged as general marine ecosystem (Qiu, 1982). The flow 中国科技论文在线 http://www.paper.edu.cn

B.-P. Han / Ecological Modelling 107 (1998) 51–62 57

Fig. 1. Food web energy model for an intertidal oyster reef ecosystem (Dame and Patten, 1981), ﬂows in Kcal·m−2 ·d−1, standing stocks in Kcal·m−2.

networks of the two ecosystems are shown as DX(C) and DX(L) are indexes with respect to

Figs. 1 and 2. Their two-dimensional distribution standing stocks. In two-dimensional space, HII of total system throughflows are shown in Tables and DT(CL) are indexes concerning throughflow, 3 and 4. The adjacency matrixes of the two and DX(CL) addresses standing stock. In three- ecosystems are presented in Table 5. dimensional space, there is only a flow index of Both ecosystems have six compartments but the diversity. For the diversity indexes of each ecosys- connectance of the first ecosystem is greater than tem, the following two relationships always hold: E E that of the second (Table 5). Since cycling paths (1) HIII HII and DT(CL) HI and DT(L); (2) exist in the first ecosystem (the cycling index is DX(CL)EDX(C) and DX(L). In other words, 0.1101 by Finn (1976) formula). The sum of the the diversity of structures and processes increases

powers of adjacency matrix (A1) tends to diverge with the number of dimensions. The four one-di- in the ﬁrst ecosystem. However,it tends to con- mensional indexes, based on Lindeman’s trophic verge in the second ecosystem. This means that dynamics, conceal detailed trophic behaviours

path diversity in the intertidal oyster reef ecosys- and structures. Similarly, ﬂow diversity (HII) ne- tem is greater. glects concrete processes of each ﬂow along paths, Table 6 shows diversity indexes of structures although it is formulated on the basis of network and processes in the two ecosystems. In the space models. According to Table 5, for the two ecosys-

of one-dimension, HI (DT(C)) and DT(L) are tems whose structural complexities are very differ-

indexes of diversity with respect to throughﬂows, ent, the values of HII are almost equal to each 中国科技论文在线 http://www.paper.edu.cn

58 B.-P. Han / Ecological Modelling 107 (1998) 51–62

Fig. 2. Food network model for a marine ecosystem (Qiu, 1982), ﬂows in ﬂows in Kcal·m−2 ·y−1, standing stocks in Kcal·m−2.

other. That is, HII can not adequately specify the The one-dimensional diversity index of standing distinction between the two ecosystems. Con- stocks, DX(C), is similar to HI (DT(C)) and is versely, DT(L), DT(CL) and HIII reveal the diver- smaller in the intertidal oyster reef ecosystem than sity resulting from all the transfers of matter-energy in the general marine ecosystem. DX(CL) and along all the paths in the ecosystems. DX(L) are greater in the intertidal oyster reef Among all diversity indexes in these ecosystems, ecosystem. In comparison with the two diversity the indexes with respect to paths in the intertidal indexes of throughflows DT(L) and DT(CL), oyster reef ecosystem are greater than those in the DX(L) and DX(CL) do not show an obvious general marine ecosystem. This may be because the difference between the two ecosystems. That is, the trophic structure of the intertidal oyster reef ecosys- diversity indexes that do not regard paths hardly tem is more complex than that of the general detect the effects of ecosystem structure on the marine ecosystem. According to the author’s previ- processes of matter or energy transfer. The distribu- ous analysis (Han, 1996a), the number of macro- tion diversity of flows embodies more directly the scopic trophic levels in the intertidal oyster reef complexity of trophic structure in ecosystems ecosystem reaches 30, when only the distribution of rather than the distribution diversity of standing total system throughflow in each compartment, stocks. which is not less than 10−4, is taken into consider- ation. Throughflow transferred by the paths with length \20 tends to be B10−3. This number is 5. Conclusion still far greater than the five levels in the general marine ecosystem. Diversity concerning standing Diversity is one of the most important concepts stock decribes the distribution structure of standing in ecology. Many ecologists pay particular atten- stock. Diversity indexes concerning flow represent tion to features of ecological diversity at different trophic structures and processes to maintain the levels of organization. Their efforts facilitate the distribution structure of standing stocks in each understanding of the features of diversity. On the ecosystem, and implicitly reflects the roles of path other hand, the meaning of diversity is so wide that diversity in the functions of ecosystems. confusion often occurs in discussions. Diver- 中国科技论文在线 http://www.paper.edu.cn

B.-P. Han / Ecological Modelling 107 (1998) 51–62 59

Table 3 Two-dimensional distribution of total system throughﬂow over all trophic levels and all compartments in an intertidal oyster reef ecosystem (Dame and Patten, 1981)

Level k Compartment i

i=1 i=2 i=3 i=4 i=5 i=6 Total

k=1 41.470 0.000 0.000 0.000 0.000 0.000 41.470 k=2 0.000 15.792 0.000 0.000 0.000 0.513 16.305 k=3 0.000 0.244 5.796 5.159 0.456 0.000 11.655 k=4 0.000 2.926 0.090 0.935 1.265 0.031 5.247 k=5 0.000 1.444 1.074 0.969 0.171 0.087 3.744 k=6 0.000 0.656 0.530 0.630 0.276 0.012 2.103 k=7 0.000 0.530 0.241 0.292 0.146 0.019 1.228 k=8 0.000 0.266 0.195 0.209 0.074 0.010 0.753 k=9 0.000 0.165 0.098 0.116 0.053 0.005 0.436 k=10 0.000 0.100 0.061 0.068 0.028 0.004 0.261 k=11 0.000 0.057 0.037 0.042 0.017 0.002 0.155 k=12 0.000 0.035 0.021 0.024 0.010 0.001 0.092 k=13 0.000 0.021 0.013 0.015 0.006 0.001 0.055 k=14 0.000 0.012 0.008 0.009 0.004 0.000 0.032 k=15 0.000 0.007 0.005 0.005 0.002 0.000 0.019 k=16 0.000 0.004 0.003 0.003 0.001 0.000 0.011 k=17 0.000 0.003 0.002 0.002 0.001 0.000 0.008 k=18 0.000 0.002 0.001 0.001 0.000 0.000 0.004 k=19 0.000 0.001 0.001 0.001 0.000 0.000 0.003 k=20 0.000 0.001 0.000 0.000 0.000 0.000 0.001 Total 41.470 22.265 8.172 8.480 2.510 0.685 83.582

sity defined at different levels of organization networks. Partitioning standing stocks and flows characterize different aspects of ecosystems. Due in an ecosystem according to the path length to the complexity of structures in an ecosystem, reveals detailed trophic processes. For flow net- the expectation that a comprehensive index of works characterizing ecosystem structures, we can diversity involving all aspects of an ecosystem can construct two-dimensional distributions of total be constructed is not realistic. In the present standing stock and total system throughflow (Ta- study, the author focuses on diversity of macro- bles 1 and 2). These Tables were used to construct trophic structures and processes in ecosystems. In overall diversity indexes in two-dimensional space Lindeman’s theory, trophic levels and pyramids for standing stocks and throughflows and the define linear food chains, i.e. one-dimensional relationship of these overall diversity to the over- structure. Distribution diversity of total standing all diversity indexes that are defined in one-dimen- stock and total system throughflow along trophic sional space. The difference between chains are two measures to indicate trophic diver- two-dimensional diversity of throughflows along a sity of an ecosystem. But for real ecosystems, their compartment axis and a trophic level and the structures are generally networks rather than lin- one-dimensional diversity of throughflows is ear chains. Lindeman’s theory can not entirely defined by the average trophic niche of all com- map trophic processes and structures in food net- partments. Flow diversity, as expressed by Eq. works. Thus, in order to overcome the drawbacks (2.4), usually involves only direct links or paths of Lindeman’s principle, Higashi et al. (1989) with a length of 1. However, the diversity of developed macroscopic trophic dynamics and gen- two-dimensional distribution of throughflow eralized Lindeman’s theory by means of unfolding (DT(CL)) includes trophic processes anticipated 中国科技论文在线 http://www.paper.edu.cn

60 B.-P. Han / Ecological Modelling 107 (1998) 51–62

Table 4 Two-dimensional distribution of total system throughﬂow over all trophic levels and all compartments in a general marine ecosystem (Qiu, 1982)

Level k Compartment i

i=1 i=2 i=3 i=4 i=5 i=6 Total

k=1 1096.000 0.000 0.000 0.000 0.000 0.000 1096.000 k=2 0.000 422.000 0.000 400.000 0.000 0.000 822.000 k=3 0.000 0.000 48.000 160.000 23.571 109.715 341.286 k=4 0.000 0.000 0.000 0.000 9.429 10.714 20.143 k=5 0.000 0.000 0.000 0.000 0.000 0.571 0.571 Total 1096.000 422.000 48.000 560.000 33.000 121.000 2280.000

by all paths. By partitioning flows between com- distribution structure of throughflows but also partments, we can discuss trophic processes in on the ways of utilizing throughflows as well as three-dimensional space. I first presented the re- on utilizing efficiency. Therefore, the analysis of

lationship between three-dimensional diversity three-dimensional diversity (HIII) may offer a

(HIII) and two-dimensional diversity (HII)as new framework to understand trophic processes suggested by Ulanowicz and Norden (1990). The and structures in complex ecosystems. The anal- difference can be expressed by the average ysis of two real ecosystems illustrates the differ- trophic niche of all compartments. Then I estab- ence between diversity indexes deﬁned by lished the expression relating three-dimensional different numbers of dimensions. Among all in-

diversity (HIII) to two-dimensional diversity dexes listed in Table 6, those that do not include (DT(CL)) (Eq. (3.30)). In this formula, three-di- the third-dimension (paths) do not show a sig-

mensional diversity (HIII) is divided into three nificant difference between the two ecosystems. parts that have ecological meaning. The first However, indexes that regard the third dimen- equals two-dimensional diversity (DT(CL)). The sion produce an evident difference. Thus, intro- second is an average of resource diversities as duction of the third-dimension is crucial in defined by Higashi et al. (1992), which expresses comparing trophic structures and processes in the ways of utilizing resources in an ecosystem. different ecosystems. The third concerns the transfer efficiency of In the analysis of two ecosystems, the applica- flows. In other words, three-dimensional diver- tion of the theoretical considerations provided by sity of throughflows depends not only on the the present contribution contains only the analysis of the overall diversity indexes defined in the different dimensions. There remains many theo- Table 5 Adjacency matrices of the two ecosystems retical problems that need to be investigated by the analysis of real ecosystems. The two ecosys- Intertidal oyster reef ecosystem A general marine tems used here have the same number of compart- (Dame and Patten, 1981) ecosystem (Qiu, 1982) ments and therefore, the effect of compartment number is excluded. How will the comparison of Æ0 1 0 0 0 1Ç Æ0 1 0 1 0 1Ç Ã Ã Ã Ã diversity between ecosystems with different num- 0 0 1 1 1 0 0 0 1 1 1 0 Ã Ã Ã Ã bers of compartments proceed? For a given Ã0 0 0 1 1 0Ã Ã0 0 0 0 1 0Ã ecosystem, whether the contributions of each A1= A2= Ã0 1 0 0 1 0Ã Ã0 1 0 0 1 0Ã compartment to each of the diversity indexes Ã0 1 0 0 0 1Ã 0 1 0 0 0 1 defined in different dimensions are the same or È É Ã ÉÃ 0 1 0 0 0 0 È0 1 0 0 0 0 not? To what extent can the analysis and comparison of the second and the third terms in Eq. 中国科技论文在线 http://www.paper.edu.cn

B.-P. Han / Ecological Modelling 107 (1998) 51–62 61

Table 6 Diversity of trophic structure and processes in the two ecosystems

One-dimension Two-dimensions Three-dimensions

HI DT(L) DX(C) DX(L) HII DT(CL) DX(CL) HIII

Intertidal oyster reef ecosystem 1.8816 2.2038 1.1604 1.5431 3.1444 2.7248 1.7164 3.9876 A general marine ecosystem 1.8865 1.5120 1.3510 1.3117 3.1349 2.1361 1.6058 3.3845

(3.30) provide important information to enhance Acknowledgements our knowledge of ecosystem performance? The answers to these questions will largely test the I am grateful to J. Titus and A. Sikder for theoretical exploration provided by this study help in editing the manuscript and two anony- and improve our understanding of trophic struc- mus referees for their valuable reviews and sug- ture and processes in ecosystems. gestions which improved this paper.

Appendix A

The proof of Eq. (3.27). The second right hand term of Eq. (3.25) can be written as G n n % % % % H =− P·jk (Pijk/P·jk) log(Pijk/P·jk) k=1 j=1 i=0 G n n % % % =− (Tj(k+1)/TST) [Fij(k)/Tj(k+1)] log[Fij(k)/Tj(k+1))(Tj(k+1)/Tj(k))] k=1 j=1 i=0 G n ! n % % % =− (Tj(k+1)/TST) [Fij(k)/Tj(k+1)] log[Fij(k)/Tj(k+1))]+ k=1 j=1 i=0 n " % [Fij(k)/Tj(k+1)] log[Tj(k+1)/Tj(k)] i=0 By use of Eq. (3.28), we get G n ! n " % % % % H =− (Tj(k+1)/TST) [−Cj(k+1)]+ [Fij(k)/Tj(k+1)] log[Tj(k+1)/Tj(k)] k=1 j=1 i=0 G n % % % H =− (Tj(k+1)/TST)[−Cj(k+1)] k=1 j=1 G n n % % % − (Tj(k+1)/TST) [Fij(k)/Tj(k+1)] log[Tj(k+1)/Tj(k)] k=1 j=1 i=0 G n % % =− (Tj(k+1)/TST)[−Cj(k+1)] k=1 j=1 G n % % − (Tj(k+1)/TST)(Tj(k)/Tj(k+1)) log[Tj(k+1)/Tj(k)] k=1 j=1

Utilizing Eq. (3.29), we have G n G n % % % % % H = (Tj(k+1)/TST)Cj(k+1)+ (Tj(k)/TST)Ej(k) k=1 j=1 k=1 j=1

The proof of Eq. (3.6)

DX(CL) =DX(C)+DXC(L)

=DX(L)+DXL(C) 中国科技论文在线 http://www.paper.edu.cn

62 B.-P. Han / Ecological Modelling 107 (1998) 51–62

Refered to Eq. (3.1),

n G % % DX(CL) =− (wki/w·i)(w·i) log[(wki/w·i)(w·i)] i=1 k=1 n G n G % % % % =− (wki/w·i)(w·i) log(wki/w·i)− (wki/w·i)(w·i) log(w·i) i=1 k=1 i=1 k=1 n G n % % % =− w·i (wki/w·i) log(wki/w·i)− (w·i) log(w·i) i=1 k=1 i=1

=DXC(L)+DX(C)

Similarly, we have,

n G % % DX(CL) =− (wki/wk·)(wk·) log[(wki/wk·)(wk·)] i=1 k=1 n G n G % % % % =− (wki/wk·)(wk·) log(wki/wk·)− (wki/wk·)(wk·) log(wk·) i=1 k=1 i=1 k=1 G n n % % % =− wk· (wki/wk·) log(wki/wk·)− (wk·) log(wk·) k=1 i=1 i=1

=DXL(C)+DX(L)

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