Two goals of aggregation are to reduce dynamical complexity (numbers of state variables) and the number of independent parameters that must be estimated. The simplest and most straightforward approach to aggregation is exemplified in the report of Totterdell et al. (1993), where discrete "synthetic" species (Lehman et al., 1975) are constructed to exemplify representative functional groups of organisms. Totterdell et al. (1993) suggest, for example, that a robust ocean ecosystem model would need to contain roughly six categories of phytoplankton (representing diatoms, coccolithophorids, nitrogen fixers, picoplankton, phytoflagellates, and dinoflagellates). The first three are included so that their roles in geochemical cycles other than the carbon cycle can be represented (silica for diatoms; carbonate, and hence alkalinity, for coccolithophorids). The other three are needed "to correctly model the seasonal pattern of the carbon and nitrogen fluxes, taking into account processes such as grazing, sinking, etc." (Totterdell et al., 1993). To this list were added four types of zooplankton (zooflagellates, microzooplankton, mesozooplankton, and salps) that had different grazing strategies and/or sinking rates of fecal material; these groups could be split or combined in certain ways depending on modeling requirements. Bacteria, detritus, dissolved organic matter, and mineral nutrients were then added to complete the model.
Two major problems with this approach were identified (Totterdell et al., 1993). (1) These models are complicated enough to have complex dynamical behaviors that are difficult to understand in terms of system structure and parameterization, and (2) parameter values may be difficult to estimate, since each of these groups is a "synthetic" species (Lehman et al., 1975) whose characteristic parameter values may vary seasonally and/or geographically. Even so, this approach represents an enormous simplification of the actual biological and geochemical complexity.
One extension of this approach would be to represent zooplankton functionality in terms of a single entity that could switch among phytoplankton prey types as they change in relative abundance (Fasham et al., 1990; Totterdell et al., 1993). This procedure diminishes dynamical complexity by requiring fewer zooplankton state variables; however the switching mechanism itself adds complexity and increases the difficulty of parameter estimation. Other objections to this approach are found in Totterdell et al. (1993).
Within the multispecies context, it may be possible to reduce the number of independent parameters that must be estimated by making certain assumptions about the structure of parameter space. Moloney and Field (1991), for example, exploited allometric (power law) relations in rate constants (e.g., growth rates, sinking rates) among species of different sizes to reduce the number of independent parameters. Their approach allows an important axis of variability (size) to be included in models, allowing different size classes to appear in different situations, while adding a minimal number of parameters. However, these models are still dynamically complex, since they can contain large numbers of phytoplankton and zooplankton size classes; this dynamical complexity can lead in turn to a large range of possible dynamical behaviors and steady-state endpoints with only minor changes in food web structure (Armstrong et al., 1994). In addition, the basic size-structured model no longer contains biogeochemically diverse taxa; the needed diversity could be added either by constructing additional size-lineages for geochemically important taxa (with an attendant increase in dynamical complexity) or by parameterizing their effects (e.g., Maier-Reimer, 1993).
Building on the above considerations, Hurtt and Armstrong (1996) have proposed the further simplification that densities of individuals in successive size classes should conform to the pattern observed empirically by Raimbault et al. (1988) and by Chisholm (1992): that to a good approximation there is equal (maximum) chlorophyll biomass in equal logarithmic size classes, and that biomass is added by adding successively larger size classes rather than by adding biomass within existing size classes. This approach allowed them to define summary growth, death, and sinking rates for the entire size spectrum; total phytoplankton biomass is then represented by a single state variable, dramatically reducing dynamical complexity. The resulting model has only four compartments (nitrate, ammonium, phytoplankton, and a "recycling" compartment), yet produces excellent fits to simultaneous time series of nitrate, chlorophyll, and productivity from the JGOFS BATS program. As in the explicit multiple-chains approach, geochemical diversity could be added either by adding extra state variables or by parameterizing the needed effects.
At present it is not clear which of these approaches is best, or whether each is best in some domain. While the estimation of physiological size-specific growth parameters is probably not feasible in the field, it is important to know whether patterns such as that proposed by Chisholm (1992) hold generally, and what the taxonomic correlates of these patterns might be. Characterizing phytoplankton biomass simultaneously by size and taxonomic composition (dictated by geochemical differences) would indicate whether patterns such as Chisholm's apply to the Southern Ocean; if they do, it may be possible to reduce dramatically both the dynamical complexity and the number of parameters to be estimated in constructing a predictive model of the Southern Ocean.
A different approach has been suggested by Flierl and Davis (1996) who propose using empirical orthogonal functions (EOF) as a basis for reducing the complexity of coupled biological/physical models. Underlying this approach is the assumption that there is a finite (and presumably small) number of modes of biological variability that characterize most of the important processes in the system. The EOFs are used to identify a basis set that is now small enough to be linked to a complex physical model.
Flierl and Davis (1996) apply this approach to a complex model of copepod population dynamics, and it worked successfully in several physical models. They identify two potential limitations to this approach. First, the details of the forcing may affect the mode reduction process. For example, rapid changes in forcing may require retention of more modes. In EOF analysis, nonlinear processes greatly complicate the procedure. In some cases, the reconstructed fields may not be positive definite. The second limitation is also common to EOF analysis in general; the EOFs often do not have a clear biological meaning. Although the EOF approach shows promise, it clearly needs much more research.
