Introduction

1.1 Models of marine ecosystems

Understanding and quantitatively describing of marine ecosystems requires an integration of physics, chemistry and biology. Coupling biology and physical oceanography in models has many attractive features: We can do experiments with models of marine systems while the real system can only be observed in the state at the time of the observation. We can also employ the predictive potential of models for applications such as environmental management or, on a larger scale, we can study past and future developments with the aid of experimental simulations. Moreover, a global synthesis of sparse observations can be achieved by using coupled three-dimensional models to extrapolate data in a coherent manner.

However, running complex coupled models requires substantial knowledge and skill. To approach the level of skill needed to work with coupled models, it is reasonable to proceed step-by-step from simple to complex problems.

What are biological models? We use the term ‘model’ synonymously for theoretical descriptions in terms of sets of differential equations that describe the food web dynamics of marine systems. The food webs are relatively complex systems, which can sketched simply as a flux of material from nutrients to phytoplankton to zooplankton to fish and recycling paths back to nutrients. Phytoplankton communities consist of a spectrum of microscopic single-cell plants, microalgae. Many microalgae in marine or freshwater systems are primary producers, which build up organic compounds directly from carbon dioxide and various nutrients dissolved in the water. The captured energy is passed along to components of the aquatic food chain through the consumption of microalgae by secondary producers, the zooplankton. The zooplankton in turn is eaten by fish, which is caught by man. Moreover, there are pathways from the different trophic levels to nutrients through respiration, excretion and dead organic material, detritus, which is mineralized by microbial activity. The regenerated nutrients can then again fuel primary production.

It is obvious that the complex network of the full marine food web cannot practically be covered by one generic model. There are many models that were developed for selected, isolated parts of the food web. For these models the links to the upper or lower trophic levels must be prescribed or parameterized effectively in an appropriate manner. In order to construct such models, we may consider a number of individuals or we can introduce aggregated state variables to characterize a system. State variables must be well defined and measurable quantities, such as concentration of nutrients and biomass or abundance (i.e. number of animals per unit of volume). The dynamics of the state variables (i.e. their change in time and space), is driven by processes, such as nutrient uptake, respiration or grazing, as well as physical processes such as light variations, turbulence and advection.

Ecosystem models can be characterized roughly by their complexity, i.e. by the number of state variables and the degree of process resolution. The resolution of processes can be scaled up or down by aggregation of variables into a few integrated ones or by increasing the number of variables, respectively. For example, zooplankton can be considered as a bulk biomass or can be described in a stage resolving manner. Models with very many state variables are not automatically better than those with only a few variables. The higher the number of variables, the larger the requirements of process understanding and quantification. If the process rates are more or less guesswork, there is no advantage to increase the number of poorly known rates and parameters. Moreover, not every problem requires a high process resolution, and the usage of a subset of aggregated state variables may be sufficient to answer specific questions. Models should be only as complex as required and justified by the problem at hand. Model development needs to be focused, with clear objectives and a methodological concept that ensures that the goals can be reached.

Alternatively, models can be characterized also by their spatial dimension, ranging from zero-dimensional box models to advanced three-dimensional models. In box models, the physical processes are largely simplified while the resolution of chemical biological processes can be very complex. Such models are easy to run and may serve as workbenches for model development. The next step is one-dimensional water column models, which allow a detailed description of important physical controls of biological processes by, for example, vertical mixing and light profiles. Such models may be useful for systems with weak horizontal advection. In order to couple the biological models to full circulation models, it is advisable to reduce the complexity of the biological representations as far as reasonable. If, for example, advection plays an important part for the life cycles of a species, the biological aspects may be largely ignored. Extreme cases of reduced biology coupled to circulation models are simulations of trajectories of individuals, cells or animals, which are treated as passively drifting particles.

The process of constructing the equations, i.e. building a model, will be described in the following chapters, starting with simple cases followed by increasingly complex models. Ecologists often are uncomfortable with the numerous simplifying assumptions that underlie most models. However, modelling marine ecosystems can benefit from looking at theoretical physics, which demonstrates how deliberately simplified formulations of cause–effect relationships help to reproduce predominant characteristics of some distinctly identifiable, observable phenomena. It is illuminating to read the viewpoint of G.S. Riley, one of the pioneers in modelling marine plankton, to this problem, as mentioned in his famous paper (Riley, 1946). He wrote: ‘…physical oceanography, one of the youngest branches in the actual years, is more mature than the much older study of marine biology. This is perhaps partly due to the complexities of the material. More important, however, is the fact that physical oceanography has aroused the interest of a number of men of considerable mathematical ability, while on the other hand marine biologists have been largely unaware of the growing field of bio-mathematics, or at least they have felt that the synthetic approach will be unprofitable until it is more firmly backed by experimental data’.

There is another important issue that deserves some consideration. The biological model equations are basically nonlinear and, in general, not analytically solvable. Thus, early attempts to model marine ecosystems were retarded or even stopped by mathematical problems. These difficulties could only be resolved by numerical methods that require computers, which were not available in an easy-to-use way until the 1980s. This frustrating situation may be one of the reasons that many marine biologists or biological oceanographers were not very much interested in mathematical models in the early years.

With the advent of easily accessible computers, the technical problems were removed. However, it seems the interdisciplinary discussion on modelling develops slowly. Some biologist are doubtful whether anything can be learned from models, but a growing community sees a beneficial potential in modelling. Why do we need models? The reasons include,

• to develop and enhance understanding,

• to quantify descriptions of processes,

• to synthesize and consolidate our knowledge,

• to establish interaction of theory and observation,

• to develop predictive potential,

• to simulate scenarios of past and future developments.

Models are mathematical tools by which we analyse, synthesize and test our understanding of the dynamics of the system through retrospective and predictive calculations. Comparison with data provides the process of model validation. Owing to problems of observational undersampling of marine systems, data are often insufficient when used for model validation. Validation of models often amounts to fitting the data by adjusting parameters, i.e. calibrating the model. It is important to limit the number of adjustable parameters, because a tuned model with too many fitted parameters can lose any predictive potential. It might work well for one situation but could break down when applied to another case. Riley stated this more than 50 years ago, when he wrote ‘…(analysis based on a model, expressed by a differential equation,)… is a useful tool in putting ecological theories to a quantitative test. The disadvantage is that it requires detailed quantitative information about many processes, some of which are only poorly understood. Therefore, until more adequate knowledge is obtained, any application of the method must contain some arbitrary assumptions and many errors due to over-simplification’ (Riley, 1947).

Because many modelers have backgrounds in physics and mathematics, ecosystem modelling is an interdisciplinary task that requires a well-balanced dialogue of biologists and modelers to address the right questions and to develop theoretical descriptions of the processes to be modelled. The development of the interdisciplinary dialogue is a process which should start at the universities where students of marine biology should...