## Modelling Constraints

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Modellers are very much concerned about the application of the correct description of the components and processes in their models. The model equations and their parameters should reflect the properties of the model components and processes as correctly as possible. The modeller must, however, also be concerned with the right description of the system properties, and too little research has been done in this direction. A continuous development of models as scientific tools will need to consider how to apply constraints on models according to the properties of the system. Several possible modelling constraints are mentioned below. The sequence reflects decreasing relations to physical properties and increasing relations to biological properties of the ecosystems. The ecological modelling constraints will only be mentioned briefly in this context. A more profound discussion will take place in Chapter 9, where the application of these constraints is the basis for development of what may be called next generation models.

The conservation principles are often used as modelling constraints. Biogeo-chemical models must follow the conservation of mass and bioenergetical models must equally obey the laws of energy and momentum conservation.

Energy and matter are conserved according to basic physical concepts that are also valid for ecosystems. This requires that energy and matter are neither created nor destroyed.

The expression "energy and matter" is used, as energy can be transformed into matter and matter into energy. The unification of the two concepts is possible by Einstein's law:

where E is energy, m mass and c the velocity of electromagnetic radiation in vacuum ( = 3x 10s m s '). The transformation from matter into energy and vice versa is only of interest for nuclear processes and does not need be applied to ecosystems on earth. We might therefore break the proposition down into two more useful propositions, when applied in ecology:

• ecosystems conserve matter,

• ecosystems conserve energy.

The conservation of matter may be expressed mathematically as follows:

where m is the total mass of a given system. The increase in mass is equal to the input minus the output. The practical application of the statement requires that a system is defined, which implies that the boundaries of the system must be indicated. Concentration, c, is used instead of mass in most models of ecosystems:

where Kis the volume of the system under consideration and assumed constant.

If the law of mass conservation is used for chemical compounds that can be transformed to other chemical compounds, the Eq. (2.31) must be changed to:

V * dc/dr = input - output + formation - transformation (MT1) (2.32)

The principle of mass conservation is widely used in the class of ecological models called biogeochemical models. The equation is set up for the relevant elements, e.g., for eutrophication models for C, P, N and perhaps Si (see J0rgensen, 1976a,b; 1982a; J0rgensen et al., 1978).

For terrestrial ecosystems, mass per unit of area is often applied in the mass conservation equation:

A * dm2 /dt = input - output + formation - transformation (MT1) (2.33)

The Streeter-Phelps model (see Chapter 3) is a classical model of an aquatic ecosystem that is based upon conservation of matter and first-order kinetics (for further details, see also Chapter 3). The model uses the following central equation:

dD/dt + K -D = L0 ■ KyK^-2"' ■ e"M* (ML^ T') (2.34)

where D = Cs - C(f); Cs = concentration of oxygen at saturation; C(t) = actual concentration of oxygen; t = time; Kt = reaeration coefficient (dependent on the temperature); L0 = BOD^ at time = 0; Kt = rate constant for decomposition of biodegradable matter; and Kx = constant of temperature dependence.

The equation states that change (decrease) in oxygen concentration + input from reaeration is equal to the oxygen consumed by decomposition of biodegradable organic matter according to a first-order reaction scheme.

Equations according to (2.32) are also used in models describing the fate of toxic substances in the ecosystem. Examples can be found in Thomann (1984), J0rgensen (1991) and J0rgensen et al. (2000).

The mass flow through a food chain is mapped using the mass conservation principle. The food taken in by one level in the food chain is used in respiration, waste food, undigested food, excretion, growth and reproduction. If the growth and reproduction are considered as the net production, it can be stated that net production = intake of food - respiration - excretion - waste food (2.35)

The ratio of the net production to the intake of food is called the net efficiency. The net efficiency is dependent on several factors, but is often as low as 10-20%. Any toxic matter in the food is unlikely to be lost through respiration and excretion, because it is much less biodegradable than the normal components in the food. This being so, the net efficiency of toxic matter is often higher than for normal food components, and as a result some chemicals, such as chlorinated hydrocarbons including DDT and PCB, will be magnified in the food chain.

This phenomenon is called biological magnification and is illustrated for DDT in Table 2.16. DDT and other chlorinated hydrocarbons have an especially high biological magnification, because they have a very low biodegradability and are only excreted from the body very slowly, due to dissolution in fatty tissue.

^ ppb wet weight

^ ppb wet weight

1200

Fig. 2.29. Increase in pesticide residues in fish as the weight of the fish increases. Top line = total residues: bottom line = DDT only (after Cox. 1970).

DDE only

Weight of fish in mg

1200

Fig. 2.29. Increase in pesticide residues in fish as the weight of the fish increases. Top line = total residues: bottom line = DDT only (after Cox. 1970).

These considerations also can explain why pesticide residues observed in fish increase with the increasing weight of the fish (see Fig. 2.29).

As man is the last link in the food chain, relatively high DDT concentrations have been observed in human body fat (see Table 2.17).

The understanding of the principle of consenation of energy, called the first law of thermodynamics, was initiated in 1778 by Rumford. He observed the large quantity of heat that appeared when a hole is bored in metal. Rumford assumed that the mechanical work was converted to heat by friction. He proposed that heat was a type of energy that is transformed at the expense of another form of energy, here mechanical energy. It was left to J.P. Joule in 1843 to develop a mathematical relationship between the quantity of heat developed and the mechanical energy dissipated.

Two German physicists J.R. Mayer and H.L.F. Helmholtz, working separately, showed that when a gas expands the internal energy of the gas decreases in proportion to the amount of work performed. These observations led to the first law of thermodynamics: energy can neither be created nor destroyed.

Table 2.16. Biological magnification (data after Woodwell et al.. 1967)

Trophic level

Concentration of DOT (mg kg dry matter)

Magnification

Water

Phytoplankton Zooplankton Small fish Large fish Fish-eating birds

Table 2.17. Concentration of DDT (mg per kg dry matter)

 Atmosphere 0.000004 Rain water 0.0002 Atmospheric dust 0.04 Cultivated soil 2.0 Fresh water 0.00001 Sea water O.OOOOOl Grass 0.05 Aquatic macrophytes 0.01 Phytoplankton 0.0003 Invertebrates on land 4.1 Invertebrates in sea 0.001 Fresh-water fish 2.0 Sea fish 0.5 Eagles, falcons 10.0 Swallows 2.0 Herbivorous mammals 0.5 Carnivorous mammals 1.0 Human food, plants 0.02 Human food, meat 0.2 Man 6.0

If the concept of internal energy, dU. is introduced:

where dQ = thermal energy added to the system; dU = increase in internal energy of the system; and dW = mechanical work done by the system on its environment.

Then the principle of energy conservation can be expressed in mathematical terms as follows:

U is a state variable which means that JdU is independent on the pathway 1 to 2.

The internal energy, U, includes several forms of energy: mechanical, electrical, chemical, and magnetic energy, etc.

The transformation of solar energy to chemical energy by plants conforms with the first law of thermodynamics (see also Fig. 2.30):

a SUNLIGHT Reflection and evaporation

Gross production (0.024) = Net production (0.020) < Respiration (0.004)

Fig. 2.30. Fate of solar energy incident upon the perennial grass-herb vegetation of an old field community in Michigan. All values in GJ m : y '.

Solar energy assimilated by plants = chemical energy of plant tissue growth +

heat energy of respiration (2.37)

For the next level in the food chain—herbivorous animals—the energy balance can also be set up:

where F = the food intake converted to energy (Joule); A = the energy assimilated by the animals; UD = undigested food or the chemical energy of faeces; G = chemical energy of animal growth; and H = the heat energy of respiration.

These considerations pursue the same lines as those mentioned in the context of Eq. (2.35), where the mass consen-ation principle was applied. The conversion of biomass to chemical energy is illustrated in Table 2.18. The energy content per g ash-free organic material is surprisingly uniform, as is illustrated in Table 2.18. Table 2.18D shows AH, which symbolizes the increase in enthalpy, defined as: H - U + p-V. Biomass can be translated into energy (see Table 2.18), and this is also true of transformations through food chains. Ecological energy flows are of considerable environmental interest as calculations of biological magnifications are based on energy flows.

 Organism Species Heat of combustion (kcal/ash-free g) Ciliate Tetrahymena pyriformis -5.938 Hydra Hydra liitoralis -6.034 Green hydra Chlorohydru viridissinia -5.729 Flatworm Dngesia tigrina -6.286 Terrestrial flatworm Bipalinm kewense -5.684 Aquatic snail Succinea ovulis -5.415 Brachiipode Gottidia pyramidala ^t.397 Brine shrimp Anémia sp. (nauplii) -6.737 Cladocera Leptodora kindtii -5.605 Copepode CaUmits helgolandicus -5.400 Copepode Trigriopus californiens -5.515 Caddis fly Pycnopsyche lepido -5.687 Caddis fly Pycnopsyche gutlifer -5.706 Spit bug Philenns leucopthalmns -6.962 Mite Tyroglypltus lintneri -5.808 Beetle Tenebrio molitor -6.314 Guppie Lebisles relicidatus -5.823

(B) Energy values in an Andropogus rirginicus old field community in Georgia

Component Green grass

Litter

Roots

Green herbs

Energy value (kcal/ash-free g)

Average

(C) Combustion heat of migratory and non-migratory birds

Sample

Fall birds

Spring birds

Non-migrants

Extracted bird fat

Fat extracted: fall birds

Fat extracted: spring birds

Fat extracted: non-migrants

Ash-free materia (kcalg)

Fat ratio (% dry weight as fat)

71.7

44.1

21.2 100.0

(D) Combustion heat of components of biomass

Material AH protein AH fat AH carbohydrate

 Eggs -5.75 -9.50 -3.75 Gelatin -5.27 -9.50 Glycogen —4.19 Meat, fish -5.65 -9.50 Milk -5.65 -9.25 -3.95 Fruits -5.20 -9.30 -4.00 Grain -5.80 -9.30 -4.20 Sucrose -3.95 Glucose -3.75 Mushroom -5.00 -9.30 -4.10 Yeast -5.00 -9.30 -4.20

Many biogeochemical models are given narrow bands of the chemical composition of the biomass. Eutrophication models are either based on a constant stoichiometric ratio of elements in phytoplankton or on an independent cycling of the nutrients, where, for instance, the phosphorus content may vary from 0.4% to 2.5%, the nitrogen content from 4% to 12% and the carbon content from 35% to 55%.

Some modellers have used the second law of thermodynamics and the concept of entropy to impose thermodynamic constraints on models; see for instance Mauers-berger (1985), who has used this constraint to assess process equations, too. The idea is that the second law of thermodynamics is also valid for ecosystems, and which implications can be deduced from the application of this law to ecological processes?

Ecological models contain many parameters and process descriptions and at least some interacting components, but the parameters and processes can hardly be given unambiguous values and equations, even bv using the previously mentioned model constraints. This means that an ecological model in the initial phase of development has many degrees of freedom. It is therefore necessary to limit the degrees of freedom in order to come up with a workable model, which is not doubtful and non-deterministic.

Many modellers use a comprehensive data set and a calibration to limit the number of possible models. This is a cumbersome method if it is not accompanied by some realistic constraints on the model. The calibration is therefore often limited to giving the parameters realistic and literature-based intervals, within which the calibration is carried out, as mentioned in Section 2.9.

But far more would maybe be gained if it were possible to give the models more ecological properties and/or test the model from an ecological point of view to exclude those versions of the model that are not ecologically possible. How could, for instance, the hierarchy of regulation mechanisms be accounted for in the models? Straskraba (1979; 1980) classifies models according to the number of levels that the model includes from this hierarchy. He concludes that we need experience with models of the higher levels to develop structurally dynamic models. This is the topic for Chapter 9.

We know that evolution has created very complex ecosystems with many feedback mechanisms, regulations and interactions. The coordinated co-evolution means that rules and principles have been imposed for cooperation among the biological components. These rules and principles are the governing laws of ecosystems, and our models should follow these principles and laws as broadly as possible.

It also seems possible to limit the number of parameter combinations by using what could be called "ecological" tests. The maximum growth rates of phytoplankton and zooplankton may, for instance, have realistic values in a eutrophication model, but the two parameters do not fit to each other, because they will create chaos in the ecosystem, which is inconsistent with actual or general observations. Such combinations should be excluded at an early stage of the model development. This will be discussed further in Chapter 9.

Figure 2.31 summarizes the considerations of using various constraints to limit the number of possible values of parameters, possible descriptions of processes and

 Are the parameters feasible according to literature? 1 Yes Does the model comply with the laws of conservations? Yes r Are the biochemical compo-tions feasible? Yes r Are the rates and concentrations at steady state feasible? * Does the model comply with ecological principles? < Yes f Does the model included the parameter combination comply with the goal function (the orientor)?

No g

Continue the model development

Fig. 2.31. Considerations for using various constraints bv development of models. The range of parameter values particularly is limited by the procedure shown.

possible submodels to facilitate the development of a feasible and workable model. The two last steps of the procedure will be presented in Chapter 9, where the so-called next generation structurally dynamic models are developed.

This requires the introduction of variable parameters, governed by a goal function (an orientor). Several possible goal functions must be introduced before a presentation of structurally dynamic models can take place.