Fig. 9.2. SDG (trophic graphs) corresponding to the sign-matrices in Fig. 9.1.

black vortex is linked to only one white vortex; therefore, this chain is qualitatively stable, i.e. if equilibrium exists then it is always stable for any values of flows between trophic levels. In case (b) the chain is qualitatively unstable since, as is easy to see, it passes the colour test. In other words, any trophic chain self-regulated either with the first or end trophic level (or both) is always stable, no matter what the values of flows between trophic levels.

9.3. Dynamic models of trophic networks and compartmental schemes

A typical example of a graphic representation of a "trophic network" is shown in Fig. 9.3. This is the "flows-storages" diagram (or compartmental scheme) of carbon flows and storages in a bog ecosystem in the temperate zone.

Let xi be a state variable, i.e. the storage of matter within the ith compartment (we shall consider only matter flows); fkj is the flow from kth to jth compartment; qt and yi are the in- and outflow into and out of the ith compartment. Then the general "flows-storages" diagram looks like the one in Fig. 9.4.

Fig. 9.3. Carbon cycle of a bog in the temperate zone (Zavalishin and Logofet, 2001). Storages in g C/m , flows in g C/m2 year. Storages: xi—plants, X2—animals, X3—bacteria and fungi, X4—dying-off biomass, excepting peat. Inflows: qi—assimilation and photosynthesis, q4—inflow with precipitation and from other ecosystems. Outflows: yi—plant metabolism and consumption by phytophagous of other ecosystems, y2—animal metabolism, y3—metabolism of bacteria and fungi, y4—outflow with sink, formation of peat and abiotic oxidation.

Fig. 9.3. Carbon cycle of a bog in the temperate zone (Zavalishin and Logofet, 2001). Storages in g C/m , flows in g C/m2 year. Storages: xi—plants, X2—animals, X3—bacteria and fungi, X4—dying-off biomass, excepting peat. Inflows: qi—assimilation and photosynthesis, q4—inflow with precipitation and from other ecosystems. Outflows: yi—plant metabolism and consumption by phytophagous of other ecosystems, y2—animal metabolism, y3—metabolism of bacteria and fungi, y4—outflow with sink, formation of peat and abiotic oxidation.

In turn, the matter balance is represented in the form of differential equations d = qt - yi + X - fit); i, k = i, .■•, n dt k-i or, by introducing the vector notations dx

where x = [xu...,xn}; y = {yi, ...,y„}; q = {qi, ..., q„}; f = {fi, ...,fn} and ft =

What kind of conformity is there between the system of Eq. (3.1) and the diagrams in Figs. 9.3 and 9.4? What could be said about the general properties of the system, if only a picture with flows and storages is known?

It is necessary to note that as a rule we deal with a static picture, a "snap-shot" of a dynamical system; therefore, we implicitly assume that the system is at steady state. It is obvious in this case q* + f * = y*

i.e. flows must be balanced.

Considering the environment as "zero" compartment we can denote qt = f0i, yi = f0. By also setting fii = 0, Eq. (3.1) is written as

where Q¿n = Ynj=0 fji is the sum of all flows coming into the ith compartment and Qout = X"=ofij is the sum of all flows going out of it. Then the equivalent of Eq. (3.3) will be (Qin)* = (Qout)*, i = 1, n. Clearly the sum Q = £n=o Qf ^n=o Qout = Ij=o Ij=ofji is the total flow of matter flowing through the system.

If the "dimension" of the network (number n) is not high then we can try to apply the methods developed by the theory of differential equations (if, certainly, we know how to construct these equations by using the flows diagram, but more about this later on). However, what do we do if the dimension is very high? One idea is to apply a thermodynamic approach. A direct application of classic thermodynamic concepts to these systems is difficult, since they differ from ensembles of weakly interacting particles (we have already written about it). Nevertheless, we will attempt to do it (see Section 9.4).

Note that the information theory concepts, which do not significantly depend on the number of parts (compartments) constituting the system and the type of interaction between them, could be applied to such a sort of systems (see also Chapter 4). One of the most popular information concepts is "organisation"; therefore we may assume that

• The better (in a certain sense) the ecosystem is organised the more effectively it uses the free energy of the environment.

Such a principle is consistent with basic principles of non-equilibrium thermodynamics. Different values, defined on the set of (independent of time) flows and storages, for instance MacArthur's diversity index, trophic diversity, ascendancy, exergy, indirect effects (Patten, 1995), etc., were suggested as measures of organisation. We shall consider in more or less detail only the first four measures; to get more familiar with the latter concept we recommend the works by Patten (1995) and Fath and Patten (1998). Nevertheless, in Section 9.4 we shall consider an example of how to use a classic thermodynamic approach.

There is a very close analogy between an ecosystem and metabolic cycle in an organism. Indeed, the metabolic cycle is an open system of step-by-step reactions, which transform input substrates into some "useful" products in such a way that the original products are regenerated. At each intermediate stage the dissipation of energy contained in molecules of substrates occurs, and also the products of transformation are secreted into the environment. At the same time a new portion of substrates can be input from the environment. It is obvious that the cycle is a "chemical machine" performing the work of transformation of some sorts of matter and energy into another (Fig. 9.5). It occurs, and this is very important, with full or partial regeneration of input substrates and dissipation of internal energy. All this is very similar to the description of an ecosystem. Perhaps there is only one distinction: the autotrophic level of any ecosystem uses directly the energy of solar photons, but this distinction is not significant in this context.

Analysis has shown that entropy increase caused by internal processes in this machine is determined by the change of the total Gibbs potential of the system and its environment

Fig. 9.5. Scheme of abstract metabolic cycle. C1,..., Cn are intermediate components of a cycle.

Cn Ci+1

Fig. 9.5. Scheme of abstract metabolic cycle. C1,..., Cn are intermediate components of a cycle.

taken with opposite sign:

where G and Genv are the Gibbs potentials of the system and the environment, the flows of energy and matter provide for the functioning of the machine. Incoming matter passes through the ecosystem along a corresponding trophic chain being stored in compartments and in partial trophic cycles within the system. The turnover time of the general cycle for the whole ecosystem is t = £k Tk, where Tk is the characteristic time of kth stage. For instance, the residence time of carbon in the kth compartment can be used as Tk. If there is a "bottle neck" as the slowest stage in the local biogeochemical cycle realised by the ecosystem (ts » Tk, k = 1,2,...; s — k) then t = Ts.

The amount of energy dissipated in the course of a single turnover is defined as rt+T rt+T

= - [G(t + T) - G(t)] - [GeIW(t + T) - GeIW(t)]. (4.2)

Since after the turnover the ecosystem comes back to the initial state (cyclic process) then G(t + T) = G(t), and

If A is useful work of our cycle and De Gm is the change of Gibbs potential of the system caused by the matter exchange between the system and the environment then, in accordance with the First Law, DE + A = DeGm. From Eqs. (4.2) and (4.3) it follows that after the system has concluded its turnover and returned to the initial state, having done some useful work, the dissipated energy is equal to the decrease of thermodynamic potential (or free energy) of the environment. If the turnover time t is rather small then the rate of energy dissipation, i.e. the function of dissipation will be:

Here, A/t can be interpreted as the working power of the system and De Gm/ t as the power of flows of matter exchange between the system and its environment; the ratio h = A/DeGm can be interpreted as the efficiency coefficient. The work A can be interpreted as the exergy of the system.

These expressions allow us to compare different ecosystems in respect of their energetic efficiency. Indeed, assume that there are two ecosystems with the same values of DGenv and A: (DGenv)1 = (DGenv)2, A1 = A2. It is obvious that if t1 < t2 then, as follows from Eq. (4.4), Diss1 > Diss2, i.e. the rate of energy dissipation in the first ecosystem is higher than in the second one under the same amount of performed work. Immediately a very attractive teleological evolutionary concept springs to mind (see also Chapter 12):

The evolution of living systems (ecosystems being one sort of them) moves in the direction of decreasing the rate of energy dissipation so that the more perfect systems have lower rates of dissipation.

In the course of evolution the ecosystem structures have become more and more complex: trophic chains became longer, trophic cycles are formed, competition communities become more complex, while the main biochemical mechanisms did not change. As a result the total time t, which is necessary for the entire completion of the local biogeochemical cycles, or, in other words, the residence time of living biomass within the ecosystem, has to increase. The conclusion is evident: since for the same values of useful work of the total biogeochemical cycle in the ecosystem (the same exergy) and the same exchange flows of matter (between the ecosystem and its environment), so that more complex ecosystems have lesser rates of energy dissipation, they have to be more perfect.

Note that there is another way to decrease the rate of dissipation: to increase the ecosystem exergy conserving the time t.

However, if we formally apply this concept to real ecosystems then grass is less perfect than forest, since the residence time of the first ecosystem is less than that of the second. Moreover, since Homo sapiens possesses the highest exergy then the most perfect ecosystem is an ecosystem with its prevalence. This is nonsense, since both grass and forest and other ecosystems are coexisting on our planet. The point is that we consider the concept to be universal, though it is not necessary that the high rate of dissipation means imperfection of the system in general. The evolutionary way is a movement to some target, but the target is attained under different constraints, so that the evolutionary perfection is most likely to be a compromise between different tendencies and does not need to match the absolute extremum of some universal criterion. In addition note that, of course, only stable systems exist, but there is no direct correspondence between stability and complexity (see Section 6.12, Chapter 6).

9.5. MacArthur's diversity index, trophic diversity and ascendancy as measures of organisation

By analogy with the species diversity index D = — JjLi p ln p where p are the frequencies of species, MacArthur (1955) has suggested using the so-called "flows diversity index"

where p(s) is a probability of energy transfer along some sth path inside a system; the summation is effected over the complete set of paths, so that £sp(s) = 1. The simplest way to calculate these probabilities is the following. First, bearing in mind that flows of matter and energy are often expressed in the same energy units, we can consider matter flows as energy ones. Secondly, as follows from Eq. (3.3)

Then p{s) = pij = fj/f, where s runs over all the indices i and j, and n n

It is obvious that the measure DF is maximal, max DF = 2 ln n when there is no hierarchy inside the network: all fj = f /n2, i.e. the total through-flow is homogenously "spread" over the network. We can say that in this case the network is "fully disorganised".

Let us consider the "partially organised" network when flows are directed from compartments with lesser numbers towards compartments with larger ones: fj ; 0 if i $ j (Fig. 9.6a). We also assume that the total through-flow is also homogenously spread over all non-zero internal flows. Then fj(i < j) = 2f/n(n — 1) and max DF = ln n + ln(n — 1) — ln 2- It is obvious that max DF < max DF (n $ 2)- The limit case of such type of organisation is a trophic chain (Fig. 9.6b). It is easy to see that the number of internal flows is (n — 1), and max DF = ln(n — 1) < max DF < max DF•

One can see that the MacArthur diversity index decreases with increase of ordering (organisation) of the network.

Another generalisation of the concept of diversity is trophic diversity or trophic diversity index (Svirezhev and Logofet, 1977). Suppose we have a community with its trophic structure represented by a trophic graph. If we take into account the input of the external resource to the system (for instance, the solar energy for the producing species), we obtain additional links between the resource and these species. Graphs of this kind, much simplified as compared with the graphs of real communities, but still keeping the principal features of these structures, are exemplified in Fig. 9.7. The graphs represent the networks shown in Fig. 9.6.

In each graph one may find linear structures, known as trophic chains. In a chain every species is connected with only two other species (the preceding and the succeeding ones), or only with the preceding species, if the chain terminates in the given species. In a sufficiently complex community one and the same species may belong to several chains. For instance, the community illustrated in Fig. 9.7a displays three chains: R ! 1! 4, R ! 1! 2 ! 4 and R ! 1! 2 ! 3- Species 2 and 4 belong to two chains.

Define the number Ij —the trophic index of the ith species in the jth chain—as the overall number of links between all the vertices in the chain, preceding the ith one (beginning from vertex R up to vertex i). For instance, the index of species 4 in the chain R ! 1 ! 2 ! 4 equals 3, whereas the index of species 1 in the same chain is 1

Fig. 9.6. Partially ordered trophic networks.

Fig. 9.6. Partially ordered trophic networks.

6/61

15/61

30/61

10/61

3/25

4/25

6/25

12/25

Fig. 9.7. Trophic graphs and the appropriate trophic pyramids maximising the trophic diversity.

(cf. Fig. 9.7a). (The vertices preceding the given one are those which may be reached when moving along the trophic chain in the direction opposite to that of the links.) If a certain ith specPies belongs to several trophic chains, then the general trophic index for this species is It = £ jIj, where the sum is taken over all the chains involving this ith species. For instance, in Fig. 9.7a I4 = 5, while I3 = 3.

If now the species population in the network is Ni, and its frequency is pi = Ni / Xi N, then the complexity of the network may be defined as I = £i piIi • The notion of the trophic frequency of ith species is defined in terms of complexity and the trophic index:

This notion takes into account not only the abundance of a given species, but also its role in the trophic structure and in the community hierarchy. Finally, by analogy with the classic definition for the species diversity, we shall define the trophic diversity as

We may consider this value as a measure of organisation of the ecosystem. If we now postulate that the trophic diversity is maximal at the equilibrium state, what are the equilibrium compositions we have to obtain? Examples of graphs from Fig. 9.7a,b are used to illustrate this. Clearly, the maxpi DT is attained for p1 = P2 = P3 = P4 = 1/4. Calculate the trophic indices for all the four species in the two cases: (a) I1 = 1, I2 = 2, I3 = 3, I4 = 5; (b) I1 = 1, I2 = 2, I3 = 3, I4 = 4. Next, by formula (5.3) one may easily find the optimal values of p*. They are depicted as diagrams in Fig. 9.7 where the area of rectangles is in proportion to p*. We see that the resulting figures resemble the pyramids of populations or biomasses really observed in most natural ecosystems. Hence, it is hoped that this description also reflects some of the really existing regularities, and that it may be useful for prime analysis of ecosystem properties such as complexity and diversity. But it is obviously insufficient for stability analysis, just as the community diversity alone clearly is not sufficient to characterise such a dynamic ecosystem property as stability.

Apparently, from the point of view of information theory, ascendancy is one of the most justified measures (Ulanowicz, 1986). The ascendancy concept considers an ecosystem as some complex transformer, which transforms the information associated with inflows into the information leaving the system with outflow. The measure of organisation in this case is the mean mutual information of two random values:

1. The hit of some hypothetical particle of through-flowing matter into ith compartment.

Assume that the probability of the first event is proportional to the fraction of incoming flows (including the exchange with environment) with respect to the total through-flows in the system. If {£} is the set of these events then Pr(^) = Qf/Q-Analogously, if {h} is the set of the second events then Pr(Hj) = Qf^/Q- The probability for events from the sets {£} and {h} to occur simultaneously is naturally defined by the fraction of fij—flow in respect to the total through-flow, Pr(£,, Hj) = fij/Q- Then the entropy of a two-dimensional distribution of this probability is naturally defined as a measure of organisation (ascendancy):

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