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Fig. 13.3. Buffer capacities of phytoplankton to changed nutrients loading (mg/l) are plotted versus the nutrient concentration in flowing water.

2. The buffer capacities for the influence of nutrients on zooplankton and planktivorous fish have a maximum at 1 mg/l and are decreasing above this concentration by increasing nutrients input.

The results are consistent with the general observations and previous model studies, referred above. The buffer capacities are generally either increasing or constant up to a nutrient level of about 1-4 mg/l, except for the influence of nutrients on phytoplankton. At about the same concentration, specific exergy has its maximum.

The results may be interpreted as follows: up to a total nutrient concentration of about 2 mg/l, the examined buffer capacities are quite constant or even increasing, except for the influence of nutrients on phytoplankton. Specific exergy, which measures the ability of the system to utilise the resources, is increasing along the same line.

3. The buffer capacities for the influence of temperature are generally decreasing by increasing nutrient input above a nutrient concentration of about 2-4 mg/l except for carnivorous fish, where the buffer capacity is consequently increasing slightly with increasing nutrient input.

In this range of nutrient loading the top-down and the bottom-up controls are working in parallel; see Sommer (1989). A nutrient loading up to this level seems therefore fully acceptable, but if the loading increases above this level, crucial buffer capacities and the specific exergy decrease.

Some buffer capacities measuring the influence of nutrient input on phytoplankton and carnivorous fish increase, and so does the exergy, but it reflects only the ability of the system to meet changes in forcing functions (in the case of the nutrient loading) by such changes in the structure that the direct influence of these changes is reduced (in this case the influence of nutrients on phytoplankton).

The results of an ecosystem integrity analysis, as has been illustrated in this case study, can only be interpreted semi-quantitatively, because the accuracy of the underlying model does not allow very precise quantifications. The case study has, on the other hand, shown that the analysis is very useful, as it enables us to assess approximately the acceptable level of nutrient loadings and to see the consequences in relation to buffer capacities if we increase the loading above this level, i.e. to predict the expected outcome of the structural changes in the system. In this range of nutrient loading the top-down control has collapsed; see Sommer (1989). A nutrient concentration above approximately 2 mg/l should consequently be omitted and measures should be taken to reduce the nutrient loadings accordingly.

Fifteen lake case studies taken from ICLARM's ECOPATH survey on various ecosystems (Christensen and Pauly, 1993) were compared in relation to exergy and specific exergy. The figures are taken from the steady-state average situations based on observations. The exergy increases with increasing eutrophication for the 15 lakes. Fig. 13.4 shows, from these 15 case studies, the specific exergy ( = exergy divided by the

Fig. 13.4. The specific exergy (kJ/g) is plotted versus the eutrophication for 15 lake studies.

total biomass versus the eutrophication) measured by the biomass of phytoplankton and macrophytes including export. The specific exergy has a maximum at a relatively low level of eutrophication, which is to be compared with a medium level at the model exercise (see Fig. 13.2) as most of the 15 examined lakes are eutrophic to hypereutrophic. Fig. 13.4 indicates that the results found by modelling studies (Figs 13.1-13.2) are also valid when lake observations are used.

13.4. Thermodynamics of controlled ecological processes and exergy

— = Fk(Ni,...,Nn-,fi(t), ...,fm(t); ai,..., Ar), k = 1;..., n (4.1)

A model of an ecosystem described by n state variables Ni can be written as the system of ordinary differential equations

N dt where f1(t) are the so-called driving functions describing processes that take place in the environment and do not depend on the ecosystem state such as temperature, solar radiation, etc. Parameters as are internal characteristics of the ecosystem, determining its functioning and its interaction with the environment such as, for instance, rates of nutrient uptake, respiration coefficients, maximal intrinsic rates of the ecosystem species, mortality coefficients, etc. Certainly, all these can depend on driving functions. It is natural to assume that values of all these parameters result in a long-term evolution of ecosystems while the maximal degree of adaptation is being attained. A measure of the adaptation is exergy. However, in order to reach this state some of the parameters have to change in the course of evolution, increasing the degree of adaptation, i.e. they are control variables. We assume for simplicity that all as are the control variables; since as vary within some natural borders binding the domain Va, aseVa Since fk(t) are given functions, Eq. (4.1) is written as a non-autonomous dynamical system:

Below, we follow the concepts of "Pontryagin's maximum principle" (Pontryagin et al., 1969).

We do not concretise the expression for exergy now; we shall do it later. Let the exergy derivative be dEx

then the current value of exergy is f

J to

If we introduce a new variable No = —Ex extended system (4.2) is written as

= Fk(N1,...,Nn; «1,..., ar; t), k = 0, 1,..., n. (4.5)

We consider one more system in relation to auxiliary variables ck:

If we take into consideration the Hamiltonian n

then all these equations are written as the following Hamiltonian system: dNk dH dck dH

The system starting at time t0 from initial point N0 moves with a(t) [ Va . At time t1 its exergy is equal to Ex1 — Ex(N1, a(t1), t1) . In accordance with one of the Pontryagin theorems about optimality, this value is maximal if such a non-zero vector-function C(t) — (Co, Ci, Cn) corresponding to the optimal functions a(t) and N(t) exists that

1. For any t [ [t0, t1] Hamiltonian (4.7) attains its maximum at the optimal point a(t):

sup [H(C(t), N(t), a(t), t)] — M(C(t), N(t), t), (4 . 9)

2. The following relationships hold:

C0(t) — constant # 0, M(C(t), N(t), t)— - Y —- Ck(t)dt. (4 .10)

Consider the system from the thermodynamic point of view. Since the total time derivative of exergy is dEx _ 9Ex n 9Ex dNk

then

One of the optimality conditions is Co(t) = constant # 0; without loss of generality we set Co(t) = _1- If in addition we set (9Ex/9Nk) = ~Ck, k = 1,..., n then Eq. (4.11) is represented as

However, in accordance with Pontryagin's maximum principle at any t [ [t0, t1] along the optimal trajectory leading to the maximal value of exergy at t1 the Hamiltonian

H(C, N, t) = maxa H(C, N, a, t). Therefore, everywhere along the optimal trajectory the rate of exergy growth, 'dEx/'dt, is maximal for the given state and parameters.

We have obtained a very important result proving the equivalency of local and global forms of the "exergy maximum principle". Of course, the equivalency has been proved for a special class of system. The problem of how large is the class is open until now, but it seems to us that a lot of chemical and ecological systems belong to this class. For instance, in the case of a chemical system, when exergy is equivalent to thermodynamic potential, G(N, t) = Yi=\ MkNk, H(C, N, a, t) = 8G(N, t)/>t, Ck = ~Mk Mk is the chemical potential), and the equations describing a state of the system are the chemical kinetic ones.

In the case of the stationary environment when the right sides in the model equations do not explicitly depend on time, the Hamiltonian also does not depend on time and the function M(C(t), N(t), t) ; 0 along the optimal trajectory. Therefore, 9Ex/9t ; 0, and Ex = Ex(N), i.e. the exergy is only a function of the state. This can be interpreted as "the adaptation is possible only in a changing environment".

Defining the exergy at time t as Ex = £n= 1 exgNi(f1(t),...,fm(t); a1,..., ar) where exg are the specific "genetic" exergies (see Chapters 5 and 12) and Ni, i = 1,..., n are the solution of Eq. (4.1), we formulate the principle of adaptation as a local extreme principle for exergy.

Any ecosystem which is unable to forecast the future state of environment and does not have a sufficiently long memory, i.e. it is a Markovian system, adjusts its interference with the environment in such a way that the maximal increment of exergy is provided.

Mathematically this principle is expressed as:

Let dEx/dt = Yj- 1 exg^(N, f, a) where f = (f1,...,fm). For the given variation of f : Dfk, k = 1,..., m the parameters a vary in such a manner that (dEx/dt) tends to maximum. Thus

8(dEx/dt) = £ X exg f >fk + X X exg 1À À = 0' (4.13)

By denoting as = £1 exg—- and <pk = Y. 1= 1 exg—- equality (4.13) is represented as

The variations Sas have to belong to a domain DVa containing the zero point. In addition, they are not to fall outside the domain of admissible as: Va.

So, the adaptation principle given by Eq. (4.14) imposes a single constraint on variations Sas, therefore the system can adapt to a changing environment by changing its internal parameters. Moreover, if the line YZ-1 asDas = K and domain DVa are mutually disjointed, this means that the environment is changing so fast that the system is not able to adapt to the change. In this case the optimal strategy is to await further steps.

Certainly, there is a large uncertainty, and to reduce it we may introduce additional relationships between Safl: the problem is that we do not know how to do it. However, we could go another way, namely to reduce this uncertainty by increasing the uncertainty in the environment. We assume that there is a set of K: Kp, p = 1,2,..., P belonging to some probabilistic distribution. For instance, this may be various temperatures typical for this site and season, etc. As a rule, P >> r, and we have an overdetermined system of linear algebraic equations r

—1 s and fp is a value of the driving vector-function for a concrete case. Analogously the value of Kp is calculated. System (4.15) can be solved by the method of least squares.

So, we obtain a single solution (in a certain sense) of Eq. (4.15), a*s(t), s = 1,..., r, which can be considered as a mean adaptive strategy. Note that this approach (in another formulation) forms the basis of the structurally dynamic modelling (see Section 13.5).

13.5. Modelling the selection of Darwin's finches

As mentioned in Section 13.1, one example is applied in this chapter to illustrate the ideas behind structurally dynamic modelling and the application of exergy maximisation for parameter estimation and improved calibrations. J0rgensen and Fath (in press) compare a model that contains the available information on the selection of the beak size of Darwin's finches, a model with three classes of beak size and the structurally dynamic model which currently changes the beak size in accordance with a maximisation of the exergy. This latter model is presented in detail below.

All three models are based upon the detailed information about Darwin's finches which can be found in Grant's book (1986, 1999). All additional references are also in the book. The models reflect, therefore, the available knowledge which in this case is comprehensive and sufficient to validate even the ability of the third model to describe the changes in the beak size as a result of climatic changes, causing changes in the amount, availability and quality of the seeds that make up the main food item for the finches. The medium ground finches, Geospiza fortis, on the island Daphne Major were selected for these modelling cases due to very detailed case-specific information found in Grant (1986, 1999). The conceptual diagram of the model in the STELLA format is shown in Fig. 13.5.

The juvenile finches are promoted to adult finches 120 days after birth, according to growth curves by Boag (1984). The mortality of the adult finches is expressed as a normal mortality rate (Grant, 1986) + an additional mortality rate due to food shortage and also caused by a disagreement between beak depth and the size and hardness of seeds.

The beak depth can vary between 3.5 and 10.3 mm (Grant, 1986, 1999). Abott et al. (1977) have published an accordance (see Fig. 13.6) between beak size and a special parameter -JDH , where D is the seed size and H the seed hardness which both are dependent on the precipitation, particularly in the months January-April (Grant and Grant, 1980; Grant, 1985). It is possible to determine a handling time for the finches for a given DH as a function of the beak depth (Grant, 1981), which explains that

Fig. 13. 5. Conceptual diagram of the presented structurally dynamic model of Darwin's finches. The model has three state variables: juvenile finches, adult finches and seed. The beak size is currently adjusted to give the highest exergy.
Fig. 13.6. The relative beak size is plotted versus where D is the seed size and H the seed hardness.

the accordance between VDH and the beak depth becomes an important survival factor. Fig. 13.6 (Abott et al., 1977) is used in the model in order to find a function called "diet", which is compared with VDH, and to find how well the beak depth fits into the VDH of the seed. This fitness function is based on information given by Grant (1981) about the handling time. It is named ofs in the model. It influences, as mentioned above, the mortality of adult finches, but also has an impact on the number of eggs laid and the mortality of the juvenile finches.

The growth rate and mortality of seeds is dependent on the precipitation, which is a forcing function known as a function of time (Grant, 1986, 1999). A function called shortage of food is calculated from the food required by the finches (Grant, 1985) and from the food available (the seed state variable). How the food shortage influences the mortality of juvenile finches and adult finches can be found in Grant (1985).

The seed biomass and the number of G. fortis as a function of time from 1975 to 1982 are known (Grant, 1985). These numbers from 1975 to 1976 (compare with Figs. 13.7 and 13.8) have been used to calibrate the coefficients determining:

1. The influence of the fitness function, ofs, on (a) the mortality of adult finches, (b) the mortality of juvenile finches, and (c) the number of eggs laid.

2. The influence of food shortage on the mortality of adult and juvenile finches is known (Grant, 1985). The influence is therefore calibrated within a narrow range of values.

3. The influence of precipitation on the seed biomass (growth and mortality).

All other parameters are known from the literature.

Fig. 13.7. The observed number of finches (•) from 1973 to 1983, compared with the simulated result (o). 1975 and 1976 were used for calibration and 1977/1978 for the validation referred to in Table 13.3.

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