Fig. 7.5. "Slow" evolution of the total biomass (N) and the mean value of the specific entropy production (d) as a function of inflow q: a0 = a1 = 1; d1 = 3, d2 = 1, d3 = 2.

example of Fig. 7.5, where d1 > d2; d2 < d3) then in the corresponding interval the value

As we shall show below, it is this kind of "disordering" which can cause the system to behave in a complex, "non-thermodynamic", even chaotic way.

7.6. The closed chains with conservation of matter. Thermodynamic cost of biogeochemical cycle

All ecosystems live only at the expense of a permanent flow of energy through the system. The energy is permanently losing value; it cannot be accumulated in its original (for instance, solar) form within the system. The situation with matter is different. Ecosystems do not consume matter, as they do with low-entropy incoming energy, but they exchange matter with the environment by means of biogeochemical cycles. Therefore, we can consider the joint system "ecosystem + environment" to be closed with respect to matter. In other words, if we measure biomass in units of matter (carbon, nitrogen, phosphorus, etc.) then we can assume that the total amount of matter contained in the biomass and the environment is constant.

Of course, we immediately encounter the "problem of scale" here. For instance, in relation to carbon, the whole atmospheric carbon is considered as an environmental one; therefore only a change in the state of the global ecosystem could be significant for atmospheric carbon. The dynamics of local ecosystems have no influence. This is explained by the high mobility of CO2: all local emissions become mixed up in the atmosphere in the course of 3-4 months. Because of this carbon is practically never a limiting element. The mobility of such nutrients as nitrogen and phosphorus is relatively low; their environmental concentrations are often determined by local conditions, and they are very often limiting factors. All these arguments allow us to assume the existence of corresponding local laws of matter conservation for these elements. Their content in the environment and in living biomass is thus naturally strictly negatively correlated.

of d increases.

The balanced equations for the system should be written as n

—jNk = qk-i,k(Nk-i,Nk) - qk,k+i(Nk,Nk+i) - Dk(Nk), k = 2;•■•> n - i;

where N0 is the concentration of the selected element (for instance, nitrogen) in the environment, Nk is the same in the biomass of kth level, and Dk(Nk) represents only dying-off biomass. It follows from these equations that nn

The general theory of such systems was developed by Svirezhev and Logofet (i978). As we have already done above, we restrict the analysis to three levels that is sufficient for understanding and interpretation.

Assuming that N0 = A - Ni - N2 - N3, qk-i,k = ak-iNk-iNk and Dk = dkNk; k = i, 2, 3 system (6.i) is written as dNi

— = Ni [(o^A - di) - «0Ni - («0 + «i)N2 - aNs], dt dN2

dt dN3

The parameter A (the total amount of matter in the system) is naturally considered as a bifurcation parameter.

The separatrices portrait of Eq. (6.3) is shown in Fig. 7.6.

The trivial equilibrium Np = {0,0,0}, which corresponds to thermodynamic equilibrium when there is no living matter in the system, is stable if A < Ap = di/a0; if the inverse inequality holds then it is unstable. The unstable manifold is the separatrix Si outgoing from Np; its equation is (dNi/dt) = Ni(a0A - di - a0Ni). It is incoming in the next equilibrium Ni = {Np = A - (di/a0), 0,0} (see Fig. 7.6). It is clear that Np > 0 for A > A0. For A > Ap = A0 + (d2/oi) = (di/o0) + (d2/oi) the equilibrium Np is unstable; its unstable manifold is the separatrix S2 situated in the plain (Ni,N2), as shown in Fig. 7.6.

Consider the other equilibrium Np = {Np,N|, 0} where Np = d2/oi, N| = (a0/ (a0 + oi))(A - Ai). For A > Ai the equilibrium Np is situated within the positive

quadrant of the plain (N1, N2). It loses its stability when A > Ap = + d3 (a1 /a0a2) = (d1 /a0) + (d2/a1 ) + (d3/a2)((a0 + a1 )/a0); its unstable manifold is the separatrix S3 outgoing from Np inwards the positive 3D orthant (see Fig. 7.6). Since system (6.3) does not have cycles (oscillations) then the separatrices S2 and S3 come into the equilibriums Np and Np, where (Svirezhev, 1987)

Np = —A— (A - A2) + A, NP = A; Np = (A - A2) (6.4)

We see that the dynamics of system (6.3) is very simple: the critical bifurcation numbers A0, A\ and Ap divide the axis A into the domains in each of which either there is no chain at all, or a chain of fixed length (one, two, three) can exist (Fig. 7.7). This is a typical bifurcation picture, which is very similar to that shown in Fig. 7.4: one solution loses its stability, a new stable solution arises, etc. when the bifurcation parameter passes over the critical value of A: A0, A\ and Ap.

Fig. 7.7. The bifurcation diagram of equilibrium solutions for the total biomass: — stable branch,--unstable branch.

Fig. 7.7. The bifurcation diagram of equilibrium solutions for the total biomass: — stable branch,--unstable branch.

If now we calculate the total chain biomass:

we can see that it increases with the growth of A and number of links, but the biomass increases slower than the total amount of matter A (Fig. 7.7).

From a thermodynamic point of view the thermodynamic equilibrium corresponds to the case when there is no living biomass matter (N* = A). The appearance of living biomass implies that some thermodynamic machine such as the biogeochemical cycle began to work, and the system started its evolution from thermodynamic equilibrium; in the system, exergy began to accumulate. In the ideal case, the process has to continue until all the matter is transferred from a non-living to a living form. This would be possible if the residence time of biomass is equal to infinity, but this is not the case: this time is in principle finite, and such a kind of infinity is provided by the biogeochemical cycles of corresponding elements.

It is obvious that if the total amount of matter in the environment is equal to zero then it is senseless to talk about a turnover of matter, even if there is an inflow of energy. However, from the point of view of linear thermodynamics, it is sufficient for a very small quantity of matter to appear (A ~ e << 1) in order to make the cycle work. But our system is non-linear in principle, and the cycle cannot work before the amount of matter in the environment exceeds the critical value of A> = d1/a0. It is natural that this value is determined by characteristics of the system realising the matter turnover. Just after this moment the system begins to move far from thermodynamic equilibrium with a velocity equal to the growth rate of A. Note that although the slopes of A and N * are the same, they are shifted from each other by the value of A> = d1/a0 = 1/a»r1. The t1 is implied as some mean residence time of living organisms forming the first (autotrophic) level. By expressing this in terms of energy units the value of A> can be considered as the cost of biological turnover for a simplest chain consisting of a single level, whereas the t1 is the time of turnover. Note that the dynamics of such a system is very poor and simple and because of that we are not yet far from thermodynamic equilibrium.

There is one curious interpretation of this result. Let A be the total amount of nitrogen in assimilated form which is turned over in the global biogeochemical cycle of this element. However (it follows from the result), for living matter to arise and the nitrogen cycle to start up it is necessary to have some initial finite quantity of ammonium and nitrates; note that the atmospheric nitrogen is biologically inert. Apparently, in the "pre-biosphere" these compounds could be formed as a result of thunderstorms from atmospheric nitrogen. It is evident that their quantity was very low, and in order for the turnover to begin to function the threshold A* = 1/a»r1 has also to be low. For this a living matter has to possess a very high rate of biomass production (the a0 is large) and a very low rate of dying-off of biomass (the t1 is also large). A good candidate for this role could be photosynthesising micro-organisms (for instance, green algae). And only after the non-biological processes (thunderstorms) have made a sufficient quantity of nitrogen compounds, which are necessary for living matter in the simplest ecosystems to arise

and exist, could the global cycle of nitrogen and other biogenic elements by the same token start.

The next increase of A results in a growth of the length of the trophic chain that can be interpreted as a complication of the ecosystem structure. The set of bifurcation values Al, A2, Ap,... can be considered in this case as the costs of sequential complication of structure.

There is one more serious problem here: the criterion of the choice of branch when the solution branches after passing over a bifurcation point. Above we used some thermo-dynamic criteria for this, but we have to say frankly that this is not obtained very well here. The passage over the first critical point A = A0 is accompanied by the branching of stable solution N(0) = 0, NO = A into two branches: stable N(1) = A — A0 and unstable Np = AO (see Fig. 7.7). It seems at first sight that the branch corresponding to the greater biomass can be selected as a true stable path of evolution. However, when we consider the passage over the second critical point A = Al, we see that greater total biomass goes to the unstable branch N(1) = A — Ap; the stable branch

corresponds to a lesser value than the total biomass (N(1) > N(2) for A > Ap). So this idea can be discounted.

We already dealt with the problem of weighing biomasses at different levels: it was assumed that these biomasses have different exergy. The exergy is ranked in relation to the energy content (enthalpy) of detritus. In the framework of our model the value of N0 could be considered as a mass of detritus. Then the total exergy of the system "chain biomass + detritus" expressed in detritus units can be written as

In accordance with the principle of exergy growth in the process of system evolution we postulate that the system always selects the branch (and correspondingly the structure of length of the chain) on which the exergy is greater than another. For instance, in the right vicinity of the first critical point Ex(0) = A < Ex^) = AO + ex1(A — AO) or (A — Ap) < ex1(A — AO). If A > AO then ex1 > 1. In other words, if the first trophic level is existing and stable then the specific exergy of its biomass must be greater than one.

Let us now consider the case of the second critical point Ap. In its right vicinity where A > Ap

a0 + ai whence it immediately follows that ex2 > i + (i + (ai/a0))(exi - i). For the third critical point we have

Ex(2) < Ex(3) = Ap - Ap + exJ «2 (A - A2) + ^ 1

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