## Types of Empirical Models

In principle, empirical models in ecology like theoretical models can be oral, physical, graphical, or mathematical, but since they represent data, the model format most often is graphs or mathematical expressions representing relations between two or more variables. These can be quantities, for example, numbers, weights, sizes, physical and chemical properties, or rates, for example, speed and metabolic rates, growth, birth, and death rates, and concentration changes. The relations between two or more parameters most often are represented as tables and graphs, and linear or nonlinear relations between parameters can be statistically treated. Supplied with appropriate uncertainty measures, such models can be named statistical models (see Statistical Prediction). In general, empirical models do not postulate process-based explanations whereas statistical models are more applied on data with more realistic process-based relations.

Allometric Models

The metabolism of the various organisms in ecosystems can be predicted from their type and sizes.

Often logarithmic relations of the type log y = log a + b log x are preferred since many relations are exponential and the ranges of parameters cover several orders of magnitude. Classical studies have showed such allometric relations for standard respiration and body masses over wide ranges for unicellular organisms and coldblooded and warmblooded animals. All organism groups increase their metabolism with body mass following a similar slope close to 0.75 in a double logarithmic plot, but the levels - the b values -differed for the three organism types. If a linear relation between metabolism and body mass existed, the slope would be 1.0 and if the linear relation was between metabolism and the surface area of the organisms the slope would be 0.67. Similar allometric relations have been

 Temperate invertebrates 0.95 \ >> r2 = 0.66 ' / ' ° J* • * o African mammals Y = 0.082B 078 prey r2 = 0.85

10 102 103 104 Prey biomass (mg wet wt. m-2)

10 102 103 104 Prey biomass (mg wet wt. m-2)

Figure 2 The relation between predator and prey biomass. From Peters RH (1991) A Critique for Ecology. Cambridge: Cambridge University Press.

found for food uptake and body mass of herbivorous and carnivorous warmblooded animals without differences between the two types. One model was I = 10.7M0'70, where I is the ingestion in joules per second and M is the mass in kilograms. For coldblooded animals, the equation was I = 0.78M0' 2 Combinations of allometric relations have been used to calculate new allometric relations, and other relations (Figure 2) are allowed to predict relations between predatory invertebrates and their prey on different climate regions, thereby increasing our understanding of functioning of ecosystems. However, one should always be aware of self-correlations leading us to trivial models, although tautologies, of course, can lead to improved understanding and inspire to new concepts.

### Resource Response Models

The overall quantity of organic matter in closed ecosystems is controlled by production and degradation. For the production of organic matter, we need energy, water, carbon, and other elements. In aquatic ecosystems, the relations between demands for phosphorus, nitrogen, and carbon compared to the availability of these elements are by far the highest compared to other elements. In terrestrial ecosystems water also is a major factor, and in all ecosystems temperature also is a regulator. Many empirical models for relations between these elements and the abundance and productivity of organisms have been made within or across ecosystems.

### Terrestrial ecosystems

Relations between net primary production in terrestrial ecosystems on a global scale and temperature show a sig-moid pattern with an increase between 0 and 20 °C, but the relation is questionable since it spans over different day lengths and temperature influences many processes other than photosynthesis. Relations between primary production and annual production show more a saturation-like pattern with a saturation value of about 2500 gCm~ yr~ occurring at about 2000mmyr~ . However, if only restricted climatic regions and specific vegetation types are considered, empirical models with much lower uncertainty can certainly be developed. And across biomes and ecosystems, similar simple relations can be expressed between biomass of herbivores and vegetation, and between carnivores and preys. But in terrestrial ecosystems, empirical models of resource response are indeed probabilistic and the ignorance of causal steps and many supplementary factors often increases their uncertainty and limits their application.

### Aquatic ecosystems

In aquatic ecosystems, we can exclude water as a resource factor when we develop models across ecosystems which will simplify the relations. Compared to lakes, marine systems are much more open systems, which makes it difficult to establish relations between loading and concentrations of nutrients. In marine ecosystems, most often nitrogen is the limiting nutrient and our missing knowledge on regulation of denitrification in marine systems is one explanation for the scarcity of empirical models. One example is the relation between loading of total nitrogen and the annual phytoplankton production (Figure 3).

Much more success has been achieved with empirical modeling when we consider phosphorus in freshwater lakes where it is the dominant controlling nutrient and many models including phosphorus have been developed (Figure 4). A suite of loading models exist to calculate the export of phosphorus from a watershed and the atmosphere

io o

to yt

Figure 3 Annual phytoplankton production as a function of nitrogen load in temperate coastal ecosystems (o), in nutrient-enriched mesocosms (*), and in Baltie, Dutch, and North American ecosystems (•). From Borum J and Sand-Jensen K (1996) Is total primary production in shallow coastal marine waters stimulated by nitrogen loading? Oikos 76: 406-411.

Figure 3 Annual phytoplankton production as a function of nitrogen load in temperate coastal ecosystems (o), in nutrient-enriched mesocosms (*), and in Baltie, Dutch, and North American ecosystems (•). From Borum J and Sand-Jensen K (1996) Is total primary production in shallow coastal marine waters stimulated by nitrogen loading? Oikos 76: 406-411.

Population, land use

Waste source model

Inflow

Inflow

Outflow

Sediment losses and feedback Phosphorus budget model

Outflow

Sediment losses and feedback Phosphorus budget model

Total phosphorus concentration

Total phosphorus concentration

Hypolimnetic oxygen concentration

Figure 4 Flow diagram indicating steps for empirical modeling of lake eutrophication. From Ahlgren I, Frisk T, Kamp-Nielsen L (1989) Empirical and theoretical models of phosphorus loading, retention and concentration vs. lake trophic state. Hydrobiologia 170: 285-303.

### Hypolimnetic oxygen concentration

Figure 4 Flow diagram indicating steps for empirical modeling of lake eutrophication. From Ahlgren I, Frisk T, Kamp-Nielsen L (1989) Empirical and theoretical models of phosphorus loading, retention and concentration vs. lake trophic state. Hydrobiologia 170: 285-303.

to a lake. From geological mapping and agricultural statistics, the contribution from cultivated and noncultivated areas can be registered; from meteorological observations the contribution from the atmosphere can be calculated and from the human population density the anthropogenic input can be calculated. Depending on the type of watershed, more elaborate submodels can be adopted including other phosphorus point sources like industry, fish ponds, and livestock or by separating the watershed in other types or mixtures of point and diffuse sources summing up the total load phosphorus and or other elements like nitrogen. Together with a hydrological budget and the morphometry ofthe lake, we can calculate the total load as amount, average inlet concentration, lake area load, or lake volumetric load.

To establish a link to the resource models we need to establish a mass balance for phosphorus since some of the phosphorus load is lost through the eventual outflow and some is lost to the sediments:

where M is the annual input of phosphorus (mg P yr—*), P is the in-lake phosphorus concentration (mgPm— ), V is the volume of the lake (m3), Q is the discharge of the outlet (m yr~ ), and S is the annual net sedimentation (mg P yr~ ). Assuming a linear relation between sedimentation and in-lake mass of phosphorus, we get the following steady-state concentration:

where s is a sedimentation coefficient (yr_1).

By a regression of in-lake phosphorus concentrations versus inlet phosphorus concentrations Pi across a number of lakes from all over the world, it was found that after logarithmic transformations the relation improved by scaling the inlet concentration with 1/(1 + -y/rw) where y/rw — V/Qis the hydraulic residence time (in years):

where qs is the water discharge height = Q/V(m yr—*) and Tw is the water residence time (year). The equation was later applied to data from a much larger study performed by OECD (Organisation for Co-operation and Economic Development) and the equation was modified to:

where RP = 1 — P/Pi is the retention coefficient. Many studies based on geographical more restricted data have been carried out and a number of empirical models have been developed. These models have been tested on material from 131 Danish lakes and the Vollenweider model and three other models were found as the best models to predict in-lake phosphorus concentrations from loading and residence time. Although such models developed on large number of lakes are successful in predicting phosphorus concentration in large populations of lakes, their success is limited when it comes to the behavior of single lakes since many lakes do not fulfil the prerequisite of steady state and are not in equilibrium with their sediments. Lakes which have received a high loading for many years have accumulated phosphorus in their sediments which may continue to release phosphorus depending on the concentration in the surface sediment. Such an internal loading may delay the response to reduced loading, and the mass balances during the recovery period will exhibit a negative net sedimentation for several years. Another study followed mass balances in nine shallow and nine deep, stratified

European lakes over many years. In the initial phase, the loading was constant followed by a phase with a gradual, significant reduction in load, preferably from point sources, and ending with a period where there were no systematic changes in load or in-lake concentrations. The mass balances showed that in all shallow lakes a net annual release of phosphorus followed the first years after reduction and that the net release rates reflected in negative net retention during the first two years after load reduction were linearly related to the sediment surface concentrations of phosphorus. In the deep lakes, seasonally net releases of phosphorus were observed but never on an annual scale. Using data from the steady-state period before (Ppre) and after (Ppost) load reduction and applying the OECD model in another form with Pi as the average inlet concentration of phosphorus, we get P = ^(Pi)p where k and p are constants. After log-transforming of P and Pi data, linear regressions on each case were performed (Figure 5). The average relation appeared as follows:

Ppost/Ppre (Pi,post/Pi,pre)

The exponent was significantly lower than the 0.82 found in the OECD study and the 0.83 which was found when the P = ^(Pi)p equation was applied on the OECD data. This shows the danger of applying empirical models developed on large data sets on the behavior of single lakes. The explanation of the weaker response in the average behavior of single lakes lies presumably in assumption of true steady state after load reduction is not fulfilled - the lakes are still influenced by internal load and are only in quasi-steady state.

Since phosphorus is the main resource factor for algal biomass, we observe a linearly increasing algal biomass with increasing phosphorus concentrations up to a certain level where self-shading, grazing by zooplankton, or limitation by other macro- or micronutrients may be

1000

03 ; All lakes

? Shallow

100 Deep

Figure 5 Regressions performed for in-lake vs. inlet concentrations of phosphorus during pre- and postrestoration steady state. From Sas H, (1989) Lake Restoration by Reduction of Nutrient Loading. Sankt Augustin: Academia-Verlag Richarz.

Table 1 Various models for phosphorus-chlorophyll relations in lakes

Linear models chl a = 1.19TP - 7.3 chl a = 0.58TP + 4.2 chl a = 0.55TP - 4.8

Logarithmic models chl = 0.073 5TP1583 chl a = 0.072 4TP145 chl a = 0.028TP0 98

Models with defined maximum levels chl a = 150(1 -e(-0 000867TP-0 000011 1TP2)) chl a = 50(1 -e(-0 0026TP-0 000102TP2)) chl a = 40.1/(1 + 130e(-0114TP))

Model based on total N and total P concentrations log(chl a) = 0.6531 log TP + 0.548 log TN - 1.517

From Ahlgren I, Frisk T, and Kamp-Nielsen L (1989) Empirical and theoretical models of phosphorus loading, retention and concentration vs. lake trophic state. Hydrobiologia 170: 285-303.

controlling the algal biomass. Many empirical models for the relation between phosphorus and algal biomass expressed as chlorophyll have been published. They are not all comparable because different time periods have been used. In the OECD study, annual averages were used, but most studies compare annual averages of phosphorus with summer (May-September, Northern Hemisphere) values of chlorophyll. In the majority of studies, linear models are generated with slopes from 0.4 to 1 in the range up to 100mgPl-1; other models are logarithmic and models with defined maximum levels or dependence of both phosphorus and nitrogen have been published (Table 1). As for the phosphorus loading models, the uncertainty can be reduced by grouping the lakes by depth and geographical position. Also, screening for nitrogen limitation may reduce uncertainty.

To investigate the application of phosphorus-chlorophyll models developed on across-lake data on the behavior of single lakes during recovery after reduction in phosphorus load, the response in chlorophyll to phosphorus concentration was made for nine shallow and nine deep lakes (Figure 6). Three of the lakes were excluded after screening for nitrogen limitation and for the other lakes, years without phosphorus limitation were excluded. After logarithmic transformation, linear regressions were performed on each lake and the average relation appeared as:

chlpost/chlpre = (Ppost/Ppre)

This exponent was not significantly different from the exponent 0.96 ± 0.12 of the OECD study. But more surprising was that if the OECD-type plot was performed on pooled data from the non-P-limited a year, an exponent of 1.10 ± 0.2 was found which was not significant from the OECD study. This shows the weak explanatory power of

Annual average P concentration in lake, P¡y (mg m 3)

Figure 6 Regressions performed for summer average of chlorophyll a vs. annual average concentrations of phosphorus. From Sas H (1989) Lake Restoration by Reduction of Nutrient Loading. Sankt Augustin: Academia-Verlag Richarz.

Annual average P concentration in lake, P¡y (mg m 3)

Figure 6 Regressions performed for summer average of chlorophyll a vs. annual average concentrations of phosphorus. From Sas H (1989) Lake Restoration by Reduction of Nutrient Loading. Sankt Augustin: Academia-Verlag Richarz.

empirical models and the danger of applying models developed across lakes on single cases. By grouping in shallow and deep lakes, an exponent for shallow lakes of 1.4 ± 0.3 was found and one of 0.6 ± 0.5 was found for deep lakes. Although the number of data was critically low, the data allowed an error analysis and maintained at least a qualitative difference in response between deep and shallow lakes. A similar study of the relations between transparency and phosphorus levels came up with an exponent of —0.34 compared to the OECD value of—0.28 ± 0.11. Since phosphorus is the main controlling factor in lakes, many other parameters have been found to relate to mean annual phosphorus concentrations (Table 2).

### Ecotoxicological Models

In ecotoxicology, empirical models are widely used both to model the fate and to model the effects of toxic substances in ecosystems. The fate and transport of inorganic substances like heavy metals can be modeled as the nutrient models above using empirically derived retention coefficients. In addition, the well-described chemistry of inorganic substances allows inclusion of theoretically based elements of chemical reactions and influence of, for example, pH and temperature.

We know about 10 million organic compounds of which more than 100 000 are registered in use by man and 20 000 of them are used in a scale where they may have a negative effect on ecosystems. And each year several hundred new chemicals are added. Both for environmental assessment of the registered chemicals and the approval and development of new chemicals, there is a high demand for simple methods of identifying possible dangerous substances and predicting the effects on ecosystems of existing and new chemicals. Quantitative structure-activity relationships (QSARs) come from pharmacology and use simple relations between chemical structures and physical properties of molecules and their ecotoxicological effects. In the fate of chemicals in ecosystems, important processes are sorption equilibria with sediments and soils, uptake of the chemicals, bioconcentration and biomagnification through food chains, detoxification and toxic effects on organisms, populations, and whole ecosystems. All these processes are influenced by the chemical structure of the molecules, and for organic molecules chemical structures like molecular weight, number of carbon atoms per molecule, and the degree of branching of the molecular skeleton can be important parts of empirical fate models. But more successful are physical descriptors like water solubility, boiling point, the Henry constant (partition between air and water), and Pow, the octanol-water partition coefficient, probably the most used, found by the

 Dependent variable Units Equation r2 n [Chlorophyll] mg m 3 Y = 0.73TP14 0.96 77 Transparency m Y = 9.8TP"028 0.22 87 [Phytoplankton] mg wet wt. m—3 Y = 30TP14 0.88 27 [Nanoplankton] mg wet wt. m—3 Y = 17TP13 0.93 23 [Net plankton] mg wet wt. m—3 Y = 8.7TP17 0.82 23 [Blue-greens] mg wet wt. m—3 Y = 43TP098 0.71 29 [Bacteria] millions ml—1 Y = 0.90TP066 0.83 12 [Crustacean plankton] mg dry wt. m—3 Y = 5.7TP091 0.72 49 [Zooplankton] mg wet wt. m—3 Y = 38TP064 0.86 12 [Microzooplankton] mg wet wt. m—3 Y = 17TP071 0.72 12 [Macrozooplankton] mg wet wt. m—3 Y = 20TP065 0.86 12 [Benthos] mg wet wt. m—2 Y = 810TP071 0.48 38 [Fish] mg wet wt. m—2 Y = 590TP071 0.75 18 Avg prim. prodn. mgCm—3d—1 Y = 10TP-79 0.94 38 Max. prim. prodn. mgCm—3d—1 Y = 20TP-71 0.95 38 Fish yield mg wet wt. m—2yr—1 Y = 7.1TP10 0.87 21

From Peters RH (1991) A Critique for Ecology. Cambridge: Cambridge University Press.

From Peters RH (1991) A Critique for Ecology. Cambridge: Cambridge University Press.

log Po

Figure 7 The effect of the octanol-water partition coefficient (Pow) on the bioconcentration factor of organic compounds by aquatic organisms. From Peters RH (1991) A Critique for Ecology. Cambridge: Cambridge University Press.

log Po

Figure 7 The effect of the octanol-water partition coefficient (Pow) on the bioconcentration factor of organic compounds by aquatic organisms. From Peters RH (1991) A Critique for Ecology. Cambridge: Cambridge University Press.

study brought relationships between area and number of individuals, number of habitats and number of species, and habitats and diversity. The strong interrelations between area and number of individuals and habitats were reflected in the lack of relation between the area and the residuals from the regressions. It was concluded that it was not possible to separate the individual importance of abundance, area size, and number of habitats for the diversity of birds.

A more successful study was carried out on plants on the Shetland Islands. In the study, it was shown that the direct effect of area and number of individuals had the same positive effect on the diversity of plants, but since the area created a linear increase in number of habitats and also an increase in number of plants per area, the area had an influence twice as great as the effect of habitat variation.