Quantification of openness and allometric principles

As we have already seen above, the surface area of a 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 utilised 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, and, on the other hand, they are all dependent on the size of the animal. 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 upon size. Naturally, the parameters are also dependent on several characteristic features of the species, but their influence is often minor compared to the size, and providing good estimates is valuable for 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. The 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 upon fundamental biochemistry and thermodynamics.

Above all, there is a strong positive correlation between size and generation time, tg, ranging from bacteria to the biggest mammals and trees (Bonner, 1965). The relationship

Fig. 3.2. Size and generation time plotted on log-log scale: (a) pseudomonas, (b) daphnia, (c) bee, (d) house fly, (e) snail, (f) mouse, (g) rat, (h) fox, (i) elk, (j) rhino, (k) whale, (l) birch, (m) fir.

is illustrated in Fig. 3.2. 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, the greater the metabolic activity. It is interesting that Bonner's relation could be obtained directly from the latter statement.

Indeed, taking into consideration the new variables: z = V/V*, where V* = 1~Gsf /r*, and t = (r*/e)t, Eq. (5.11) can be written in a dimensionless form:

Since the right side of the equation does not depend on any parameters, then the dimensionless characteristic time of the system must be equal to 1, Tch < 1. Since the length of generation can be considered as such a type of dimension characteristic time then (r/e)tg < 1. From the expression for V*, we have r* = sf/(V*)1-g; substituting it into the previous expression, we obtain

Finally, setting y = 2/3 and bearing in mind that the characteristic size L* ~ (V*)1/3, we get tg ~ L*, i.e. Bonner's relation.

Note that, since the generation time is a characteristic value both for organism and for population the relation between the organism description and population one, then Bonner's relation allows us to construct a bridge between these descriptions. Indeed, the g g most popular models in ecology are Malthusian ones, dN/dt = aN, and logistic, dN/dt = aN(1 — (N/K)), where N is a population size, and the intrinsic rate a < 1/tg. But since tg ~ L* and the weight W* ~ V* = L*3, then a = A/(W*)1/3 where A = const. In the logarithmic scale, the relation will be written as log a = log A — (1/3)log W*. This implies that a is related to the size of the organism but, as shown by Fenchel (1974), actually falls into three groups: unicellular, poikilo- and homeotherms (see Fig. 3.3).

The analogous allometric relations (principles) are expressed in the following equations giving the respiration, feed consumption and ammonia excretion for fish when the weight, W, is known:

Respiration = ARW0,80, Feed consumption = AFCW0 65, (6.3)

Ammonia excretion = Aae W0 72. This is also expressed in Odum's equation (E.P. Odum, 1959, p. 56):

where k is roughly a constant for all species, equal to about 5.6 kJ/g2/3 day, and r is the specific metabolism (metabolic rate) per unit weight.

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 the biochemical activity. The surface quantitatively determines the contact with the environment, and through that the possibility of taking up food and excreting waste substances.

Fig. 3.3. Intrinsic rate of natural increase against weight for various animals.

The same relationships are shown in Figs. 3.4-3.6, where biochemical processes involving toxic substances are applied as illustrations. These figures are constructed from data in the literature and, as can be seen, the excretion rate and uptake rate (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 is dependent on the surface.

These considerations are based on allometric principles (Peters, 1983; Straskraba et al., 1997), which in 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 organisation, characterised by a rate that is ultimately determined by its size. Note that the degree of openness plays a significant role in the establishment of all these relations.

Intuitively, it is clear that the degree of openness must be proportional to the area available for exchange of energy and matter, relative to the volume, i.e. it must be inversely proportional to space co-ordinate L. 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 3.1 illustrates the relationship between hierarchical level, openness, and the four scale hierarchical properties presented in Simon (1973). The openness is expressed here 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

Fig. 3.4. Excretion rate of Cd (1/24 h) plotted against the size of various animals: (1) Homo sapiens, (2) mice, (3) dogs, (4) oysters, (5) clams, (6) phytoplankton.
Fig. 3.5. Uptake rate (Mg/g 24 h) plotted against the size of various animals (CD): (1) phytoplankton, (2) clams, (3) oysters (J0rgensen, 1984).

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 to make the possible supply of energy sufficient. These considerations

Fig. 3.6. BCF (Biological Concentration Factor) denoted CF for Cd versus size: (1) goldfish, (2) mussels, (3) shrimps, (4) zooplankton, (5) algae (brown-green).

Table 3.1

The relationship between hierarchical levels, the approximate magnitudes of their openness and approximate values of the typical four scale hierarchical properties (energy portions/volume, space scale, time scale, and behavioural frequencies)

Table 3.1

The relationship between hierarchical levels, the approximate magnitudes of their openness and approximate values of the typical four scale hierarchical properties (energy portions/volume, space scale, time scale, and behavioural frequencies)

Hierarchical level

Openness (m 1)

Energy (kJ/m3)

Space scale (m)

Time scale (s)

Dynamics (g/m3 s)

Molecules

109

109

109

< 10 3

104-106

Cells

105

105

1025

101-103

1-102

Organ

102

102

1022

104-106

1021-10~3

Organism

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