Sigmoid Relationships between Phosphorus, Algal Biomass, and
Algal Community Structure
Susan Watson and Edward McCauley
Ecology Division, Department of Biological Sciences, University of Calgary, Calgary, Alta. T2N 1N4, Canada
and John A. Downing
Department de sciences biologiques, University de Montreal, Montreal (Quebec) H3C 3}7, Canada
Watson, S E.. McCauley, and J. A. Downing. 1992. Sigmoid relationships between phosphorus, algal biomass
and algal community structure. Can. J. Fish. Aquat. Sci. 49: 2605-2610.
o.omass,
It has long been recognized that there is a positive relationship between total phytoplankton biomass and eutroph.cat.on. Recent independent studies demonstrated that algal biomass (chlorophyll) actually responds in a non
linear, s.gmoidal fash.on w.th increasing phosphorus levels among lakes. Chlorophyll has been considered (bv
some authors) as an inconsistent estimate of algal biomass. Using a wide range of published data we first dem
onstrate that the s.gmoidal nature of the phosphorus-biomass relationship is quite robust, anc1 noUimply gen
erated by a systematic variation in the relationship between algal chlorophyll to cell volume ratio and nutrient
levels We show that the s.gmoid relationship with total phosphorus persists whether algal biomass is meaured
by chlorophyll or b.ovolume We hypothesize that this nonlinearity actually reflects an underlymg systeS
variation m one or more of the components of total phytoplankton biomass. In this paper, we examine two
functional size groups and show that the large inedible fraction exhibits a strong, nonlinear response to increasing
nutrient levels, while the small ed.be algae do not vary systematically with phosphorus. We hypothesize that
this discontinuous shift in the ratio of edible to inedible phytoplankton should be accompanied by concomi an
shifts in the structure of the herbivore community.
On sait depuis longtemps qu'il existe un rapport positif entre la biomasse phytoplanctonique totale et I'eutroph.sat.on. Des etudes independantes r<§centes confirmed que la biomasse algale (chlorophylle) re"aeit en fait
selon une courbe non lin^aire sigmoVdale a ('augmentation du taux de phosphore dans les lacs. Certains auteurs
cons.derent la mesure de la chlorophylle comme etant une evaluation peu coherente de la biomasse aleale
D apres un large eventail de donnas publiees, nous demontrons d'abord que la nature sigmoidale de la relation
phosphore-b.omasse est tres marquee et n'est pas simplement le r&ultat d'une variation systematique dans le
rapport entre la chlorophylle algale et le volume cellulaire ainsi que la teneur en elements nutritifs Nous demon
trons que le rapport sigrnoTdal avec le phosphore total subsiste, que la biomasse algale ait ete mesuree d'apres
la chlorophylle ou le volume des bio-elements. Notre hypothese est la suivante : cette relation non lineaire reflete
en n§alite une variation systematique sous-jacente d'un ou de plusieurs elements de la biomasse phytonlancto
nique totale. Dans le present rapport, nous etudions deux groupes de taille fonctionnels, et montrons aue la
grande fraction non comestible presente une forte reaction non lineaire a I'augmentation des taux d'ei^ments
nutr.t.fs, tand.s que la petite fraction d'algues comestibles ne varie pas de facon systematique par rapport au
phosphore. Nous pensons que cette variation discontinue dans le rapport phytoplancton comestible - phyto-
plancton non comestible doit etre accompagnee par des variations concomittantes dans la structure de la commu-
naute herbivore.
Received January 30, 1992
Accepted June 25, 1992
(JB390)
The observation that phytoplankton biomass rises with
phosphorus concentrations in lake water has been a staple
component of limnological theory for three decades (e.g.
Sakamoto 1966; Dillon and Rigler 1974; Jones and Bachmann
1976; Schindler 1977, 1978; Nicholls and Dillon 1978; Smith
1982; Canfield 1983; McCauley et al. 1989). The theoretical
idea that algal growth is limited by phosphorus availability has
led to the belief that the relationship should be a power function
with constant exponent. When power functions of chlorophyll
to total phosphorus are fitted for different geographical areas
or for different suites of data, exponents are not always equiv
alent (e.g. Nicholls and Dillon 1978; Janus and Vollenweider
1981; Straskraba 1986; McCauley et al. 1989; Prairie et al.
1989). Recent studies have shown, however, that variation
Can. J. Fixh. Aquat. Sci., VoL 49, 1992
Recu le 30 Janvier 1992
Accept le 25 juin 1992
among lakes in chlorophyll a (Chi a) with total phosphorus (TP)
can be better described by a sigmoidal relationship than by sim
ple linear models (McCauley et al. 1989; Prairie et al. 1989)
This observation is significant in terms of lake management and
might account for discrepancies among the linear models
derived from geographically discreet of relatively small data
sets (e.g. Nicholls and Dillon 1978; Paloheimo and Zimmer
man 1983; Bierhuizen and Prepas 1985; Dillon et al. 1988).
Furthermore, it suggests that explanations for the phosphorus-
chlorophyll relationship may be incomplete.
Although the measurement of chlorophyll is one of the most
convenient and widely applied methods used to estimate algal
biomass, there has been some debate concerning its validity as
a measure of phytoplankton biomass (e.g. Tolstoy 1979; Can2605
field 1983; Lampou el al. 1982; White et al. 1988). In partic
ular, the relationship between Chi a and algal biovolume is
thought to be highly variable and generally unpredictable (e.g.
Saraceni et al. 1978; Dillon et al. 1988; Aleya and Amblard
1989). This has been attributed to species- or size-specific dif
ferences in cell chlorophyll content and/or to the influence of
environmental factors, such as light (e.g. SteeleandBaird 1965;
Ruggiu et al. 1979; Desortova 1981; White et al. 1988; Pridmore el al. 1989) and nutrient levels (e.g. Steele and Baird
1965; Ahlgren 1970; Nicholls and Dillon 1978; Saraceni et al.
1978; White et al. 1988) on the ratio of chlorophyll to cell
volume.
The existence of a sigmoid relationship of chlorophyll with
TP raises two important possibilities: (1) the relationship
between phytoplankton biomass and phosphorus is really lin
ear, but chlorophyll is an inconsistent measure of plant bio
mass, or (2) chlorophyll is a good measure of plant biomass
but our theoretical explanation for the form of the relationship
between plant biomass and phosphorus in lakes is overly sim
plistic. We therefore begin this paper by presenting evidence to
show that the curviltnearity in the Chi a - TP relationship is
not an artefact of the lack of consistency in the chlorophyll to
biovolume ratio of cells. We do this by comparing the relation
ship between algal biovolume and TP among lakes with pre
vious results on the Chi a - TP relationship.
The sigmoidal nature of the algal-TP relationship suggests
that there are several important features of the nutrient-algal
relationship which have yet to be explained. For example, why
does the rate of change of algal biomass per change in total
phosphorus increase at very low nutrient levels and then decline
at very high nutrient levels? Many hypotheses could be
advanced to explain both of these features. One explanation
might be that the nonlinearities can be accounted for by sys
tematic variation in the biomass of functional algal groups.
Another possibility is that systematic variation in average
growth or loss rates of algae with TP might yield nonlinear
algal-nutrient relationships (McCauley et al. 1989). Much
emphasis has been placed recently on how properties of planktonic food webs might influence algal biomass (e.g. Kerfool
and DcMott 1980; Carpenter et al. 1985, 1987; Kcrfoot et al.
1985; Dorazio et al. 1987; Kerfoot 1987; Vanni 1987;
McCauley et al. 1989; Vanni and Temte 1990; Elser and Gold
man 1991) either via direct or indirect interactions, and a fur
ther explanation for nonlinear plant biomass - phosphorus rela
tionships might be related to differential responses of algal
groups that are either susceptible or insusceptible to grazing by
planktonic herbivores. This study tests the hypothesis thai nonlinearities in the algal-phosphorus relationship are consistent
with changes in growth or loss rates of functional algal groups
mediated by shifts in the trophic structure of lakes of differing
Downing 1985; Leibold 1989). These two groups have been
labelled "edible" and "inedible", respectively. They not only
display different short-term responses to manipulation of her
bivores, but they appear to respond differently to variation
among lakes in TP (Watson and McCauley 1988).
Ecological theory also predicts very different responses for
[he two groups with enrichment. Classical predator-prey theory
(e.g. Rosensweig 1972; Leibold 1989; McCauley et al. 1989)
predicts that if the edible group is regulated by herbivores, then
enrichment should lead only to an increase in herbivore biomass
which subsequently suppresses edible algal density to preenriebment levels. Edible algal biomass should therefore be
invariant among lakes differing in enrichment. Other predic
tions are made by theory (Fretwell 1977; Oksanen et al. 1981)
that takes into account the potential effects of species at other
trophic levels (e.g. carnivorous zooplankton, planktivorous or
carnivorous fish). For example, Oksanenet al. (1981) predicted
that if productivity of algal prey increases with enrichment, then
prey biomass should vary as a step function with increasing lake
fertility (e.g. Leibold 1989, fig. 1). The discontinuities in the
predicted relationship are produced by the development of addi
tional trophic levels and their subsequent influence on the plantherbivore interaction.
In this paper, we examine differences in algal community
structure among lakes to test whether the sigmoidal pattern
observed in the biomass-TP relationship can be accounted for
by changes in the relative abundance of edible or inedible
phytoplankton.
fertility.
While taxonomic considerations are undeniably important
ton were defined as cells or colonies <35-5O |xm in maximum
linear dimension, while netplankton or inedible algae were
Data and Analyses
We tested our hypotheses using a wide range of published
data on algal biomass, chlorophyll concentrations, and phos
phorus concentrations. Mean epilimnetic summer values were
used wherever possible, since the inclusion of spring and fall
data could affect the relationship beiwecn some algal size frac
tions and TP (e.g. Watson and McCauley 1988). These would
also represent values which were close to "equilibrium" val
ues, and eliminate some of the variation around temporal or
short-term fluxes. However, some individual observations were
also included (16%), although where these were reported over
one or more growing seasons for a particular lake, the mean
(summer) values were calculated. Where necessary, values were
converted to micrograms per litre. For algal biomass, we used
only estimates of wet weight or biovolume that had been arrived
at using the Utermohi technique (Vollenweider 1969). Where
biovolume was given, it was converted into micrograms wet
weight per litre (Vollenweider 1969). A total of 67 different
sources of data were used from 362 different lakes over one or
more stations or years.1 Nannoplankton or edible phytoplank
(e.g. Lampert 1981; Infante and Abella 1985). the general cate
defined as cells or colonies >35-50 \L\n. The number of obser
gorization of the algal community into functional groups thai
vations
span taxonomic boundaries is supported by extensive measure
(Table I) because complete data for algal community structure
ments of zooplankton feeding rates (e.g. Bums 1968; Gliwicz
were not available from all studies.
used
to
test
each
hypothesis
varied
considerably
1977; Porter 1977; McCauley and Downing 1985; Hawkins and
Following the approach used by McCauley el al. (1989), the
Lanipert 1989) and field manipulations of herbivorous zoo-
shape of each of the curves was examined using two methods:
plankton (e.g. Porter 1972; Gliwicz 1975; Briand and McCauley
(3) multiple regression analysis (Draper and Smith 1981) with
1978; McCauley and Briand 1979; Leibold 1989; Vanni and
variable selection by backward elimination (Mocking 1976) and
Temte 1990). Nannoplanklon are typically consumed by her
bivorous zooplankton, and netphinkton are relatively insuscep
'A complete set oi tubular data is available at a nominal charge from
tible to grazing (e.g. Porter 1972; Gliwicz 1975; Briand and
[he Depository of Unpublished Data, National Research Council of
McCauley 1978; McCauley and Briand 1979; McCauiey and
Canada, Ottawa, Ont. K1A 0S2, Canada.
26G6
Can..!. Fish. Aquat. Sri., Vol. 49. 1992
(2) robust locally weighted sequential smoothing (LOWESS:
Cleveland 1979). Logarithmic transformations were used to sta
bilize residual variability. The multiple regression analysis was
used to examine whether relationships were linear or nonlinear
by assessing the pattern of residuals and the significance of
higher-order terms of the independent variable (i.e. TP).
LOWESS is a nonparametric smoothing technique that yields
an unbiased estimate of the form of the relationship between
two variables, which is not constrained by assumptions about
the form of the relationship (Cleveland 198S). It is an especially
powerful tool for studying the dependence of y on x when
'*.. .the signal is embedded in noise'' (Cleveland 1985). Because
LOWESS fitting is "model free", it has the flexibility to
describe many patterns including those with discontinuous
derivatives. LOWESS also includes a robust fitting procedure
that "guards against deviant points distorting the smooth
points" (Cleveland 1979). Because fitted trends are locally
weighted, so-called "outliers" have negligible effects on
LOWESS fits. All LOWESS analyses were performed with
delta = 0 (the smoothing function considers every single data
point), n - steps = 2 (number of iterations), and / = 0.5
(weighting factor). The values of «-steps and / were chosen
based on the recommendations of Cleveland (1985).
Results and Discussion
CD
E
o
m
"5
*-<
o
en
o
1.5
0.0
2 .5
mass and TP is shown to be robustly sigmoid when either Chi a
(3-1000 jig/L; Fig. 1A), both curves show a similar acceler
ation phase (Fig. IB). It is important to note that the nonlinear
relationship remains significant if the seeming "outliers" at the
extreme ends of the data range are removed and the data analCan. J. Fish. Aquat. Sci.. Vol. 49. 1992
1.5
2.0
2.5
3.0
1
1
1
1——i
1
T
—i—
3.5
5.0
B
2 .0
(01
lorphyl
-
4.5
-
4.0
-
3.5
-
3.0
CO
1 .5
1 .0
E
o
S
To
at
sz
o
o
or biovolume is used to measure total algal biomass. In fact,
the relationship between phytoplankton biovolume and TP is
more markedly sigmoidal than that between chlorophyll and TP
(Fig. IB). Over the range of TP where there are the most data
1.0
log10 Total Phosphorus
log1
Total algal biomass varies nonlinearly with TP. The para
metric analysis and LOWESS fit yield comparable results,
showing a sigmoidal relationship between TP and total biomass
similar to that exhibited by Chi a (Fig. 1; Table 1). There are,
however, slight quantitative differences in the curves derived
from parametric and model-free methods. These differences
probably arise from constraints inherent in the parametric tech
nique (i.e. the use of polynomial equations to fit the data).
Below, we first discuss the LOWESS results and then compare
them with the parametric models.
In very oligotrophic systems (i.e. TP < 5 |xg/L), increases
in TP correspond to very little change in total biomass
(Fig. 1A); however, once TP exceeds this level, the rate of
increase of algal biomass accelerates dramatically with increas
ing TP. Over the range of TP from ~6 to 60 jxg/L, algal bio
mass appears to increase at a relatively constant rate, and this
rate decreases markedly once TP values surpass 65-75 |xg/L.
The polynomial fit yields a qualitatively similar pattern, but the
level of TP at which the rate of change of algal biomass begins
to decrease in eutrophic lakes differs substantially from that
described by the LOWESS results (Fig. 1 A). According to the
parametric model, the rate of change in algal biomass begins
to decline at ~300-500 jxg TP/L. In oligotrophic lakes, the
results are virtually identical for the two techniques.
It appears, therefore, that the sigmoid relationship between
TP and phytoplankton biomass (Chi a) found by McCauley
et al. 1989) and Prairie et al. (1989) is not an artefact of those
authors' use of chlorophyll to estimate algal biomass. This is
demonstrated by the fact that the relationship between algal bio
0.5
0 .5
0.0
'
-0
5
i
0.0 • 0.5
i
i
t
i
i
1.0
1.5
2.0
2.5
3.0
3 .5
4.0
2.5
4. 5
login Total Phosphorus
Fig. I. Relationship between total phosphorus and total algal biomass
among lakes. (A) Algal biomass estimated from algal biovolume,
showing parametric (dotted curve) and LOWESS (solid curve) fits;
(B) LOWESS fits for algal biomass estimated from biovolume (solid
curve) and Chi a (dotted curve) (McCauley et al. 1989) versus total
phosphorus. All variables in |xg/L.
ysis is repeated. In addition, variation in the Chi a to biovolume
ratio does not account for observed nonlinearities in the rela
tionship between total algal abundance and TP. Inspection of
the relationship between the Chi a to biomass ratio and TP shows
that this ratio varies in a nonlinear but also nonsystematic fash
ion with phosphorus (Fig. 2). In fact, the general trend followed
by this curve is actually opposite to that which would be
expected if the relationship between TP and algal biomass were
primarily influenced by this ratio. Neither this, therefore, nor
the large amount of residual variation about the curve could
account for the sigmoidal nature of the phosphorus-biomass
curves. This nonsystematic variation of the Chi a to biovolume
ratio with TP may explain some of the discrepancies among
studies (e.g. Tolstoy 1979; Dillon et al. 1988) attempting to
find a consistent pattern in the relationship using smaller datasets covering a limited range of nutrient levels.
The two algal size categories, designated as edible and in
edible algae, respond very differently to variation in TP. Both
the parametric and LOWESS analyses show that over the range
of TP covered by available data, inedible algal biomass
increases nonlinearly with TP by more than 40-fold (Fig. 3;
Table 1). Edible algae, on the other hand, do not vary system
atically with TP. It appears that the overall shape of the curve
of total algal biomass is primarily influenced by changes in
2607
Table I. Partial regression coefficients for the effect of total phosphorus (TP) on log ,„ total, edible, and inedible algal biomass (estimated from
biovolume), found using least squares multiple regression analysis. The number of observations (/r). the coefficient of determination (r), and
the overall F-value are also given, together with the general shape of the corresponding curve obtained using locally weighted robust sequential
smoothing (LOWESS; Cleveland 1979) and the range of TP covered by each dataset. Previous results from a similar analysis of the relationship
between TP and total biomass estimated as chlorophyll a (Chi a) (McCauley et al. 1989) are included for comparison. TP, algal biomass
(biovolume), and Chi a measured in jig/L.
Parametric model
Variable
dog.o)
Total biomass
Edible biomass
Inedible biomass
Total Chi a
LTP
(LTP)2
(LTP)J
Intercept
-0.92
-1.44
NS
-0.83
1.24
1.82
1.33
0.34
-0.26
-0.54
-0.46
-0.12
2.82
2.92
1.85
0.03
«...
451
101
101
875
LOWESS
Prob > F
227
9
39
431
0.60
0.22
0.44
0.71
0.0001
0.0001
0.0001
0.0001
model"
TP range
*S
1-1300
1-200
1-200
1-14093
S
*S
(McCauley et al. 1979)
*S = sigmoidal, *S = strongly sigmoidal.
0.025
0.020 )
v>
v>
CD
E
|
0-012
■>
"§■ 0.008
o
s:
U
0.004
0.000
0.5
1.0
1.5
2.0
2.5
3.0
0.5
3.5
1.0
1.5
2.0
2.5
2.0
2.5
log10 Total Phosphorus
log10 Total Phosphorus
Fig. 2. LOWESS fit of the relationship between total phosphorus
(jig/L) and Chi a to biomass ratio.
large algae that cannot be ingested by zooplankton rather than
by edible phytoplankton.
Under very oligotrophic conditions (i.e. <5 |xg/L), both
inedible and edible algal biomasses are nearly constant or even
decline slightly with increasing TP (Fig. 4). From approxi
mately 5 to 30 fig TP/L, edible algal biomass appears to
increase slightly and then levels off, while inedible phytoplank
ton biomass increases rapidly. There is considerable variation
in edible or inedible biomass at a given level of TP, and assess
ing "significance" of patterns is somewhat problematic. How
ever, the LOWESS analysis shows a discontinuous change over
this TP range for both edible and inedible algal biomasses which
are measured independently, and the data density is high in the
region where most change occurs.
The markedly different responses to increased nutrient levels
exhibited by the two algal size fractions seem to be related to
their susceptibility to grazing by herbivores. This indicates that
the nonlinearity of the total biomass curve may be attributable
to factor(s) other than differential responses to nutrient levels.
The discontinuities and shifts in community structure are per
haps best illustrated by Fig. 5. Below levels of —8-10 (xg
TP/L, total biomass is dominated by edible algae. Above this
range, there is a transition zone where both the edible and ine
dible biomasses are similar in relative abundance, but the ine
dible fraction exhibits a much more pronounced rate of change
(Fig. 4). With increasing concentrations of TP (i.e. TP >
—30 (xg/L) the inedible algae rapidly become more dominant,
2608
E
o
3
p
0.0
0.5
1.0
1.5
loglo Total Phosphorus
Fig. 3. Relationship between total phosphorus and algal size fractions
(biomass; estimated from biovolume), showing parametric (dotted
curve) and LOWESS (solid curve) fits. (A) Edible algal biomass;
(B) inedible algal biomass. All units in |Ag/L.
until at TP > —50 |xg/L the phytoplankton biomass consists
almost entirely of this larger size fraction (Fig. 5).
If the discontinuous change in edible-inedible community
structure results from mechanisms proposed by Oksanen et al.
(1982), then we should see concomitant changes in the rela
tionship between average herbivore biomass and TP. Specifi
cally, herbivore biomass should increase with TP among lakes
with TP < 5 ng/L. and in the region of 5 < TP < 30 |xg/L,
there should be a discontinuous change in herbivore biomass
among lakes. Alternatively, the rate of change of zooplankton
Can. J. Fish. Aquat. Set., Vol. 49. 1992
0.5
1.0
1.5
2.0
2.5
3.0
3.5
log1o Total Phosphorus
to
(0
a
E
o
3
75
biomass could remain the same, and systematic changes in zooplankton community structure could occur. The discontinuous
change in edible algae would occur if the herbivorous zooplankton shifted to species with lower attack rates over this range of
TP, thereby yielding a higher "equilibrium" concentration of
prey (McCauley et al. 1989). Unfortunately, existing studies of
among-lake variation in zooplankton biomass (McCauley and
Kalff 1981; Hanson and Peters 1984; Pace 1984) have not
'looked for patterns at this scale of comparison, nor have obser
vations been analysed using model-independent techniques such
as LOWESS. We are currently investigating these possibilities.
We recognize that the division of the phytoplankton com
munity into only two functional size groups may limit the
potential interpretation of the phosphorus-biomass curve.
Analyses based on size and those based on taxonomic divisions
may not be mutually exclusive: some of the major taxonomic
groups comprise primarily smaller individuals (e.g. Chrysophyta, Cryptophyceae) and others larger algae (e.g. Cyanophyta). Furthermore, as we noted previously, the response of
the grazer community to certain dominant taxonomic algal
groups could in tum influence the overall shape of the phos
phorus-biomass curve. In a subsequent paper, we examine the
relationships between major taxonomic groups and phosphorus
enrichment and compare them with those of the major size
groups that we have described here.
Acknowledgements
o
We thank Sophie Lalonde and Chris Briggs for technical help. This
research was supported by NSERC operating grants to E.M. and J. A.D.
log10 Total Phosphorus
Fig. 4. Relationship between total phosphorus and total (solid curve),
edible (broken curve), and inedible (dotted curve) algal biomass. (A)
Parametric fits; (B) LOWESS fits. All units in |Ag/L. Note that the
calculated intercept for the edible fraction is higher than that of the
total biomass simply because the lakes represented in each dataset
differ among fractions.
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