Fig. 13.8. The observed number of seeds (full line) compared with the simulated results (dotted line) from 1975 to 1983. The two first years, 1975/1976 observations were used for calibration, while the next 2 years are used for the validation referred to in Table 13.3.

This model is structurally dynamic, since it uses a current adjustment of the beak depth according to optimisation of a goal function. An exergy index, which measures the distance of the system (or rather the system as described by the model) from thermodynamic equilibrium, is used as a goal function. Exergy is calculated as ex^N + exfN2 , where exf refers to the information carried by the seed (i = 1) and by the finches (i = 2), and Ni refers to the densities of the seed and the finches. The application of this goal function presumes that the ecosystem tends to move as far as possible away from thermodynamic equilibrium under the prevailing conditions. The model was calibrated by the use of the data from 1975 and 1976. The precipitation and the beak depth (initial value in 1975-1976 4.5 mm) are known as a function of time according to Boag and Grant (1981, 1984). If these functions give the approximately right beak size as a function of time, the right number of seeds and G. fortis for the validation period, then the model is validated satisfactory. The discrepancy between predicted and observed beak size, number of seeds and G. fortis can be used as a measure of the model validation. The model is validated by the use of the data from 1977/1978; the number of finches, the biomass of seeds and the beak depth are used. These two years were used as it is considered to be most important that the model can simulate the shift in numbers that takes place during this period.

The model equations and the corresponding numerical algorithm are represented in Table 13.2 (in STELLA format).

The validation results can be seen from a comparison between the observations and the simulated results in Figs. 13.7 and 13.8 and Table 13.3. Notice that the number of parameters to be calibrated is only five, while a model that would contain n age classes would have 5n parameters.

Table 13.2

The equations for the presented structurally dynamic model of Darwin's finches finches(i)/di = (promotion-mort_finches) INIT finches = 1170

promotion = IF(TIME > 120)THEN(juv_finches)ELSE(0)

mort_finches = (0.00125 + 0.0022 * (1-ofs) + 0.0026 * shortage_of_food) * finches juv_finches(i)/di = (eggs-promotion-mort_jf) INIT juv_finches = 0

eggs = IF(TIME < 35)THEN(finches * (0.039-(1-ofs) * 0.05))ELSE(0) promotion = IF(TIME > 120)THEN(juv_finches)ELSE(0)

mort_jf = IF(juv_finches > 50) THEN(0.0155 * juv_finches + 0.007 * (1-0.006 * shortage_of_food)ELSE(0)

seed(i)/di = (growth-mortality-feed) INIT seed = 8000

growth = 7.9 * rain mortality = 0.0028 * seed-14 * (rain)A2

feed = finches * 0.024 * 0.055 + juv_finches * 0.015 * 0.1

DH = 50-12 * rain exergy = seed * 120 * 20 + (finches * 16 + juv_finches * 9) * 370

shortage_of_food = IF(finches * 5 + juv_finches * 2 > seed)THEN(1)ELSE(0) diet = GRAPH(beak)

(7.00, 4.00), (7.56, 4.25), (8.11, 4.50), (8.67, 4.75), (9.22, 5.00), (9.78, 5.26), (10.3, 5.55), (10.9, 5.80), (11.4, 6.10), (12.0, 6.50) obs_finches = GRAPH(TIME)

(0.00, 200), (30.4, 230), (60.8, 257), (91.2, 288), (122, 317), (152, 346), (182, 380), (213, 360), (243, 335), (274, 320), (304, 305), (335, 290), (365, 272) obs_seed = GRAPH(TIME)

(0.00, 3000), (30.4, 3100), (60.8, 3250), (91.2, 3400), (122, 3550), (152, 3800), (182, 4100), (213, 3820), (243, 3630), (274, 3480), (304, 3350), (335, 3180), (365, 3000) rain = GRAPH(TIME)

(0.00, 2.40), (33.2, 2.00), (66.4, 2.90), (99.5, 2.30), (133, 2.20), (166, 1.80), (199, 0.92), (232, 0.43), (265, 0.45), (299, 0.5), (332, 0.4), (365, 1.70)

Table 13.3

Validation of the model

Percent average deviation between observed and simulated value 11.6

Linear regression value for observed vs. simulated finches (R2) °.971

Linear regression value for observed vs. simulated seed (R2) 0.811

Simulated beak size at end of 1977 (observed value = 9.96 mm) 9.96

Simulated beak size at mid-1978 (observed value = 10.04 mm) 10.04

Number of parameters 5

13.6. Exergy of the global carbon cycle: how to estimate its potential useful work

Now we come back from geological times to the current state of the biosphere in order to estimate its capability to maintain the biological evolution. But at the beginning we would like to bear in mind several estimations connected with capabilities of the climate machine (see, for instance, Section 5.3).

Lorenz (1955) has introduced the concept of "available potential energy of the atmosphere", A, which maintains its general circulation. He defined it as the difference between the total potential energy, P, and the minimum total potential energy, which could result from any adiabatic redistribution of mass. This value shows how much of the mechanical work can be performed by the atmosphere in an ideal case. In accordance with Lorenz, the ratio A/P = 1.75%. If we take into account the internal energy of the biosphere, U, then A/(P + U) = 0.5%. In reality, the kinetic energy will still be less: about 10% of A: This means that in spite of the fact that the potential energy of the atmosphere is enormous, only its contemptible part is able to perform mechanical work, in particular, to "turn on" the machine of global circulation.

This result has stimulated us to muse upon: what percentage of the biosphere's NPP should be engaged in performing the useful work? Under this value we may imply, for instance, the maintenance of the animal kingdom (including humans) or the measure of ability for developing and reconstructing the biosphere.

How to estimate this work? Before that we shall talk about some data on the global carbon cycle. A studied system is the biota; its state is described by the total amount of carbon contained in the living matter of biota. Such compartments as the atmosphere and pedosphere, as well as the ocean containing inorganic carbon are considered as the environment of the system—the biota. A sum of carbon contained in the terrestrial and oceanic biota (living matter) is equal to = 610 Gt, in vegetation + 3 Gt, in marine biota = 613 Gt (den Elzen et al., 1995). The atmosphere and the biota are connected by inflow, which is equal to the annual net primary production NPP = 60 Gt for terrestrial + 6 Gt for oceanic biota. We see that the contribution of the oceanic part to the total amount of carbon contained in the biota, and to the exchange of carbon between the atmosphere and the biota compartment, is relatively small; in spite of it the net production of phytoplankton is equal to approximately 35-40 Gt. Actually the biological turnover of carbon within the surface ocean is practically closed. Since zooplankton grazes up to 80% of phytoplankton and its metabolism is very high, it plays a main role (with bacteria) in the process of decomposition of organic matter. Chemical (not biological) mechanisms play a main role in the carbon exchange and establishing equilibrium between the atmosphere and the ocean. So, when we consider the whole biogeochemical cycle we could be confined by the consideration of its terrestrial part, so that NPP = 60 Gt and C = 610 Gt.

The global biogeochemical cycle of carbon is considered as a closed thermodynamic system. In favour of this hypothesis we should say: (a) anthropogenic emission of carbon dioxide today is sufficiently small in comparison to biological production and (b) if the biosphere is at the equilibrium then the amount of carbon, which is equal to the annual net production, must be released into the atmosphere as a result of decomposition of the dead organic matter contained in the litter and humus. The latter process is spontaneous.

The characteristic time of this process is 1 year. Certainly, when a new equilibrium is established, the amount of carbon in humus is also changed but this process is much slower.

We assume that the biosphere operates in a periodical regime (with a 1-year period). At the beginning the vegetation accumulating green biomass, by the same token, disturbs the equilibrium between the biota (biosphere) and the atmosphere and accumulates the exergy. Then the new biomass goes to the litter and soil and approximately the same amount of old biomass is decomposed with the emission of carbon dioxide into the atmosphere. As a result the accumulated exergy dissipates (but within another "decomposing" system), and the biosphere comes back to the primary equilibrium state. An annual cycle is completed. Certainly, two such stages, separated from each other, represent an ideal scheme; in reality they are overlapping.

The exergy accumulated during an annual cycle is equal to (see Section 5.4) Ex = cb ln(Cb/C0) - (Cb - C0) where Cb = C0 + NPP. Assuming NPP = AC P C0 the expression for exergy is re-written as


Substituting the corresponding numerical values into Eq. (6.1) we get the estimation of exergy to be equal to 2.95 Gt C. Note that in this model the exergy is estimated in carbon units, like biomass.

It is necessary to say a few words about the energy cost of carbon contained in biomass. If we take into account only the "free" carbon or the carbon dioxide then the energy cost (contents) of 1 g C is equal to RT/mc = 8.4 X 287/12 < 200 J. On the other hand, the enthalpy estimation of the energy cost of 1 g C of biomass is 42 kJ. This example shows that the factor RT in the expression for chemical exergy ought to be used with care. For instance, when we calculate the "genetic" exergy, it is estimated in relation to detritus energy contents (detritus enthalpy), ~ 18.5 kJ/g. If we take into account that carbon content in detritus is about 46% (Bazilevich, 1993) then the energy contents of 1 g C in detritus is equal to ~ 40 kJ; that is very close to the standard value for living biomass.

Keeping in mind the interpretation of exergy as a quantity of mechanical (or useful) work, which can be performed by the system, the quantity Ex = 2.95 Gt C can be considered as an upper limit of the work performed by the terrestrial biosphere. This is only 4.9% of the whole energy flow of carbon cycle contained in the NPP; the rest is spent to support the turnover of carbon.

How can this work be interpreted? We think that this is maintenance of metabolism of the terrestrial superior animals (including humans), although the terrestrial cycle can be closed without terrestrial animals, since the main decomposers here are soil protists and microorganisms. It is interesting that the value 4.9% practically coincides with the estimation of the fraction of energy, which passes from autotrophic to the next trophic level in terrestrial ecosystems, and is approximately 5-10% (Odum, 1971a,b). There is also another hypothesis: this work can be used for the development and reconstruction of the biosphere.

Considering the total amount of carbon in the system, 610 Gt, as an analogue of its potential energy, we see that only Ex/C° = 2.95/610 < 0.48% of it can be considered as an "available potential energy". It is interesting that this value practically coincides with Lorenz's estimation of such type of value for the atmosphere, 0.5%.

Calculating the exergy residual (see Section 5.4)—a measure of the system's ability to evolve—we get

It is interesting that the ratio Rreal = Ex/SEx < 30.4, i.e. it is equal to the lower estimation of weighting factor in the specific "genetic" exergy for plants (see Table 5.1). If we take into account that almost all biomass of the biosphere is the biomass of plants, then we obtain a very surprising coincidence!

Calculations by the VECODE vegetation model under doubling CO2-scenario (carbon amount in the atmosphere increases by two times, ((C°°)1 = 1500 Gt) gives an increase of carbon in the vegetation of 30% ((C0)1 = 793 Gt), and an increase of production of 25% (P1 = 75 Gt) (Cramer et al., 2001). By substituting these values into Eqs. (6.1) and (6.2) wegetEx1 < 3.5 GtCand SEx1 < 0.11 Gt C. The ratio Rmodel(Ex/SEx) < 31.8 is almost unchanged in comparison with the real value, 30.4. Repeating the same calculations for the Moscow Biosphere Model (Svirezhev et al., 1985)—P2 = 70.6 Gt and (Cb)2 = 678 Gt— we get Ex2 < 3.67 GtC, SEx2 < 0.127 GtCand (Ex/SEx)2 < 28.9. We see that the ratio Ex/SEx is practically the same for both models and the real cycle. It is natural to expect this if we assume that this ratio reflects the genetic nature of plants, which of course has not changed with climate change.

The ratio R can be used as a criterion of thermodynamic consistency for different "mechanistic" models of the global carbon cycle. In an "ideal" model, if a modelled biosphere does not undergo some kind of catastrophe, the ratio value, which is calculated using model data, must be approximately equal to its current value, 30.4. If the model ratio differs from the current one then this can indicate the possible thermodynamic inconsistency of the model. For instance, the degree of the thermodynamic inconsistency defined as TI = lRmodei — Rreall/Rreal is the same for both the Moscow Biosphere Model, 4.9%, and for VECODE, 4.6%.

When this value is applied, the specific exergy is found to be 520 kJ/g, which can be compared with the expected value, namely 30 X 18.4 kJ/g = 550 kJ/g, where 30 is the ¡3-value for some average plant; see Table 5.1. The two values are very close, which may be considered in favour of calculating the weighting factors based on the genetic information.

The external factors steadily change species composition. The introduction of exergy calculations in ecological models makes it possible to describe this shift in the species composition, as will be presented later. Exergy is, so to speak, a measure of the survival as it accounts for biomass and information. Properties that give the best survival can be determined by testing those values of models' parameters that provide the highest exergy.

The exergy weighting factors have also been used in an attempt to express the evolution quantitatively by multiplication of the number of families and the weighting factors of the most developed species. The number of families expresses the possibilities that the ecosphere offers to utilise the available resources and ecological niches—we could call it the width of the biological information—and the weighting factor expresses the amount of feedback and regulation mechanisms the most advanced species have—we could call it the depth of the biological information. The product of the two would, therefore, be a relative measure of the overall increase of the biological information. Fig. 11.7 demonstrates this quantification of the evolution.

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