O zJi oinoout

The value of As characterises the degree of complexity of a "flows -storages" diagram, since it has a minimal value As = 0 if and only if every compartment connects with others and the environment by equal bilateral flows (the system is maximally disordered), and the ascendancy is maximal when the diagram is a closed ring of sequentially linked compartments. The latter is considered as an example of maximal order (Ulanowicz, 1995). Comparing Ulanowich's measure, As, with MacArthur diversity index, DF, we see that despite their formal similarity they fundamentally differ from each other: if the ascendancy increases with the growth of organisation then, vice versa, the MacArthur index decreases.

Ascendancy was estimated for many systems, in particular, for the Cone Spring ecosystem (Ulanowicz, 1995), but since these diagrams resulted from single measurements it is difficult to say something about the degree of organisation of these systems (except a trivial confirmation of their diagrammatic tangle). Nevertheless, Aoki (1995, 1998) has shown that there is a positive correlation between the value of ascendancy and the degree of eutrophication for several lakes in Japan. So, we can say today that the extremity of ascendancy is not yet confirmed by reliable observations, although it is implicitly implied in many ecological works. Developing his method, Ulanowicz (1998) assumed that the increase of ascendancy in the course of evolution is a tendency which could be masked and even not realised, since the function of ascendancy is not a strictly maximised goal function.

Note that definition (5.5) has two significant defects:

1. it does not take into account storages,

2. it is not dynamic, i.e. it operates only with a static picture.

Although there are a lot of attempts to overcome these defects, the concept of dynamic organisation is still not developed. Ulanowicz and Abarca-Arenas (1997) have attempted to introduce the storages xi into the definition of ascendancy. They defined the probabilities Pr(£), Pr(^) and Pr(£, Hj) as Pr(£) = xjx, Pr(Hj) = xj/x and Pr(£, Hj) = fij/Q where x = £f=0 xi is the total biomass of all compartments including the environment. Then

Unfortunately, this definition contradicts the general properties of probability, since the average probability Pr(^) = Yj=0 P*(£n Hj) = QT — xi/x; analogously for Pr( Hj). In addition, we have to introduce some fictive biomass of environment; for this we have to assume (without any reasonable arguments) that the ratio of the total exchange flow to the environment storage is equal to the same ratio of the total internal flow to the total storage of the system.

However, the main problem is how to formulate functional dependences between the change of storages and flows. Of course, we could do it with the help of differential equations, but immediately the problem arises: how to define the ascendancy in this case, and how to define the corresponding probabilities? This is not trivial, since on the one hand, there is the discrete set of states, and on the other hand, the continuous time. This problem is simpler for a finite time interval (Pahl-Wostl, 1995). If we divide the whole interval by r subintervals and denote the flow from ith to jth compartment at the kth subinterval as fk, then the measure of temporal organisation (the integral ascendancy) can be represented as i r n n ( fk)2Q

Q k=1 i=0 j=0 JijJi-Jj nhere the point • denotes the summation over the corresponding index, and Q = Xk=1 Y!n=0 Y.j=0fkj is the integral (over all subintervals) total through-flow.

Note that despite a visible diversity of these definitions they are all based on the basic definition of information contained in two- and three-letter words (see Section 4.3, Chapter 4).

9.6. How exergy helps to organise the ecosystem

There are also a lot of attempts to use the exergy as a measure of organisation. The standard definition of exergy in our case is Ex = RT0[xK + x ln(x/xeq) — (x — xeq)] where x = Y.j=1 xi, xi are the storages, K = Yn=1 Pi ln(pi/P;q) is Kullback's measure and pi = xi/x, p®q = x®q/xeq. It is postulated that the ecosystem evolves striving to increase its own exergy, which attains its maximum in a steady state. The main problem when we want to test the statement is how to estimate the equilibrium values x®q: in fact, we observe the state, which is far from thermodynamic equilibrium. For the particular case (lakes in Japan), Aoki (1993) has suggested a witty idea based on the ergodic paradigm: he has replaced the temporal evolution by the spatial one. He used the values of compartmental storages in the least eutrophic lake as a reference state. Calculating the values of exergy for other lakes, he has got the increase of exergy with growth of eutrophication. The degree of eutrophication was estimated by experts. Note, however, that there is an underwater stone here: by giving the origin milestone (oligotrophic lake) we implicitly give the way.

Nevertheless, we think that the exergy concept (especially in its "genetic definition") plays a very important role in the ecosystem evolution organising such a system's property as stability. In order to clarify our point of view we have to keep in mind a very important result in the theory of Lotka-Volterra equations (Svirezhev and Logofet, 1978).

If we assume that the dynamics of the system are described by Lotka-Volterra equations then dx; n

In studying systems of type (6.1), V. Volterra considered two special classes, conservative and dissipative systems. System (6.1) is said to be conservative (in the Volterra sense) if there exists a set of positive numbers a1, —, an such that n n

If the quadratic form F(x1, —, xn) is positive definite and the system is called dissipative. If the quantity Ex = £n=1 is interpreted as the total weighted biomass of the system then the meaning of the above-given definitions is that interactions between compartments, given by the matrix of y^, do not affect the course of the total weighted biomass in the conservative case, but do hamper biomass increase in the dissipative case. Using the Lyapunov method, we can prove (Svirezhev and Logofet, 1978) that the equilibrium N* is locally Lyapunov stable (but not asymptotically) in a conservative system, and globally (in the entire positive orthant) asymptotically stable in a dissipative system (see Chapter 6). But if one or several co-ordinates of the model's equilibrium become negative, the corresponding species disappear from the system. Thus, the condition of existence and Lyapunov stability of a feasible equilibrium N* (the equilibrium where all the co-ordinates are positive) is not only a sufficient condition, but also a necessary one for ecological stability in both conservative and dissipative systems. Ecological (or Lagrange) stability is interpreted as a boundedness of solutions from below and above.

However, the definition of dissipative (in Volterra's sense) systems does not give us their characterisation, i.e. some testing criterion, and a search of such a1, —, an is a serious problem. Redheffer (1985a) named it as the problem of Volterra's multipliers. The problem arises in different fields, when a sign-definite quadratic form as Lyapunov's function is sought. Redheffer (1985b) has proved the theorem about the reduction of an n-dimensional case to a three-dimensional one, which already has a well-known characterisation. He has suggested a computational algorithm for such a kind of reduction.

Our experience in an ecological modelling (in particular, a modelling of multispecies fish pond ecosystems; Svirezhev et al., 1984) has shown that very often, when we used Lotka-Volterra equations and, after calibration (as a result we obtained an estimation of the coefficients yij), we tried to use the function of dissipation Diss = Yj=\ Xj=i GjNNj as Lyapunov's function, or to apply Prigogine's theorem using the expression Diss as the rate of internal entropy production, then our attempts were often unsuccessful. This quadratic form was not sign-definite. However, a crazy idea comes to mind: what if we consider the specific "genetic" exergy exj,..., exn as Volterra's multipliers a1,..., an? We have tested the idea in the two particular models mentioned above and have got positive results. Nevertheless, this is only a speculative hypothesis for now, which needs further testing. If the hypothesis would be confirmed then we can say that Volterra's multipliers succeed to obtain from profound "genetic" reasons. This means that the conceptual construction of the model, in general, responds to the paradigm of evolution towards a global stable equilibrium.

So, we see that the different extreme principles and measures of organisation have their own place in the analysis of static compartmental schemes, but all attempts to do a step towards dynamic models and dynamic organisation are at present "not very successive". But the situation is not desperate; for one partial but rather wide class of compartmental models we know how to construct dynamic models by using only static compartmental diagrams (Logofet, 1997; Svirezhev, 1997c; Zavalishin andLogofet, 1997).

9.7. Some dynamic properties of trophic networks

We assume that flows between two compartments depend only on their storages, fki = fki(xk, xi) and fik = fik(xi, xk); inflows qt are either constants or explicitly dependent on time; outflows are proportional to storages, so that yi = mixi. We also assume that every intercompartmental flow belongs to one of these three types: (1) donor type, fki = akixk; (2) recipient type, fki = Bkixk'; (3) Volterra's "prey-predator" type, fki = ykixkxi. The latter assumption allows us to express the values of these coefficients as functions of flows and storages at the equilibrium: aM = fki/x*k, Bki = fH/x*, yM = fki/x*kxk and mi = yk/xk. Note that we can consider the expression for input flow in more general form, q'i = qi + Dixi. However, we can include the term 8ixi into the expression for output flow, yi = (mi — 8i)xi, and by the same token reduce the problem to the preliminary formulation. A dynamic system corresponding to a given diagram is written as dx

where matrices B and C are the matrices with elements

IByll =

afi 2 Ptj for i

yji yy 0

for i

and diag{x} is the diagonal matrix with elements x1;...,xn. Of course, there is a certain degree of arbitrariness in the construction of these equations: types of intercompartmental flows are given proceeding from ecological expert arguments, but, on the other hand, this arbitrariness brings a certain flexibility into the constructive algorithm. Note that if the total inflow is non-negative, and there is at least a single strictly positive coefficient among non-negative coefficients mi then all the solutions in Eq. (7.1) will be bounded, i.e. they can always be interpreted from the ecological point of view.

System (7.1) linearised in the vicinity of equilibrium x* is written as dAx/di = AAx where Ax = x — x * and A is the so-called Jacobi's matrix. Knowing it, we can judge about stability of the system, its sensitivity in relation to variations of inflows and, generally speaking, about some general dynamic properties of trophic networks.

An elementary "particle" of any network is the pair of compartments, ith and jth, and its "sort" is defined by characteristics of the flows linking them. It is obvious that the ith compartment may be a donor in relation to the jth one, and then the flow ij is a donor type, whereas the jth compartment may be a recipient in relation to the jth one, hence the flow ji belongs to a recipient type, etc. In this case, a sort of {i, j}-pair is defined as a "donor-recipient" (DR). Analogously, other sorts of {i,j}-pair are defined as the following: "donor-donor" (DD), "donor-Volterra's type" (DV), "recipient-donor" (RD), "recipient-recipient" (RR), "recipient-Volterra's type" (RV), "Volterra's type-donor" (VD), "Volterra's type-recipient" (VR) and "Volterra's type-Volterra's type" (VV). Naturally, the elements of Jacobi's matrix depend on the "sort" of selected pair.

Let i — j. Then the values of non-diagonal elements are determined as:

xj xj f.. - f.. f.. (RD) : Aj = Ai - ßij= fL~ffiL ; (RR) : Aj = -ßij = - -j ;

(VD) : Aj = Ai - Jijx* = f^L ; (VR) : Aj = - jf = -ffL ;

Let i =j. Then ah = —mi + Y.k-t (fiki— Aik) + Xk-i (Jkt — Gik)x* = —mi + Y.k-ipu where the items Pik are determined as:

(RD) : Pk = 0; (RR) : Pik = •% ; (RV) : Pik = % ;

(VD) : Pk = -fpk ; (VR) : Pik = ffki-^ ; (VV) : Pik = .

Generally speaking, the Jacobi matrix can be represented as A = (M — Q)diag{1/x *} where M is a matrix depending on the equilibrium flow matrix F * = kfjII, Q and diag{1/x *} are diagonal matrices with elements q1,..., qn and 1/x1,..., 1/xn.

Each non-diagonal element in M can accept one of the values: 0, f*, —f*, f* — f*, and the diagonal elements are combinations (rather complex) of different flows. In particular, if all intercompartmental flows belong to Volterra-type then M = (F*)T — F* is an anti-symmetric matrix and the Jacobi matrix is represented as kAjk =

It is known that in this case the matrix A is semi-stable (in the sense ReA(A) # 0), but it can always be made stable by a small "movement" of parameters. In fact, we have to assume that any observed equilibrium is stable since it exists. On the other hand, calculating the Jacobi matrix by our observations, we can obtain that some ReA(A) will be positive, i.e. the corresponding equilibrium will be unstable. Thus, there is a contradiction between the observability of this system and its instability (or more correctly, the instability of its equilibrium). How to resolve this contradiction? What kind of incorrectness could generate it? Firstly, there is some non-coincidence between real and observed values of storages and flows. Secondly, we assume that the system is in equilibrium, while it is not true. And thirdly, we have wrongly defined the system structure, i.e. the number of variables, type of flows, etc. Each of these reasons separately and all these together can provoke this contradiction. Unfortunately, we cannot say what the causal reasons for this contradiction are if we have only the information about storages and flows. At least, we can formulate the following statement: if the matrix A is unstable then the observed equilibrium cannot exist. The latter gives us a good method for the selection of observed data.

How to estimate the system response in relation to change in the inflow? We assume that the vector q changes slowly, quasi-stationary, so that the system is remaining at the equilibrium manifold x* = x*(q)- Then, by applying to the equality (3.3) the implicit function theorem, we obtain

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