the assumptions and rationales of a computer model of phytoplankton population dynamics

The assumptions and rationales of a computer model of phytoplankton population dynamic?

John T. Lehman2 Haskins Laboratories, Department of Biology, Yale University, New Haven, Connecticut

Daniel B. B&kin School of Forestry and Environmental Studies, Yale University

Gene E. Likens Section of Ecology and Systematics, Cornell University, Ithaca, New York

Abstract Predictions of phytoplankton growth dynamics and nutrient assimilation by a computer

simulation model are consistent with studies of field and laboratory populations. The model simulates population dynamics and gross physiology of phytoplankton species in the epilimnion of a lake where algal growth is subject to temperature, light, and nutrient constraints and includes luxury consumption, end-product inhibition of both carbon fixation and nutrient uptake, and species-specific differential efficiencies of nutrient assimilation. C : P, C : N, and N : P ratios of the algal cells respond to changes in external nutrient conditions, and nutrient storage by the cells permits biological effects of nutrient pulses to be evident long after assimilation of dissolved nutrients forces the pulses to decline. Species succession results when abundances of specific taxa decline due to such factors as sinking or grazing, which assume overriding importance when cell division rates arc slowed by chemical or physical limitations. The physiological tenets and limiting assumptions of the model have been used to formulate patterns of competition among spe- ties for light and nutrients in natural systems and to predict temporal changes in plankton biomass and spccics composition.

Patten in 1968 reviewed the major plankton productivity models that had been proposed and concluded that they justify Levins’ ( 1966) axiomatic denial of simulta- neous generality, precision, and reality. The pace of limnological investigations has nonetheless prompted continuing attempts to survey and synthesize our knowledge of lacustrine ccosys terns. Models should be realistic and precise enough to deal with ur- gencics of watershed management and cultural eutrophication but must maintain the aspect of generality valued by theoreticians. They must serve in twin roles as repositories for hypotheses already supported by empirical observations and as testing grounds for new hypotheses. The models themselves

IA contribution to the II&bard Brook Ecosys- tcm Study. Financial support was provided by grants from the National Science Foundation to G. E. Likens and F. H. Bormann.

2 Present address: ‘Department of Zoology NJ- 15, University of Washington, Seattle 98195.

must be completely tentative, composed of connected hypotheses each of which is sub- jcct to change in light of fresh data. The models must interact with experimental results from the field and laboratory, and if their theoretical framework is sound, they should help us to cvaluatc each new datum as it is produced.

The early models of Riley (1946) and Riley ct al. ( 1949)) by incorporating physiological factors that contribute to marinc plankton productivity, permitted phytoplankton distribution and abundance to be predicted over depth and time. Subsequent attempts to describe pelagic dynamics more accurately have rcfincd and extended the formulative equations of these early papers. Chen and Orlob (1972) and Male (1973), among many others, have used computer simulations to predict plankton population dynamics and patterns of productivity. These models usually treat the phytoplankton as a single unit, without regard for dif-

LIMNOLOGY AND OCEANOGRAPHY 343 MAY 1975, V. 20(3)

344 Lehrmm et al.

ferences among species in the relation between maximum photosynthesis and light intensity or for species-specific differences in nutrient uptake efficiencies. There has been little effort to simulate the seasonal periodicity of individual species and little attempt to model changes in species composition during cultural eutrophication. Be- lieving that such efforts would serve both theoretical and practical aims, we prepared a simulation of plankton growth dynamics at the spccics level.

In this model, we examine the consequences, in terms of community dynamics and species interactions, of a series of established relations in phytoplankton physiology. Although the simulator treats growth primarily as cell division, cell size is not necessarily held constant, because cell quotas of carbon and other elements are al- lowed to vary. Some of the general assumptions explicit in this model are outlined below.

Each cell possesses a maximum division rate possible during optimal conditions;

Growth rate can be reduced from the maximum due to suboptimal temperature, light, or nutrient conditions;

Each species can be characterized by a light intensity that saturates photosynthesis. Greater intensities progressively reduce carbon fixation due to inhibition or photo- oxidation effects;

Each species can be characterized by an optimal temperature and by upper and lower thermal limits;

Each species possesses a growth limiting, nutrient-specific minimum cell nutrient content and a finite nutrient-specific storage capacity beyond this minimum;

Nutrient uptake of each species can be characterized by Michaelis-Menten equations with both maximum rates of nutrient uptake and nutrient-specific half-saturation constants;

Actual uptake velocities are dependent on both internal and external nutrient concentrations;

Cells divide at rates determined by temperature and cellular nutrient contents;

Phvsiological death is insignificant ex-

cept under very suboptimal growth conditions;

Cells are characterized with species-specific sinking velocities and with species- specific light extinction coefficients.

The algal growth simulator is written in APL, a language that is very compact: operations requiring involved subroutines in FORTRAN or ALGOL are reduced to single characters in APL. Alternate hypotheses or expansions of existing hypotheses can often be tested within a matter of min- utcs. Such ease of formulation and func- tional versatility help to focus emphasis on the essence of the model, rather than on the technical aspects of its implementation.

Numerical results are achieved by time- step solutions of difference equations. Be- cause important events in nutrient dynamics occur over periods much shorter than a single day, calculations are performed on an iterative basis over intervals not exceeding 4 h. On an IBM 370/I%, average execution time is 0.18 s species-l day-l with a calculation interval of 3 h. During each interval, cell nutrient quotas are increased by uptake and diluted by cell division. The principal environmental variables that need be specified for this model are initial concentrations and daily ( or hourly) rates of supply of nutrients per liter of epilimnion. Some simple mass-balance equations have not been spelled out here in equation form but will be evident to the interested modeler.

,

We thank L. Provasoli and G. E. Hutch- inson for helpful discussion and criticism.

The moclel

The model simulates the growth of algae in the epilimnion of a lake during a 360- day year. Growth, tabulated as increases in numbers of cells per liter, is computed for each species with regard to light, temperature, and nutrient relations. Cell loss through sinkagc and physiological death is also computed.

Temperature-The epilimnetic temperature is assumed constant over 24 h and is computed once per day to determine species-specific relative growth rates. The annual temperature variation is represented

Phytoplankton dynamics 345

Table 1. Half-saturation constants for N, P, and S4 ,uptake (PM) reported for marine and freshwater plankton algae.

Cyclotella NO3 nana

0.4-l .9

1.8

0.35

0.5

0.4

0.21

0.17

1.4

0.6

0.7-1.3

1.0

1.0

0.42

0.29

0.6

0.4

0.6-l .6

Carpenter & Guillard 1971 Leptocylindricus NO, 1.25 Eppley et al. 1969

MacIsaac 4 Dugdalc 1969

Caperon C Meyer 1972

Eppley et al. 1969

danicus “If; 0.7

NO3 1.7

Nil4 0.5

NO3 3.0

NH4 7.5

NO3 0.6

NH4 1.1

NO 3 2.6

NH4 2.0

NO3 3.6

NH4 4 .9

PO4 16.

Rhizosolenia stolterfothii

NH4 Dunaliella NO3

tertiolectaNH4

NO3

NH4

Asterionella NO3 j aponica

Rhizosolcnia robusta

Caperon G Flcyer 1972

Eppley et al. 1969

Ditylum brightwellii

Coscinodiscus 1 ineatus Eppley & Thomas 1969

Eppley et al. 1969 Coscinodiscus wailesii NH4

blonochrysis NO3 lutheri NH4

No3 NI I4

Fragilaria NO3 pinnata

Caperon Fc Meyer 1972 Euglena gracilis

Cyclotella nana

Mum 1966

po4 0.58 Fuhs et al, 1972 Eppley et al. 1969

Thalassiosira fluviatilis po4 1.72

Carpenter 4 Guillard 1971

Chlorella pyrenoidosa

Nitzschia actinastreoides

PO4 4.-s.

PO4 1.0

Jeanjean 1969

Miiller 1972

PO4 0.6 Rhee 1973

PO4 1.1 Lehman unpublished

Bellochia sp. NO3

Coccochloris NO3 stagnina

PhaeodactylumN03 tricornutum

Anabaena N03 cylindrica N02


NO2

Chaetoceros NO3 gracilis NH4

Gonyaulax NO3 polyedra NI I4

Gymnodinium NO3 splendens NH4

Coccolithus NO3 huxleyi N1f4

Skeletonema NO3 costatum NH4

Isochrysis NO3 galbana

0.1-0.9

0.31 Caperon C Meyer 1972

2.6 Ketchum 1939b Scenedesmus sp.

70.

40.

25,

lfattori 1962 Pediastrum

duplex

Dinobryon cylindricum

D. sociale var. americanum

Nitzschia actinastreoides

PO4 0.8

Knudsen 1965

Eppley et al. 1969

PO4 0.5

0.2

0.4

9.5

5.5

3.8

1.1

0.1

0.1

0.45

0.8

0.1

si 3.5 Miiller 1972

Thalassiosira Si 1.4-2.9 Paasche 1973a pseudonana Si 1.39 Paasche 1973b

Thalassiosira Si 3.37 decipiens

Skeletonema Si 0.80 costatum

Licmophora sp. Si 2.58

Ditylum Si 2.96

brightwellii

by a cosine curve during ice-free months and by a constant function during ice cover:

minimum yearly epilimnetic temperature, and Tmin + Tnlax =

T = Tmin netic temperature. maximum yearly cpilim- When the model is in-

+ 0.5T,,,,[l- cos 24D - Dm)/(Dc - Dm)] tegrated with data from a particular lake,

for D, < D < D,, (1) this temperature curve can be replaced by

T = Tlllill otherwise, where D = day of the a Fourier scrics expansion conforming to

year, D,, = day that ice cover melts, D, = actual temperature measurements (Birge

day that ice cover is established, TIllin = ct al. 1927).

Temperature dependent relative growth

346 Lehman et al.

Table 2. Maximum rates of nutrient uvtake (Fmoles cell-’ h-‘) reported for ‘marine and freshwater plankton algae or calculated from available data.

Cyclotella N03 4.-9. x1o-g Carpenter FI nana Guillard 1971

Biddulphia NH4 2.1 x1o-8 Lui & Roels 1972 aurita

N03 0.9 x1o-8

Dunaliclla NH4 2.G-10.6 x10a8 Caperon 8 tertiolecta NO_ 2.2-3.7 x~O-~ Meyer 197 2

Monochrysis 2.1-3.8 x~O-~ luthcri

NO, 1.4 x10 -8

Coccochloris NO: 4.-g. x10-lo stagnina

Cyclotclla NO3 0.3-l.G x1o-8 nana

Ditylum N03 1.2 xlo-6 Epplcy E brightwellii Coatsworth 1968

Euglena po4 2,4 x~O-~ Blum 1966 gracilis

Phaeodactylum PO4 0.7-l .4 x10 -’ Kuenzler & tricornutum Kctchum 1962

Astcrionella formosa po4 8. x10-’ Mackercth 1953

Scenedesmus PO4 1.2 x1o-8 Rhee 1973

Pediastrum PO4 4. x10 -8 Lehman duplex unpubl ishcd

Navicula Si 1. x1o-8 Busby E pelliculosa Lewin 1967

Nitzschia Si 2.5-5.7 x1o-8 Lcwin & alba Chcn 1968

Si 3.4 x1o’g Paaschc 1973b Skclctoncma costatum

Thalassiosira oscudonana

Si

Licmophora sp. Si

Ditylum brightwellii

Thalassiosira dccipiens

Si

Si

2.6 x10-’

7.7 x1o-8

1.46 x10-’

rates ( TD ) are reprcsentcd by skewed normal distributions:

TD = exp{-2.X (T - To&( L - T,,t) I”> for T > Topt,

TD=e~~~-~~(~-To~t)/(T~~-T,t)l”) ’ opt* (2)

Growth rate is highest at the optimal tem- pcrature ( Topt) and declines as temperature deviates from the optimum toward either higher (T,J or lower (Tn) limits. The limits arc set arbitrarily at some fraction (here 10%) of the maximal growth rates under optimal conditions. The distribution used is a somewhat inexact approach to the

Arrhenius equation of enzyme activity that Johnson et al. ( 1954) suggested may fit the growth patterns of microorganism populations undergoing exponential increase. Data on temperature limits for algae are from laboratory studies of growth rates of individual species (e.g. Thomas 1966; Smayda 1969; Fogg 1965) and field data (Klotter 1955). Hutchinson ( 1967) compiled ranges and optima for many common species of phytoplankton.

Nutrient uptaJce---Nutrient uptake by plankton algae generally follows Michaelis- Mcnten kinetics (e.g. Caperon 1967; Dug- dale 1967; Droop 1968) ; half-saturation constants and maximal velocities of nutrient uptake for a variety of species have been determined (Tables 1 and 2). The range in values indicates that algae differ markedly in their efficiencies of nutrient uptake. Or- ganisms from pelagic, nutrient-poor locali- tics are generally characterized by low half- saturation constants and thus by very efficient nutrient uptake. MacIsaac and Dugdale ( 1969) found that half-saturation constants for inorganic nitrogen uptake were an order of magnitude lower among phytoplankton assemblages from “oligotrophic” than from “eutrophic” regions of the ocean, Eppley et al. ( 1969) have shown how such differences in uptake efficiencies, when translated to specific growth rates, can control the outcome of competition between potentially sympatric species, and how the patterns of competition can be al- tered by physical conditions or by the type of nutrient supplied.

In formulation of our model, we assume that nutrient uptake depends on external substrate concentration in standard Mi- chaclis fashion, i.e. the plot of uptake velocity u ( pmoles cell-i h-i) versus external nutrient concentration S ( PM ) yields a rectangular hyperbola:

u = [ (Qm - Q)/(Qm - kQ)]u,,,S/(k, + S), (3)

where urnal = maximum uptake velocity, k, = half-saturation constant of nutrient uptake, Q = quantity or quota of nutrient contained per cell, kg = minimum cell nutrient content (below which division cannot pro-


Table 3. Minimum cell nutrient quotas (pmoles cell-‘) of N, P, and Si for some marine and freshwater phytoplankton.

Phosphorus :

Asterionclla f ormo sa

Asterionclla j aponica

Cyclotella nana

Nitzschia actinastreoidcs

Phaeodactylum tricornutum


Scencdcsmus quadricauda

Scencdcsmus sp.

Thalassiosira fluviatilis

A. formosa

Gymnodinium

Dinobryon

Anabaena

Silicon:

Navicula pclliculosa

Nitzschia alba

Astcrionclla formosa

Fragilaria crotonensis

Thalassiosira pseudonana

Nitrogen:

Isochrysis galbana

Astcrionella formosa

Gpnodinium

Dinobryon

Anabaena

3 x1o-g (35~) Milller 1972

4 x1o-g rso-59)

2 x1o-g

3 x10-9

4.5 x10-9

1.7 x10-9

12.5 x10-’

Rhcc 1973

Fuhs et al. 1972

3 x10 -8

1.1 x10 -8

0.5 x10 -9

2.S x10 -9

Grim 1939*

0.5 x10 -7

3 x10 -7

2 x10 -G

1.8 x10 -6

4 x19-6

2 x1o-8

Busby 6 Lcwin 1967

Lcwin 4 Chcn 1968

llughcs G Lund 1962

Grim 1939*

Paaschc 1973a

3 x1rl-8

6 x~O-~

Droop 1973

Grim 1939”

3.9 x10 -7

1.8 x~O-~

1 x10 -7

2 x10 -9 blackereth 1953

1.5-3. x10 -9 i&ller 1972

1.5 x1o-g

0.9 x10 -9 Fuhs 1969

*These quantities are not necessarily the minimum.

teed) (Table 3), and Qfn = maximum nutrient storage capacity of a cell. The function includes a feedback factor to account for end-product inhibition of uptake by nutrients that accumulate in the cells. Lux- ury consumption of nutrients (Ketchum 1939a) is a natural consequence of this model whenever high external nutrient levels product uptake rates in excess of

90 t

80 -

OV I I I I 1 0 I 2 3 4 5

RELATIVE NUTRIENT CONCENTRATION (k 8” 1.0)

l?ig. 1. Relative velocities of nutrient uptake in response to both external and internal nutrient regimes. Q and ko rcfcr to ccl1 nutrient contents, as cxplaincd in the text.

the dilution caused by ccl1 division. Ki- netic evidcncc ( Rhec 1973) that internal phosphate stores act as noncompetitive in- hibitors of phosphate uptake provides the mechanism for limiting luxury consumption that Droop (1973) sought on an intuitive basis. Bccausc empirical data for inhibition constants are limited to Rhee’s measurements for Scenedesmus and these constants are difficult to mcasurc, we use a simplification in the model to ensure that uptake rates decline as ccl1 nutrient stores increase. Each population is characterized by a limiting nutrient store capacity ( Qlrt ) ; maximal rates of nutrient uptake diminish as internal nutrient stores ( Q ) approach the maximal limiting quantity. Uptake rates are greatest when cell nutrient contents equal a minimum value ( k, ) below which cell division cannot proceed. Figure 1 shows predicted relative nutrient uptake rates based on equation 3. Each curve corresponds to a diffcrcnt cellular nutrient content ( Q), and uptake velocities decline as cells become progressively less starved. Q,n can be estimated as the maximum Q approached in nutrient excess, and it can bc obtained from

348 Lehman et al.

published data for some species. Line- weaver-Burk transformation of the curves in Fig, 1 shows that the simplification used is superficially indistinguishable from genuine noncompetitive inhibition.

Data by Fuhs (1969), Droop (1973), and Paasche ( 1973a), among others, show that ccl1 division rates, plotted as functions of internal nutrient contents, empirically re- semble rectangular hyperbolas; they are treated as such in the model, using Droop’s ( 1973) derivation of the kinetic relations. The nutrient dependence of cell division ( ND ) is given by

ND= (Q-W/Q> (4)

with Q constrained to the range [k@, Qm]. ND thus has a lower limit of 0 and ap- proachcs 1 asymptotically, although in prac- tice the function is truncated at Q = Qm, The model thus recognizes uptake rates as functions of both internal and external nutrient levels, but cell division as dependent solely on internal concentrations. This permits formulation of situations in which dissolved phosphate and nitrate disappear within hours following artificial enrichment of a lake, but produce long term growth effects among the phytoplankton that absorb them (e.g. Schindler et al. 1971). The distinction bctwecn uptake and division rates enables changes in cell nutrient quotas like those observed by Droop ( 1968) to be reproduced by this model and also predicts variable intracellular nutrient ratios, depending on external nutrient conditions and the growth regime.

Three nutrients are considered in the model, treated simply as dissolved inorganic P, N, and Si. Additional nutrients, trace metals, and vitamins could be immediately included without programing changes if the appropriate species-specific kinetic parameters were known. For purposes of il- lustration, however, and for reproducing much nutrient-related population dynamics, the present arrangement is sufficient.

Photosynthesis-The treatment of photosynthesis is similar to that of Talling (1957) or Vollenweider ( 1965)) but the photosynthetic function used is that of Steele ( 1962).

Light intensity (1) and its attenuation with depth (2) are determined from Beer-Lam- bert predictions based on a vertical light path:

I = IOexp(-EZ), (5)

where I0 = surface intensity in langleys per minute ( g-cal cm-2 min-l). The light extinction coefficient (E) is a sum of contributions from the lake water itself and from plankton, the latter term being density dependent and species-specific, The parameters used to rclatc photosynthesis to light intensity for different taxa follow observations (e.g. Ryther 1956; Brown and Richard- son 1968) of species-specific intensity dependent saturation and inhibition:

P ( 1) = %,,, ( I&t) exp ( 1 - W,,t ) , ( 6) where P,,,,, = maximum photosynthetic rate, occurring at Iopt, and Iopt = light intensity at which photosynthetic rate is saturated. Both the appearance of surface inhibition and the correlation dcmonstratcd by Rodhe ( 1965) between light attenuation and photosynthesis are predicted by the model. Though Rodhe stressed the use of the intensity of the most penetrating spectral com- poncnt in each particular water mass as a correlate with photosynthesis, our model ignores spectral modification with depth and treats light intensity in terms of photo- synthetically active radiation ( PHAR : 380- 720 nm: Strickland 1958). The success of the treatment is partly due to the factors in- ducing Hutchinson’s (1957, p. 392) remark that spectral modification below the top mctcr of penetration is often sufficiently small for a mean vertical extinction coefficient for white light to be useful.

Incident light intensity at full noon sun is represented by a cosine function of period 1 year:

I ,nnx =I,+1,[1-cos~(D+8)/180]. (7)

This function predicts a minimum insolation of I, at winter solstice (22 December) and a maximum of I, + 21, at summer solstice (22 June). WC have used I,, = 0.10 ly min-’ (PHAR) and I, = 0.25, corresponding approximately to expectations for north

temperate regions (ca. 42”N). A similar curve for photopcriod predicts daylengths varying from 9 to 15 h over the year. Inci- dent light intensity from sunrise to sunset is likewise rcprcsented by a cosine function. Phytoplankton populations are assumed to be uniformly distributed throughout the epilimnion.

Phytoplankton dunamics 349 d remb,:

, .‘j I

rates generally increase in cells entering the stationary growth phase as nutrient limitation becomes increasingly pronounced. Fogg ( 1965-j, Fulls ct al. ( 19r12), and others have shown that mean cell carbon contents generally increase over the same period, mainly as lipid or carbohydrate stores. Car- bon quotas are raised to the 0.67 power in accord with the empirical “surface law” of respiration, which claims that most of a cell’s cncrgy is expcndcd at its surface to maintain its integrity vis a vis the environment, A possible alternative to this equation is that suggested by Bannister (1974) for nutrient saturated phytoplankton.

Photosynthetic rates, determined from the average light intensity within each meter interval below the surface, are inte- grated over the calculation interval of several hours, The model includes allowances for end-product inhibition by carbohydrate or lipid reserves stored during previous periods of intense fixation. Each species is characterized with a minimum, growth limiting carbon content (Co) and a maximum carbon store capacity (C,). The carbon stores ( C ) of active cells will lie somewhere between these extremes. Pho- tosynthetic carbon fixation, P(I,C), reduced by end-product inhibition, is given by:

The carbon dependence of cell division rate ( CD) is given by:

CD = (C - C,.),‘C, (10)

a quantity analogous to that used for other nutrients.

P(W) = [(Cnt-C>/(C,,a-Co)lP(I), (f-3) where P( 1) is the function of light intensity described in equation 6. This function is analogous to that for nutrient uptake (cqua- tion 3), since carbon must be treated not simply as a source of energy but also as a nutrient clement. Further reduction of carbon fixation rates, resulting from CO2 limitation caused by rapid removal of inorganic carbon from solution during periods of intense photosynthesis, is critical in soft-water lakes at high cell densities; it is treated by Lehman et al. (in press).

CeZZ division-The temperature, carbon, and nutrient dcpendenccs (TD, CD, and ND) described above were defined as normalized functions which take on values only bctwcen 0 and 1. Actual calculated cell division rates (p) for each species are given as

Respiration-Cellular carbon stores are increased by photosynthetic gains and de- creased by respiratory losses. Respiration rate (R) is assumed to vary with both carbon stores and water temperature:

,u=p,,TD CD r NDi, (11)

where p,n = the species-specific maximum division rate, approached asymptotically under optimal growth conditions, and TD, CD, and NDi are the calculated relative growth dcpcndences for temperature, carbon, and P, N, and Si. A multiplicative nutrient dependence ( Vcrduin 1964; Droop 1973) is used in our simulations, rather than Liebig’s classic law of the minimum, Both alternatives were readily examined with the model, since the opposing hypotheses involve changing only a single character in the program. They give similar results when only one factor is limiting, but equation 11 is more versatile and permits simul- taneous limitation by more than one ele- mcnt, although it perhaps predicts it to be more severe than is realistic. The true relation may lie s0mewher.e between equation 11 and Liebig, as Bloomfield et al. (1974)

Strickland’s (1960) note that respiration have suggested.

I-l = Rrl,x( c/c,)“.“7 - expW3(T - Topt)/(Tll - ~o,t)l, (9)

where R,,,, = the maximum respiration rate obtained at T = Tol,t. This treatment of respiration, a modification of that used by Riley (1946) based on his light and dark bottle studies (Riley 1943)) is suggested by

350 Lehman et al.

Mortality-Physiological death is assumed to be insignificant under optimal cell growth conditions. Cushing ( 1955, 1959) calculated natural mortality of actively dividing marinc diatom populations to be less than 1% of either daily production or grazing rates. Under extreme suboptimal conditions when growth rate falls below some small fraction of maximum, set by us arbitrarily at 5%, physiological death becomes significant. The longer a population remains at these conditions, the greater the mortality effects become, until some maximal fraction (M,,,) of the population dies each day.

M = M,,,[ 1 - exp(-kSG)] (12)

where SG = number of days at suboptimal conditions (p/pm < 0.05)) and k = (In 2) divided by the number of days at subopti- ma1 conditions until M increases to half of M rnax-

Since the quantitative aspects of physiological death are so poorly understood, we have simply lumped them together in terms of their gross effects. Treatment of mortality in this fashion avoids the difficulty of determining separate contributions of temperature, photo-oxidation, or nutrient starvation, We believe that a suitably flcx- iblc mortality function of this sort is pref- erable to constant mortality rates, but it still does not explain such nonobvious and poorly studied aspects of cell mortality as, for instance, time-lagged effects of temperatures above the Grenzoptimum like those demonstrated for Melosira by Rodhe (1948). The treatment of physiological death must therefore be regarded as a working simplification introduced to explain certain gross similarities of mortality resultant from a variety of possible causes and to simulate phenomena about which very little is known.

Dead cells that do not sink release their nutrient contents to an epilimnion particulate nutrient pool; we assume that decom- position rates back to the dissolved pool are proportional to particulate pool size because of the swift reproductive potential

of bacterial populations. The nutrient stores of cells that do sink are assumed to be lost: some fraction is lost to deep sediments, the remainder is redistributed to the mixed layer at vernal and autumnal overturns.

Mean epilimnion depth, or the depth of uniform surface mixing, can vary over the year according to any predetermined pattern specified by the user. Stewart (1965) il- lustrated the variety of annual temperature profiles possible in dimictic temperate lakes, depending on weather conditions. Empirical depths were originally fit by Fourier series, but comparative runs showed that results are almost identical if the points are simply connected by a series of straight lines, the method we now use.

Treatment of ccl1 sinkage from the epilimnion is simplified by the assumption of uniform cell distribution. Hutchinson (1967) tabulated maximal sinking rates for some freshwater species; Smayda (1970) pre- scnted linear regressions of sinking, rates on mean diameters of living and senescent individual cells and maximum diameters of palmelloid colonies, albeit for marine species. If the vertical distribution of each species is assumed to be uniform, changes in population numbers can be given by

dN/dt=[pln2-(V/D)-M]N,(13)

where N is cell concentration, V is sinking rate, D is the mean epilimnetic depth, and /A and M arc the previously defined growth and death rates. The assumption of thor- ough mixing is obviously an utter simplification. Rilcy ct al. ( 1949) proposed a more realistic method based on eddy turbulence, and some workers have attempted discrete simulations based on their approach (Male 1973). The adoption of such methods was, however, unnecessary for the types of biological questions posed here.

Results

The implications of the assumptions stated explicitly at the outset encompass enough empirical characteristics of phytoplankton communities to provide the model with realism and precision, but not sacri-


Table 4. Principal parameters used in describing model species according to the scheme described in the text. Published values are used when possible. Otherwise published ratios (e.g. Gkln 1939; Par- sons et al. 1961) are used to estimate parameters.

Iopt (ly min-1)

k, (MO

Green Blue-green Diatom Dinoflagellate Chrysophyte

0.045 0.03 0.1 0.2 0.06

P 1 s"

1 5"

0.6 N 2 l-5 0.75 si 0 0 7 0 0

s (~10~ poles cell-1 h-l) P 10 N 30 Sl 0

10 0

3

7'o"o

10 kQ (x109 poles cell-l)

P 4.5 1 N 90 20 4: 200 si 0 0 2000 0

10 100

0

1 10

0

0*5 10

0

s (X108 poles cell-l) P 7e2 1.6 3-2 16 0.8 N 72 16 32 160 8 si 0 0 1000 0 0

P (1~10~ poles C cell-l h-l) 7 1.5 3 15 1.0 max

R (z&O7 pmoles C cell-l hY1) 1.4 0.3 0.6 3 0.36 max

-1 kC (~10~ jxnoles C cell ) 7 l-5 3 15 1.8

Cm (x106 poles C cell-l) 7 1.5 3 15 1.8

pm (divisions day") 3 2.5 3 1.5 1.5

V (m day-l) 0.8 0.2 2.5 0.5 0.5 -1 Ed (x107 liter cell m-1) 1.2 1 3 30 0.3

fice its generality in the process. The en- displaying photosynthetic maxima lowest tities treated here are composed of values in the water column are those which toler- gleaned and synthesized from the literature ate or require subdued light intensities. ( Table 4) ; they arc necessarily composite Patterns of photosynthesis by a single spe- due to the lack of sufficient data to charac- cies change as light intensity varies from terize any single real species with all the sunrise to noon ( Fig. 3). Light limited in parameters necessary. To this extent they the early morning hours, the population are hypothetical and their characteristics progressively shows saturation and surface are subject to the needs and improved data of each user; we hope to, show that efforts

inhibition effects as incident light intensity increases.

to characterize ecologically important phy- Daily photosynthesis by each

species has therefore both a time and a toplankton with regard to certain defined depth component, and comparative com- parameters may be rewarded by enhanced putcr runs have shown that estimates of predictability for natural communities. Fig- ure 2 shows the predicted photosynthetic

species productivity based on mean daily

profiles of four hypothetical species in wa- insolation intensities (e.g. Male 1973; Hy- droscience 1973) rather than on actual

ter of moderate clarity ( E = 0.4 m-l) at a daily integrals may overestimate photosyn- surface light intensity of 0.36 ly rnml. The thesis by 59% or more when severe midday different patterns of surface inhibition and inhibition is present. For instance, in a 9-m attenuation with depth are ascribable to the water column, with e = 0.3 m-l, noon in- different Iopt values assigned to each. Taxa tensity = 0.4 ly min-l, and daylength = 12

352 Lehman et al.

I,= 0.36 ly mln-’

Em0.4m’1

OOJ; I 2 3 4 5 6 7 DEPTH ( m 1

ool * 0 I 2 3 4 a 6 7 DEPTH (ml

Fig. 2. Photosynthesis-depth profiles for four Fig. 3. Photosynthesis-depth profiles for a sin- hypothetical species with Iopt values listed in Table gle species with Iopt = 0.1 ly min? PHAR. Pro- 4. The profiles illustrate differential extents of files are computed at successive hours after sun- surface inhibition and depth attenuation between rise, corresponding to increasing incident light species. intensities ( IO).

11, the two methods give results for daily photosynthesis differing by 60% for the diatom species used here (IoPt = 0.1 ly min-I).

Just as incident light intensities can control species-specific fixation rates, water transparency can profoundly influence rcal- ized rates of photosynthetic carbon fixation by algal cells and can completely alter com- petitive relationships between species in a uniformly mixed water column of finite depth. At moderately high light intensities, and in water of high transparency (low E), the dinoflagellate and the diatom display the highest normalized daily fixation rates; at successively higher values of E, the green and the blue-green increase relatively and eventually surpass them. The results imply that regardless of the incident light level, if large plankton biomass generates extensive self-shading, species which saturate at low light intensities become competitively superior. In fertilized Canadian shield lakes the species dominating at times of peak biomass are saturated at 0.03 ly mind1 or less ( Schindler and Fee 1973).

Depressed afternoon fixation rates- When end-product inhibition of carbon fix-

ation is added to these light intensity effects, photosynthesis becomes a function of the physiological state of the cells as well as of their external physical environment. Figure 4 shows changes in net and gross carbon fixation and cellular carbon contents over three 24-h cycles for a hypothetical green alga in near-optimal light and nutrient conditions. Carbon fixation is maximal in the morning of each day and declines in the afternoon, even though light intensities are identical in each period. Comparison of the gross carbon fixation curve with that of cell carbon, and reference to equation 8, rcvcals that the afternoon depression is due to end-product inhibition by accumulated reserves. Cell carbon reaches a maximum value just before sunset and a minimum just before sunrise of each day. Afternoon depression of net carbon fixation rates is further cn- hanced due to the correction for respiration, which increases in the model in proportion to the cell carbon quota. Carbon reserves that accumulate within cells in daylight product increased rates of respiration and cell division which themselves act to reduce

Phytoplankton dynamics

I I I I I I I I IO 20

HO& &&I MI~&GHT 60 70 80

Fig. 4. Net ( 0) and gross ( 0 ) photosynthesis by a hypothetical green alga cles, and corresponding cell carbon contents ( A ). Depressed afternoon rates are duct inhibition (gross ) and partly to respiratory losses (net ).

during three 24-h cy- due partly to end-pro-

cell quotas. The resulting pattern is one of maximal photosynthesis in the morning, preceding the period of maximal respiration and growth. A ncccssary consequence of the assumptions used here is that cells taken from the light and placed in the dark in the presence of adequate nutrients continue to divide, at an ever-decreasing rate, until they exhaust their carbon reserves. The photosynthetic capacity (P,,,,) of Phaeo- dactylurn tricornutum progressively declines during, sustained exposure to optimal light intensities ( Griffiths 1973), even without any significant decrcasc of cell chloro- phyll; when the cells are incubated in the dark for increasing periods, the measurable P max increases. Our model predicts those results precisely, principally on the basis of end-product inhibition of photosynthesis by accumulated carbon reserves.

The model is not constrained to produce depressed afternoon fixation rates, but these arc a consequence of near-optimal light regimes. If light intensities are especially low for instance, a low P : R ratio may keep carbon stores from accumulating to inhibi- tory levels and no pronounced afternoon depression will occur. The ability of the model to account for cell shading at high cell densities is shown in Fig,. 5. Afternoon depression due to accumulated carbon stores is reduced as carbon fixed per cell declines during a simulated population expansion.

Steady state growth relations-To determine whether cell division rates predicted by the model could be related to the extcr- nal nutrient concentration directly and thereby avoid the necessity of treating nutrient storage, we carried out simulations

Lehman et al.

1 I I I I

20 40 60 80 100 HOURS AFTER MIDNIGHT

Fig. 5. Net photosynthesis ( 0 ), and population density ( cal green alga.

(01, cell carbon A) of a hypotheti-

in which small inocula ( 500 cells liter-l) were incubated for 5 days at each of a range of nutrient concentrations. The interval was short enough so that the cells were exposed to stable light and temperature regimes. One nutrient was tested at a time, with all others in great excess, and population densities were so low that uptake did not significantly affect nutrient concentrations in the medium. The interval was more than adequate, however, to assure that the internal nutrient stores of the cells stabilized at a level determined by uptake kinetics and cell division rates. At the end of the 5 days division rates were calculated for the P-limited cells, based on their internal phosphate stores, and these rates arc plotted for three model species against phosphate concentrations in the medium (Fig. 6); curves for nitrogen from analogous simulations are qualitatively the same. The points plotted in Fig. 6 can be fit by Mi- chaelis-Menten equations specified in terms of t.~ and S. The appropriate half-saturation constants, calculated from these data, and called K, values, are 0.09, 0.32, and 0.03 PM for the green, blue-green, and chryso-

phyte. These values are not an explicit part of the model framework, i.e. no K, values are programed, but they ca.re predicted by the model for cells growing in steady state nutrient regimes, The values are all about one order of magnitude lower than the corresponding half-saturation constants of phosphate uptake (k,) for each species. The K, values for growth are valid, however, only for cells in equilibrium with a constant environment, e.g. those in a chemostat. Attempts to relate cell division rates directly to external nutrient levels without considering the moderating influ- ences of luxury consumption and cellular nutrient reserves prove inadequate when the cells are exposed to continually fluctu- ating chemical environments, as is the case in natural ecosystems.

Table 5 lists the C : P and N : P molar ratios computed for the cells in the phosphate-limitation experiment ( Fig. 6). Cells under extreme phosphate starvation showed many-fold increases in C : P and N : P ratios. Most of the increase was due simply to phosphate depletion, but storage capa- bilities for other nutrients also played an important role. Cells whose division rates are slowed by low phosphate concentrations continue to photosynthesize and to absorb nitrogen salts until end-product inhibition forces a balance among carbon fixation, nutrient uptake, and the dilution rates at- tending respiration and cell division. For example, if C and N contents of the starved ( 0.01 ,xM P ) cells had been the same as for the normal cells, then the C : P ratio of the green alga would have been 228 and the N : P ratio 70.

Fuhs et al. (1972) found that P-starved Thalassiosira pseudonana accumulated cx- cess carbohydrate reserves and that cell carbon quotas increased markedly during phosphorus deplction. Cell quotas of nitrogen, on the other hand, did not increase in ThaZassiosira and even declined slightly. This discrepancy between model prediction and empirical fact can be traced to our conceptual treatment of nutrient interactions. That is, the equations presented above implicitly assume that uptake mech-


CHRYSOPHYTE

GREEN

c5 0.8

9 BLUE-GREEN

$ 0.7 V

$! 0.8

d

g 0.5

H 1 0.4 0 w 2 0.3

2 g 0.2

0.1

0.0 II 11111 I Ill I I l III III

0.0 02 0.4 0.8 0.8 1.0 1.2 1.4 1.8 1.8 2.0

PHOSPHATE CONCENTRATION IN THE MEDIUM (NM / LITER )

Fig. 6. Steady state relative growth curves for three hypothetical species with dissimilar half- saturation constants for phosphate uptake. K, (growth) constants were derived from curves fitted to points generated by the model as described in the text.

anisms for different substances are com- little over a wide range of phosphate con- pletely independent, being determined centrations, but carbon reserves accumu- only by internal and external supplies of late as described in relation to Table 5. We individual nutrients. If nitrogen uptake is lack precise data on the influence of single made dependent on cell phosphorus co,n- nutrient deplction on the kinetic param- tent as well as ccl1 nitrogen by the same etcrs for uptake and storage of other nu- relation used to determine cell division trients, but our model could incorporate (ND), simulation results (Table 6) con- new data with a minimum of effort. Ket- form closely to those measured by Fuhs et chum ( 193%) and Rhee ( 1974) empha- al. ( 1972). Nitrogen quotas change very sized the importance of such nutrient inter-

Table 5. Predicted C : P and N : P atomic ratios of cells experiencing phosphate limitation. Nutri- ent assi,milation mechanisms are assumed mutually independent.

Dissolved P

(Pa

0.01 0.03 0.1 0.3 1.0

Green

523 320

‘2 34

C:P N:P

Blue-green Chrysophyte Green Blue-green Chrysophyte

592 3 94 51-5 70 ;: 347 37

g 26

176 18 51 I.4 76 7 10 8

356 Lehman et al.

Table 6. Cell nitrogen quotas as multiples of kQ for cells growing in different phosphate concentrations. Nitrogen uptake is assumed dependent on cell phoSPhorus quotas.

)JMP Green lxfltm Chrysophyte

0.01 4.86 5.15 4.39 0.1 5.07 5-30 4.41 1.0 5.19 5.36 4.42

actions, and their results arc reproduced if phosphorus uptake is made dependent on cell nitrogen quotas.

Many workers have observed dramatic increases in starch or lipids in cells under nutrient limitation (see Fogg 1965; Strick- land 1960). Since increased carbon quotas generally imply increased cell volumes, the model similarly predicts the common phenomenon of cell enlargement in nutricnt- poor media or stationary phase cultures (Prakash et al. 1973).

Luxury consumptioni-The pattern of nutrient assimilation and growth observed by Ketchum (1939a) and Kuenzler and Ket- chum (1962), involving luxury consumption of dissolved nutrients by algal cells, follows directly from the precepts of this model. The predicted consequences of suspending P-starved cells in a phosphate-containing medium or, alternatively, of sudden phosphate enrichment of a P-limited lake, are shown in Fig. 7. High cell concentrations were used to dramatize the results and nitrogen concentrations were held at great excess. The initial phosphorus contents of the cells were adjusted to i&, i.e. to the growth-rate limiting minimum cell phosphorus content. The cells immediately responded to the external enrichment by ab- sorbing the compound, as shown by a decrease in the phosphate concentration in the medium and an increase in the phosphate contents of the cells. Cell division, on the other hand, underwent an apparent lag phase of 8 h before measurable population expansion. Di- vision actually proceeded throughout the “lag” phase at an ever-increasing rate but became graphically perceptible only after 8 h. Because cellular nutrient quotas (Q)

,-

I-

,

HOURS AFTER P ADDITION

Fig. 7. Luxury consumption of phosphate by a hypothetical green alga, shown as an apparent lag period in cell division during which cell P quotas steadily accumulate.

are determined by the balance between uptake velocity and cell division, Q evcn- tually declines, but only after dissolved phosphate is reduced to a very low level. At the same time, uptake rates, reflected in the rate of phosphate depletion from the medium, decline markedly. Cell division continues, however, fueled by the phosphate r&serves accumulated by luxury consumption, Division rates successively decline during the period of decreasing Q, as shown by the almost linear shape of the population growth curve, which deviates from the exponential shape expected if cell concentrations had been lower or the initial P supply larger. With no additional phosphate enrichments, the population size would cvcntually reach a maximum value when division is halted by P starvation. This maximum would be only transitory due to two processes: the rate of population depletion in natural systems due to sinkage or grazing would exceed the waning rate of cell division; and physiological death would begin to operate on the senescent cells. The second factor is beyond doubt responsible for population declines in laboratory stationary phase cultures, but physi-


ological death may not bc of comparable importance in natural systems. Densities of most populations decline due to sinkage although the cells are still perfectly viable.

This point is particularly applicable to diatoms, characterized by high sinking speeds due to their siliceous frustulcs, as the following simulation run will show. Phosphorus starved populations were ad- justcd to low initial cell densities (1,000 cells liter-l), with the exception of the diatom, which was set at 2 x lo6 cells liter-l, the concentration of Syneclra acus var. an- gustissima in Linsley Pond on 17 April 1937 ( Hutchinson 1944). Initial nutrient concentrations were, as given by Hutchinson, 0.003 mg PQ4-P liter-l and 4 mg SiOz liter-l. Nitrogen was set at 0.12 mg NOs-N liter-l. Thermal data compiled by Riley ( 1940) indicate that Linsley Pond is generally stratified by mid-April, and mean cpilim- nion depth was accordingly set at 5 m in the model. Nutrient input rates to the cpi- limnion were assumed to be low: 0.01 rug Pod-P liter-l day-“, 0.5 pg N03-N liter-’ day-l, and 3.5 ,ug SiOa liter-l day-l. All species were assigned the same temperature relations to reduce the number of variables and to make interpretation easier. The physical parameters of the simulation cor- responded to the interval 17 April to 20 June. Hutchinson ( 1944) reported that Synedra reached peak population size on 30 April, subsequently declined, and was replaced by Dinobryon divergent which reached a maximum on 1 June. This same replacement of a diatom by a chrysophyte species occurred in the simulation run (Fig. 8).

The model can thus be used to dcvclop hypotheses for the species composition change in nature. In both the model and Linsley Pond, phosphate concentrations plummeted to less than 0.001 mg liter-1 as the diatom population reached maximal size. As external concentrations dropped, uptake rates and cell nutrient quotas declined, causing division rates to’ slow, Once division rates slowed in the model diatom, the observable reduction in population numbers was due to sinkage. Diatoms need

Fig. 8. Simulated succession in Linsley Pond, 1937. Actual cell counts of Synedra ( 0 ) and f)li;wqbyn ( x 10 ) ( 0 ) are after Hutchinson

high turbulence and rapid division rate to produce population expansions in pelagic cnvironmcnts. When either prerequisite for suspension is curbed, populations tend to decline. At its population maximum, the diatom was dividing at almost 0.7 divisions day-l, a growth rate which just balanced sinking losses. Division rate then declined slowly due to continued nutrient deficiency, and cells could no longer be reproduced as fast as they were lost.

Because of their more efficient uptake of dissolved phosphorus and slower sinking rates, chrysophytes continued to increase after the dcclinc of the diatom. The other species included, green, blue-green, and dinoflagellate, never achieved substantial population densities because of the low nutrient levels following the diatom bloom, The expanding chrysophytes eventually ex- hausted the remaining phosphate, their division rates declined, and loss rates assumed overriding importance. With the aid of the model we can thus understand mechanis- tically how cell division rate, turbulence, sinking speed, and, though not treated here,

Lehman et al.

CHRYSOPHYTE

MARCH APRIL

Simulated succession in Linsley Pond, 1938, using the initial conditions of Hutchinson ( 1944) and Riley ( 1940 ).

grazing rates, operate in species succession. When either of the first two quantities declines, loss rates take command. Low nutrient levels can thus cause the decline of a pelagic diatom population by slowing division even before cells become excessively starved.

The exact patterns of seasonal succession predicted by our model naturally depend on the initial conditions specified. When biological conditions ( Hutchinson 1944) were combined with chemical and physical conditions ( Riley 19140) for Linsley Pond on 23 February 1938, and the same rates of nutrient addition assumed as in the previous example, the prcdictcd pattern of succession once again agrees well with that reported by II-Iutchinson ( 1944) (P’ig. 9,). The blue-green population ( Oscillatoria in Linsley Pond) started at 9 X lo6 cells liter-“, rose to a maximum in early March, and thereafter declined irregularly due to a combination of sinkage and physiological death. It was succeeded by the bloom of a green ( Scenedesmus in Linsley Pond), which subsequently gave way to a chrysophyte ( Dinobryon) in phosphate depleted waters, Diatoms, started at the same concentration as the green and chrysophyte (1,000 cells liter-l), never achieved substan-

tial numbers in the model: they were similarly depauperate in the lake (Hutchinson 1944 ) .

The pattern of succession predicted by the model was due to interspecific competition for the waning supply of dissolved phosphorus; similar competition may have been at least partly responsible for the actual events in nature. Both simulations dealt with periods near spring overturn when nutrient supply was variable and grazing pressure low. Under similar conditions, this model faithfully reproduces patterns of succession in a variety of lakes, but corroboration between observed and predicted events declines in later months unless additional sources of competition, and grazing pressure, are included (Lehman et al. in press).

Temperature effects-The previous ex- amples ignored temperature for simplicity’s sake, but empirical distinction of Kiikefor- men from Wiirmeformen (Findenegg 1943) implies that thermal effects may bcl quite pronounced among freshwater plankton. Apart from their explicit effects on division rates, the temperature dependency functions used in this model affect the physiological condition and nutritional composition of cells. Growth of small inocula (500 cells liter-l) was simulated for periods of 3 days in the presence of excess nutrients and a favorable light regime, with only the epilimnion temperature varied to simulate a series of suboptimal, optimal, and supraoptimal temperatures. The cxperimen t was analogous to placing a series of laboratory cultures with the same medium at different temperatures but under identical light regimes. The average carbon quotas of the cells were calculated for the third day of each run, since by that time carbon quotas had stabilized into regular rhythmic patterns (see Fig. 4). These are listed in Ta- ble 7 together with calculated temperature dependenccs of cell division ( TD ) . Cells at low temperatures had larger carbon reserves than cells under more optimal conditions, because carbon fixed at low temperatures is not immediately used in growth but is stored, At optimal temperatures, how-


Table 7. Cell carbon quotas as multiples of kc for a hypothetical chrysophyte uncler several temperature regimes.

Temp ("C) TD

0 0.1

5 0.36 10 0.775 15 1.0 20 0.1

25 0.001

C f kc

2::

2 3-8 2.3

ever, the fixed carbon is rapidly depleted by division and respiration, and at supraoptimal temperatures, abnormally high respiration rates dispose of the carbon almost as fast as it is fixed. Increased reserves at suboptimal temperatures were also observed for the other nutrients (N, I’, and Si). Another result that emerged was that rates of nutrient uptake and carbon fixation were considerably lower at suboptimal temperatures, even though those rates were not explicitly specified as temperature dcpen- dent. The larger nutrient reserves resulting from slow division rates reduced uptake, through end-product inhibition. Cells at low temperatures are therefore characterized by slow growth, low uptake and fixation rates, and large cell nutrient quotas. Cells at supraoptimal light intensities pho- tosynthesizc at very rapid rates, even though ccl1 division is slowed, because high respiration prevents carbon stores from accumulating.

Discussion

The model described here, based on a few simple empirically and conceptually supported relations concerning the physiology of algal cell growth, can be used re- ciprocally with experimental data and can serve as a tool in hypothesis testing. We propose it as an easily expandable start for future, more extensive models of lake ccosys tcm dynamics, Its conceptual design and its coding in APL make changes easy and create a flexible tool for testing new ideas. By its own existence and operation it suggests experiments that can refine it and add to its realism. The model adopts a mechanistic view of competition and

niche space, in that both are immediate conscqucnces of the physiology of the organisms themselves and of the ambient physicochemical environment. This approach has the advantage of being based on primary observations, for which empirical tests can easily bc formulated, rather than on derived indices whose real analogs may be vague and difficult to distinguish. It has incorporated, synthesized, and formal- ized the results of diverse observations of algal physiology and has produced experimentally testable predictions. It has also made obvious those areas where informa- tion is particularly lacking, as in the factors influencing physiological death, and the types and methods of nutrieint interaction.

The incorporation of nutrient storage permits the model to mimic such subtle observations from field and laboratory as luxury consumption and starvation induced changes in cellular nutrient ratios. The nutrient storage and growth relations used imply the existence of time lags and differ- enccs in magnitude between rates of substrate assimilation and of cell division. It does this in a manner that suggests how phosphate and nitrate added to lakes can disappear from solution within hours and how large algal standing crops can develop in equilibrium with low concentrations of dissolved nutrients, This simulated rapid uptake and storage likewise helps predict the empirical rates of nutrient removal from the epilimnion of a fertilized lake resulting from cell sinkage. Schindler et al. (1973) estimated losses as high as 8% of epilimnion P per day, virtually all of which is contained in phytoplankton cells. Simi- lar sinking losses are predicted by this model in certain instances (Lehman ct al. in press),

Differences of as much as an order of magnitude between half-saturation constants of cell division rates ( K, ) and those of phosphate uptake (k,), empirically demonstrated by Rhee (1973), follow from the assumptions of this model. The ability of submaximal uptake rates to support Aear- maximal growth is due, as Rhee suggests, to the cell’s ability to store excessive P ac-

360 Lehman et al.

cumulated during periods of nutrient abundance, to the direct dependency of its growth rate on internal rather than external concentrations, and to the dependency of nutrient uptake on both, Simulation models that measure growth rates purely as functions of immediate external nutrient concentrations are thus inaccurate unless steady state conditions arc a genuine fea- ture of the region or period considered. Koonce ( 1972) recognized the importance oE cellular nutrient quotas in determining division rates, but his formulative equations do not seem entirely accurate in light oE recent physiological data. Bierman et al. ( lQ74) have similarly proposed a model of phytoplankton growth that separates uptake from division; they showed that it could account for luxury consumption and time lags, whereas models that considered external concentrations alone could not.

Carbon is treated in our model much like other macronutrients, with the notable exceptions that its uptake is light dependent and its loss involves respiration. This treatment has facilitated the simulation of many empirical observations of physiological and ecological interest. For example, the model predicts reduced net photosynthesis in the afternoon caused by increased respiration and end-product inhibition. Similar empirical observations made by Harris ( 1973 ) and Harris and Lott ( 1973) were ascribed entirely to increased photo- respiration during the afternoon.

We have also modeled the excretion of organic carbon by algal cells. In some simulations, excretion was assumed proportional to the cell carbon quota and the production of extracellular material followed under different growth regimes. Cells ex- creted carbon during exponential growth in quantities that increased during periods of active photosynthesis, mimicking results of empirical studies (Fogg et al. 1965; F’ogg 1966). Excretion increased during periods of nutrient limitation, as in stationary phase cultures ( Guillard and Wangersky 1958; Marker 1965). An implication of the results generated by the model under the above assumption is that plankton algae in oligo-

trophic conditions might be expected to ex- crete a greater percentage of the carbon they fix than would species in eutrophic waters. Fogg ct al. (1965) in fact claimed that “oligotrophic” species tended to ex- crete greater percentages of the carbon fixed than did “eutrophic” species. The moderating process appears to be competition for fixed carbon between growth pro- ccsses and losses by excretion.

The phenomenon of phytoplankton succession is a direct consequence of the limiting assumptions of this model. Species abundance increases during periods optimal for cell growth, and ccl1 densities decline due to factors that assume prominence when division rates are slowed by physical or chemical limitation. Simple grazing functions have been coupled with the model, and some of the consequences are mentioned by Lehman et al. (in press), By the same argument used by O’Brien ( 1974) we conjecture that grazing is as important as sinkage in causing population crashes in nature. IIis thesis is that species- specific removal of cells, whether by grazing, death, or sinking, can have major effects on species dominance. All three factors can cause exponential declines in population numbers when division rates fall below critical maintenance levels. In Fig. 8, this critical level was 0.7 divisions day-l for the diatom, as slower division rates could not maintain stable populations in the epilimnion in the face of sinkagc. If grazing pressure had been superimposed on the diatom, the critical division rate might have been greater yet. Reasoning of this sort argues against the importance of physiological death in most natural plankton communities. In fact, Williams ( 1972) cited the absence of proof of severely nutrient depleted cells in nature as evidence that other factors, physical or biological, reduce most populations before extreme nutrient starvation can occur. We retain the assumption of physiological death in the model however, because of indications that it could be of widespread significance in some casts (e.g. Jassby and Goldman 1974).

The conceptual treatment outlined here


is applicable to all pelagic communities where competition for light and nutrients plays a role in phytoplankton succession. It is not restricted to the elements C, N, P, and Si, but is equally applicable to vitamins and tract metals, which Provasoli ( 1969)) Goldman ( 1972)) and Patrick et al. (1969) advocated as possible detcrmi- nants of algal succession, The main difficulty with modeling these trace substances is the lack of data regarding uptake parameters and cell quotas. Droop (1968, 1973) compiled kinetic data for iron and B12, but they arc restricted to a very few species. Furthermore, seasonal variations in the concentrations and fluxes of these trace substances in natural waters are less well studied than are supply rates of other com- pounds.

By comparison, data for N, P, and Si arc widely available, but kinetic parameters are nonetheless wholly lacking for most freshwater species of ecological interest. The simulations of succession in Linsley Pond (Figs. 8 and 9) were conducted using composite species that may bear only hypothetical similarity to, those populations actually in the lake. Predicted cell counts of the chrysophyte, for instance, are aImost an order of magnitude higher than those actually achieved by Dinobryon in both 1937 and 1938, though the predicted peaks coincide well. Recently, uniform labeling experiments with 83P (Lehman unpub- lishcd) have shown that the Dinobryon cell phosphorus quotas used in the runs, calcu- latcd from Grim’s (1939) data, are actually an order of magnitude too low. Correction for this error makes the predicted and observed cell counts coincide much more exactly. The ultimate predictive value of the model thus cannot be properly gauged until enough parameters are measured to replace the hypothetical species used here with defined entities, a goal attainable only through concerted empirical studies,

It is sometimes argued that detailed models will prove too expensive to provide predictions of general usefulness for real prob- lems, such as those of eutrophication, On the contrary, we find that, correctly formu-

latcd, a model of considerable detail can predict general patterns precisely without extcnsivc or costly simulations. Moreover, the results presented here imply that simple models that treat all phytoplankton as a single unit without regard for species-spc- cific differences in nutrient uptake, luxury consumption, and division rates ignore the implications that species composition holds for water quality and secondary production ( Hutchinson 1973).

Coefficients such as P,,, and IoBt are treated as constants here even though they arc not strictly so in nature. The same is equally true of half-saturation “constants” and other such quantities used by physiolo- gists. Although this treatment is simplistic when compared to nature, we have tried to investigate the consequences of conncctcd hypotheses, which, though they can be improved as adaptations are better understood, arc nonetheless basic to a predictive model of phytoplankon dynamics. If a model such as this is used not only as a working research tool, but as a repository for experimentally defined relations, it can incorporate each new dependency as it is quantitatively described, and the accuracy of its predictions should increase in proportion to our understanding of the intricacies of l$custrine systems,

In the ultimate sense, Levins (1966) is correct in claiming that no model can si- multaneously satisfy all demands for si- multancous reality, generality, and precision. Different models, however, possess different degrees of intrinsic order, order attained at the expense of energy used to provide their conceptual and data bases. WC hope that the predictions of our model can stimulate the research needed to, reduce the entropy of these bases. In this way, the generality of the model will not be sacri- ficed, but its precision and reality may im- prove enormously.

References

BANNISTER, T. T. 1974. A gcncral theory of steady state phytoplankton growth in a nutrient saturated mixed layer. Limnol. Ocean- ogr. 19; 13-30.

BIEIIMAN, V. J., Jn., F. II. VERIIOFF, T. L. POUL-

362 Lehman et al.

SON, AND M. W. TENNEY. 1974. Multi- nutrient dynamic models of algal growth and species competition in eutrophic lakes, p. 89-109. In E. J. Middlebrooks et al. [eds.], Modeling the eutrophication process. Ann Arbor Sci.

BINGE, E. A., C. JUDAY, AND H. W. MAXI-I. 1927. The temperature of the bottom deposits of Lake Mendota; a chapter in the heat exchanges of the lake. Trans. Wis. Acad. Sci. 23: 187-231.

BLOOMFIELD, J. A., R. A. PARK, D. SCAVIA, AND C. S. ZAI-IORCAK. 1974. Aquatic modeling in the eastern deciduous forest biome, U.S.- International Biological Program, p. 139-158. In E. J. Middlebrooks et al. [eds.], Modeling the eutrophication process. Ann Arbor Sci.

BLUM, J. J. 1966. Phosphate uptake by phosphate starved Euglena. J. Gen. Physiol. 49: 1125-l 137.

BROWN, T. E., AND F. L. RICHARDSON. 1968. The effect of growth environment on the physiology of algae: Light intensity. J. Phycol. 4: 38-54.

BUSBY, W. F., AND J. LEWIN. 1967. Silicate uptake and silica shell formation by synchro- nously dividing cells of the diatom Navicula pelliculosa (Breb.) Hilse. J. Phycol. 3: 127- 131.

CAPERON, J . 1967. Population growth in micro- organisms limited by food supply. Ecology 48 : 715-722.

AND J. MEYER. 1972. Nitrogen-limited growth of marine phytoplankton. 2. Uptake kinetics and their role in nutrient limited growth of phytoplankton. Deep-Sea Res. 19 : 619-632.

CAIXPENTER, E. J,, AND R. R. L. GUILLARD. 1971. Intraspecific differences in nitrate half-saturation constants for three species of marine phytoplankton. Ecology 52 : 183-185.

CHEN, C. W., AND G. T. ORLOB. 1972. Ecologic simulation for aquatic environments. Water Resour. Eng. Walnut Creek, Calif. 156 p.

CUSI-IING, D. I-1. 1955. Production and a pelagic fishery. Fish. Invest. Ser. 2 18( 7 ).

1959. On the nature of production in the sea. Fish. Invest. Ser. 2 22( 6). DROOP, M. R. 1968. Vitamin B,s and marine

ecology. 4. The kinetics of uptake, growth and inhibition in Monochrysis Zutheri. J. Mar. Biol. Assoc. U.K. 48: 689-733.

-. 1973. Some thoughts on nutrient limitation in algae. J, Phycol. 9: 264-272.

DUGDALE, R. C. 1967. Nutrient limitation in the sea: Dynamics, identification, and significance. Limnol. Oceanogr. 12 : 685-695.

EPPLEY, R. W., AND J. L. COATSWORTEI. 1968. Nitrate and nitrite uptakes by DityZum brightwellii: Kinetics and mechanisms. J. Phycol. 4: 151-156.

J. N. ROGERS, AND J. J. MCCARTHY. 1969. Half-saturation constants for uptake

of nitrate and ammonium by marine phytoplankton. Limnol. Oceanogr. 14 : 912-920.

-, AND W. I-1. TEIOMAS. 1969. Comparison of half-saturation constants for growth and nitrate uptake by marine phytoplankton. J. Phycol. 5: 375-379.

FINDENEGG, I. 1943. Untersuchungen iiber die 0kologie und die Productionverhaltniss des Planktons in Ktirtner Secngebiete. Int. Rev. Gesamten Hydrobiol. 43 : 368-429.

FOGG, G. E. 1965. Algal cultures and phytoplankton ecology. Univ. Wis.

-. 1966. The extracellular products of algae. Oceanogr. Mar. Biol. Annu. Rev. 4: 295-212.

- C. NALEWAJKO, AND W. D. WATT. 1965. Ex&acellular products of phytoplankton photosynthesis. Proc. R. Sot. Land. Ser. B 162 : 517-534.

Funs, G. W. 1969. Phosphorus content and rate of growth in the diatoms CycZoteZZa nunu and Thalassiosira f Zuviutilis. J. Phycol. 5: 312321.

-, S. D. DEMMERLE, E. CANELLI, AND M. &EN. 1972. Characterization of phosphorus-limited algae (with reflections on the limiting nutrient concept ), p. 113-132. In G. E. Likens [ed.], Nutrients and eutrophication. Am. Sot. Limnol. Oceanogr. Spec. Symp. 1.

GOLDMAN, C. R. 1972. The role of minor elements in limiting the productivity of aquatic ecosystems, p. 21-33. In G. E. Likens [ed.], Nutrients and eutrophication. Am. Sot. Lim- nol. Oceanogr. Spec. Symp. 1.

GRIFFITI-IS, D. W. 1973. Factors affecting the photosynthetic capacity of laboratory cultures of the diatom Phueoductylum tricornut,um. Mar. Biol. 19: 117-126.

GRIM, J. 1939. Beobachtungen am Phytoplank- ton des Bodensees (Obersee) sowie deren rechnerische Auswertung. Int. Rev. Gesam- ten Hydrobiol. 39 : 193-315.

GUILLARD, R. R. L., AND P. J. WANGERSKY. 1958. The production of extracellular carbohydrates by some marine flagellates. Limnol. Ocean- ogr. 3: 443-454.

HARRIS, G. P. 1973. Diel and annual cycles of net plankton photosynthesis in Lake Ontario. J. Fish. Res. Bd. Can. 30: 1779-1787.

AND J. N. A. LOTT. 1973. Light inten- sit; and photosynthetic rates in phytoplankton. J, Fish. Res. Bd. Can. 30: 1771-1778.

HATTORI, A. 1962. Light-induced reduction of nitrate, nitrite, and hydroxylamine in a blue- green alga, Anabaena cylindricu. Plant Cell Physiol. 3 : 355-369.

HUGEIFS, J. C., AND J. W. G. LUND. 1962. The rate of growth of AsterioneZZu formosa Hass. in relation to its ecology. Arch. Mikrobiol. 42 : 117-129.

HUTCHINSON, G. E. 1944. Limnological studies in Connecticut. 7. A critical examination of


the supposed relationship between phytoplankton pcriodicity and chemical changes in lake waters. Ecology 25: 3-26.

-* 1957. A treatise on limnology, v 1. Wiley.

-. 1967. A treatise on limnology, v 2. Wiley.

-. 1973. Eutrophication. Am. Sci. 61: 269-279.

HYDROSCIENCE. 1973. Limnological system an- alysis of the Great Lakes. Phase 1-Prelimi- nary model design. 431 p.

JASSBY, A. D., AND C. R. GOLDMAN. 1974. Loss rates from a lake phytoplankton community. Limnol. Oceanogr. 19 : 618-627.

JEAN JEAN, R. 1969. Influence de la carence en phosphore sur les vitesses d’absorption du phophate par les Chlorclles. Bull. Sot. Fr. Physiol. Veg. 15: 159-171.

JOIINSON, F. II., H. EYRING, AND M. J. POLISSAR. 1954. The kinetic basis of molecular biology. Wiley.

KETCHUM, B. II. 1939a. The development and restoration of deficiencies in the phosphorus and nitrogen composition of unicellular plants. J, Cell. Camp. Physiol. 13: 373-381.

-. 1939b. The absorption of phosphate and nitrate by illuminated cultures of Nitx- schia closterium. Am. J. Bot. 26: 399-407.

KLOTTER, II. E. 1955. Die Algcn in den Seen dcs siidlichcn Schwarzwaldes 2. Eine ijkolo- gische-floristische Studie. Arch. Hydrobiol. Suppl. 22, p. 106-252.

KNUDSEN, G. 1965. Induction of nitrate reduc- tase in synchronized cultures of Chlorella pyrenoidosa. Biochim. Biophys. Acta 103 : 495-502.

KOONCE, J. F. 1972. Seasonal succession of phytoplankton and a model of the dynamics of phytoplankton growth and nutrient uptake. Ph.D. thesis, Univ. Wis., Madison.

KUENZLER, E. J., AND B. I-1. KIZTCIIUM. 1962. Rate of phosphate uptake by Phaeodnctylum tricornutum. Biol. Bull. 123 : 134-145.

LEH~MAN, J. T., D. B. BOTKIN, AND G. E. LIKENS. In press. Lake eutrophication and the limiting CO, concept: A simulation study. Proc. Int. Assoc. Theor. Appl. Limnol. 19.

LEVINS, R. 1966. The strategy of model build- ing in population biology. Am. Sci. 54: 421-431.

LEWIN, J., AND C. CIIEN. 1968. Silicon mctabo- lism in diatoms. 6. Silicic acid uptake by a colorless marine diatom, Nitzschia alba Lewin and Lewin. J. Phycol. 4: 161-166.

LUI, N. S. T., AND 0. A. ROELS. 1972. Nitrogen metabolism of aquatic organisms. 2. The assimilation of nitrate, nitrite, and ammonia by Biddulphia aurita. J. Phycol. 8: 259-263.

MACISAAC, J, J., AND R. C. DUGDALE. 1969. The kinetics of nitrate and ammonia uptake by natural populations of marine phytoplankton, Deep-Sea Res. 16: 45-57.

MACKERETI-I, F. J. 1953. Phosphorus utilization by Asterionella formosa. J. Exp. Bot. 4: 296-313.

MALE, L. M. 1973. A temporal-spatial model for studying nutrient cycling dynamics of a phytoplankton production system. Part l-development of model. Center Quant. Sci. Forestry, Fish., Wildl. Univ. Wash. Quant. Sci. Pap. 35. 29 p.

MARKER, A. F. II. 1965. Extracellular carbohydrate liberation in the flagellates Isochrysis galbana and Prymnesium parvum. J. Mar. Biol. Assoc. U.K. 405: 755-772.

MUELLER, H. VON. 1972. Wachstum und Phos- phatbedarf von Nitzschia actinastreoioks ( Lemm. ) v. Goor in statischer und homokon- tinuierlichcr Kultur unter Phosphat-limitier- ung. Arch. Hydrobiol. Suppl. 38, p. 399- 484.

O’BRIEN, W. J. 1974. The dynamics of nutrient limitation of phytoplankton algae: A model reconsidered. Ecology 55 : 135-141.

PAASCI-IE, E. 1973a. Silicon and the ecology of marine plankton diatoms. 1. Thalassiosira pseudonana Hasle and Heimdal ( Cyclotellu nana Hustedt ) grown in a chemostat with silicate as the limiting nutrient. Mar. Biol. 19: 117-126.

-. 1973b. Silicon and the ecology of marine plankton diatoms. 2. Silicate uptake kinetics in five diatom species. Mar. Biol. 19: 262-269.

PARSONS, T. R., K. STEPIIANS, AND J, D. H. STRICK- LAND. 1961. On the chemical composition of eleven species of marine phytoplankters. J. Fish. Res. Bd. Can. 18: 1001-1016.

PATJXICK, R., B. CRUM, AND J, COLES. 1969. Temperature and manganese as determining factors in the presence of diatom or blue- green algal floras in streams. Proc. Natl. Acad. Sci. U.S. 64: 472-478.

PATTEN, B. C. 1968. Mathematical models of plankton production. Int. Rev. Gesamten Hydrobiol. 53 : 357-408.

PRAKASII, A., L. SKOGLUND, B. RYSTAD, AND A. JENSEN. 1973. Growth and cell-size distribution of marine planktonic algae in batch and dialysis cultures. J. Fish. Res. Bd. Can. 30 : 143-155.

PIXOVASOLI, L. 1969. Algal nutrition and eutrophication, p. 574-593. In Eutrophication: Causes, consequences, correctives. Natl. Acad. Sci. Publ. 1700.

RIIEE, G-Y. 1973. A continuous culture study of phosphate uptake, growth rate and polyphos- phate in Scenedesmus sp. J. Phycol. 9:495- 506.

-. 1974. Phosphate uptake under nitrate limitation by Scenedesmus sp. and its ecological implications. J. Phycol. 10: 470-475.

RILEY, G. A. 1940. Limnological studies in Connecticut. Part 3. The plankton of Linsley Pond. Ecol. Monogr. 10: 279-306.

364 Lehman et al.

-. 1943. Physiological aspects of spring diatom flowerings. Bull. Bingham Oceanogr. Collect. 8( 4) : 53 p.

-. 1946. Factors controlling phytoplankton populations on Gcorges Bank. J. Mar. Res. 6: 54-73.

-, H. STOMMEL, AND D. F. BKMPUS. 1949. Quantitative ecology of the plankton of the western North Atlantic. Bull. Bingham Oceanogr. Collect. 12 (3) : 169 p.

Romq W. 1948. Environmental requirements of freshwater plankton algae. Experimental studies in the ecology of phytoplankton, Symb. Bot. Upsal. 10: 149 p.

-. 1965. Standard correlations between pelagic photosynthesis and light. Mem. 1st. Ital. Idrobiol. 18( suppl. ) : 365-381.

RYTIIER, J. H. 1956. Photosynthesis in the ocean as a function of light intensity. Limnol. Oceanogr. 1: 61-70.

SCHINDLER, D. W., F. A. J. ARMSTRONG, S. K. HOLMGREN, AND G. J. BRUNSKILL. 1971. Eutrophication of Lake 227, Experimental Lakes Area, northwestern Ontario, by the addition of phosphate and nitrate. J. Fish. Res. Bd. Can. 28: 1763-1782.

AND E. J. FEE. 1973. Diurnal variation of ‘dissolved inorganic carbon and its use in estimating primary production and CO2 in- vasion in Lake 227. J. Fish. Res. Bd. Can. 30: 1501-1510.

- II. KLING, R. V. SCIIMIDT, J. PROKOPO- WI&I, V. E. FROST, R. A. REID, AND M. CAPEL. 1973. Eutrophication of Lake 227 by addition of phosphate and nitrate: The second, third, and fourth years of enrichment 1970, 1971, and 1972. J. Fish. Res. Bd. Can. 30: 1415-1440.

SMAYDA, T. J. 1969. Experimental observations on the influence of temperature, light, and salinity on cell division of the marinc diatom Detonula confermcea ( Cleve ) Gran. J. Phy- col. 5: 150-157.

-I 1970. The suspension and sinking of phytoplankton in the sea. Oceanogr. Mar. Biol. Annu. Rev. 8: 353-414.

STEELE, J. 1% 1962. Environmental control of photosynthesis in the sea. Limnol. Oceanogr. 7 : 137-150.

STEWART, K. M. 1965. Physical limnology of some Madison lakes. Ph.D. thesis, Univ. Wis., Madison.

STIUCKLAND, J. D. I-1. 1958. Solar radiation penetrating the ocean, a review of requirements, data and methods of measurement, with particular reference to photosynthetic ;;p;vity. J. Fish. Res. Bd. Can. 15:

-. 1960. Measuring the production of marine phytoplankton, Bull. Fish. Rcs. Bd. Can. 122. 172 p.

TALLING, J. F. 1957. The phytoplankton population as a compound photosynthetic system. New Phytol. 56: 133-149.

THOMAS, W. II. 1966. Effects of temperature and illuminancc on cell division rates of three species of tropical oceanic phytoplankton. J. Phycol. 2: 17-22.

VERDUIN, J . 1964. Principles of primary productivity: Photosynthesis under completely natural conditions, p. 221-238. In D. F. Jackson [ed.], Algae and man. Plenum.

VOLLENWEIDER, R. A. 1965. Calculation models of photosynthesis-depth curves and some implications regarding day rate estimates in primary productivity measurements. Mem. 1st. Ital. Idrobiol. 18( suppl. ) : 425-457.

WILLIAMS, R. B. 1972. Nutrient levels and phytoplankton productivity in the estuary, p. 59- 89. In R. I-1. Chabreck [cd.], Coastal marsh and estuary management. La. State Univ.

Submitted: 29 May 1974 Accepted: 2 January 1975

the assumptions and rationales of a computer model of phytoplankton population dynamics

Documents