For all the main model types available today, the characteristics of the model type, the advantages, the disadvantages (mostly expressed as a limitation of the application), and the area of application are given below. Although catastrophe theory and chaos theory were mentioned and included in Figures 1 and 2, these types are not included in the overview below, because they are considered mathematical tools that in principle can be applied as useful tools in the development of several different model

r s0

Stochastic SDBBD Statist.

Fuzzy Catastr. Model type

Pop. dyn. Biogeochem.

Figure 1 The number of model papers published in the period 1975-82 classified into different model types: stochastic models spatial distribution models and structurally dynamic models (SDBBDs), models based on the use of statistics, fuzzy models, models using catastrophe theory, population dynamic models, and biogeochemical models.

400 350 300

100 50

Stoc. SDM Stat. Fuzzy Cat.

PD BGC Spat. Model type

IBM ANN

Figure 2 The number of model papers published in the period 2000-06 classified by model type: stochastic models, structurally dynamic models, statistical models, fuzzy models, models based on the use of catastrophe theory, population dynamic models, biogeochemical models, spatial distribution models, individual-based models, artificial neural network models, models developed by the use of artificial intelligence, models using chaos theory, and static models.

types. Furthermore, statistical models will also not be mentioned because statistics is considered as a tool that can be applied in ecological modeling to give better process description. If a model is based entirely on statistics, it is a so-called black box model, because it has no causality. Black box models can hardly be used to uncover new ecological knowledge. They can be used as a management tool, but as they are not based on ecology but on statistics, they can hardly be denoted ecological models. As ecological modeling is a strong tool in the science ecology, where the focus to a high extent is on causality, the use of black or dark gray box models is avoided to the extent possible.

This model type is still widely used, as can be seen by a comparison of Figures 1 and 2; see above. From 1975 to 1982, this type was used in 62.5% of the model publications in Ecological Modelling, while it was applied in 32% of the model publications in the journal from 2000 to 2006. The model type generally applies differential equations to express the dynamics. Changes in state variables are expressed as the results of the incoming minus the outgoing substances and the model is therefore based on conservation principles. The process equations are based usually on causality, but can in principle also be a result of a statistic analysis of data. The model type has some clear advantages that make it attractive still to use this model type for the development of many models. The advantages are as follows:

• most often based on causality;

• based on mass or energy conservation principles;

• easy to understand, interpret, and develop;

• software is available, for instance, STELLA;

• easy to use for predictions.

The disadvantages are as follows:

• can hardly be used for heterogeneous data;

• requires relatively good data;

• is difficult to calibrate when it is complex and contains many parameters;

• cannot account for adaptation and changes in species composition.

The advantages and disadvantages define the so-called area of application: for description of the state of an ecosystem, when a good data set is available. A developed model may be applied on different ecosystems of the same type, although calibration and validation should always be carried out for each case study. The model will often but not always take many processes and several state variables into account and require therefore in most cases a good data set. The model type has been extensively applied in environmental management as a powerful tool to understand the reactions of ecosystems to pollutants and to set up prognoses.

Due to the limitations of this model type, it has not been used in more than 1.8% of the publications in Ecological Modelling during the last 6 years. The model type is a biogeochemical or bioenergetic dynamic model where all the differential equations are set to zero to obtain the values of the state variables corresponding to the static situation.

The advantages are as follows:

• requires smaller databases than other types;

• is excellent to give a worse-case or average situation;

• the results are easily validated (and verified).

The disadvantages are:

• does not give any information about dynamics and changes over time;

• prediction with time as independent variable is not possible;

• can only give average or worse-case situations.

This model type will often be used when a static situation is sufficient to give a proper description of an ecological system or to take environmental management decisions.

This model type is rooted in the Lotka-Volterra model that was developed in the 1920s. Numerous papers have been published about the mathematics behind this model and a number of deviated models. The mathematics of these equation systems is not very interesting from an ecological modeling point of view, where the focus is a realistic description of ecological populations. Population dynamic models may include age structure, which in most cases is based on matrix calculations. Population dynamic models were represented in 31% of the model papers in Ecological Modelling in 1975-82, while they were applied in 25% in the period 2000-06. The number of population dynamic papers is however 5 times greater in the latter period than in the former period, which illustrates that ecological modeling has developed significantly from the 1970s to today. The minor reduction in percentage is due to the application of a wider spectrum of different model types today.

The advantages are as follows:

• fitted to follow the development of a population;

• age structure and impact factors can easily be considered;

• easy to understand, interpret, and develop;

• most often based on causality.

The disadvantages are as follows:

• the conservation principles are sometimes not applied;

• limited to population dynamics;

• require a good database;

• difficult in some situations to calibrate;

• require a relatively homogeneous database.

This model type is typically used to keep a track on the development of a population. The most applied unit is the number of individuals, which can of course easily be translated into biomass. Effects of toxic substances on the development of populations can easily be covered by increasing the mortality and decreasing the growth correspondingly. The model type is extensively used in management of fishery and national parks.

In these models, the parameters, corresponding to the properties of the biological modeling components, change over time to account for adaptation and changes in species composition. It is possible either to use knowledge or artificial intelligence to describe the changes in the parameters. Used most often, however, is a goal function to find the changes of the parameters. Eco-exergy has often been used as goal function in SDMs. By minor changes of the parameters it may be due to adaptation to the changed conditions, but for major changes it is most probably a change in the state variables (i.e., a shift in the species composition), that is causing the changed parameters. It may also be possible to use the approach to a major change in the ecological network, although no reference to this application of the structurally dynamic modeling approach is yet available.

SDMs are applied much more today than 25-30 years ago. In the period 1975-82, only 1.5% of the model papers were about SDMs, while 8% of the model papers were about SDMs in 2000-06.

This model type has the following advantages:

• able to account for adaptation;

• able to account for shift in species composition;

• can be used to model biodiversity and ecological niches;

• the parameters determined by the goal functions (a) do not need to be calibrated and (b) are relatively easy to develop and interpret.

The disadvantages of this model type are as follows:

• selection of a goal function needed;

• usually computer time consuming;

• information about structural changes required;

• no available software, programming often needed.

This model type should be applied whenever it is known that structural changes take place. It is also recommended for models that are used in environmental management to make prognoses resulting from major changes in the forcing functions (impacts).

Fuzzy models may either be knowledge based (called the Mamdani type) or data based (called the Sugeno type). The Mamdani-type models are based on a set of linguistic expert formulations, and they are applied when no data are available. The Sugemo type applies an optimization procedure and it is applied when only uncertain data are available.

Fuzzy models were only represented in 0.5% of the model papers in Ecological Modelling from 1975 to 1982, while this model type was found in 1.8% of the model papers in the period 2000-06.

This model type has the following advantages:

• can be applied on semiquantitative (linguistic formulations) information;

• can be applied for development of models, where a semiquantitative assessment is sufficient.

This model type has the following disadvantages:

• can hardly be used for more complex model formulations;

• cannot be used where numeric indications are needed;

• fuzzy models based on data are black box models;

• no software available to run this type of models, although there are provisions in MATLAB to run fuzzy models.

The application of this model type is obvious. It should be applied when the data set is fuzzy or only semiquantitative expert knowledge is available, provided of course that the semiquantitative results are sufficient for the ecological description or the environmental management.

These types of models are able to give relationships between state variables and forcing functions based on a heterogeneous database. It is a black box model and is therefore not based on causality; but it gives in most cases very useful models, that can be applied for prognoses, provided that the model has been based on a sufficient big database that allows to find the relationships and to test it afterward on an independent data set. This model type was not applied before 1982 in ecological modeling, but 3% of the papers published in Ecological Modelling presented ANNs in the period 2000-06.

ANNs including self-organizing maps have the following advantages:

• may be used where other methods must give up;

• give a good indication of the certainty due to the application of a test set;

• can be used on a heterogeneous data set;

• give a near-optimum use of the data set.

The disadvantages can be summarized in the following points:

• no causality unless algorithms are introduced or a hybrid between ANN and a normal model is applied;

• cannot replace biogeochemical models based on the conservation principles;

• the accuracy of predictions is sometimes limited.

The advantages and disadvantages of this model type indicate where it would be advantageous to apply ANN, namely where ecological descriptions and understandings are required on the basis of a heterogeneous database, for instance data from several different ecosystems of the same type. It is also often applied beneficially when the database is more homogeneous, for instance, when the focus is on a specific ecosystems, although the modeler should seriously consider to use biogeochemical dynamic models due to their causality. ANN is, however, faster to use and the time-consuming calibration that is needed for biogeochemical models is not needed.

This model type can be regarded as a reductionistic approach, deriving the properties of a system from the properties and interactions among elements ofthe system. The model type was developed because all the biological components in ecosystems have different properties. Within the same species, the differences are minor and are therefore often neglected in biogeochemical models, but the differences among individuals of the same species may sometimes be important for the ecological reactions. For instance, individuals may have different sizes, which gives a different combination of properties as it is known from the allometric principles. The right combination may be decisive for growth and/or survival in certain situations, as it is known by all modelers. Consequently, a model without the differences among individuals may give a completely wrong result.

Cellular automata are systems of cells interacting in a simple way but displaying complex overall behavior. They are usually characterized by a few salient features. Cellular automata form a class of spatiodynamical models where time, space, and states are discrete. IBMs are often using the cellular automaton approach, although there are IBMs that are not cellular automaton models. Furthermore, there are cellular automaton models that are not IBMs, but models that should belong to the next type, spatial models. They are treated here as one type, because IBMs are frequently based on cellular automaton models.

These types of models were not represented in Ecological Modelling in the period 1975-82, while 5% of the model papers were about IBMs in 2000-06.

Advantages of this type of model are as follows:

• able to account for individuality;

• able to account for adaptation within the spectrum of properties;

• software is available, although the choice is more limited than for biogeochemical dynamic models;

• spatial distribution can be covered.

The disadvantages are as follows:

• if many properties are considered, the models get very complex;

• can be used to cover the individuality of populations, but it cannot cover mass and energy transfer based on the conservation principle;

• requires a large amount of data to calibrate and validate the models.

As mentioned under the characteristics above, we know that the individuals have different properties and that may sometimes be crucial for the model results. In such cases, the IBMs are absolutely needed and the cellular automata can often be considered a proper ecological modeling approach.

As the individual differences may be crucial for the model results, the spatial differences of the forcing functions, nonbiological state variables, and biological state variables may be decisive for the model results, too. Furthermore, it may be required to obtain model results that reveal the spatial differences, because they may be needed to understand the ecological reactions or to make a proper environmental management. Models that give the spatial differences must of course also consider the spatial differences in the processes, forcing functions, and state variables. It can therefore be concluded that there is an urgent need for inclusion of the spatial differences in ecological models. Therefore, it is not surprising that Ecological Modelling has published almost 250 papers about spatial modeling from 2000 to 2006. This model type was not represented in Ecological Modelling in the period 1975-82, while as much as 10% of the model papers were about spatial models in the period 2000-06.

There are a number of possibilities to cover the spatial differences in the development of an ecological model. It is not possible to cover them all; but as mentioned under IBMs, cellular automata may be used in this context. Geographic information system (GIS) is another possible approach that, however, can also be considered a convenient method to present the model results. For aquatic ecosystems, the ultimate spatial model would give a three-dimensional (3-D) description of the processes, forcing functions, and state variables. It is often a question about a good description of the hydrodynamics.

Spatial models offer the following advantages:

• cover spatial distribution, that is often of importance in ecology;

• the results can be presented in many informative ways, for instance, GIS.

The disadvantages are as follows:

• require usually a huge database, giving information about the spatial distribution;

• calibration and validation are difficult and time consuming;

• a very complex model is usually needed to give a proper description of the spatial patterns.

The spatial models are applied whenever it is required that the results include the spatial distribution, because it is decisive or the spatial distribution is crucial for the model results.

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