Modelling Microbial Food Webs

MARINE ECOLOGY PROGRESS SERIES Published December 31 Mar Ecol Prog Ser

REVIEW

Modelling microbial food webs

Keith Davidson*

Swansea Algal Research Unit, School of Biological Sciences, University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, Wales, UK

ABSTRACT M~crobialprocesses are now perceived to be an important component of the marlne food chaln Mathematical models present a quantltatlve means of analys~nglnterspecles lnteract~onswithin food webs Although detalled models of ind~v~dualcomponents of the m~crobialfood web (bacteria, phytoplankton, protozoan predators) have been denved, then incorporation into large-scale food web models has yet to become commonplace Current models have confirmed the general importance of m~crob~alfood webs and that the speclflc inclusion of protozoan predators In models is often necessary to simulate the observed dynamics Simulat~onshave lnd~catedthat the effect of the microb~alfood web on the food chain may vary greatly depend~ngon imposed physical or b~ologicalconditions The manner in wh~chnutrient regeneration from protozoan predators and ~tssubsequent reutilisation by bacteria and phytoplankton 1s simulated IS an Important cons~deratlonModels employ equat~onsof varying sophist~cat~on,but deta~ledrecent formulations use a dual model currency of C and W Models have also Increased our understanding of the Interaction between grazlng and nutrient cycl~ngand the different behav~ourof food chains and webs

KEYWORDS: Mathematical models M~crobialloop . Protozoan predators

INTRODUCTION Phytoplankton are grazed by heterotrophic dinoflagellates, ciliates and tintinnids, as well as metazoan graz- Since Azam et al. (1983) proposed the existence of a ers such as copepods and rotifers. Processes such as 'microbial loop' the microbial food web has become mixotrophy (Boraas 1988), nutrient regeneration in increasingly recognised as an important component of both refractory and non-refractory forms (Flynn & the aquatic food chain. A representation of the micro- Davidson 1993), cannibalism by flagellates (Fenchel bial food web is given in Fig. 1. Bacteria and phyto- 1982), prey selection (Stoecker et al. 1986, Flynn et al. plankton compete for mineral nutrients, and are 1996), the action of viruses (Suttle 1994) or more com- preyed on by a variety of protozoans which are them- plicated N cycling pathways (Jumars et al. 1989) fur- selves ingested by mesozooplankton predators. Bacte- ther complicate the picture. The dynamics of the food ria utilise dissolved organic matter (DOM;a large frac- web and the relative fractions of nutrients retained, tion of which is released from phytoplankton) and lost and recycled remain uncertain. compete with the phytoplankton for mineral nutrients. As discussed by Pommeroy (1991) food webs are The major consumers of bacteria are thought to be inherently difficult to analyse as they contain too many small (<5 pm) non-pigmented flagellates (Capriulo et parameters to be amenable to simple calculus, but too al. 1991), however, other protists also prey on bacteria. few to allow the implementation of statistical tech- niques. However, mathematical models provide tools - which allow investigation of complex dynamical sys- 'Present address. Department of Statistics & Modelling Science, University of Strathclyde. Livingstone Tower. 26 as food webs (Duck'ow Richmond St, Clasuow G1 IXH,Scotland. UK. 1991). Models can be used both to analyse a system, E-mail. [email protected] perhaps to gain a better understanding of a particular

O Inter-Research 1996 Resale of full artlcle not pei-mltted Mar Ecol Prog Ser 145. 279-296, 1996

POM 4

8 mixotrophic P protists

rnixotrophic phytoplankton Fig 2 Rectangular hyperbola as described by Eq. (1). p,,,, maximum specific growth rate; K?:half-saturat~on constant, S. concentration of nutnent In the medium

The major success of the model has been In describing the growth of bacterial populations, although it has also been applied to steady state algal growth (Grover Fig. 1. Representation of the major pathways within the microbial loop. POM: particulate organic matter; DOM: dissolved 1991). Numerous studies have produced modified organic matter models incorporating further physiological and/or physical factors. For example, Ramkrishna et al. (1967) compared modified Monod models and more detailed interaction such as the utilisation of alternative nutri- formulations which included a degree of structure in ent sources by phytoplankton, and also to predict the the protoplasmic mass and Billen & Fontigny (1987) future behaviour of an ecosystem under a variety of included the uptake of amino acids and dissolved pro- conditions. teins. Models of marine ecosystems have been reviewed However, more complete investigation of bacterial by Fasham (1993) and (for the North Sea) Fransz et al. cell physiology, perhaps in ord.er to determine the flux (1991). Such models are often quite general and do not of C and N in a food web, requires more detailed for- contain a realistic representation of the microbial food mulations. Recently, Thingstad and co-workers investi- web. Here, I review models of the major components of gated, as part of a larger study, aspects of the microbial the microbial food web (bacteria, phytoplankton, pro- food web (Thingstad & Pengerud 3.985, Pengerud et al. tozoan predators) in isolation (models of monocultures) 1987, Thlngstad 1992). They produced a model of bac- and in combination (models of inter-species interac- teria in aquatic environments which related growth to tions and food webs). the concentration of both C and N in the medium using a combination of 2 rectangular hyperbolas:

MODELS OF BACTERIA

Monod (1942) proposed a model for microbial where KN and K, are the half-saturation constants for growth which related the specific growth rate of the uptake of N and C, respectively. population to the concentration of the yield limiting Bacterial models using quota type equations (Cap- nutrient in the culture medium by a rectangular hyper- eron 1968a, b, Droop 1968) allowed intracellular nutri- bola of the form: ent levels to be considered. For example, Thingstad (1987) investigated the effect of variable N:C and P:C ratios in the medium. Uptake (V) of C was now descnbed as: where p is the specific growth rate with maximum pmax at infinite S, S IS the concentration of nutrient in the medium and K, the half-saturation constant (Fig. 2). The dynamic behaviour of Monod's model in chemo- where l'-,,, is the maximum uptake rate and QC is the stat systems was studied by Koga & Humphrey (1967) carbon quota with maximum Q,,, and minimum Q,,,. who found it to be overdamped for all realistic parame- Analogous equations were written for N and P. Cell ter values. It remains the most common method of sim- number growth was then made a function of the nutri- ulating bacterial growth. ent with cell quota closest to its minimum. Davidson: Modelling microbial food webs 281

Further refinement to bacterial models includes the work of Fasham et al. (1990)and Ducklo\v & Fasham (1992) who included a relationship to balance the uptake of dissolved organic nitrogen (DON) and ammonium [and related this to the utilisation of dissolved organic carbon (DOC)].A more descriptive model of bacterial growth relating to pelagic systems is that of Hopkinson & Vallino (1994) which simulates nutrient Flg. 3. Schematic representation of the Caperon/Droop con~positionof the cell. (quota) model of algal growth. A: uptake rate per unit bio- As highlighted by Fasham et al. (1990) one of the mass; S: extracellular nutrient concentratlon, CS: cell quota; cntical factors in a food web model containing bacteria pmar.maximum speclflc rate of biomass increase. S: biomass. is the fraction of primary production which cycles Extracellular nutrient IS converted to intracellular nutrient as a hyperbol~cfunction of S. Q intracellular nutrient per unit through the bacterial component. A potentially impor- biomass. Biomass increase is a hyperbolic function of Q tant factor in a model is therefore the loss of nutrient from a cell through nutrient regeneration. Bacterial N regeneration was simulated by Goldman et al. (1987) Unfortunately, the quota model has been less suc- by relating it to the C:N ratio of both the substrate and cessful in simulat~ngtransient growth. Droop (1975) the bacteria and the gross growth efficiency. This achieved reasonable success in simulating batch cul- equation was first used in aquatic systems by Fenchel ture data for Monochrysis lutheri only if the lag phase & Blackburn (1979)and has also been used for bacteria was neglected. Furthermore, Burmaster (1979) found by Lancelot & Billen (1985) (a similar equation for pro- the quota model incapable of adequately describing tozoa will be discussed at greater length below). the changes in cell number density following step changes in dilution in a chemostat for phosphate- limited M. lutheri. MODELS OF PHYTOPLANKTON Recently Davidson et al. (1993) found the quota model to be inadequate to represent C, N and cell Quota type models numbers in their laboratory batch cultures of Isochrysis galbana. They derived a nutrient processing model The inability of the Monod model to describe the which simulated semi-independently the dynamics of experimental chernostat data of Caperon (1968a, b) C and cell numbers by introducing a processing time and Droop (1968)for 2 green flagellates Isochrysis gal- during which newly absorbed N is unavailable to pro- band and Pavlova (Monochrysis)lutheri, respectively, mote cell division. This allowed simulation of both time led them to seek a new explanation of algal growth, lags prior to cell division and variations in C cell-' with the Caperon/Droop (quota) model (Fig. 3). These time without significant increase in model complexity authors suggested a compartment model in which compared to a quota formulation. steady state algal growth was a function of the nutrient Quota models or similar formulations such as the within the cells. Caperon (196813) proposed a function nutrient processing model are relatively simple but of the form: maintain a degree of biological realism and are easily parameterisable. These models are therefore are potentially useful as the phytoplankton sub-model in larger ecosystem models. where p,,, is the maximum achievable specific bio- A variety of other phytoplankton growth models mass growth rate, p, Q is the cell quota defined as have been derived. These range from empirical rela- intracellular nutrient per unit biomass, Q. is the rnini- tionships (Collos 1986) to models which represent a mum value of Q, and Kq is the half-saturation constant considerable degree of cell physiology (Shuter 1979). for growth. This equation reduces to Droop's (1968) However, as discussed below, many of these models equation on making the assumption that K, = Q,. are not easily incorporated into large ecosystem simu- The quota model has been successful in predicting lation models. (at least in steady state) the growth of various species limited by a range of single nutrients including vitamin B12, phosphate, s~llcate,nitrate, ammonium and iron. 'Compartment' phytoplankton models The basic model has been modified to account for numerous other facets of growth and environmental con- Multiple nutrient pool models incorporate a chain of ditions such as dual nutrient limitation and the effect of intracellular compartments. A number of such models light limitation (see the review by Tett & Droop 1988). have been reviewed by Cunningham & Maas (1978). 282 Mar Ecol Prog Ser 145: 279-296, 1996

Later formulations include those of Demanche et al. of these equations relating them specifically to micro- (1979) who attempted to model the dual use of nitrate bial growth (by simulating a chemostat) were proposed and ammonium and Riegman & Muir (1984) who by Bungay & Bungay (1968). related growth rate to the nutrient in the first of 2 intracellular compartments. These models are quite descriptive in nature and a major drawback is that Bacterialgrazer models parameterisation is no easy task as the model compartments are not always physically or biochemically iden- An early modelling study by Canale (1970) investi- tifiable within the cell. gated the ingestion of bacteria by comparing the dy- Attempts have been made to simulate transient cell namics of the Lotka-Volterra equations with 2 alterna- number data by introducing a time lag into the growth tive models. These were first, an equation to describe function of a quota type model (Caperon 1969, Cun- the rate of change of the yleld limiting nutrient, and ningham 1984). However, such a lag introduces second, saturating Monod type functions for both prey dynamical instabilities and formally violates nutrient (bacteria) and predator (protozoan) growth. Canale conservation (Grenney et al. 1973). (1973) then attempted to simulate batch and continuous cultures of a bacterium and a ciliate. Although the Lotka-Volterra model was capable of simulating the Growthlirradiance models general response of the system, it was unable to repro- duce transients. Sudo et al. (1975) extended this work Phytoplankton growthhrradiance models seek to by introducing the possibility of bacterial flocculation relate growth rate to indicators of nutritional status and deflocculation. They found that the introduction of such as the chlorophy1l:C ratio or to photosynthetic such an inhomogeneity was necessary to allow simu- parameters such as optical cross section and quantum lation of their experimentally observed oscillations. yield for photosynthesis (Laws & Bannister 1980, Kiefer Comparison between the Lotka-Volterra model and & Mitchel 1983, Laws & Chalup 1990 and others). As a 'double Monod' type model was also made by poi.nted out by Cullen (1990) these models involve Tsuchiya et al. (1972). They found the steady states numerous parameters not all of which can be routinely reached for a particular parameter set by the Monod measured, and fall strictly in the domain of steady state type model were independent of initial conditions, growth. They are therefore, as yet, unsuitable for use whereas those of the Lotka-Volterra model were not. in predator/prey or ecosystem modelling studies. A variety of factors which could potentially affect the Before they can be more widely applied, the robust- dynamics and stability of food web models have been ness of the underlying assumptions in transient condi- studied using bacteria/grazer models. For example, tions requires testing. Such work is now being under- Jost et al. (1973) introduced an 'S' shaped ingestion taken, e.g. Cullen et al. (1993). function. The inclusion of respiration and death into the equations was proposed by Fredrickson et al. (1973) in a model which extended the food chain to MODELS OF PROTOZOA include primary producers, herbivores and carnivores. These factors were also considered by Villareal et al. Protozoan micro- and dinoflagellates present a link (1975) and Wilcox & MacCluer (1979). Fredrickson which allows the transfer of primary production up the (1977) deta~lsvarious factors which affect the dynamics food chain. They can exhibit autotrophic or hetero- of mixed cultures of microorganisms, including preda- trophic growth or mixotrophy and are able to regener- tion, and other interactions such as commensalism and ate nutrients at high weight specific rates. amensalism. Micro- and dinoflagellates have occasionally been Nisbet et al. (1983) suggested the introduction of modelled as part of the phytoplankton (Andersen et al. predator endogenous metabolism into 'double Monod' 1987, Andersen & Nival 1988).Predatory heterotrophic models would increase stability. More recently, a num- protozoa are usually modelled in terms of their preda- ber of papers by Sambanis (see Sambanis & Fredrick- tor/prey interactions and are discussed below. son 1988) developed a model to account for the coexistence of ciliates and bacteria in continuous culture. Saprotrophy (the growth of bacteria on products of PREDATORIPREY INTERACTIONS lysis) and bacterial wall growth were highlighted as potential means of imparting stability in the predator- The traditional equations to represent predator/prey prey cultures interactions are the Lotka-Volterra equations derived Applications of Lotka-Volterra equations to marine by Lotka (1925) and Volterra (1926). Modified versions food chains include Thingstad & Sakshaug (1990) who Davidson: Modelling microbial food webs 283

investigated nutrient enrichment in food webs and Stoecker & Evans (1985) who considered competition by 2 ciliate grazers feeding on a dinoflagellate. Umorin where I is the ingestion rate with maximum I,,,, y is a (1992) formulated a model for competition between constant, EPN is the effective food concentration and bacterivorous flagellates and bacterivorous ciliates in the subscript i denotes the copepod feeding category. continuous culture. He found it necessary to allow fla- The study found a number of factors including the tem- gellates to feed on both dissolved organic matter and poral scale of a particular event (e.g. the frequency of bacteria to prevent their elimination from the system. nutrient input) to be important in determining the The ability of relatively simple model formulations dynamics. to simulate experimental data sets of bacterial/grazer Frost (1987),who assessed the effect of 2 types of graz- systems indicated that modelling of more complete ers: micro- and mesozooplankton. His model was tested ecosystems was indeed feasible. However, model agalnst field data from the subarct~cPacific Ocean. Var- instability encountered when stable single species ious grazing hypothesis were implemented and tested. (Monod) models were combined to represent preda- Herbivorous microzooplankton were predicted to be the tor/prey situations pointed to potential problems in the major grazers of the phytoplankton with mesozoo- derivation of large-scale models when output may not plankton existlng as indiscriminate grazers. easily be related to the initial model assumptions. A number of more biologically detailed models of phytoplankton/grazer interactions have been derived. In general, these models investigate the effect of par- Phytoplanktonlgrazer models ticular aspects of cell physiology on the system. Car- penter & Kitchel (1984) developed a model which took Grazing of phytoplankton by protozoa and zoo- account of phytoplankton size distribution. Cell growth plankton has also been studied. Most of the models in was related to C content and the half-saturation to cell this category consider nutrients, phytoplankton and radius. The model predicted variability in herbivorous herbivores (N-P-H models) and investigate a particular zooplankton size and abundance was a function of interact~onbetween species without considering the food web interactions not nutrient supply. This led to growth dynamics of the individual species in particu- variations in primary productivity of up to 2 orders of larly great deta~l.A number of models specifically magnitude. The work was extended by Bartell et al. sought to examine grazing response and therefore (1988) who found variations in productivity were tackled the problem of potential model instability men- determined primarily by grazing and nutrient regener- tioned in the previous section. These included: ation. Steele & Henderson (1981), who compared hyper- The choice of model currency in terms of carbon, bolic and 'S' shaped grazing functions. nitrogen cell numbers or a combination of these was Franks et al. (1986), who recognised that saturating highlighted by Davidson et al. (1995) who modelled functional responses (such as rectangular hyperbolas) the interaction between a phytoplankter and a dinofla- have a potentially destabilising effect on a model. They gellate. Phytoplankton growth was described in terms proposed the use of a modified version of the function of C, N and cell numbers, based on the model of David- proposed by Mayzaud & Poulet (1978),in which graz- son et al. (1993). The model's ability to simulate labora- lng rate did not saturate at high prey dens~tiesas it tory experimental data was found to be significantly gave a more stable response than a saturating function better if phytoplankton C cell-' was allowed to vary, in transient conditions. An alternative method to rep- indicating a degree of cell structure was required in resent ingestion was suggested Berrymann (1992) and the model. others who proposed that ingestion was a function of the predator/prey ratio rather than simply prey density. Further work Investigating various ingestion func- Multiple trophic levels tions (Type I and I1 functional response and prey- dependent vs ratio-dependent grazing) was carried Phytoplankton-based food chains have recently out by Carpenter et al. (1993) by applying them to a been used to investigate the interaction between dif- field data set. ferent trophic levels. Thingstad & Sakshaug (1990) Hofmann & Ambler (1988), who produced a model in used steady state Lotka-Volterra equations to repre- which copepod grazing on phytoplankton was con- sent phytoplankton-based food chain webs. In the food sidered in detail. Two phytoplankton size classes and web model, 2 size classes of phytoplankton were in- 5 copepod development stages were included. Inges- cluded. Large algae could be ingested by copepods tion of phytoplankton was based on a modified Ivlev and small algae by protozoa. By analytically determin- function: ing steady states the authors investigated nutrlent 284 Mar Ecol Prog Ser 145: 279-296, 1996

enrichment and cycling through sedimentation and flagellates, ciliates and dissolved substrate. 'Selectivity mlxing. In a linear food chain, enrichment sustained an coefficients' were introduced to distinguish ciliate increasing number of trophic levels. In the food web a grazing on bacteria and flagellates. The model con- dominant linear food chain of small algae, protozoa, firmed phagotrophy by heterotrophic flagellates to be copepods and fish developed until nutrient levels were an important process as its inclusion was necessary to sufficient to allow the growth of large phytoplankton. generate the observed decline in bacterial density. The The control of a particular population was thought model of Pace et al. (1984)included protozoan grazers to be a function of the entire ecosystem rather than in a complex system of 17 compartments and 73 fluxes. limitation by a single nutrient or grazing by a single Their simulations indicated that a representation of the predator. microbial food web was a necessary component of the Frost & Franzen (1992) extended Thingstad & Sak- model. Subsequent models have investigated particu- shaug's (1990) work by considering grazing and iron lar aspects of the microbial food web. limitation in the control of phytoplankton. They used chemostat type equations as an analogue for an equatorial upwelling zone. Phytoplankton, herbivores and Predation by protozoa 2 species of secondary carnivores were included. Graz- ing control was a necessary feature of the model, but a Thingstad & Pengerud (1985) simulated the micro- combination of grazing and nutrient limitation gener- bial loop using both Monod and quota type models ated the observed sub-optimal phytoplankton growth representing chemostat conditions. They examined the rates under optimal conditions. Armstrong (1994)then effect of culture conditions and combinations of bacte- developed a model with multiple food chains, based on ria, algae and protozoa. The protozoan predators were different size classes using allometric relations. He found to be important to the system both in terms of found that multiply linked food chains were more real- thelr predatory behaviour and their ability to regener- lstic than independent food chains and that nutrients ate nutrients. The use of a quota model, although com- could limit the number of classes but grazing would plicating the analysis, introduced a degree of flexibility limit the size of each class. into the model. The authors proposed that still more Evans & Parslow (1985) produced an N-P-H model complicated models would be necessary to account for which made a significant advance by introd.ucing sea- more realistic growth cond~tions. sonality and an equation for the change in the depth of The importance of prey ingestion by protozoa was the mixed layer. Ingestion of phytoplankton was made highlighted again recently by Ducklow (1994). He a hyperbolic function of prey d.ensity above a threshold compared the model of Fasham et al. (1990) (which concentration. The authors investigated the dynamics contained many components of the microbial loop but of their model by removing levels of complexity and not an explicit representation of protozoa) to a revised studying a simplified version of the model. They found version of the same model (Ducklow & Fasham 1992) wind driven mixing and grazing to be responsible for which did incorporate a protozoan compartment. The the seasonal cycle of phytoplankton and that the shal- dynamics of the 2 models differed with respect to lowing of the mixed layer was not always necessary to ammonium regeneration. The mean predicted bacter- initiate a spring bloom. The form of the annual cycle ial biomass was found to be lower when protozoans was closely related to the midwinter phytoplankton were explicitly modelled (Fig. 4). growth rate, a difficult parameter to measure experimentally. The model was further extended by Evans (1988) to include an extra phytoplankton group. Nutrient cycling

In addition to Thingstad & Pengerud (1985) a num- MODELS OF THE MICROBIAL FOOD WEB ber of other studies have found regeneration and cycling of nutrients to be important. Parsons & Kessler Many aquatic ecosystem models do not, as yet, in- (1986) found the recycling of organic material to be an clude a detailed description of the microbial food web important pathway in (and heterotrophic flagellates to (which I will define as a minimum of: bacteria, phyto- be an important component of) their model of the plankton and protozoan predators). I will restrict my upper m~xedlayer. review to the relatively few models which do contain Moloney et al. (1986) employed a more detailed rep- microb~alcomponents. resentation of the microbial food web. Microbial com- Initial models were quite descriptive in design. partments were found to be important wi.th bacteria These included the model of Laake et al. (1983).The acting as mineralisers of N to sustain phytoplankton model (based on Monod kinetics) contained bacteria, growth, and zooflagellates and large protozoa regen- Davidson: Modelling m~crob~a!food webs

1 slze class model observed, with considerable N being made available 2 slze class model - by the action of microheterotrophs at the pycnocline. -- Fasham (1985) developed a flow analysis model which incorporated both C and N under steady state conditions. He assumed phytoplankton and bacteria competed for nutrients. Simulations indicated the importance of nutrient regeneration from both protozoa and zooplankton.

Physiological models of nutrient regeneration

A number of more detailed physiological models have been developed to investigate the temporal effects of variation in N regeneration rate. Day of Year Caron & Goldman (1988) and Caron et al. (1990) related protozoan nutrient regeneration to the nutri- Fig. 4. Comparison of the models of Fasham et al. (1990):the tional status (C:N) ratio of the prey. They developed an 1 predator size class model, and Ducklow & Fasham (1992): the 2 predator size class model. Bacterial biomass measured equation similar to that of Goldman et al. (1987) for as mm01 N m-3 was lower in the 2 size class model, where pro- bacteria which was discussed above: tozoan predators were explicitly represented. Redrawn from Ducklow (1994)

erating significant quantities of N during a phyto- where Eis the excretion rate, R the respiration rate and plankton bloom. If no regeneration of nutrients was GGE the gross growth efficiency. allowed, a single phytoplankton peak and low proto- Landry (1993) modified Eq. (6)to account for nutrient zoan biomass were predicted by the model (Fig 5a). assimilat~on efficiency. Hoi'vever, as pointed out by However, if N cycling was permitted, 2 phytoplankton Caron & Goldman (1993), difficulties exist in parame- peaks due to new and regenerated production (the 2 terising such a modification. peaks shown in Fig 5b, respectively) were predicted Anderson (1992) and Davidson et al. (1995) produced as well as a considerably higher protozoan yield. models for nutrient regeneration based on intracellular Newall et al. (1988) applied Moloney et al.'s (1986) nutrient status. Anderson's (1992) model, which is model to determine the effect of different conditions on applied to bacteria, protozoa and copepods, assumes nutrient cycling. Differences in the behaviour of mod- that non-nitrogenous substrates are preferentially els of the pycnocline and the euphotic zone were used for respiration, thereby conserving nitrogen for

phytoplankton phytoplankton A Drotozoa ---- protozoa ----

I I I I I I I 0.0 2.5 5.0 7.5 10.0 12.5 15.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 rime (days) tlme (days)

Fig. 5. Output of Moloney et al.'s (1986) model (a)without and (b) with N rem~neralisationincluded, showing phytoplankton and protozoan N content. Redrawn from Moloney et al. (1986) 286 Mar Ecol Prog Ser 145: 279-296, 1996

growth. In contrast, based on their experimental re- using a model by Bratbak & Thingstad 1985, see also sults with the dinoflagellate Oxyrrhis marinain which Ietswaart & Flynn 1995) using a theoretical matrix they observed N to be continually regenerated, David- model. Grazing by protozoa on bacteria and subse- son et al. (1995) related N regeneration to an optimal quent rapid recycling of nutrients was found to reduce C:N ratio, uslng the equation: competitive pressure from the bacteria and created conditions in which the phytoplankton equilibrium density increased. A carbon-based model (of a salt marsh estuary) was where m, and p, define the rate of C respiration, N is used by Wright et al. (1987) to address the linkkink cellular nitrogen, Ox is the concentration of the question in steady state. Substrate concentration, bac- dinoflagellate and O is the optimal C:N ratio (6.5 for teria, heterotrophic microflagellates and zooplankton Oxyrrhis manna).If the C:N ratio is higher than opti- were simulated using Monod type equations along mal, excess C is respired or voided and if it is lower with growth yields. Using steady state solutions the then excess N is regenerated. relative roles of substrate and grazing control of bacte- Eq. (7) simulated their data sets better than Eq. (6) rial densities were investigated. This model predicted (Fig. 6) when both were incorporated in a dynamic that either grazing or substrate limitation would occur model framework. The major difference between the 2 independently of the other, with the former acting as a formulations is that Caron & Goldman's (1988) equa- link to higher trophic levels and the latter acting as a tion uses a constant GGE which is unrealistic in tran- sink. (The development of other models of estuarine sient situations. microbial food webs was reviewed by Christian & Wetzel 1991.) King (1987) formulated a steady state model of the Predationlnutrient regeneration in food webs open ocean to examine the interaction between grazing and nutrient cycling. The model predicted that the The simple phytoplankton/grazer models discussed importance of nutrient cycling would increase with the above i.ndicated that primary producers may be limited number of trophic levels and the excretion to ingestion simultaneously by grazing and nutrients. Such an ratio of the predators. interaction cannot be studied by simple observation. Lancelot et al. (1991a, b) combined 3 previously The models detailed below have been used to study published models. For phytoplankton bacterioplank- this interaction and have further confirmed the impor- ton and physical conditions. They then investigated tance of nutrient regenera.tion within ecosystems. carbon cycling in the marginal ice zone of the Scotia- Stone (1990) was able to explain the seemingly Weddell Sea. Further work (Lancelot et al. 1993) found paradoxical release of extracellular organic carbon by that the phytoplankton bloom was controlled by wind phytoplankton under nutrient stress (first examined mixing and protozoan grazing.

Alternative ecosystem model structures

A variety of other modelling strategies have been employed to represent ecosystems incorporating microbial food webs. As pointed out by Fasham (1993) not only the values of state variables but also the inter- compartment flows are of interest when analysing !he output of a model. Microbial models which use network analysis to estimate the flux of nutrients between compartments include Vezina & Platt (1988)and Duck- low et al. (1989). Descriptive models which attempt to represent in some detail the perceived major food web interactions TIME [days) include that of Taylor & Joint (1990). Their model in- Fig. 6. Oxyrrhismarina. Comparison of predicted N content cluded both nitrate and ammonium as well as DOC using a dynamic predator prey model ~ncludingEq. (6) (solid and detritus The model predicted that bacteria were lines) and Eq. (7) (dashed line) to simulate protozoan nutnent unimportant in determining the efficiency of the food regeneration. Three values of 0. marinagross growth effl- ciency (GGE], bracketmy the experimentally observed value. web in steady state stratified conditions. Baretta- were used with Eq. (6). Redrawn from Davidson et al. (1995) Bekker et al. (1994)produced a model similar to that of Davidson: Modelling microbial food webs 287

Taylor & Joint (1990) but considered transient rather Models of individual species may also be analytical than steady state conditions by incorporating a stan- or descriptive In character. Analytical models are use- dard organ~smin terms of C, N and P. Heterotrophic ful to predict features such as growth rates and cell flagellates were included as the link between bacteria yields. These may be of use to biotechnologists for pre- and microzooplankton, whereas Taylor & Joint (1990) dicting culture behaviour, or to ecologists for predict- made microzooplankton graze bacteria directly. ing the expected density 01- biomass content of a spring Size-based models have been proposed by Moloney bloom given a known nutrient loading Descriptive & Field (1991a).These authors grouped populations on models seek to answer (or pose new) questions about the basis of size, with predators acting on the size class particular aspects of physiology. They often contain a below. Their model uses a common structure for each d.egree of biochemical speculation and are of most use size class with parameters calculated using allometric in the analysis of laboratory datasets. relationships. Using this model Moloney et al. (1991) Whole ecosystem models are not readily produced found the microbial food web to be more important as from a combination of single species models as com- a carbon source in oceanic rather than coastal ecosys- plenty becomes too great to allow an analysis of tems. Furthermore, the microbial food web was found model behaviour. Instead, analytical models are de- to contribute on average in excess of 75 % of regener- rived which seek to represent the interaction be- ated N. Further developments of this model have con- tween biotic and abiotic compartments. The art in tinued (eg. Moloney 1992, in which chain forming producing such models is to include sufficient detail diatoms are included, and Painting et al. 1993, who to produce a realistic simulation while keeping the look at species succession in the southern Benguela model simple enough that one can hope to analyse upwelling region). and understand the reasons for a particular model prediction.

DISCUSSION Models of single species The perceived importance of the microbial loop indicates that inclusion of protozoa in ecosystem models Simple analytical and descriptive physiological may often be necessary. However, the simulation of models of bacteria and phytoplankton have been pro- protozoa as a separate entity from the zooplankton in duced. Early formulations were simple in design, ecosystem models remains relatively rare. relating cell number yields and growth rates to para- Mathematical models may be either mechanistic or meters such as extracellular nutrient concentration. empirical in design. However, the complexity of a However, some of these formulations have proved microbial food web and indeed of its component parts remarkably enduring and representations of both bac- is such that mechanistic models are almost exclusively terial and algal steady state growth have been pro- employed. The design of a particular model may vary duced in the forin of the Monod and quota models, greatly and depends on the particular purpose of the respectively, whlch are adequate for many purposes. modelling exercise, as modelling of an ecosystem as a These models are capable of predicting nutrient whole and modelling of the physiology of the individ- uptake rates, cell (or biomass) growth rates and cell ual biotic and abiotic components are carried out with (or biomass) yields. different objectives and often using different ap- A common implementation of the quota model has proaches. been to predict growth rates (Tett & Droop 1988) of Mechanistic microbial models may be subdivided algal cultures. However, the poor transient behaviour into 2 general categories: descriptive and analytical. of the quota model (Grover 1991, Davidson et al. Descriptive models attempt to represent the system of 1993) indicates that models with a better temporal interest in a detailed manner including as many as response are required to represent short-term tempo- possible of the factors which are involved. In contrast, ral fluctuations or the response of a population to a analytical models represent the system in as simple a pulse of nutrient (perhaps due to an upwelling event). manner as is possible, with the condition that the Notwithstanding this shortcoming, the quota model model is still sufficiently detailed to capture the impor- may well be adequate to represent phytoplankton tant features of the d.ynan1ics. Good analytical m.odels growth in steady state or slowly varying transient are not easily formulated, as difficult decisions must be conditions, but as yet has rarely been used in this made regarding what should and should not be in- manner (exceptions are Sharples & Tett 1994 and Tett cluded in the model. However, they potentially pro- & Walne 1995). duce results which are easier to interpret than those Table 1 shows that quite simple bacterial and obtained from descriptive models. phytoplankton models are often adopted in food web 288 Mar Ecol Prog Ser 145: 279-296, 1996

Table 1. Microbial food web component of models. assim: assimilation; C: carbon; conc: concentration; const: constant; dept: dependent; DOC: dissolved organic carbon; DON: dissolved organic nitrogen; egest: egestion; excr: excretion; ext: extracellular; fn. funct~on;graz: grazing; hyp: hyperbolic; mort: mortality; N: nitrogen; phyto: phytoplankton; pred: predator; prop: proportional; respn: respiration

Model Bacterial growth fn Phyto growth fn Protozoan growth fn Ingestion fn Regenerahon fn

Laake et al. (1983) Modified Monod Not modelled Modified monod As growth fn Not modelled

Thingstad & Pengerud Combinvd Monod Monod Yield X hyp Hyp fn of Const per (1985) (Model l) (C and \) lngestlon bacterial numbers bacteria eaten

Thingstad & Pengerud Quota Monou Fn of cell hyp fn of Fn of protozoan (1985) (Mode1 2) nutrients bacterial numbers C:N rat10

Parsons & Kessler Monod type fn Monod type fn Hyp fn ot As growth fn Const fraction (1986) modified for loss modified for loss bacterial numbers (zooplankton by resp by resp only)

Moloney et al. (1986) Asymptotic ins of Hyp fn of Const X prey X As growth fn const fraction (flow model) 'N uptake - losses N uptake - losses pred

Wright et al. (1987) Monod type Not modelled Monod type AS growth fn dot modelled

Fasham et al. (1990) Fn of ext N + Fn of light and Not modelled Hyp fn and food Const fraction DON - losses (NH4. Nos) explicitly preference

Taylor & Joint (1990) Monod fn of Combined fn of Prop to food COILL As growth fn Const fraction DOC NO? and NH,

Lancelot et al. Fn of substrate Intracellular pool Not modelled Temperature dept Not modelled (1991a, b) conc hnear equation

Moloney & Field Hyp uptake Hyp uptake Ingestion - (excr + Michaelis Menten Fn of resp rate (1991a, b) fn - losses fn - losses egest + pred)

Ducklow & Fasham Fn of DON and Fn of NO,, NH, Assirn efficiency X Fn of prey density Const rate (1992) NH4uptake m~xedlayer graz - excr and feeding depth and light preference

Baretta-Bekker et al. Fn of uptake, respn, As for bacteria As for bacteria Const X prey Fn of cell (1994) mort, excr and graz density nutrient levels

Davidson et al. (1995) Not modelled Fn of cell Fn of cell Hyp fn of prey Fn of cell nutrient status nutrient status nutrient status

models. This is at least partly based on the desire to studied to date are those most easily cultured in the keep models as simple as possible. However, the in- laboratory and not necessarily those of greatest envi- clusion of phytoplankton as a homogeneous group ronmental importance). simulated by the sin~plestpossible equation may be More physiologically detailed models which seek too large a generalisation. For example, sub-group- to explain the mechanisms of phytoplankton photo- ings such as the picophytoplankton are an important synthesis have also been developed. Such models part of the microbial food web but are rarely modelled are, in general, only applicable to steady state con- as a separate entity. However, when this has been ditions and therefore not yet particularly suitable to undertaken (Evans 1988, Ducklow & Fasham 1992) simulate real data. Moreover, their high degree of model predictions have deviated from those previ- complexity makes them unlikely candidates for inclu- ously obtained. sion in ecosystem models unless drastic simplification Further study of ecologically important bacteno- can be achieved. The inclusion of an adequate re- plankton and phytoplankton species of different size presentation of light limitation of phytoplankton and physiology is required to produce sufficiently growth in ecosystem models is, however, important detailed datasets to allow us to determine if models of and simplified photosynthesis models have been in- particular species should be routinely included in our corporated into models of ecosystems (Fasham et a1 simulations (phytoplankton species which have been 1990). Ddvidson: Modelling nlicrobial food webs 289

Models of ecosystems turnover times should not be combined within a single mode compartment. Following Platt et al. (1981) ecosystem models have generally attempted to minimise complexity by i-educ- ing the number of compartments in a model. Indeed, it Ingestion of prey would be an impossible task to include the almost infinite number of details which could influence a food Sensitivity analysis indicates that the function used web. to simulate ingestion is one of the most important in A potential problem when developing ecosystem determining model performance. Therefore, the lack models is that complex models may not behave as the of rigor applied to the choice of ingestion function may sum of their component parts and aspects of a model result in serious shortcomings in the final model. For- which may be unimportant in one particular situation tunately, unlike many other potentially attractive mod- may become significantly more important under differ- ifications to a model, the incorporation of alternative ent conditions. For example, the cell quota model has ingestion functions requires little increase in model been found to adequately represent batch culture complexity. phytoplankton growth, given certain assumptions Many models choose to make ingestion of prey (Droop 1979), but fails to represent datasets in which by heterotrophic protozoa a function (usually hyper- nutrient is available in pulses (Davidson & Cunning- bolic) of prey density above a threshold concentration ham 1996). A second example is preferential ingestion (Table 1). However, the choice of a hyperbola often of a particular prey species. This will not be evident in seems to be based on historical precedent rather than studies incorporating only a single prey species but direct observation. There remains a lack of basic labo- may influence food web dynamics in mixed culture or ratory data on microflagellate feeding responses with natural populations. which to construct a robust hypothesis. The studies of Alternative models of the microbial food web can Franks et al. (1986),Berryman (1992) and Carpenter et produce significantly different results. The model of al. (1993) have questioned the use of hyperbolic inges- Lancelot et al. (1991a) predicted ?l'%, of primary pro- tion functions. The effects of selective predation duction to pass through the web. However, Jones & (Fasham et al. 1990). prey rejection and cannibalism by Henderson (1987) calculated that the bacteria-proto- predators (Flynn et al. 1996), the form of the upper zoan pathway was only capable of increasing zoo- predatory closure (Steele & Henderson 1992) and links plankton food by 7%. The large discrepancy in the 2 to higher trophic levels (Ducklow 1994) may all be sig- results indicates how the dominance of d~fferent nificant factors and require more careful consideration microbial processes under different conditions may than they have received up to now. drastically affect the overall behaviour of a pelagic Threshold prey concentrations are commonly incor- food web. Furthermore, it highlights the importance of porated in models (Steele & Henderson 1992) and have carrying out a sensitivity analysis to determine which a stabilising effect on predator/prey interactions parameters the model is most sensitive to, and how (Moloney & Field 1991a found them to damp the oscil- large an error may be introduced by poorly con- latory response of the model). However, the need to strained parameter values. However, such analysis include a threshold prey concentration may indicate a does not tell us why a particular parameter is impor- fundamental deficiency in the model. This is because a tant. An alternative method, adopted by Evans & model should be able to predict observed prey thresh- Parslow (1985), is to remove complexity from a model olds rather than require them to be externally imposed. reducing it to a simpler form. The dynamics of this One model which achieved a stable predator/prey re- reduced model and the influence of particular parame- lationship without requiring the parameterisation of a ters can then be studied by numerical or if possible threshold value was that of Armstrong (1994) in which analytical means. a piecewise linear grazing function was employed. A number of studies have approached the problem The studies of Evans (1988) and Armstrong (1994) of producing succinct food web models by deriving highlight the potential inadequacy of simple food analytical equations which represent the interaction chains to represent aquatic ecosystems. Evans (1988) between trophic levels based on cell size. Moloney & found that extending the model of Evans & Parslow l Field (1991a, b) found this approach allowed them to (1985) to include 2 phytoplankton categories caused parameterise their model based on allometric relation- the periodicity of certain aspects of the simulation to ships. Similar formulations have been produced by change. However, although multiple food chains with Thingstad & Sakshaug (1990) and Armstrong (1994). numerous predator/prey interactions may be more However, this approach is hindered by the observation realistic, it is questionable whether they will be mathe- (Fasham 1993) that organisms with very different matically tractable or parameterisable. 290 Mar Ecol Prog Ser 145: 279-296, 1996

Nutrient cycling models include a degree of biochemical speculation and further refinement will be necessary to allow them Caron (1991) highlighted the importance of proto- to be generalised sufficiently for inclusion in ocean zoan excretion (compared to that by metazoan zoo- models. plankton) as a result of their high weight specific meta- bolic rates. Regeneration of nutrients and their subsequent re-utilisation is included in the majority of Phytoplanktonlbacteria the models (Table 2). The simplest approach assumes that a constant fraction of ingested nutrient is regener- A number of models have predicted simultaneous ated. However, the rate of release may be dependent limitation of primary production by grazing and nu- on the nutrient status of the food and the physiological trient availability (Thingstad & Sakshaug 1990, Frost & status of the predator. Caron (1991) noted that the Franzen 1992, Armstrong 1994). In a food web the degree of starvation may also be important in deter- interaction between bacteria and phytoplankton fur- mining the rate of release of a particular nutrient. The- ther complicates thls interaction and requires careful oretical (King 1987, Newell et al. 1988, Thingstad & consideration. Taylor & Joint (1990) found bacteria Sakshaug 1990, Frost & Franzen 1992) and simulation to be unimportant in determining microbial loop (Fasham et al. 1990, Moloney & Field 1991a, b) models dynamics. Yet, other studies including Pace et al. have confirmed the importance of nutrient cycling and (1984),Stone (1990) and Painting et al. (1993)reached highlighted its stabilising influence. These models the opposite conclusion. Physical and biological condi- have led to attempts to derive more detailed physio- tions (or inadequate conceptual models) may be logical representations of N cycling (Caron & Goldman responsible for this apparent contradiction: Taylor & 1988, Anderson 1992, Davidson et al. 1995). Such Joint's (1990) model was applied to stratified systems

Table 2. Models ~ncludlngnutnent regeneration. N-P-H: nutrient-phytoplankton-herbivore;ML: microbial loop; FW: food web

I I Model Type N-regeneration function

Steele & Henderson (1981) N-P-H Both phytoplankton and zooplankton regenerate nutrients as a constant fraction of intake Carpenter & Kitchell (1984) N-P-H Constant fraction of that ingested Thingstad & Pengerud Constant per bactena ingested (1985) (Model 1) Thingstad & Pengerud Function of protozoan C:N ratio (1985) (Model 2) Franks et al. (1986) N-P-H Unassimilated grazing returned to nutrient pool Moloney et al. (1986) ML Constant fractlon of that ingested Parsons & Kessler (1986) No N regeneration from zooflagellates, zooplankton regenerate a constant fraction of that ingested Frost (1987) N-P-H Constant fractlon of that ingested Bartell et al. (1988) N-P-H Constant fraction of that ingested Hoffmann & Ambler (1988) N-P-H Copepod nutnent regeneration made decreasing function of food increasing concentration Thlngstad & Sakshaug (1990) N-P-H Constant fractlon of that ingested Moloney & Field (19913, b) FW Constant fraction of standing stock fo~eacll blze class Fasham et al. (1990) FbV Constant fraction of that ingested Taylor & Jolnt (1990) ML Constant fraction of that ingested Frost & Franzen (1992) N-P-H Constant fraction of that ingested Baretta-Bekker et al. (1994) ML If nutnent content of cell exceeds a defined level regeneration occurs at a rate proportional to the excess Klng (1987) Constant fraction of that ingested Vezina & Platt (1988) Constant fraction of that ingested Caron & Goldman (1988) Function of gross growth efficiency, respiration rate and C:N ratio of predator and prey Anderson (1992) Regeneration of nitrogenous substrates required for respiration Davidson et al. (1995) Nutrient regeneration occurs to ma~ntainoptimal C:N rat10 Davidson:Modelling microbial food webs 291

in steady state while the other models simulated tran- timescale of frontal eddies to be too short to allow the sient situations where rapid changes occ.ur. development of a large copepod population, which did There remains uncertainty about the behaviour of occur under different conditions. The frequency of bacteria in terms of remineralisation and cons.umption upwelling has also been modelled: Painting et al. (Thingstad 1992). Models have shown that this may be (1993) found either short diatom-based chains or the related to the interaction with phytoplankton. For microbial food web to dominate the simulations example, Ducklow & Fasham (1992) found bacteria to depending on the imposed conditions. act as net remineralisers in netplankton dominated The above discussion indicates that care must be simulations but as net utilisers in picoplankton domi- exercised when interpreting model predictions, espe- nated simulations. cially those from steady state models without explo- ration of the effect of varying physical conditions on the model. Extending a biological model to accommo- Interaction with the environment date physical processes may not always be practical. However, estimation of the effect of physical forcing The physical components of coupled physical/bio- within a sensitivity analysis would at least indicate if logical models containing a representation of the there was cause for concern and more detailed model- microbial food web remain relatively modest. The inte- ling. gration of more realistic physical representations have been undertaken in other ecosystem models (Tett & Walne 1995). Parameterisation Modelling physical effects may be necessary during rapidly changing transient situations when the micro- An obstacle to simulating field data is parameterisa- bial food web may be particularly important. Models of tion of the models. microbial processes which have included representa- Many of the models reviewed attempt to describe tions of physical conditions have in general indicated a field datasets either qualitatively or quantitatively need for coupled physical/biological models. Examples (Table 3). Unfortunately, it proved impractical to make include Evans & Parslow (1985), who simulated varia- a quantitative comparison of the parameter values tions in the mixed layer depth. Hoffmann & Ambler used in the models reviewed, because of the variety of (1988), who included frontal eddies and bottom intru- different modelling strategies and the different situa- sions, Newel1 et al. (1988). who simulated the euphotic tions to which the models were applied. Although a zone and the pycnocline, and Moloney et al. (1991), vast amount of data pertaining to microbial processes who found that in coastal situations or upwelling the has been collected, there remain surprisingly few time fraction of the primary production which reached large course field datasets which are suitable for the para- heterotrophs (>25 pm) was more than an order of mag- metensation and testing of models (Ducklow 1994). nitude greater than in oceanic situations. The model Moreover, laboratory experimentation has concen- therefore predicted the microbial loop to be an ineffi- trated on a relatively few species which are easy to cul- cient pathway to metazoa, but found it to be of signifi- ture. cance especially in oceanic conditions (where large au- A further problem is that the results of models which totrophs do not persist) as the only available C pathway. seek to simulate transient events may be sensitive to Most microbial models are currently point process the inltial conditions. Small variations in initial popula- models which take no account of spatial scale, with the tion levels may cause large discrepancies in the model cells being assumed to exist within a uniforn~ly mixed output. Uncertainties in initial conditions can poten- photic zone. However, models which have incorpo- tially be overcome by simulating a number of annual rated transport across the thermocline of nutrients or cycles and only considering results after the effect of cells (Evans & Parslow 1985, Fasham et al. 1990) have the initial conditions has died away. However, accurate found these processes also to be important. Models of simulation of an ecosystem which may be subject to cell transport through turbulent mixing or advection variable temperature, light and physical forcing is no within the water column exist (Tett & Walne 1995), but easy task and may require the derivation of a much have yet to be applied to microbial food webs. How- larger and more complicated model than one would ever, turbulent flow regimes may influence the micro- desire. Furthermore, computing constraints on such a bial food web as protozoans and their relatively model may not allow the spatial or temporal resolution smaller prey may be physically separated under such necessary to look at a short-term transient phenome- conditions. non. A more realistic solution to the problem lies in The timescale of the particular event to be modelled obtaining more detailed and more closely temporally is also relevant. Hoffman & Ambler (1988) found the resolved datasets than currently exist. Mar Ecol Prog Ser 145: 279-296, 1996

Table 3 Application of microbial loop models to field situations. N-P-H: nutnent-phytoplankton-herbivore; ML: microbial loop; FW: food web

Model Type Region and characterist~c(if any)

Pace et al. (1984) FW Southeastern USA, continental shelf food web Newel1 & Linley (1984) FW Western English Channel Evans & Parslow (1985) N-P-H Eastern Subarct~cPacific and North Atlantic, with and without spring bloom, respectively Moloney et al. (1986) ML English Channel Frost (1987) N-P-H Eastern Subarctic Pacific, high nutnent/low phytoplankton Jones & Henderson (1987) FW Scottish east coast Wright et al. (1987) ML Parker Estuary, USA, salt marsh Hoffmann & Ambler (1988) N-P-H Southeastern USA continental shelf (no data presented), frontal eddies and bottom intrusions Vezina & Platt (1988) English Channel, tidal front; Celtic Sea Fasham et al. (1990) Station S near Bermuda Taylor & Joint (1990) Celtic Sea, Porcupine Bight Christian & Wetzel (1991) Sapelo Island, salt marsh estuary Lancelot et al. (1991a. b. 1993) Scotia-Weddell Sea, Antarctica, marginal Ice zone Moloney & Field (1991) Agulhas Bank, coastal stratified; Benguela, coastal upwelling; southeastern Atlantic, ol~gotrophic Frost & Franzen (1992) N-P-H Eastern Pacific, equatonal upwelling zone Baretta-Bekker et al. (1994) ML Danish coastal waters, enclosure experiments

Although particular features such as upwelling or sage of nutrients through the microbial loop to higher open oceans are addressed by different studies which trophic levels. In order to determine whether the incor- in general give consistent results, it is uncommon for poration of multi-nutrient interactions is a worthwhile different modelling approaches to be applied to simu- next step in ecosystem modelling, it is first necessary to late the same region. Exceptions are Vezina & Platt formulate physiological models which can be tested (1988) and Taylor & Joint (1990) who simulated the under a range of conditions (this in turn will require Celtic Sea, and Evans & Parslow (1985) and Frost accurate experimental determinations of nutrient (1987) who investigated the high nitrate levels in the uptake and utilisation rates). Should these models indi- subarctic Pacific. Detailed comparison of a number of cate the importance of intracellular processes, then model structures for a limited number of geographical effort may be applied to producing simple and succinct regions of different character would provide valuable verslons for inclusion in ecosystem models. information on the relative importance of microbial processes. Coupled models of C and N flux

FUTURE DEVELOPMENTS Many of the current ecosystem simulation models represent biomass with a single variable (such as car- Advances in single species models bon, nitrogen or cell numbers). The transfer of this quantity from 1 level of the ecosystem is then tracked. As yet, most compartment type models of cell growth Those models which do use multiple currencies gener- depend on a single yield limiting nutrient. However, ally assume a constant conversion factor, for example the concept of a single yield limiting nutrient may be between chlorophyll and N. However. the C:N ratio of misleading Due to predation and nutrient cycling the components of the food web may vary with time as primary producers probably never reaches a state of organisms do not necessarily process C and N at the yield limitation. Phytoplankton and bacterial growth same rate (Vezina & Platt 1988) and the biomass and will be governed by the temporal and spatial availab~l- nutrient content of prey cells ingested by a predator ity of a number of rate limiting nutrients. The cell's may vary. Moreover, the nutrient cycling models of ability to take up and process these nutrients will Caron & Goldman (1988),Anderson (1992) and David- therefore determine growth rate and the rate of pas- son et al. (1995)all found it necessary to represent both Davldson: Modelling nlicrobial food webs 293

C and N when calculating protozoan nutrient regener- important. Furthermore, organic matter may have ation. The danger of using cell numbers as the single varylng turnover tlmes and particulates may sink out representation of biomass was recognised by Williams of the photic zone. (1971), and the papers of Burmaster (1979) and Droop (1979) indicate that the use of cell numbers as an indi- cator of biomass may produce spurious model results. Stochastic effects Coupled models of C and N flux and the ability to represent changes per cell in these quantities therefore Most models of food webs are deterministic. The seems to be a minimum requirement if we are to accu- effect of non-deterministic variability on food web rately simulate the global C and N cycles. processes is therefore, as yet, poorly studied (Kremer 1983). However, uncertainty in initial conditions and parameter values may be significant especially over Inclusion of loss processes short timescales in rapidly changing environments. Moreover, the possibility of parameters (such as the Although cell growth has been studied in detail, cell maximum ingestion or growth rate) changing during death has received less attention. If cell death is the duration of the simulation may significantly affect included in a model it is usually as a constant loss fac- model predictions. tor. However, death will be more significant for older cells and may cause shifts in the age structure of a population which will potentially affect growth dynamics CONCLUSION in transient situations. Other loss processes such as the action of viruses, cell lysis and the interaction with In general, experimental evidence indicating the grazers farther up the food chain remaln poorly under- importance of microbial processes has been confirmed stood and therefore inadequately modelled. by modelling studies. However, the relative importance of various processes such as grazing and nutrient cycling under different conditions remains uncertain. Mixotrophs Simulation modelling has clarified the particular interactions which give rise to different dynamical behav- The above analysis highlights algal/bacterial compe- iour of the food web In different geographical regions tition for mineral nutrients and predation pressure as but we are still some way from a complete model important aspects of microbial food webs. Mixotrophic framework which can cope with a variety of physical protists (see Fig. 1) that compete for nutrients are capa- and biological conditions. A significant increase in the ble of grazing their competitors. They are therefore in number of abiotic and biotic compartments in ecosys- the pos~tionto alter the dynamics of a food web inter- tem models is unlikely to increase our understanding action. As yet, little modelling effort has been focused of model dynamics and system behaviour, although on the mixotrophs, at least in part due to the lack of particular features may require inclusion should the information on their physiology. biology dictate. Models retaining the same number of compartments but including coupled representations of C and N flux would be a significant advance. This is Time lags in nutrient cycling not necessarily at odds with the goal of minimising complexity in ecosystem models but does require more A better understanding of the production and elegant formulations than currently exist. dynamics of dissolved and particulate organic matter and the availability of nutrients to bacteria and phyto- Acknou~ledgemenls. This work was carried out while the author was in receipt of a grant from NERC. plankton is necessary as this is relevant to the rate and efficiency of nutrient cycling. The rate of regeneration of nutrients is weight specific and it 1s a significant LITERATURE CITED process for microflagellate grazers (Caron 1991). The Andersen A, Nival P (1988) A pelagic ecosystem model simu- rate of nutrient cycling (and hence the rate of transfer lating production and sedimentation of biogenic particles: of biomass through the food web) may be significantly role of salps and copepods. Mar Ecol Prog Ser 44:37-50 affected by the form in which N is regenerated. The Andersen V, Nival P, Harris R (1987) blodelling of a plank- availability of N to phytoplankton as amino acids, NO3, tonic ecosystem in an enclosed water column. J Mar Biol ASS UK 67:407-430 NH, or NO2 will determine the rate of uptake and the Anderson T (1992) Modelling the influence of food C:N ratio energetic cost of processing of N; in addition, the spa- and respiration In growth and excretion of marine plank- tial availability of nutrients (Moloney 1992) may be ton and bacteria. J Plankton Res 14:1645-1671 Mar Ecol Prog Ser 145: 279-296, 1996

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This review was submitted to the editor Manuscript first received: January 29, 1996 Revised version dccepted: October 29. 1996