## Quantification Of Openness And Allometric Principles

All process rates are in physics described as proportional to a gradient, a conductivity or inverse resistance and to the openness, compare for instance with Fick's laws of diffusion and Ohm's law. The import and export from and to an ecosystem is, therefore, dependent on the differences between the ecosystem and the environment, as well as of openness. For instance, the rate of the reaeration process of a water stream can be expressed by the following equation:

where Ra is the rate of reaeration, Ka a temperature constant for a given stream, A the area= V/d , V the volume, d the depth, Cs the oxygen concentration at saturation, and C the actual oxygen concentration. Ka is here the "conductivity" or inverse resistance. The faster the water flow in the stream, the higher is Ka. (Cs-C) is the gradient and A, the area, is the openness. Numerous expressions for rates in nature follow approximately the same linear equation.

The surface area of the species is a fundamental property. The surface area indicates quantitatively the size of the boundary to the environment. Flow rates are often formulated in physics and chemistry as area times a gradient, which can be utilized to set up useful relationships between size and rate coefficients in ecology. Loss of heat to the environment must for instance be proportional to the surface area and to the temperature difference, according to the law of heat transfer. The rate of digestion, the lungs, hunting ground, etc. are, on the one hand, determinants for a number of parameters (representing the properties of the species), and on the other hand, they are all dependent on the size of the organism. It is, therefore, not surprising that many rate parameters for plants and animals are highly related to the size, which implies that it is possible to get very good first estimates for most parameters based only on the size. Naturally, the parameters are also dependent on several characteristic features of the species, but their influence is often minor compared with the size, and good estimates are valuable in many ecological models, at least as a starting value in the calibration phase. It is possible, however, to take these variations into account by the use of a form factor = surface/volume. This form factor may vary considerably among species.

The conclusion of these considerations must, therefore, be that there should be many parameters that might be related to simple properties, such as size of the organisms, and that such relationships are based on fundamental biochemistry and thermodynamics (Figures 2.2-2.6).

Above all there is a strong positive correlation between size and generation time, T, ranging from bacteria to the biggest mammals and trees (Bonner, 1965). This relationship can be explained by use of the relationship between size (surface) and total metabolic action per unit of body weight mentioned above. It implies that the smaller the organism

Ihour 1 day 1 week 1 month 1 year 10 years 100 years

### Generation Time

Figure 2.2 Length and generation time plotted on log-log scale: (a) Pseudomonas, (b) Daphnia, (c) bee, (d) housefly, (e) snail, (f) mouse, (g) rat, (h) fox, (i) elk, (j) rhino, (k) whale, (l) birch, and (m) fir (Peters, 1983). Reproduced from Jorgensen, 2000a.

Ihour 1 day 1 week 1 month 1 year 10 years 100 years

### Generation Time

Figure 2.2 Length and generation time plotted on log-log scale: (a) Pseudomonas, (b) Daphnia, (c) bee, (d) housefly, (e) snail, (f) mouse, (g) rat, (h) fox, (i) elk, (j) rhino, (k) whale, (l) birch, and (m) fir (Peters, 1983). Reproduced from Jorgensen, 2000a.

the greater the metabolic activity. The per capita rate of increase, r, defined by the exponential or logistic growth equations is again inversely proportional to the generation time:

where N is the population size, r the intrinsic rate of growth, and K the environmental carrying capacity. This implies that r is related to the size of the organism, but, as shown by Fenchel (1974), actually falls into three groups: unicellular, heterotherms, and homeotherms (see Figure 2.3).

The same allometric principles are expressed in the following equations, giving the respiration, food consumption, and ammonia excretion for fish when the weight, W, is known:

Respiration = constant X W080 (2.21)

Food consumption = constant X W0 65 (2.22)

Ammonia excretion = constant X W0 72 (2.23)

It is also expressed in the general equation (Odum, 1959, p. 56):

where k is roughly a constant for all species, equal to approximately 5.6 kJ/g day, and m the metabolic rate per unit weight W.

Figure 2.3 Intrinsic rate of natural increase against weight for various animals. After Fenchel (1974). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

Figure 2.3 Intrinsic rate of natural increase against weight for various animals. After Fenchel (1974). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

Similar relationships exist for other animals. The constants in these equations might be slightly different due to differences in shape, but the equations are otherwise the same. All these examples illustrate the fundamental relationship in organisms between size (surface) and biochemical activity. The surface determines the contact with the environment quantitatively, and by that the possibility of taking up food and excreting waste substances.

The same relationships are shown in Figures 2.4-2.6, where biochemical processes involving toxic substances are applied as illustrations. The excretion rate and uptake rate

Length

Figure 2.4 Excretion of Cd (24 h)-1 plotted against the length of various animals: (1) Homo sapiens, (2) mice, (3) dogs, (4) oysters, (5) clams, and (6) phytoplankton (Jorgensen 1984).

### Length

Figure 2.4 Excretion of Cd (24 h)-1 plotted against the length of various animals: (1) Homo sapiens, (2) mice, (3) dogs, (4) oysters, (5) clams, and (6) phytoplankton (Jorgensen 1984).

Length

Figure 2.5 Uptake rate (^g/g (24h)-1) plotted against the length of various animals (Cd): (1) phytoplankton, (2) clams, (3) oysters. After Jorgensen (1984). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

### Length

Figure 2.5 Uptake rate (^g/g (24h)-1) plotted against the length of various animals (Cd): (1) phytoplankton, (2) clams, (3) oysters. After Jorgensen (1984). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

Length

Figure 2.6 Biological concentration factor (BCF) denoted CF for Cd versus length: (1) goldfish, (2) mussels, (3) shrimps, (4) zooplankton, (5) algae (brown-green). After Jorgensen (1984). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

### Length

Figure 2.6 Biological concentration factor (BCF) denoted CF for Cd versus length: (1) goldfish, (2) mussels, (3) shrimps, (4) zooplankton, (5) algae (brown-green). After Jorgensen (1984). Source: Fundamentals of Ecological Modelling by Jorgensen and Bendoricchio.

(for aquatic organisms) follow the same trends as the metabolic rate. This is of course not surprising, as excretion is strongly dependent on metabolism and the direct uptake dependent on the surface.

These considerations are based on allometric principles (see Peters, 1983; Straskraba et al., 1999), which with other words can be used to assess the relationship between the size of the units in the various hierarchical levels and the process rates, determining the need for the rate of energy supply. All levels in the entire hierarchy of an ecosystem are, therefore, due to the hierarchical organization, characterized by a rate which is ultimately constrained by their size.

Openness is proportional to the area available for exchange of energy and matter, relative to the volume = the inverse space scale (L-1). It may also be expressed as the supply rate = k gradient-area relative to the rate of needs, which is proportional to the volume or mass. An ecosystem must, as previously mentioned, be open or at least non-isolated to be able to import the energy needed for its maintenance. Table 2.3 illustrates the relationship between hierarchical level, openness, and the four-scale hierarchical properties presented in Simon (1973). The openness is here expressed as the ratio of area to volume.

For the higher levels in the hierarchy approximate values are used. As we move upwards in the hierarchy, the exchange of energy (and matter) becomes increasingly more difficult due to a decreasing openness. It becomes increasingly more difficult to cover needs, which explains why energy density, time scale, and dynamics decrease according to the inverse space scale or openness, or expressed differently as the rates are adjusted

### Box 2.2 Basic elements of hierarchy theory

Many of the allometric characteristics described in Section 2.6 are based on correlations between body size and other biological or ecological features of the organisms. These interrelationships are frequently comprehended as basic components of ecological hierarchies and basic objects of scaling procedures. Thus, they are highly correlated to hierarchy theory.

Following Simon (1973), hierarchy is a heuristic supposition to better understand complex systems, and following Nielsen and MÃ¼ller (2000) hierarchical approaches are prerequisites for the definition of emergent properties in self-organized systems. Hierarchy theory (Allen and Starr, 1982, O'Neill et al., 1986) or the holarchy principle (Kay, 1984) represents an integrative concept of ecosystem-based classification and conception, which is compatible with most of the existing approaches to ecological system analysis. The theory has been developed by Simon (1973), Allen and Starr (1982), and O'Neill et al. (1986) and recently there have been several applications in ecosystem analysis and landscape ecology.

The fundamental unit of hierarchy theory is the holon, a self-regulating open (sub)system (see Figure 2.6). Holons function as autonomous entities and are also components of superior organizational units. They incorporate all inferior subsystems and are parts of higher level systems themselves. Thus, on a specific level of resolution, a biological system consists of interacting entities and is itself a component of a higher organizational unit. Hierarchies are partly ordered sets, in which the subsystems are interacting through asymmetric relationships. These interactions produce an integral activity of the whole, where the variations of the whole complex are significantly smaller than the sums of the variations of the parts. In contrast, the degrees of freedom of single processes are limited by the higher hierarchical level. Controlling functions (constraints) determine the basis for systems organization: microscopic reactions are coordinated at the macroscopic level. O'Neill et al. (1986) defined the interacting constraints of a specific level of an ecosystem as its environmental limits, while the dynamics of lower levels, which generate the behavior of the higher level, are defined as the biotic potential of the system.

The distinction of hierarchical levels has to be determined by the observer as does the definition of the investigated system. Criteria of the levels' differentiation are:

(a) The spatial extent of higher levels is broader than the extent of lower levels. Thus, distinguishing levels is connected with distinguishing spatial scales.

(b) Higher levels change more slowly than lower levels. Significant changes require longer periods on higher levels.

(c) Higher levels control lower levels. Under steady-state conditions they assert the physical, chemical, and biological limits the system of interest can operate within.

(continued)

(d) Higher levels can contain lower levels (nested hierarchies). Accordingly, the spatial and temporal constants of system behavior are important criteria of differentiation. Scale is defined as a holon's spatial and temporal period of integrating, smoothing, and dampening signals before they are converted into messages (Allen and Starr, 1982).

(e) Signals (including fluxes of energy and matter) are filtered in hierarchies. The way a holon converts or ignores signals defines its functional environment and its scale.

All of these assumptions refer to steady-state conditions. The hierarchy of an ecosystem thus continuously develops and its complexity rises during phases of orientor optimization (see Chapters 6, 7, and 9). Whenever phase transitions appear, the hierarchy is broken and the system is enabled to adapt to the changing constraints by forming a new structure.

 Hierarchical Openness1'3 Energy2 Space Time Dynamics3 level (A/V m-1) (kJ/m3) scale1 (m) scale1 (s) (g/m3 s) Molecules 109 109 10-9 <10-3 104-106 Cells 105 105

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