Structurally dynamic modelling

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Dynamic models whose structure changes over time are based on non-stationary or time-varying differential or difference equations. We will refer to these as structurally

Figure 1.3 Log-log plot of the ratio of nitrogen to phosphorus turnover rates, R, at maximum exergy versus the logarithm of the nitrogen/phosphorus ratio, log N/P. The plot is consistent with Vollenweider (1975).

Figure 1.3 Log-log plot of the ratio of nitrogen to phosphorus turnover rates, R, at maximum exergy versus the logarithm of the nitrogen/phosphorus ratio, log N/P. The plot is consistent with Vollenweider (1975).

dynamic models. A number of such models, mainly of aquatic systems (J0rgensen, 1986, 1988, 1990, 1992a,b; J0rgensen and Padisak, 1996; Coffaro et al., 1997; J0rgensen and de Bernardi, 1997, 1998), but also as population dynamic models (J0rgensen, 2002) and terrestrial systems (J0rgensen and Fath, 2004), have been investigated to see how structural changes are reflected in free-energy changes. The technicalities of parameter fitting aside, this overall result means that the system structure must change if its eco-exergy storage is to be continually maximized. Changes in parameters, and thus system structure, not only reflect changes in external boundary conditions, but also mean that such changes are necessary for the ongoing maximization of exergy. For all models investigated along these lines, the changes obtained were in accordance with actual observations (see references). These studies therefore affirm, in a general way, that systems adapt structurally to maximize their content of exergy. The shortcomings of assessing the exergy content of an ecosystem have been discussed above. At least in models the applicability of the exergy calculations has shown their more practical use, which can be explained by a robustness in the model and eco-exergy calculations. It is most probably important to have different weighting factors for organisms that are developed radically different and it is probably also important that the model focuses on the very problem that causes the structural changes, but whether the exergy contributions calculated are too high or too low is not important for a description of the structural changes.

It is noteworthy that Coffaro et al. (1997), in his structural dynamic model of the Lagoon of Venice, did not calibrate the model describing the spatial pattern of various macrophytes species such as Ulva and Zostera, but used exergy-index optimization to estimate parameters determining the spatial distribution of these species. He found good accordance between observations and model, as was able by this method without calibration to explain more than 90% of the observed spatial distribution of various species of Zostera and Ulva.

We need a number of different complementary approaches to explain ecosystems which are not surprising as much simpler physical phenomena; light, for instance, needs two different descriptions, namely as waves and as particles. Several ecosystem theories have been presented in the scientific literature during the last 2-3 decades. At the first glance they look very different and seem to be inconsistent, but a further examination reveals that they are not so different and that it should be possible to unite them in a consistent pattern. This was already stated in the book Integration of Ecosystem Theories: A Pattern, in its first edition in 1992. Now it has been published as a third edition (J0rgensen, 2002). This unification of the various ecosystem theories has been widely accepted among system ecologists since 1998/1999, but as a result of two meetings in year 2000—one in Italy, Porto Venere late May, and one in Copenhagen, early June in conjunction with an American Society of Limnology and Oceanography (ASLO) meeting—it can now be concluded that a consistent pattern of ecosystem theories has been formed. Several system ecologists agreed on the pattern presented below, as a working basis for further development in system ecology. Further steps towards a more comprehensive and more consistent ecosystem theory are taken with this volume. This is of utmost importance for the progress in system ecology, because with a theory in hand it will be possible to explain many rules that are published in ecology and applied ecology which again explain many ecological observations. Moreover, by a good theory in hand it is possible to calculate and predict what otherwise would require a lot of expensive observations and trial and error experiments. We should, in other words, be able to attain the same theoretical basis that characterizes physics: a few basic laws, which can be used to deduce rules that explain observations.

An important contribution to a clear pattern of the various ecosystem theories came from the network approach used often by Patten (see, for instance). Fath and Patten (2001) and Fath et al. (2004) have shown by a mathematical analysis of networks in steady state (representing, for instance, an average annual situation in an ecosystem with close to balanced inputs and outputs for all components in the network) that the sum of through-flows in a network (which is power) is determined by the input and the cycling within the network. The input (the solar radiation) again is determined by the structure of the system (the stored exergy, the biomass). Furthermore, the more structure the more maintenance is needed. Therefore, more exergy must be dissipated, the greater the inputs are. Cycling, on the other hand, means that the same energy (exergy) is utilized better in the system, and therefore more biomass (exergy) can be formed without an increase of the inputs. It has been shown previously that more cycling means increased ratio of indirect to direct effects, while increased input doesn't change the ratio of indirect to direct effects, as both effects increase by the same factor.

Fath and Patten (2001) used these results to determine the development of various variables used as goal functions (exergy, power, entropy, etc.). An ecosystem is of course not setting goals, but a goal function, or orientor maybe a better word to use in this context, can be used to describe the direction of development an ecosystem will take. Their results can be summarized as follows:

1. Increased inputs (more solar radiation is captured) means more biomass, more exergy stored, more exergy degraded, therefore also higher entropy dissipation, more through-flow (power), increased ascendency, but no change in the ratio of indirect to direct effects or in the retention time for the energy in the system = total exergy/ input exergy per unit of time.

2. Increased cycling implies more biomass, more exergy stored, more through-flow, increased ascendency, increased ratio of indirect to direct effects, increased retention, but no change in exergy degradation.

Almost simultaneously J0rgensen et al. (2000) published a paper which claims that ecosystems show three growth forms:

I. Growth of physical structure (biomass) which is able to capture more of the incoming energy in the form of solar radiation, but also requires more energy for maintenance (respiration and evaporation). II. Growth of network, which means more cycling of energy and matter. III. Growth of information (more developed plants and animals with more genes), from r-strategists to K-strategists, which waste less energy but usually carry more information.

These three growth forms may be considered an integration of E.P. Odum's attributes which describe changes in ecosystem associated with development from the early stage to the mature stage. Nine of the most applied attributes associated to the three growth forms should be mentioned:

1. Ecosystem biomass (biological structure) increases.

2. More feedback loops (including recycling of energy and matter) are built.

3. Respiration increases.

4. Respiration relative to biomass decreases.

5. Bigger animals and plants (trees) become more dominant.

6. The specific entropy production (relative to biomass) decreases.

7. The total entropy production will first increase and then stabilize on approximately the same level.

8. The amount of information increases (more species, species with more genes, the biochemistry becomes more diverse).

9. r-strategists are replaced by K-strategists.

Growth form I covers attributes 1, 3 and 7. The biomass increases according to attribute 1, which implies that also the respiration increases, because it costs more exergy to maintain more biomass. This means that also the entropy production will increase.

Growth form II covers 2 and 6. When the network increases, there will be more feedback mechanisms available for regulation of the network. The energy and mass will thereby circle to a higher extent which means that more biomass can be supported with the same total input and output of the eco-exergy.

Growth form III covers the attributes 4, 5, 7, 8 and 9. Bigger and more developed species will take over according to growth form III. It implies more biomass relative to the respiration and while the total entropy production is not changed, the specific entropy production is decreased.

Five of the presented hypotheses to describe ecosystem growth and development are examined with respect to the three growth forms:

A. The entropy production tends to be minimum (this is proposed by Prigogine, 1947, 1955 and 1980, for linear systems at steady non-equilibrium state, not for far from equilibrium systems). It is applied by Mauersberger (1983, 1995) to derive expressions for bioprocesses at a stable stationary state.

B. Natural selection tends to make the energy flux through the system a maximum, so far as compatible with the constraints to which the system is subject (H.T. Odum,

1983, 1988, 1996). This is also called the maximum power principle.

C. Ecosystem will organize themselves to maximize the degradation of exergy (Kay,

D. A system that receives a through-flow of exergy will have a propensity to move away from thermodynamic equilibrium, and if more combinations of components and processes are offered to utilize the exergy flow, the system has the propensity to select the organization that gives the system as much stored exergy as possible (J0rgensen and Mejer, 1977, 1979; J0rgensen, 1986,1988, 1990,1992a, 2002).

E. Ecosystem will have a propensity to develop towards a maximization of the ascendency (Ulanowicz, 1986).

The usual description of ecosystem development illustrated, for instance, by the recovery of Yellowstone Park after fire, an island born after a volcanic eruption, reclaimed land and so on, is well covered by E.P. Odum (1969a, 1969b, 1971): at first, the biomass increases rapidly which implies that the percentage of captured incoming solar radiation increases and also the energy needed for the maintenance. Growth form I is dominant in this first phase, where not only the exergy stored increases (more biomass, more physical structure to capture more solar radiation), but also the through-flow (of useful energy), exergy dissipation and the entropy production increases due to increased need of energy for maintenance.

Growth forms II and III become dominant later, although an overlap of the three growth forms takes place. When the percentage of solar radiation captured reaches about 80%, it is not possible to increase the amount of captured solar radiation further (due in principle to the second law of thermodynamics).

Further growth of the physical structure (biomass) therefore does not improve the exergy balance of the ecosystem. In addition, all or almost all the essential elements are in the form of dead or living organic matter and not as inorganic compounds ready to be used for growth. The growth form I will therefore not proceed, but growth forms II and III can still operate. The ecosystem can still improve the ecological network and can still replace r-strategists by K-strategists, small animals and plants by bigger ones (Cope's Law: the later descendent may be increasingly larger than their ancestors—for instance, the horse today is much bigger than the horse fossils from 20 to 30 million years ago) and less developed by more developed.

Growth forms II and III require, however, not more exergy for maintenance. Exergy degradation is therefore not increasing but is maintained on a constant level, or expressed differently: specific exergy degradation and specific entropy production are decreasing with growth forms II and III. It is in accordance with Aoki (1998, 2006). He has shown that respiration per biomass or entropy production per biomass is increasing with increasing trophic diversity up to a trophic diversity of 3-4 and afterwards decreasing rapidly with further increase of the trophic diversity up to 7 for aquatic ecosystems. The results by Aoki are in accordance with the actual observations, which have been applied to develop ECOPATH models (Christensen and Pauly, 1993; Christensen, 1995).

The accordance with the five descriptors + specific entropy production and the three growth forms based on this description of ecosystem development is shown in Table 1.3.

Based upon the results, it is possible to formulate the following hypothesis which unites the five hypotheses.

Table 1.3 Accordance between growth forms and the proposed descriptors

Hypothesis

Growth form I Growth form II Growth form III

Table 1.3 Accordance between growth forms and the proposed descriptors

Hypothesis

Growth form I Growth form II Growth form III

Exergy storage

Up

Up

Up

Power/through-flow

Up

Up

Up

Ascendency

Up

Up

Up

Exergy dissipation

Up

Equal

Equal

Retention time

Equal

Up

Up

Entropy production

Up

Equal

Equal

Exergy/biomass = specific

Equal

Up

Up

exergy

Entropy/biomass = specific

Equal

Down

Down

entropy production

Ratio indirect/direct effects

Equal

Up

Up

Ecosystem develoPment in all Phases will move away from thermodynamic equilibrium and select the comPonents and the organization that yields the highest flux of useful energy through the system and the most exergy stored in the system. This corresPonds also to the highest ascendency.

Ecosystem development is accomplished by the three growth forms:

1. Growth of biomass (physical structure) which implies that more exergy is degraded due to an increased demand for maintenance energy. An ecosystem tends according to growth form I to reach the highest possible rate of exergy captured (which is in the order of 80% of the incoming solar radiation) and thereby also of exergy degradation. This growth form is therefore best measured by a determination of the exergy degradation rate.

2. Growth of the number of network linkages and thereby of recycling of matter and energy which implies a better utilization of the incoming energy, and therefore an increase in through-flow and exergy storage without an increase in exergy dissipation. It means that specific exergy degradation and specific entropy production are decreasing.

3. Growth of the number of components in the network and replacement of r-strategists and small organisms with K-strategists and bigger organisms. It implies the same changes take place as observed by growth form II, namely increased through-flow, increased exergy storage and decreased specific exergy degradation and entropy production.

The possibilities for an ecosystem to use growth form number one are limited, because of the amount of elements that is present.

The element present in the smallest amount relative to the demand will stop growth of biomass.

Growth form I could be stopped even before, when it captures about the 80% of the solar radiation that is physically possible. As it will be discussed in Chapter 2, we are very far from the limits for growth forms II and III. The evolution has therefore to a high extent played on these two growth forms: growth form III (more and more complex organisms) and growth form II (more and more different organisms are linked in a more and more complex network).

In the paper by J0rgensen et al. (2000), Figure 1.4 was presented to illustrate the concomitant development of ecosystems, exergy captured (most of that being degraded) and exergy stored (biomass, structure, information). The points in the figures correspond to ecosystems on different stages of development (see Table 1.4).

Debeljak (2001) has shown that he gets the same shape of the curve when he determines exergy captured and exergy stored in managed forest and virgin forest on different stages of development (see Figure 1.5). The exergy captured was determined as in Table 1.4 by measurement of the temperature of the infrared radiation, while the exergy storage was determined by a randomized measurement of the size of all trees and plants. The stages are indicated on the figure, where also pasture is included for

o ra

-200

200 400 600 800 1000 1200 1400 1600 1800 Eco-exergy in MJ/m2

Figure 1.4 The exergy captured (taken from Kay and Schneider, 1992, % of solar radiation) is plotted versus the exergy stored, unit MJ/m2, calculated from characteristic compositions of the eight focal ecosystems. The values from Table 1.4 are applied to construct this plot. Notice that exergy utilization is parallel (proportional) to the energy absorbed.

Table 1.4 Exergy utilization and storage in a comparative set of ecosystems

Ecosystem

Exergy utilization (%)

Exergy storage (MJ/m2)

Quarry

6

0

Desert

2

0.75

Clear-cut forest

49

60

Grassland

59

94

Fir plantation

70

360

Natural forest

71

540

Old-growth deciduous forest

72

920

Tropical rain forest

70

1650

comparison. Catastrophic events such as storm or fire may cause destructive regeneration, as described in Holling's cycle; see J0rgensen et al. (2007) and Chapter 7 for a detailed description. Destructive regeneration has happened several times during the evolution.

45 -

0 200 400 600 800 1000

Eco-exergy in MJ/m2

Figure 1.5 The plot shows the result by Debeljak (2001). He examined managed forest and virgin forest in different stages. Gap has no trees, while the virgin forest changes from optimum to mixed to regeneration and back to optimum, although the virgin forest can be destroyed by catastrophic events such as fire or storms. The juvenile stage is a development between the gap and the optimum. Pasture is included for comparison.

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