or ex3 > 1 + ( 1 + .. «2 .. (exi - 1) + „ «V (ex2 - 1)

A typical example of a simple trophic chain is the chain

Phytoplankton ! zooplankton ! fish.

Such a type of ecosystem (a warm ecosystem in Silver Springs, FL) was analysed by Svirezhev and Logofet (i978). Using H. Odum's experimental data they estimated the ratios (ai/a0) < i0, (a2/ai) < 8.5, (a2/a0) < 80. Substituting these ratios into the corresponding inequalities we get:

ex2 > i + i0(exi - i), ex3 > i + 9.5[0.9(exi - i) + 0.09(ex2 - i)].

Looking in Table 5.i we find: exi = 3.4 for phytoplankton, ex2 = 29-43 for zooplankton, and ex3 = 28i - 337 for fish. By comparing these values with our inequalities we can see that they are satisfied:

exi = 3.4 > i, ex2 = 29 4 43 > i + ii X 2.4 = 26.4, ex3 = 28i 4 337 > i + 9.5[0.9 X 2.4 + 0.09(28 4 42)] < 45 4 57. The bifurcation diagram for the total exergy is shown in Fig. 7.8.

7.7. Complex behaviour: cycles and chaos

At first sight it seems that the dynamics of the trophic chain remains relatively simple even if its length is 4,5,..., i.e. the value of A is rather large, and the system is sufficiently far from thermodynamic equilibrium. But this is not true: even in a chain of length 4, when A > A3r > Ap, cycles and oscillations may arise, i.e. a complex dynamic behaviour arises (Svirezhev, i987). The cycle encloses the equilibriumNp = {Np, Np, Np, Np}, which loses its stability for A > Ac3r. In this case, we have the right to expect that the exergy, which monotonously increases at a certain rate up to this point, has to begin to increase at a lesser rate (an unstable branch corresponds to lesser exergy).

Up to this point we have studied systems with weak non-linearities, when the flows from one level to another were proportional to the product of their biomasses. However, in reality we deal with more complex functions, for instance, qk k+1 = Vk(Nk)Nk+1 where the so-called "trophic functions" Vk(Nk) seem to look as shown in Fig. 7.9.

It is possible to prove that chains with such types of trophic functions cycles arise even in the simplest chain with a length of two (and naturally also in longer chains). Apparently, the cyclic dynamics is typical for all closed trophic chains if there is a so-called "enrichment effect", i.e. an abrupt increase in the total amount of nutrients. This point of view explains such dynamic phenomena as oscillations of the biomasses of phytoplankton, zooplankton and fish in closed lakes and ponds as a result of the so-called "anthropogenic eutrophication", when the fertilisers washed out from fields and household waste waters

increase abruptly the amount of nutrients (nitrogen and phosphorus) in these water-body systems. Since the specific mean exergy (the total exergy divided by the total amount of matter in the system) in this moment has to decrease, then the fracture on the curve of monotonously growing exergy can be a good indicator to warn about this ecological hazard (for details, see Section 7.8).

The next complex dynamic effect may be "dynamic chaos" (Svirezhev, 1983, 1987). Note that in mathematical ecology this is the rule rather than an exotic exception; the problem is how to make up an adequate model to describe it. For instance, a closed, three-level trophic chain with non-linear trophic functions can be considered, perhaps, as the simplest object that demonstrates dynamic chaos. A model of this system can be represented as:

= N3[-d3 + -2(N2)], -,(Nk) = akNk/(Kk + Nk); k = 1,2.

This system, where d1 = 0.1; d2 = d3 = 0.2; 0.30 # a0 = a # 0.37; a1 = a1 = 1; K1 = K2 = 5, was studied numerically; the total amount of matter in the system, A, and the per capita rate of resource consumption by the species of the first trophic level, a, were selected as bifurcation parameters. In the plane (A, a) the two curves A+1 (a) and AM(a) were constructed, which separate the domains with different dynamic behaviour (Fig. 7.10). The values of these curves at the point a = 0.34 are equal to A+1 (0.34) = 36.251 and Am (0.34) = 36.24.

If we now move along the line ab (at constant a = 0.34) in the direction of increase of the parameter A then the behaviour of the system varies by means of the doubling of cycles from regular to stochastic regimes. The transition does not occur directly: there exists a

Fig. 7.10. The domains of different dynamic behaviours: I—regular, II—stochastic, III—"pre-stochasticity" domain.

"pre-stochasticity" domain III, in which there are both regular and stochastic trajectories, and parallel with the "strange attractor" which results in the doubling of cycles, there is a stable limit cycle. This regime, also named as "pre-turbulence" or "metastable chaos", is characterised by the existence of both chaotic and regular trajectories, which are attracted to the stable cycle. The latter are very similar to chaotic trajectories; but in spite of this they are regular. It is interesting that the pre-stochasticity regime is typical for Lorenz's attractor, but in the latter the process of stochastisation differs from Feigenbaum's mechanism of the doubling of cycles. Our "ecological strange attractor" occupies an intermediate position between Lorenz's and Feigenbaum's attractors. It is curious that the phase volume of our system contracts; in other words, our system (like Lorenz's) is dissipative in spite of the conservation of matter taking place within it.

By moving further along ab we get into the domain of stochasticity, where chaos arises. This is domain II, or more correctly, its lower boundary, as its upper one and its geometry were not studied in detail. The chaotic trajectories are shown in Fig. 7.ii.

Let us set the question: would the property of dynamic stochasticity be conserved with the growth of the chain length? It has been proven that the existence of a strange attractor for chains of arbitrary length (longer than three) follows from the existence of the strange attractor for a closed trophic chain of a length of three (Svirezhev, i987).

In contrast, there are only regular trajectories (cycles) in open chains, in which a dead biomass is neither decomposed nor returned to the resource compartment. Naturally there is a constant inflow of an external resource, which is equal to zero for the closed chain. Now let the chain be partly closed, i.e. a certain part of matter contained in the dead biomass returns to the resource level. A model of such a system can be represented, for instance, as dN0 3

= -dkNk + Vk-iNk - VkNk+i, Vk = , k = i, 2, 3. (7.2)

It is interesting to find such a threshold value A* of the "closure" parameter A (A = 0 for open system and A = i for fully closed one) so that if A < A* then the dynamic chaos arises in system (7.2).

Naturally, other formulations are also possible. We believe that models of mathematical ecology constitute a favourable field to look for different strange attractors and dynamic stochastic behaviour corresponding to them, but it is necessary to remember the biological adequacy of the model being used.

7.8. What kind of exergy dynamics takes place when the enrichment and thermal pollution impact on the ecosystem?

Let us carry out the following two numerical experiments with system (7.i):

i. We shall effect a slow, quasi-stationary increase of the total amount of matter in the system (A) starting from the value of A < di / a0 = 0.i/0.34 < 0.29, when there

is no biomass at all, until A = 36.5 when dynamic chaos arises; a0 = const = 0.34. In the same way we simulate the enrichment phenomenon in closed water-body systems.

2. At the constant value of A = 36.5 we shall effect a slow, quasi-stationary increase in the rate of nutrient uptake by phytoplankton (a0) starting from the value of a0 < d1 /A = 0.1/36.5 < 0.27 X 10_2, when there is no biomass, until a0 = 0.34, when dynamic chaos arises. In the same way we simulate the effect of thermal pollution, since the value of a0 increases with the growth of temperature. The a0 can be interpreted as a0 = Mmax/Kph where Max is the maximal growth rate of phytoplankton and Kph is the Michaelis-Menten constant in the expression for the growth rate Mph = MmaxNo/(Kph + No).

If we interpret the chain as consisting of phytoplankton, zooplankton and fish then the total exergy is represented as

Ex(t) = [A — N1 (t) + N2 (t) + N3 (t)] + 3.4N1 (t) + 38N2 (t) + 330N3 (t). (8.1)

Since the exergy defined by Eq. (8.1) depends on the value of A (this can mask dynamic effects connected with exergy) then it may be more correct to use such a value as the specific mean exergy, ex = Ex/A. It is obvious that this value is dimensionless and does not explicitly depend on A. Another macroscopic characteristic is the total biomass of the chain, N(t) = N1 (t) + N2(t) + N3(t), but instead of that we shall use such a dimensionless value as h = N/A, which could be termed as a "utilisation coefficient". Indeed, the greater is the coefficient h, the greater part of the matter is concentrated in the biomass. At the limit when the h is close to one, practically the whole matter is in the biomass, and the nutrient which is formed as a result of dead organic matter is almost instantly consumed by phytoplankton. In this case we can say that the system is almost ideally adapted to the environment.

If the value of h determines the degree of adaptation then the value of ex determines the quality of adaptation. In other words, if the h is an index of the efficiency of the trophic chain then the ex is an index of its organisation or an index of biomass quality. Indeed, if the specific exergy determines the quality of one biomass unit of the corresponding species, then the greater its mean value is, the higher is the percentage of "quality" biomass in the total chain biomass.

The specific mean exergy (in dimensionless units) is calculated with respect to time until a stationary regime for each given value of A is established (Fig. 7.12). Until A < 9.455 (the first picture) the dynamics remains very simple and regular and the equilibrium is very quickly established so that there is not a problem to estimate the limit value of ex. One can see that it increases with the growth of A (see Fig. 7.14a). At A = 9.447 a first cycle arises: the system begins oscillating. The specific mean exergy also oscillates (the second picture), but the oscillations are very simple. Starting with this value of A the ex begins decreasing; moreover, the decrease is monotonous in relation to the increase of A (Fig. 7.14a). Finally at A = 36.24 the dynamic chaos arises, but the behaviour of ex (also chaotic) is simpler than the behaviour of the state variables N1, N2, and N3 (compare the third section of Figs. 7.11 and 7.12). The specific mean exergy continues decreasing. Note when we speak about the dependence of ex on A we imply that we operate with the temporal mean of the value, obtained as a result of averaging over one of the limit sets: point, cycle or strange attractor. For this we average the ex over a rather long piece of established trajectory. All the same methods are used when we deal with the utilisation coefficient h and the dependences of ex and h on a0.

Analogous pictures are constructed for the utilisation coefficient h (Fig. 7.13).

One can see that these pictures are qualitatively similar to Fig. 7.12 (the same regular behaviour for A = 5, oscillations at A = 9.447 and non-periodic, irregular impulses at A = 36.5, which is characteristic for chaos), but all dynamic effects are apparently weaker than in the previous case. As the value of A increases the temporal mean of h very quickly reaches saturation at the level very close to one (Fig. 7.15a). The saturation occurs when the system dynamics becomes sufficiently complex (oscillations and chaos). However, in

Fig. 7.12. Dynamics of the specific mean exergy (ex = Ex/A) for three different dynamic regimes: (a) regular regime without oscillations (A = 5), (b) doubling oscillations (A = 9.447), (c) chaos (A = 36.5). The exergy and the total amount of matter are measured in dimensionless units.

contrast to the previous case the occurrence of complex dynamics is difficult to detect if the graph h(A) is used.

Let us once more look at Fig. 7.15a. We can see that when there is chaos then the utilisation coefficient is very close to one (about 0.98-0.99). This implies that the resource is almost completely used; it suggests that the complex dynamics of ecosystems (doubling oscillations and, especially, chaos) is one form to which they adapt, allowing them to use (almost completely) a trophic resource. However, this adaptation is paid for by a fall in biomass quality: the fall of exergy is accompanied by a reduction of fish biomass and increase of the biomass of plankton (especially phytoplankton). A compromise between these tendencies is reached at the boundary between the quiet regular dynamics (A < 9.447) and the situation when the dynamics is suddenly complicated (A > 9.447). At the "boundary" point A = 9.447, on the one hand, the degree of resource utilisation is already rather high; on the other hand, the biomass quality (the specific mean exergy) is still high (at this point the ex = 75.3 is maximal). As we have already mentioned, the sharply delineated fracture on the graph in Fig. 7.14a can be used as a detector to indicate that we have to expect the occurrence of new regimes in the ecosystem.

It seems that this state is optimal: there is an almost complete utilisation of the resource here, and everything is quiet. However, at any time we have to expect the appearance of such annoying (from the anthropocentric point of view) events as big oscillations and

Models of Ecosystems: Thermodynamic Basis and Methods utilisation coefficient r\ = N/A

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