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Differential phytoplankton sinking- and growth-rates: An eigenvalue analysis
EcologicaIModelling, 9 (1980) 233--245 233 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
DIFFERENTIAL PHYTOPLANKTON SINKING- AND GROWTH-RATES: AN EIGENVALUE ANALYSIS JERALD L. SCHNOOR and DOMINIC M. Di TORO * Environmental Engineering, The University of Iowa, Iowa City, IA 52242 (U.S.A.) • Environmental Engineering and Science, Manhattan College, Bronx, N Y 10471 (U.S.A.)
(Accepted for publication in revised form 29 August 1979)
ABSTRACT Schnoor, J.L. and Di Toro, D.M., 1980. Differential phytoplankton sinking- and growthrates: an eigenvalue analysis. Ecol. Modelling, 9: 233--245. An eigenvalue analysis of the vertical phytoplankton biomass equation is applied to calculate the differential sinking- and loss-rates of phytoplankton for different taxonomic groups in Lake Lyndon B. Johnson (LBJ) (Texas) and in Lake Erie. The analysis includes factors determining the phytoplankton composition, including losses due to turbulent mixing and to sinking, and a death-term to account for endogenous decay and predation. Gross growth-rates were obtained from community and individual 14C production data (autoradiography) and from biomass measurements. Green algae produced the largest gross growth-rates, 1--1.75 d - 1 , and the largest calculated sinking-rates, 1--3 m d - 1 . The lowest gross growth-rates (< 0.5 d - 1 ) and sinking-rates (< 0.1 m d- 1 ) were determined for blue-green algae and phytoflagellates, while diatoms were intermediate, with growthrates generally in the range 0.25--0.75 d - 1 and calculated sinking-rates of 0.1--1.0 m d--1.
INTRODUCTION P h y t o p l a n k t o n u n d e r g o losses in lakes b y sinking a n d m i x i n g o u t o f t h e e u p h o r i c z o n e , a n d b y d e a t h or e n d o g e n o u s d e c a y . In o r d e r to i n v a d e or c o e x i s t , t h e y m u s t h a v e gross r a t e s o f g r o w t h e q u a l t o or g r e a t e r t h a n t h e i r loss-rates. T h e f a c t o r s i n f l u e n c i n g p h y t o p l a n k t o n g r o w t h a n d c o m p o s i t i o n are b o t h biological a n d p h y s i c a l . G r o w t h - r a t e s are i n f l u e n c e d b y environm e n t a l c o n d i t i o n s ( n u t r i e n t s , light, a n d t e m p e r a t u r e ) , b u t t h e y are also f u n c t i o n s o f t h e p h y t o p l a n k t o n t a x a , genus, a n d species, as are t h e sinking-rate a n d d e a t h - r a t e . Physical f a c t o r s i n c l u d e vertical m i x i n g or e d d y d i f f u s i v i t y , a n d t h e d e p t h o f t h e e u p h o r i c z o n e . T h e results o f all t h e s e variables d e t e r m i n e s t h e o b s e r v e d n e t g r o w t h - r a t e o f t h e species. Munk and Riley (1952) have addressed the importance of phytoplankton sinking-rates in a q u a n t i t a t i v e m a n n e r b y c o n s i d e r i n g n u t r i e n t t r a n s p o r t . Bella ( 1 9 7 0 ) d e m o n s t r a t e d t h e e f f e c t s o f sinking as c o m p a r e d t o m i x i n g in l a k e p h y t o p l a n k t o n c o m m u n i t i e s , a n d J a s s b y a n d G o l d m a n ( 1 9 7 4 ) cal-
234 culated loss-rates (sinking plus mixing plus death) from depth-averaged primary production and biomass measurements. Smayda (1974) and Titman and Kilham (1976) have measured the sinking-rates of freshwater phytoplankton directly. In this paper, an eigenvalue analysis of the vertical p h y t o p l a n k t o n biomass equation is applied to calculate the allowable sinking- and loss-rates of taxonomic groups of p h y t o p l a n k t o n in Lake L y n d o n B. Johnson (LBJ) (Texas), and in Lake Erie. The analysis includes the factors determining p h y t o p l a n k t o n composition, including losses due to turbulent mixing and to sinking, and a death-term to include predation and endogenous decay. This work places the factors determining p h y t o p l a n k t o n composition in an analytical and quantitative framework. Lake L y n d o n B. Johnson (LBJ) is the third in a series of seven reservoirs on the Colorado River in the Highland Lakes Chain of Central Texas. It has a surface area of 25.8 km 2 , a volume of 0.171 km 3 , a mean depth of 6.63 m (with a m a x i m u m depth of 26 m), and a mean annual hydraulic-detention time of approximately 80 days. Lake LBJ receives waters from the unimp o u n d e d Llano River, and the hypolimnetic" release from Inks Dam on the Colorado River. It does n o t receive any major waste or agricultural discharges, although the nutrient loadings are sufficient to classify the reservoir as eutrophic, according to the m e t h o d of Vollenweider (1968) or Dillon and Rigler (1974). In 1972 and 1973, Schnoor and Fruh (1976) noted t h a t the total phosphorus loadings were 1.6 and 2.6 g - 2 y r - 1 , and the total nitrogen loadings were 14 and 27 g - 2 y r - 1 , respectively. Anoxic conditions are normally present in the hypolimnion of Lake LBJ from June through September. Lake Erie is the shallowest of the five Laurentian Great Lakes. It has a mean depth of 17.7 m and a surface area of 25,700 km 2 , which puts it twelfth in size among the world's lakes. Lake Erie can be divided into three basins: Western, Central, and Eastern. The Western basin is the shallowest with a mean depth of 7.3 m, and it receives the major inflow from Lake Huron, via the Detroit and St. Clair Rivers. It is the most productive and eutrophic of the three basins. Anoxic conditions usually occur near the bott o m of Lake Erie in the Central basin during August or September.
THEORETICAL DEVELOPMENT One can locate regions of parameter space which allow particular phytoplankton species to invade and grow successfully. For given conditions in a lake or reservoir, it is interplay between the gross growth-rate G, the sinkingrate w, and the death-rate D, which determines the net growth-rate p and the ability of a particular alga to exist in the c o m m u n i t y . An analysis of the mass-conservation equation for p h y t o p l a n k t o n biomass P is presented below: the analysis incorporates the physical and biological
235 variables previously discussed. For a one-dimensional, vertical regime, the phytoplankton equation is (Riley et al., 1949): aP/at --a/az(E(aP/az))
(1)
+ (a/az)(wP) = (V -D)P
where E is the vertical eddy-diffusivity, z the depth, t is time, w the average sinking velocity, G the gross growth-rate, and D is a first-order loss-rate for losses due to death, endogenous decay, predation, and washout. It has been shown by Di Toro (1974) that when the biomass increases, the solution to Eq. (1) asymptotically approaches the form P ( z , t ) = P ( z ) exp(pt)
(2)
where p is the net observable growth-rate of the population and is independent of depth. Exponential growth-rates, independent of depth, have been observed in a number of lakes (Di Toro, 1974). Therefore, Eq. (1) may be replaced by Eq. (3) which, for E and w constant with depth, becomes - - E ( d 2 P / d z 2) + w ( d P / d z ) = IV(z) - - (D + p ) ] P
(3)
A two-layer approximation is used (Riley et al., 1949): the gross growthrate is vertically-averaged in the euphoric zone, and assumed to be zero below: thus G(z) =G
for0~
and G(z) = 0
for L ~< z (layer 2)
One may apply the boundary conditions: --E(dP1/dz) + wP1 = 0
at z = 0
P1 = P2 a n d - - E ( d P 1 / d z ) + w P 1 = --E.(dP2/dz ) + w P 2
(4) at z = L
(5)
and P2-*0
asz-*¢¢
(6)
where the subscripts 1 and 2 refer to the top layer (the euphotic zone) and the bottom layer respectively. The first boundary condition specifies that phytoplankton mass cannot pass through the surface of the lake. The second states that the concentration and mass-flux at the euphotic-zone depth L must be equal in both layers; and the last boundary condition requires that the phytoplankton concentration approach zero as the depth approaches infinity. Equation (3) may be solved in each of the two layers. In the layer representing the euphotic zone, the general solution is: Pl(z) = B x exp(wz/2E)
cos(~z -- X)
where B x and k are arbitrary constants, and a = ( w / 2 E ) x/[4(V -- (D + p ) ) E / w z] -- 1
(7)
236 The b o t t o m layer has the general solution: P2(z) = B1 exp(g2z)
iS)
where g2 = ( w / 2 E ) [ 1 - - x / 1 + 4(D + p ) E / w 2 ] .
Equations (7) and (8) can be combined, together with the boundary conditions, to yield an eigenvalue equation for p, the net growth-rate. In terms of dimensionless numbers, the final eigenvalue equation is given by w L / E = [tan-l(1/(~) + t a n - l ( x / ~
+ 4n)/~)]/(fl/2)
(9)
where w L / E is the Peclet number, L is the mean d e p t h of the euphotic zone, (3 = ~/4n(~ -- 1) -- 1 (dimensionless), n = (D + p ) E / w 2 (dimensionless), and = G / ( D + p) (dimensionless). The Peclet number gives a ratio between the rates of advective and diffusive transport (i.e., the greater the Peclet number, the more dominant is sinking as a transport mechanism). Equation (9) can be used to quantify the required growth-rate and the allowable sinking- and death-rates for an individual to grow. If one knows the environmental parameters E and L and the gross growth-rate G, it is possible to find a sinking-rate and/or death-rate that allows net growth of the population. Strictly speaking, this analysis is an approximation of the actual conditions in a lake, since the solution is developed for temporally constant coefficients and describes the asymptotic behavior which is approached after times of the order of 1 / p . However, it provides a quantitat!ve framework within which a very complicated problem can be analyzed directly. The average gross growth-rate in the euphoric zone, G, of a c o m m u n i t y or a species can be measured as the ratio of its production to its biomass (activity coefficient), while the net growth-rate p is simply the observed logarithmic rate of biomass increase from time tl to time t2, that is, p = In [P(z, t2)/ P(z, t l ) ] / ( t ~ - - tl). The difference between these rates, G - - p , is a measure of all the losses in the lake due to sinking, death, washout and predation, and mixing o u t of the euphotic zone. RESULTS AND DISCUSSION Near-surface biomass and productivity data for Lake LBJ is presented in Table I. An autoradiographic technique of Davenport (1975) and Maguire and Neill (1971) allowed delineation of p h y t o p l a n k t o n b y taxa, species and individual. Three taxa axe given, together with carbon-14 community-production measurements. Wet biomass is reported by Davenport (1975) by multiplying cell counts b y the mean biovolume per cell, and assuming a density of 1 g cm -~. The gross growth-rate G is estimated assuming a ten-hour photoperiod, a 10% carbon to wet-weight ratio (Vollenweider, 1969), and a 50% ratio for the average euphotic-zone to near-surface productivity. The latter assumption is supported b y Davenport's (1975) depth-integrated field
237 TABLE 1 Lake L y n d o n B. J o h n s o n (LBJ) * Sampling date
9-23-73
10-9-73
11-17-73
12-8-73
1-11-74
2-9-74
Taxa **
Wet b i o m a s s (P, m g 1- z )
Production (mg C m - 3 h - 1 )
Production/ biomass *** (G, d - 1 )
G D BG
0.181 0.187 0.690
1.76 1.59 3.52
0.485 0.425 0.255
Total
1.225
6.87
0.28
G D BG
0.163 0.112 0.130
3.36 0.96 4.24
1.02 0.43 1.635
Net growth-rate (]2, d - 1 )
--0.0066 --0.0320 --0.1045
Total
0.405
8.55
1.05
--0.0692
G D BG
0.027 0.0077 0.0000
0.980 ---
1.82
--0.0463 --0.0686
Total
0.038
0.980
1.29
--0.0607
G D BG
0.0654 0.0105 0.0000
2.43 ---
1.85
0.0425 0.0148 0.0000
1.60
0.0394
Total
0.0759
2.43
G D
0.102 0.0534
--
BG
0.0000
--
Total
0.182
3.06
0.842
0.0257
G
0.0856
2.21
D BG
0.0881 0.0000
5.77 0.00
1.29 0.327
--0.00608 0.0173 0.0000 --0.00015
0.0131 0.0478 0.0000
Total
0.174
2.79
0.803
3-9-74
G D BG
0.141 0.254 0.000
4.59 3.64 0.00
1.62 0.72
Total
0.395
8.23
1.04
0.0293
4-6-74
G D BG
0.0973 O.0606 0.0000
3.42 1.36 0.00
1.76 1.12
--0.0133 --0.0511 0.0000
Total
0.171
4.84
1.41
--0.0299
G D BG
0.160 0.0158 0.0000
5.00 1.21 0.00
1.56 3.83
0.0713 --0.192 0.0000
Total
0.204
6.21
1.52
0.0256
4-13-74
0.0179 0.0378 0.0000
(continued)
238 TABLE I (continued) Sampling date
4-27-74
5-4-74
5-27-74
6-19-74
6-28-74
7-12-74
7-31-74
8-9-74
Taxa **
Wet b i o m a s s (P, m g 1 - 1 )
G D BG
0.0925 0.0178 0.0000
Total
0.238
G D BG
0.143 0.0185 0.0000
Total
0.169
G D BG
0.599 0.0511 0.0386
Total
0.668
G D BG
Production (mgCm--3 h-1)
1.86 0.228 0.000 10.5
Production/ biomass*** (G, d - 1 )
Net growth-rate (U, d - - l )
1.01 0.64 --
--0.0393 0.0085 0.0000
2.21
0.0109
4.93 0.268 0.000
1.72 1.22 --
0.0621 b0.0055 0.0000
5.20
1.54
--0.0488
18.4 0.148 0.0000
1.54 1.45 --
0.0623 0.0442 --
18.6
1.39
0.0597
0.640 0.0138 0.0163
20.6 0.0 0.266
1.60 -0.81
0.0047 --0.0935 --0.0616
Total
0.723
20.9
1.45
0.0056
G D BG
0.390 0.352 0.0442
1.78 0.27 --
--0.0124 --0.0116 -0.0206
Total
0.502
10.3
1.03
-0.0138
G D BG
0.224 0.0406 0.0204
10.6 0.906 0.0
2.36 1.12 --
-0.0398 0.0102 -0.0552
Total
0.344
11.5
G D BG
0.115 0.0515 0.509
Total
0.715
G D BG
0.255 0.0290 0.165
11.7 0.221 0.213
2.30 0.38 0.065
0.0883 -0.0638 -0.1249
Total
0.531
12.2
1.15
-0.0330
9.96 0.192 0.00
1.67
--0.0270
4.35 0.00 4.43
1.88 -0.43
-0.0349 0.0125 0.1693
8.78
0.61
0.0385
* Biomass a n d near-surface p r o d u c t i o n data are f r o m D a v e n p o r t (1975). ** G = green algae, D = d i a t o m s , BG = blue-green algae. * * * T h e ratio o f p r o d u c t i o n t o b i o m a s s (or gross g r o w t h - r a t e G) is e s t i m a t e d a s s u m i n g a t e n - h o u r p h o t o p e r i o d , a 10% c a r b o n t o w e t - w e i g h t ratio, a n d a 50% ratio for average e u p h o t i e - z o n e t o near-surface p r o d u c t i v i t y .
239 data from 10 June, 1974. Net growth-rates are the observed logarithmic rates of biomass increase between t w o sampling dates. Table II includes the integral-production and biomass data for Lake Erie's Western, Central and Eastern basins in 1970. The production data are from shipboard incubator measurements by Glooschenko (1974). These data approximate the integral production in the euphotic zone (15 m) of Lake Erie. Wet-biomass data are from Munawar and Munawar (1976), and represent depth-average values from samples at 1 and 5 m. The ratio of production to biomass, or gross growth-rate G, is time-averaged between the two consecutive dates used in the calculation of p. A ten-hour photoperiod and a 10% carbon to wet-weight ratio were assumed. Note that both the mean annual biomass and the productivity decrease from west to east in Lake Erie, apparently due to the large nutrient loadings of the shallow Western basin. In general, it can be shown from Tables I and II that c o m m u n i t y production is somewhat lower in Lake LBJ than in Lake Erie; however, the standing crop of biomass is so much smaller in Lake LBJ that the ratio of production to biomass, or gross growth-rate G, is significantly larger there than in Lake Erie. This fact indicates that the p h y t o p l a n k t o n c o m m u n i t y of Lake LBJ sustains greater losses (due to mortality, grazing aud/or sinking) than that of Lake Erie. It must be noted that the net growth-rates in Tables I and II are n o t constant over consecutive sampling periods (periods of the order of 1 p), so that the asymptotic solution presented by Eq. (9) is truly a coarse approximation. More frequent sampling observations might alleviate this problem. The ranges of the net versus the gross average c o m m u n i t y growth-rates for Lake Erie and Lake LBJ are presented in Fig. 1. Note that even though the algae have very large growth potential (G = 0.2--1.8 d - l ) , the observed net growth-rate p of the c o m m u n i t y is always between +0.08 d - 1 over the sampling intervals (bi-weekly to monthly). Therefore, relatively small differentials in loss-rates and/or growth-rates (of the same order as p) between species of algae can determine whether the species will grow or not. For a particular lake under steady conditions, the euphotic-zone depth and vertical eddy-diffusivity are relatively constant. They may be estimated using light and temperature measurements. For Lake LBJ, the approximate depth of the euphotic zone (5% of light) and the approximate eddy diffusivity are 4 m and 1.6 m 2 d -1 (Schnoor and Fruh, 1976), while in Lake Erie, the values are 15 m and I m 2 d -1, respectively (Di Toro and Connolly, 1978). Figure 2a gives directly the interrelationships between gross growth-rate, death-rate (including grazing, endogenous decay, and washout), and sinkingrate. It was constructed from trial-and-error solutions to Eq. (9) specifically for Lake LBJ. Every lake would have a similar b u t unique plot, constructed from knowledge of the vertical eddy-diffusivity E, and the euphotic zone depth L. A distinct trade-off occurs between the t w o loss-rates due to sinking and death. A n y algae capable of sustaining a gross average growth-rate greater
240 T A B L E II Lake Erie * Approximate Julian day 1970
Wet b i o m a s s , (P, m g 1 - 1 )
Average p r o d u c t i o n (mg C m -3 h -1 )
Production/ b i o m a s s ** (G, d - 1 )
Net growth-rate (tl, d - 1 )
13 7 0.8 4.7 4.3 8.0 4.3 5.4 2.6 1.7
4.92 28.6 15.1 38.2 23.7 48.0 32.6 28.9 17.9 4.9
0.223 1.15 1.35 0.682 0.576 0.680 0.648 0.612 0.488
--0.0200 --0.0868 0.0571 --0.00356 0.0214 --0.0222 0.00813 --0.0203 --0.0152
5.2
24.3
0.71
99 130 155 186 211 240 268 296 332
1.6 2.4 1.0 4.4 3.7 5.0 5.7 2.3 2.0
6.15 9.54 5.85 9.85 11.38 17.54 21.54 20.92 12.31
0.391 0.491 0.405 0.266 0.330 0.365 0.644 0.763
Mean
3.1
12.8
0.46
99 130 155 186 211 240 268 296 332 350
1.5 3.0 4.0 1.0 3.8 1.7 4.1 1.8 1.0 1.1
3.38 7.08 7.08 5.54 14.46 9.54 12.31 9.85 6.15 8.00
0.231 0.207 0.366 0.468 0.471 0.431 0.424 0.581 0.671
Mean
2.3
8.34
0.43
W e s t e r n basin 99 130 155 186 211 240 268 296 332 350 Mean Central basin 0.031 --0.0350 0.0478 -0.00693 0.0104 0.00468 --0.0324 --0.00388
E a s t e r n basin 0.0224 0.0115 -0.0447 0.0534 --0.0277 0.0314 -0.0294 -0.0163 0.00340
* B i o m a s s d a t a are f r o m M u n a w a r a n d M u n a w a r ( 1 9 7 6 ) , a n d s h i p b o a r d average-production data from Glooschenko (1974). ** T h e r a t i o o f p r o d u c t i o n t o b i o m a s s (or gross g r o w t h - r a t e G ) is e s t i m a t e d a s s u m i n g a t e n - h o u r p h o t o p e r i o d a n d a 10% c a r b o n t o w e t - w e i g h t ratio.
241 0 L A K E LBJ A ERIE WESTERN B A S I N
0.10
•
ERIE CENTRAL BASIN
D ERIE EASTERN B A S I N
0.08 T .~ 0.06 0.04
0 D
0.02
=o
0
0
A
= ~ , ,~ 0:5 ' ~
0 o
o
o
,
O
o,
1~
1.5
2.0
-0.02 I~m •
z -0.04
0
0
O
0
--0,06
0 0
--0.08 A
--0.10 ~ G
ROWTH RATE {G), day- 1
a=
Fig. 1. Observed net growth-rate vs. average gross growth-rate (ratio of production to biomass) for Lake LBJ, 1973--74, and Lake Erie, 1970.
than the contour values shown in Fig. 2a can maintain growth. For Lake LBJ, diatoms and blue-green algae demonstrated gross growth-rates generally between 0.25 and 0.75 d - 1 , while green algae showed significantly higher values at 1.25--2.4 d -1. Therefore it is possible to conclude that on the average, green algae sustain greater loss-rates due to sinking and death than blue-green algae and diatoms in Lake LBJ. During the active-growth phase, the death-rate of a phyt0plankton species is expected to be relatively small and constant. Although the literature is somewhat sparse, typical phytoplankton death-rates seem to be of the order of 0.05--0.25 d - 1 . Riley, Stommel and Bumpus (1949) report a range of 0 . 0 5 - 0 . 2 0 d - 1 for a population isolated from a light source at 20°C. Sorokin (1959) gives endogenous respiration-rates of a pure strain of Chlorella as 2.0 mm 3 02 mm - s cells hr - ~ at 25°C, which is approximately equivalent to 0.24 d - 1 . Jassby and Goldman (1974) calculated an absolute maximum deathrate due to predation as 0.20 d - ~ in Castle Lake. A case is presented in Fig. 2b where a small and constant death-rate of 0.05_ d - 1 is assumed. A most interesting feature of Fig. 2b is that the required G for continued growth becomes a narrow-banded function of w, the sinkingrate. To a great extent, differential P/B ratios imply differential sinking-rates and vice-versa, provided that death-rates are as small as 0.05--0.25 d - ~ . If the concentration_ of biomass is increasing, and one knows the ratio of production to biomass (G) for that plankton, direct estimation of w is possible. Estimates of planktonic sinking-rates are presented in Fig. 3a for Lake LBJ. Three taxa are shown. Only data where the net growth-rate is positive are presented, as required for an eigenvalue analysis. Figure 3a indicates that
242
~
A
B 4
~ 5
E
E
< 3 ~: 2 _z 2
g
v z
z
o
0
0.2 0.4 0.6 0.8 1.0 1.2 DEATH RATE (D), day-1
1
1.0 2.0 3,0 4.0 5.0 GROWTH RATE (G), day 1
Fig. 2. ( A ) L o s s - r a t e s d u e t o s i n k i n g a n d d e a t h , as a f u n c t i o n o f t h e a v e r a g e g r o w t h - r a t e f o r L a k e L B J ; p = 0. ( B ) S i n k i n g - r a t e vs. average g r o s s g r o w t h - r a t e f o r n e t g r o w t h - r a t e c o n tours; Lake LBJ, assuming D = 0.05 d -1 .
diatoms and blue-green algae settle more slowly than do the green algae prevalent in Lake LBJ. When the death-rate is low and constant, D = 0.05 d -1, the range of sinking-rates of diatoms and blue-green algae is indicated to be 0.5--2.5 m d -1 and for green algae, 2.5--3 m d -1. If the death-rate were 0.25 d -1, the sinking-rate of diatoms and blue-green algae would be 0.1--2 m d -1, while t h a t of green algae would be 2--2.75 m d -1. Of course, the sinking-rate (and net growth-rate) of any individual plankton or species may vary over a considerably broader range. The work of Davenport (1975) shows that species net growth-rates vary from +0.6 to --0.3 d -1, while comm u n i t y measurements are considerably less variable, ranging between +0.07 d -1 in Lake LBJ. The sinking-rates in Fig. 3 are not normalized to a reference temperature. Cold-water diatoms would actually sink faster in mid-summer (10--25 ° C), due to the effect of temperature on viscosity. This is one reason why diatoms may be f o u n d most frequently in winter and spring. For example, Fig. 3a shows t h a t a diatom with G = 0.5 d -1 and growing at p = 0.03 d -1 would completely stop growing (p = 0) if its sinking-rate increased from 1.1 to 1.25 m d -1. This increase in sinking-rate corresponds to an increase in temperature of just 5°C, according to Stokes Law. Titman and Kilham (1976) report a range of sinking-rates for four pureculture freshwater p h y t o p l a n k t o n , from 0.08 to 1.87 m d -~. They f o u n d the sinking-rate to be a marked function of the nutrient status of the plankton, which would allow early-spring species to sink at their lowest possible rate. Smayda (1974) reports a range of sinking-rates for two freshwater diatoms of 0.26--0.76 m d -1. Ultimately it is the interplay between differential sinking-rates and differential death-rates t h a t determines p h y t o p l a n k t o n composition. Therefore, calculations were made assuming constant sinking-rates (0.5 and 1.5 m d -1) to assess the effects of differential death-rates in Lake LBJ (Fig. 3b). High
243
Sinkingrate,
! I
w = 3.25 m/day
|
0.06 0.04 0.02
il, ,.]°r,°:q
O 0
0.5
/.J
,t
I, °
,
1.0
1.5
2.0
,
2.5
GROWTH RATE (G), day- 1
B
i
b-
I °°I
0.06
="
JI
o~ 0.5
Death rate,
o=, 5~day
1 Y:
il ....... o
0.04
=o
o
GREENS
o ELffE-GREEN$
i I
0.O2
o
• •
,I
z 0
II , ,-II, 0.5 ,.0 ,.5 ,.0 GROWTH RATE (G), day-1
0
Sinkingrete~/
0.10 ~- C
0.o~11/I O.
0.04
O02
0
MIXED FLAGELLATE D O M I N A 7lED
o.
~
w'''°m
, ,.5
i
11/:/ I , ll" "
I".l',;
i
12
• I
eA I I ,1, 0.25 0.50 0.75 1.00 1.25 GROWTH RATE (G), day-1
Fig. 3. (A) Observed net growth-rate vs. average gross growth-rate with constant sinkingrate contours, L a k e LBJ, 1973--74. Solid line, assuming D = 0.05 d--l; dashed line, assuming D = 0.25 d - 1 . (B) Observed net growth-rate vs. average gross growth-rate with constant sinking-rate contours, L a k e LBJ, 1973--74. Solid line, assuming w = 0.5 m d - I ; dashed line, assuming w = 1.5 m d -I. (C) Observed net growth-rate vs. average gross growth-rate with constant sinking-rate contours, Lake Erie, 1970. Solid line, assuming D = 0.085 d--l; dashed line, assuming D = 0.25 d -1.
values of the death-rate are required to keep the net growth-rate in the observed range. It would seem that differential sinking-rates are the primary mechanism by which differential growth-rates are compensated in Lake LBJ. The calculated sinldng-rates (Fig. 3a) are in near agreement with literature values, while no death-rates as high as 1--1.5 d -z (Fig. 3b) have been reported. Since the net growth-rates (p) observed in nature are so low, it is still true that a slight increase in the death-rate will stop phytoplankton from growing. Both w and D are important in determining species composition, but it is the variability in w which primarily balances the variability in G of the assemblage in Lake LBJ. Autoradiographic analysis has not been performed for Lake Erie, but there were times of the year in 1970 when flagellates or diatoms dominated the system, 80% or more in proportion. For these cases, community produc-
244 tivity and growth may be taken as an approximation of taxanomic productivity and growth. A plot of c o m p u t e d sinking-rates for values of p and G in Lake Erie is given in Fig. 3c. There are n o t enough data to be certain, b u t it appears that phytoflagellates and diatoms sink more slowly than mixed communities (including green algae) in Lake Erie. The range indicated for the sinking rates, 0.1--2 m d - 1 , is slightly lower, but is in general agreement with those of Lake LBJ and literature values. However, differential deathand grazing-rates probably play a larger role in regulating p h y t o p l a n k t o n composition in Lake Erie than Lake LBJ, since relatively small death-rates can compensate for the lower G values found there. The eigenvalue equation (9) provides an analytical solution which includes the factors determining p h y t o p l a n k t o n composition. Therefore it can be applied to many water-quality management problems, to examine w h y a nuisance bloom occurs. For example, Wyatt (1974) has shown that one red tide-organism, Gymnodinium breve, occurs after stable periods of stratification (low E). The low loss due to mixing during stable stratification m a y provide the only conditions under which this relatively slow-growing (low G) dinoflagellate can become dominant; b u t it is necessary to consider other factors (mixing, growth, death, grazing and sinking) simultaneously. Similarly, the problem of blue-green algae blooms in the late fall m a y occur when losses due to mixing are small, and a low G is maximized by favorable temperatures. Of course, gas-vacuole formation can provide very low or even negative sinking-rates in the case of blue-green algae, and nitrogen fixation_and a high CO2 efficiency may provide relative advantages in terms of G. CONCLUSION The application of an eigenvalue analysis proved useful in quantifying differences in loss-rates among taxa of phytoplankton. For the systems studied, blue-green algae and phytoflagellates had low gross growth-rates (less than 0.5 d -1) and correspondingly low sinking-rates (less than 0.1 m d - l ) . Green algae produced the greatest growth potential (G = 1--1.75 d -1) and the greatest calculated sinking-rates (1--3 m d - l ) . Diatoms were intermediate, with growth-rates generally between 0.25 and 0.75 d -1, and calculated sinking rates between 0.1 and 1.0 m d -1. For Lake LBJ and Lake Er!e, b o t h sinkingand death are important in determining p h y t o p l a n k t o n composition, b u t in Lake LBJ it seems that the variation in growth-rates is primarily compensated by differential sinking-rates rather than by differential death-rates. ACKNOWLEDGEMENTS The authors wish to thank J.B. Davenport for correspondence of data and Donald J. O'Connor for helpful comments and review. A u t h o r Schnoor was a U.S. National Science Foundation Postdoctoral Fellow during the tenure of this research.
245 REFERENCES Bella, D.A., 1970. Simulating the effect of sinking and vertical mixing on algal population dynamics. J. Water Pollut. Control Fed., 42: R140--R152. Davenport, J.B., 1975. Phytoplankton succession and the productivity of individual algae species in a central Texas reservoir. Thesis, University of Texas, Austin, TX. Dillon, P.J. and Rigler, F.H., 1974. A test of a simple nutrient-budget model predicting the phosphorus concentrations in lakewater. J. Fish. Res. Board Can., 31: 1771--1778. Di Toro, D.M., 1974. Vertical interaction in phytoplankton populations -- an asymptotic eigenvalue analysis. Proc. Conf. Great Lakes Res., 17th, Internat. Assoc. Great Lakes Res., pp. 17--27. Di Toro, D.M. and Connolly, J.P., 1978. Final report on Lake Erie to U.S. Environmental Protection Agency, Contract R803--030. Glooschenko, W.A., Moore, J.E., Munawar, M. and Vollenweider, R.A., 1974. Primary production in Lakes Ontario and Erie: a comparative study. J. Fish. Res. Board Can., 31 : 253--263. Jassby, A.D. and Goldman, C.R., 1974. Loss-rates from a lake phytoplankton community. Limnol. Oceanogr., 19: 618--627. Maguire, B., Jr. and Neill, W.E., 1971. Species and individual productivity in phytoplankton communities. Ecology, 52: 903--907. Munawar, M.M. and Munawar, I.F., 1976. A lakewide study of phytoplankton biomass and its species composition in Lake Erie. J. Fish. Res. Board Can., 33: 581--600. Munk, W.H. and Riley, G.A., 1952. Absorption of nutrients by aquatic plants. Bull. Bingham Oceanogr. Collect., 11: 2,215--240. Riley, G.A., Stommel, H. and Bumpus, D.F., 1949. Ecology of plankton. Bull. Bingham Oceanogr. Collect., 12(3): 1--169. Schnoor, J.L., and Fruh, E.G., 1976. Nutrient and dissolved oxygen dynamics of a short detention-time Texas Reservoir. Rep. CRWR 134. University of Texas, Austin, TX, 313 PP. Smayda, T.J., 1974. Some experiments on the sinking characteristics of two freshwater diatoms. Limnol. Oceanogr., 19: 628--635. Sorokin, C., 1959. Tabular comparative data for the low- and high-temperature strains of Chlorella. Nature (London), 184: 643--644. Titman, D. and Kilham, P., 1976. Sinking in freshwater plankton: some ecological implications of cell nutrient-status and physical-mixing processes. Limnol. Oceanogr., 21 : 409--417. Vollenweider, R.A., 1968. Scientific fundamentals of the eutrophication of lakes and flowing waters. OECD, Paris, DAS/CSI/68.27,183 pp. Vollenweider, R.A. (Editor), 1969. A manual on m e t h o d , for measuring primary production in aquatic environments. IBP Handbook No. 12, Blackwell Scientific, Oxford and Edinburgh. Wyatt, T., 1974. Red tides and algal strategies. In: M.B. Usher and M.H. Williamson (Editors), Ecological Stability. Wiley, New York, NY, pp. 35--39.
DIFFERENTIAL PHYTOPLANKTON SINKING- AND GROWTH-RATES: AN EIGENVALUE ANALYSIS JERALD L. SCHNOOR and DOMINIC M. Di TORO * Environmental Engineering, The University of Iowa, Iowa City, IA 52242 (U.S.A.) • Environmental Engineering and Science, Manhattan College, Bronx, N Y 10471 (U.S.A.)
(Accepted for publication in revised form 29 August 1979)
ABSTRACT Schnoor, J.L. and Di Toro, D.M., 1980. Differential phytoplankton sinking- and growthrates: an eigenvalue analysis. Ecol. Modelling, 9: 233--245. An eigenvalue analysis of the vertical phytoplankton biomass equation is applied to calculate the differential sinking- and loss-rates of phytoplankton for different taxonomic groups in Lake Lyndon B. Johnson (LBJ) (Texas) and in Lake Erie. The analysis includes factors determining the phytoplankton composition, including losses due to turbulent mixing and to sinking, and a death-term to account for endogenous decay and predation. Gross growth-rates were obtained from community and individual 14C production data (autoradiography) and from biomass measurements. Green algae produced the largest gross growth-rates, 1--1.75 d - 1 , and the largest calculated sinking-rates, 1--3 m d - 1 . The lowest gross growth-rates (< 0.5 d - 1 ) and sinking-rates (< 0.1 m d- 1 ) were determined for blue-green algae and phytoflagellates, while diatoms were intermediate, with growthrates generally in the range 0.25--0.75 d - 1 and calculated sinking-rates of 0.1--1.0 m d--1.
INTRODUCTION P h y t o p l a n k t o n u n d e r g o losses in lakes b y sinking a n d m i x i n g o u t o f t h e e u p h o r i c z o n e , a n d b y d e a t h or e n d o g e n o u s d e c a y . In o r d e r to i n v a d e or c o e x i s t , t h e y m u s t h a v e gross r a t e s o f g r o w t h e q u a l t o or g r e a t e r t h a n t h e i r loss-rates. T h e f a c t o r s i n f l u e n c i n g p h y t o p l a n k t o n g r o w t h a n d c o m p o s i t i o n are b o t h biological a n d p h y s i c a l . G r o w t h - r a t e s are i n f l u e n c e d b y environm e n t a l c o n d i t i o n s ( n u t r i e n t s , light, a n d t e m p e r a t u r e ) , b u t t h e y are also f u n c t i o n s o f t h e p h y t o p l a n k t o n t a x a , genus, a n d species, as are t h e sinking-rate a n d d e a t h - r a t e . Physical f a c t o r s i n c l u d e vertical m i x i n g or e d d y d i f f u s i v i t y , a n d t h e d e p t h o f t h e e u p h o r i c z o n e . T h e results o f all t h e s e variables d e t e r m i n e s t h e o b s e r v e d n e t g r o w t h - r a t e o f t h e species. Munk and Riley (1952) have addressed the importance of phytoplankton sinking-rates in a q u a n t i t a t i v e m a n n e r b y c o n s i d e r i n g n u t r i e n t t r a n s p o r t . Bella ( 1 9 7 0 ) d e m o n s t r a t e d t h e e f f e c t s o f sinking as c o m p a r e d t o m i x i n g in l a k e p h y t o p l a n k t o n c o m m u n i t i e s , a n d J a s s b y a n d G o l d m a n ( 1 9 7 4 ) cal-
234 culated loss-rates (sinking plus mixing plus death) from depth-averaged primary production and biomass measurements. Smayda (1974) and Titman and Kilham (1976) have measured the sinking-rates of freshwater phytoplankton directly. In this paper, an eigenvalue analysis of the vertical p h y t o p l a n k t o n biomass equation is applied to calculate the allowable sinking- and loss-rates of taxonomic groups of p h y t o p l a n k t o n in Lake L y n d o n B. Johnson (LBJ) (Texas), and in Lake Erie. The analysis includes the factors determining p h y t o p l a n k t o n composition, including losses due to turbulent mixing and to sinking, and a death-term to include predation and endogenous decay. This work places the factors determining p h y t o p l a n k t o n composition in an analytical and quantitative framework. Lake L y n d o n B. Johnson (LBJ) is the third in a series of seven reservoirs on the Colorado River in the Highland Lakes Chain of Central Texas. It has a surface area of 25.8 km 2 , a volume of 0.171 km 3 , a mean depth of 6.63 m (with a m a x i m u m depth of 26 m), and a mean annual hydraulic-detention time of approximately 80 days. Lake LBJ receives waters from the unimp o u n d e d Llano River, and the hypolimnetic" release from Inks Dam on the Colorado River. It does n o t receive any major waste or agricultural discharges, although the nutrient loadings are sufficient to classify the reservoir as eutrophic, according to the m e t h o d of Vollenweider (1968) or Dillon and Rigler (1974). In 1972 and 1973, Schnoor and Fruh (1976) noted t h a t the total phosphorus loadings were 1.6 and 2.6 g - 2 y r - 1 , and the total nitrogen loadings were 14 and 27 g - 2 y r - 1 , respectively. Anoxic conditions are normally present in the hypolimnion of Lake LBJ from June through September. Lake Erie is the shallowest of the five Laurentian Great Lakes. It has a mean depth of 17.7 m and a surface area of 25,700 km 2 , which puts it twelfth in size among the world's lakes. Lake Erie can be divided into three basins: Western, Central, and Eastern. The Western basin is the shallowest with a mean depth of 7.3 m, and it receives the major inflow from Lake Huron, via the Detroit and St. Clair Rivers. It is the most productive and eutrophic of the three basins. Anoxic conditions usually occur near the bott o m of Lake Erie in the Central basin during August or September.
THEORETICAL DEVELOPMENT One can locate regions of parameter space which allow particular phytoplankton species to invade and grow successfully. For given conditions in a lake or reservoir, it is interplay between the gross growth-rate G, the sinkingrate w, and the death-rate D, which determines the net growth-rate p and the ability of a particular alga to exist in the c o m m u n i t y . An analysis of the mass-conservation equation for p h y t o p l a n k t o n biomass P is presented below: the analysis incorporates the physical and biological
235 variables previously discussed. For a one-dimensional, vertical regime, the phytoplankton equation is (Riley et al., 1949): aP/at --a/az(E(aP/az))
(1)
+ (a/az)(wP) = (V -D)P
where E is the vertical eddy-diffusivity, z the depth, t is time, w the average sinking velocity, G the gross growth-rate, and D is a first-order loss-rate for losses due to death, endogenous decay, predation, and washout. It has been shown by Di Toro (1974) that when the biomass increases, the solution to Eq. (1) asymptotically approaches the form P ( z , t ) = P ( z ) exp(pt)
(2)
where p is the net observable growth-rate of the population and is independent of depth. Exponential growth-rates, independent of depth, have been observed in a number of lakes (Di Toro, 1974). Therefore, Eq. (1) may be replaced by Eq. (3) which, for E and w constant with depth, becomes - - E ( d 2 P / d z 2) + w ( d P / d z ) = IV(z) - - (D + p ) ] P
(3)
A two-layer approximation is used (Riley et al., 1949): the gross growthrate is vertically-averaged in the euphoric zone, and assumed to be zero below: thus G(z) =G
for0~
and G(z) = 0
for L ~< z (layer 2)
One may apply the boundary conditions: --E(dP1/dz) + wP1 = 0
at z = 0
P1 = P2 a n d - - E ( d P 1 / d z ) + w P 1 = --E.(dP2/dz ) + w P 2
(4) at z = L
(5)
and P2-*0
asz-*¢¢
(6)
where the subscripts 1 and 2 refer to the top layer (the euphotic zone) and the bottom layer respectively. The first boundary condition specifies that phytoplankton mass cannot pass through the surface of the lake. The second states that the concentration and mass-flux at the euphotic-zone depth L must be equal in both layers; and the last boundary condition requires that the phytoplankton concentration approach zero as the depth approaches infinity. Equation (3) may be solved in each of the two layers. In the layer representing the euphotic zone, the general solution is: Pl(z) = B x exp(wz/2E)
cos(~z -- X)
where B x and k are arbitrary constants, and a = ( w / 2 E ) x/[4(V -- (D + p ) ) E / w z] -- 1
(7)
236 The b o t t o m layer has the general solution: P2(z) = B1 exp(g2z)
iS)
where g2 = ( w / 2 E ) [ 1 - - x / 1 + 4(D + p ) E / w 2 ] .
Equations (7) and (8) can be combined, together with the boundary conditions, to yield an eigenvalue equation for p, the net growth-rate. In terms of dimensionless numbers, the final eigenvalue equation is given by w L / E = [tan-l(1/(~) + t a n - l ( x / ~
+ 4n)/~)]/(fl/2)
(9)
where w L / E is the Peclet number, L is the mean d e p t h of the euphotic zone, (3 = ~/4n(~ -- 1) -- 1 (dimensionless), n = (D + p ) E / w 2 (dimensionless), and = G / ( D + p) (dimensionless). The Peclet number gives a ratio between the rates of advective and diffusive transport (i.e., the greater the Peclet number, the more dominant is sinking as a transport mechanism). Equation (9) can be used to quantify the required growth-rate and the allowable sinking- and death-rates for an individual to grow. If one knows the environmental parameters E and L and the gross growth-rate G, it is possible to find a sinking-rate and/or death-rate that allows net growth of the population. Strictly speaking, this analysis is an approximation of the actual conditions in a lake, since the solution is developed for temporally constant coefficients and describes the asymptotic behavior which is approached after times of the order of 1 / p . However, it provides a quantitat!ve framework within which a very complicated problem can be analyzed directly. The average gross growth-rate in the euphoric zone, G, of a c o m m u n i t y or a species can be measured as the ratio of its production to its biomass (activity coefficient), while the net growth-rate p is simply the observed logarithmic rate of biomass increase from time tl to time t2, that is, p = In [P(z, t2)/ P(z, t l ) ] / ( t ~ - - tl). The difference between these rates, G - - p , is a measure of all the losses in the lake due to sinking, death, washout and predation, and mixing o u t of the euphotic zone. RESULTS AND DISCUSSION Near-surface biomass and productivity data for Lake LBJ is presented in Table I. An autoradiographic technique of Davenport (1975) and Maguire and Neill (1971) allowed delineation of p h y t o p l a n k t o n b y taxa, species and individual. Three taxa axe given, together with carbon-14 community-production measurements. Wet biomass is reported by Davenport (1975) by multiplying cell counts b y the mean biovolume per cell, and assuming a density of 1 g cm -~. The gross growth-rate G is estimated assuming a ten-hour photoperiod, a 10% carbon to wet-weight ratio (Vollenweider, 1969), and a 50% ratio for the average euphotic-zone to near-surface productivity. The latter assumption is supported b y Davenport's (1975) depth-integrated field
237 TABLE 1 Lake L y n d o n B. J o h n s o n (LBJ) * Sampling date
9-23-73
10-9-73
11-17-73
12-8-73
1-11-74
2-9-74
Taxa **
Wet b i o m a s s (P, m g 1- z )
Production (mg C m - 3 h - 1 )
Production/ biomass *** (G, d - 1 )
G D BG
0.181 0.187 0.690
1.76 1.59 3.52
0.485 0.425 0.255
Total
1.225
6.87
0.28
G D BG
0.163 0.112 0.130
3.36 0.96 4.24
1.02 0.43 1.635
Net growth-rate (]2, d - 1 )
--0.0066 --0.0320 --0.1045
Total
0.405
8.55
1.05
--0.0692
G D BG
0.027 0.0077 0.0000
0.980 ---
1.82
--0.0463 --0.0686
Total
0.038
0.980
1.29
--0.0607
G D BG
0.0654 0.0105 0.0000
2.43 ---
1.85
0.0425 0.0148 0.0000
1.60
0.0394
Total
0.0759
2.43
G D
0.102 0.0534
--
BG
0.0000
--
Total
0.182
3.06
0.842
0.0257
G
0.0856
2.21
D BG
0.0881 0.0000
5.77 0.00
1.29 0.327
--0.00608 0.0173 0.0000 --0.00015
0.0131 0.0478 0.0000
Total
0.174
2.79
0.803
3-9-74
G D BG
0.141 0.254 0.000
4.59 3.64 0.00
1.62 0.72
Total
0.395
8.23
1.04
0.0293
4-6-74
G D BG
0.0973 O.0606 0.0000
3.42 1.36 0.00
1.76 1.12
--0.0133 --0.0511 0.0000
Total
0.171
4.84
1.41
--0.0299
G D BG
0.160 0.0158 0.0000
5.00 1.21 0.00
1.56 3.83
0.0713 --0.192 0.0000
Total
0.204
6.21
1.52
0.0256
4-13-74
0.0179 0.0378 0.0000
(continued)
238 TABLE I (continued) Sampling date
4-27-74
5-4-74
5-27-74
6-19-74
6-28-74
7-12-74
7-31-74
8-9-74
Taxa **
Wet b i o m a s s (P, m g 1 - 1 )
G D BG
0.0925 0.0178 0.0000
Total
0.238
G D BG
0.143 0.0185 0.0000
Total
0.169
G D BG
0.599 0.0511 0.0386
Total
0.668
G D BG
Production (mgCm--3 h-1)
1.86 0.228 0.000 10.5
Production/ biomass*** (G, d - 1 )
Net growth-rate (U, d - - l )
1.01 0.64 --
--0.0393 0.0085 0.0000
2.21
0.0109
4.93 0.268 0.000
1.72 1.22 --
0.0621 b0.0055 0.0000
5.20
1.54
--0.0488
18.4 0.148 0.0000
1.54 1.45 --
0.0623 0.0442 --
18.6
1.39
0.0597
0.640 0.0138 0.0163
20.6 0.0 0.266
1.60 -0.81
0.0047 --0.0935 --0.0616
Total
0.723
20.9
1.45
0.0056
G D BG
0.390 0.352 0.0442
1.78 0.27 --
--0.0124 --0.0116 -0.0206
Total
0.502
10.3
1.03
-0.0138
G D BG
0.224 0.0406 0.0204
10.6 0.906 0.0
2.36 1.12 --
-0.0398 0.0102 -0.0552
Total
0.344
11.5
G D BG
0.115 0.0515 0.509
Total
0.715
G D BG
0.255 0.0290 0.165
11.7 0.221 0.213
2.30 0.38 0.065
0.0883 -0.0638 -0.1249
Total
0.531
12.2
1.15
-0.0330
9.96 0.192 0.00
1.67
--0.0270
4.35 0.00 4.43
1.88 -0.43
-0.0349 0.0125 0.1693
8.78
0.61
0.0385
* Biomass a n d near-surface p r o d u c t i o n data are f r o m D a v e n p o r t (1975). ** G = green algae, D = d i a t o m s , BG = blue-green algae. * * * T h e ratio o f p r o d u c t i o n t o b i o m a s s (or gross g r o w t h - r a t e G) is e s t i m a t e d a s s u m i n g a t e n - h o u r p h o t o p e r i o d , a 10% c a r b o n t o w e t - w e i g h t ratio, a n d a 50% ratio for average e u p h o t i e - z o n e t o near-surface p r o d u c t i v i t y .
239 data from 10 June, 1974. Net growth-rates are the observed logarithmic rates of biomass increase between t w o sampling dates. Table II includes the integral-production and biomass data for Lake Erie's Western, Central and Eastern basins in 1970. The production data are from shipboard incubator measurements by Glooschenko (1974). These data approximate the integral production in the euphotic zone (15 m) of Lake Erie. Wet-biomass data are from Munawar and Munawar (1976), and represent depth-average values from samples at 1 and 5 m. The ratio of production to biomass, or gross growth-rate G, is time-averaged between the two consecutive dates used in the calculation of p. A ten-hour photoperiod and a 10% carbon to wet-weight ratio were assumed. Note that both the mean annual biomass and the productivity decrease from west to east in Lake Erie, apparently due to the large nutrient loadings of the shallow Western basin. In general, it can be shown from Tables I and II that c o m m u n i t y production is somewhat lower in Lake LBJ than in Lake Erie; however, the standing crop of biomass is so much smaller in Lake LBJ that the ratio of production to biomass, or gross growth-rate G, is significantly larger there than in Lake Erie. This fact indicates that the p h y t o p l a n k t o n c o m m u n i t y of Lake LBJ sustains greater losses (due to mortality, grazing aud/or sinking) than that of Lake Erie. It must be noted that the net growth-rates in Tables I and II are n o t constant over consecutive sampling periods (periods of the order of 1 p), so that the asymptotic solution presented by Eq. (9) is truly a coarse approximation. More frequent sampling observations might alleviate this problem. The ranges of the net versus the gross average c o m m u n i t y growth-rates for Lake Erie and Lake LBJ are presented in Fig. 1. Note that even though the algae have very large growth potential (G = 0.2--1.8 d - l ) , the observed net growth-rate p of the c o m m u n i t y is always between +0.08 d - 1 over the sampling intervals (bi-weekly to monthly). Therefore, relatively small differentials in loss-rates and/or growth-rates (of the same order as p) between species of algae can determine whether the species will grow or not. For a particular lake under steady conditions, the euphotic-zone depth and vertical eddy-diffusivity are relatively constant. They may be estimated using light and temperature measurements. For Lake LBJ, the approximate depth of the euphotic zone (5% of light) and the approximate eddy diffusivity are 4 m and 1.6 m 2 d -1 (Schnoor and Fruh, 1976), while in Lake Erie, the values are 15 m and I m 2 d -1, respectively (Di Toro and Connolly, 1978). Figure 2a gives directly the interrelationships between gross growth-rate, death-rate (including grazing, endogenous decay, and washout), and sinkingrate. It was constructed from trial-and-error solutions to Eq. (9) specifically for Lake LBJ. Every lake would have a similar b u t unique plot, constructed from knowledge of the vertical eddy-diffusivity E, and the euphotic zone depth L. A distinct trade-off occurs between the t w o loss-rates due to sinking and death. A n y algae capable of sustaining a gross average growth-rate greater
240 T A B L E II Lake Erie * Approximate Julian day 1970
Wet b i o m a s s , (P, m g 1 - 1 )
Average p r o d u c t i o n (mg C m -3 h -1 )
Production/ b i o m a s s ** (G, d - 1 )
Net growth-rate (tl, d - 1 )
13 7 0.8 4.7 4.3 8.0 4.3 5.4 2.6 1.7
4.92 28.6 15.1 38.2 23.7 48.0 32.6 28.9 17.9 4.9
0.223 1.15 1.35 0.682 0.576 0.680 0.648 0.612 0.488
--0.0200 --0.0868 0.0571 --0.00356 0.0214 --0.0222 0.00813 --0.0203 --0.0152
5.2
24.3
0.71
99 130 155 186 211 240 268 296 332
1.6 2.4 1.0 4.4 3.7 5.0 5.7 2.3 2.0
6.15 9.54 5.85 9.85 11.38 17.54 21.54 20.92 12.31
0.391 0.491 0.405 0.266 0.330 0.365 0.644 0.763
Mean
3.1
12.8
0.46
99 130 155 186 211 240 268 296 332 350
1.5 3.0 4.0 1.0 3.8 1.7 4.1 1.8 1.0 1.1
3.38 7.08 7.08 5.54 14.46 9.54 12.31 9.85 6.15 8.00
0.231 0.207 0.366 0.468 0.471 0.431 0.424 0.581 0.671
Mean
2.3
8.34
0.43
W e s t e r n basin 99 130 155 186 211 240 268 296 332 350 Mean Central basin 0.031 --0.0350 0.0478 -0.00693 0.0104 0.00468 --0.0324 --0.00388
E a s t e r n basin 0.0224 0.0115 -0.0447 0.0534 --0.0277 0.0314 -0.0294 -0.0163 0.00340
* B i o m a s s d a t a are f r o m M u n a w a r a n d M u n a w a r ( 1 9 7 6 ) , a n d s h i p b o a r d average-production data from Glooschenko (1974). ** T h e r a t i o o f p r o d u c t i o n t o b i o m a s s (or gross g r o w t h - r a t e G ) is e s t i m a t e d a s s u m i n g a t e n - h o u r p h o t o p e r i o d a n d a 10% c a r b o n t o w e t - w e i g h t ratio.
241 0 L A K E LBJ A ERIE WESTERN B A S I N
0.10
•
ERIE CENTRAL BASIN
D ERIE EASTERN B A S I N
0.08 T .~ 0.06 0.04
0 D
0.02
=o
0
0
A
= ~ , ,~ 0:5 ' ~
0 o
o
o
,
O
o,
1~
1.5
2.0
-0.02 I~m •
z -0.04
0
0
O
0
--0,06
0 0
--0.08 A
--0.10 ~ G
ROWTH RATE {G), day- 1
a=
Fig. 1. Observed net growth-rate vs. average gross growth-rate (ratio of production to biomass) for Lake LBJ, 1973--74, and Lake Erie, 1970.
than the contour values shown in Fig. 2a can maintain growth. For Lake LBJ, diatoms and blue-green algae demonstrated gross growth-rates generally between 0.25 and 0.75 d - 1 , while green algae showed significantly higher values at 1.25--2.4 d -1. Therefore it is possible to conclude that on the average, green algae sustain greater loss-rates due to sinking and death than blue-green algae and diatoms in Lake LBJ. During the active-growth phase, the death-rate of a phyt0plankton species is expected to be relatively small and constant. Although the literature is somewhat sparse, typical phytoplankton death-rates seem to be of the order of 0.05--0.25 d - 1 . Riley, Stommel and Bumpus (1949) report a range of 0 . 0 5 - 0 . 2 0 d - 1 for a population isolated from a light source at 20°C. Sorokin (1959) gives endogenous respiration-rates of a pure strain of Chlorella as 2.0 mm 3 02 mm - s cells hr - ~ at 25°C, which is approximately equivalent to 0.24 d - 1 . Jassby and Goldman (1974) calculated an absolute maximum deathrate due to predation as 0.20 d - ~ in Castle Lake. A case is presented in Fig. 2b where a small and constant death-rate of 0.05_ d - 1 is assumed. A most interesting feature of Fig. 2b is that the required G for continued growth becomes a narrow-banded function of w, the sinkingrate. To a great extent, differential P/B ratios imply differential sinking-rates and vice-versa, provided that death-rates are as small as 0.05--0.25 d - ~ . If the concentration_ of biomass is increasing, and one knows the ratio of production to biomass (G) for that plankton, direct estimation of w is possible. Estimates of planktonic sinking-rates are presented in Fig. 3a for Lake LBJ. Three taxa are shown. Only data where the net growth-rate is positive are presented, as required for an eigenvalue analysis. Figure 3a indicates that
242
~
A
B 4
~ 5
E
E
< 3 ~: 2 _z 2
g
v z
z
o
0
0.2 0.4 0.6 0.8 1.0 1.2 DEATH RATE (D), day-1
1
1.0 2.0 3,0 4.0 5.0 GROWTH RATE (G), day 1
Fig. 2. ( A ) L o s s - r a t e s d u e t o s i n k i n g a n d d e a t h , as a f u n c t i o n o f t h e a v e r a g e g r o w t h - r a t e f o r L a k e L B J ; p = 0. ( B ) S i n k i n g - r a t e vs. average g r o s s g r o w t h - r a t e f o r n e t g r o w t h - r a t e c o n tours; Lake LBJ, assuming D = 0.05 d -1 .
diatoms and blue-green algae settle more slowly than do the green algae prevalent in Lake LBJ. When the death-rate is low and constant, D = 0.05 d -1, the range of sinking-rates of diatoms and blue-green algae is indicated to be 0.5--2.5 m d -1 and for green algae, 2.5--3 m d -1. If the death-rate were 0.25 d -1, the sinking-rate of diatoms and blue-green algae would be 0.1--2 m d -1, while t h a t of green algae would be 2--2.75 m d -1. Of course, the sinking-rate (and net growth-rate) of any individual plankton or species may vary over a considerably broader range. The work of Davenport (1975) shows that species net growth-rates vary from +0.6 to --0.3 d -1, while comm u n i t y measurements are considerably less variable, ranging between +0.07 d -1 in Lake LBJ. The sinking-rates in Fig. 3 are not normalized to a reference temperature. Cold-water diatoms would actually sink faster in mid-summer (10--25 ° C), due to the effect of temperature on viscosity. This is one reason why diatoms may be f o u n d most frequently in winter and spring. For example, Fig. 3a shows t h a t a diatom with G = 0.5 d -1 and growing at p = 0.03 d -1 would completely stop growing (p = 0) if its sinking-rate increased from 1.1 to 1.25 m d -1. This increase in sinking-rate corresponds to an increase in temperature of just 5°C, according to Stokes Law. Titman and Kilham (1976) report a range of sinking-rates for four pureculture freshwater p h y t o p l a n k t o n , from 0.08 to 1.87 m d -~. They f o u n d the sinking-rate to be a marked function of the nutrient status of the plankton, which would allow early-spring species to sink at their lowest possible rate. Smayda (1974) reports a range of sinking-rates for two freshwater diatoms of 0.26--0.76 m d -1. Ultimately it is the interplay between differential sinking-rates and differential death-rates t h a t determines p h y t o p l a n k t o n composition. Therefore, calculations were made assuming constant sinking-rates (0.5 and 1.5 m d -1) to assess the effects of differential death-rates in Lake LBJ (Fig. 3b). High
243
Sinkingrate,
! I
w = 3.25 m/day
|
0.06 0.04 0.02
il, ,.]°r,°:q
O 0
0.5
/.J
,t
I, °
,
1.0
1.5
2.0
,
2.5
GROWTH RATE (G), day- 1
B
i
b-
I °°I
0.06
="
JI
o~ 0.5
Death rate,
o=, 5~day
1 Y:
il ....... o
0.04
=o
o
GREENS
o ELffE-GREEN$
i I
0.O2
o
• •
,I
z 0
II , ,-II, 0.5 ,.0 ,.5 ,.0 GROWTH RATE (G), day-1
0
Sinkingrete~/
0.10 ~- C
0.o~11/I O.
0.04
O02
0
MIXED FLAGELLATE D O M I N A 7lED
o.
~
w'''°m
, ,.5
i
11/:/ I , ll" "
I".l',;
i
12
• I
eA I I ,1, 0.25 0.50 0.75 1.00 1.25 GROWTH RATE (G), day-1
Fig. 3. (A) Observed net growth-rate vs. average gross growth-rate with constant sinkingrate contours, L a k e LBJ, 1973--74. Solid line, assuming D = 0.05 d--l; dashed line, assuming D = 0.25 d - 1 . (B) Observed net growth-rate vs. average gross growth-rate with constant sinking-rate contours, L a k e LBJ, 1973--74. Solid line, assuming w = 0.5 m d - I ; dashed line, assuming w = 1.5 m d -I. (C) Observed net growth-rate vs. average gross growth-rate with constant sinking-rate contours, Lake Erie, 1970. Solid line, assuming D = 0.085 d--l; dashed line, assuming D = 0.25 d -1.
values of the death-rate are required to keep the net growth-rate in the observed range. It would seem that differential sinking-rates are the primary mechanism by which differential growth-rates are compensated in Lake LBJ. The calculated sinldng-rates (Fig. 3a) are in near agreement with literature values, while no death-rates as high as 1--1.5 d -z (Fig. 3b) have been reported. Since the net growth-rates (p) observed in nature are so low, it is still true that a slight increase in the death-rate will stop phytoplankton from growing. Both w and D are important in determining species composition, but it is the variability in w which primarily balances the variability in G of the assemblage in Lake LBJ. Autoradiographic analysis has not been performed for Lake Erie, but there were times of the year in 1970 when flagellates or diatoms dominated the system, 80% or more in proportion. For these cases, community produc-
244 tivity and growth may be taken as an approximation of taxanomic productivity and growth. A plot of c o m p u t e d sinking-rates for values of p and G in Lake Erie is given in Fig. 3c. There are n o t enough data to be certain, b u t it appears that phytoflagellates and diatoms sink more slowly than mixed communities (including green algae) in Lake Erie. The range indicated for the sinking rates, 0.1--2 m d - 1 , is slightly lower, but is in general agreement with those of Lake LBJ and literature values. However, differential deathand grazing-rates probably play a larger role in regulating p h y t o p l a n k t o n composition in Lake Erie than Lake LBJ, since relatively small death-rates can compensate for the lower G values found there. The eigenvalue equation (9) provides an analytical solution which includes the factors determining p h y t o p l a n k t o n composition. Therefore it can be applied to many water-quality management problems, to examine w h y a nuisance bloom occurs. For example, Wyatt (1974) has shown that one red tide-organism, Gymnodinium breve, occurs after stable periods of stratification (low E). The low loss due to mixing during stable stratification m a y provide the only conditions under which this relatively slow-growing (low G) dinoflagellate can become dominant; b u t it is necessary to consider other factors (mixing, growth, death, grazing and sinking) simultaneously. Similarly, the problem of blue-green algae blooms in the late fall m a y occur when losses due to mixing are small, and a low G is maximized by favorable temperatures. Of course, gas-vacuole formation can provide very low or even negative sinking-rates in the case of blue-green algae, and nitrogen fixation_and a high CO2 efficiency may provide relative advantages in terms of G. CONCLUSION The application of an eigenvalue analysis proved useful in quantifying differences in loss-rates among taxa of phytoplankton. For the systems studied, blue-green algae and phytoflagellates had low gross growth-rates (less than 0.5 d -1) and correspondingly low sinking-rates (less than 0.1 m d - l ) . Green algae produced the greatest growth potential (G = 1--1.75 d -1) and the greatest calculated sinking-rates (1--3 m d - l ) . Diatoms were intermediate, with growth-rates generally between 0.25 and 0.75 d -1, and calculated sinking rates between 0.1 and 1.0 m d -1. For Lake LBJ and Lake Er!e, b o t h sinkingand death are important in determining p h y t o p l a n k t o n composition, b u t in Lake LBJ it seems that the variation in growth-rates is primarily compensated by differential sinking-rates rather than by differential death-rates. ACKNOWLEDGEMENTS The authors wish to thank J.B. Davenport for correspondence of data and Donald J. O'Connor for helpful comments and review. A u t h o r Schnoor was a U.S. National Science Foundation Postdoctoral Fellow during the tenure of this research.
245 REFERENCES Bella, D.A., 1970. Simulating the effect of sinking and vertical mixing on algal population dynamics. J. Water Pollut. Control Fed., 42: R140--R152. Davenport, J.B., 1975. Phytoplankton succession and the productivity of individual algae species in a central Texas reservoir. Thesis, University of Texas, Austin, TX. Dillon, P.J. and Rigler, F.H., 1974. A test of a simple nutrient-budget model predicting the phosphorus concentrations in lakewater. J. Fish. Res. Board Can., 31: 1771--1778. Di Toro, D.M., 1974. Vertical interaction in phytoplankton populations -- an asymptotic eigenvalue analysis. Proc. Conf. Great Lakes Res., 17th, Internat. Assoc. Great Lakes Res., pp. 17--27. Di Toro, D.M. and Connolly, J.P., 1978. Final report on Lake Erie to U.S. Environmental Protection Agency, Contract R803--030. Glooschenko, W.A., Moore, J.E., Munawar, M. and Vollenweider, R.A., 1974. Primary production in Lakes Ontario and Erie: a comparative study. J. Fish. Res. Board Can., 31 : 253--263. Jassby, A.D. and Goldman, C.R., 1974. Loss-rates from a lake phytoplankton community. Limnol. Oceanogr., 19: 618--627. Maguire, B., Jr. and Neill, W.E., 1971. Species and individual productivity in phytoplankton communities. Ecology, 52: 903--907. Munawar, M.M. and Munawar, I.F., 1976. A lakewide study of phytoplankton biomass and its species composition in Lake Erie. J. Fish. Res. Board Can., 33: 581--600. Munk, W.H. and Riley, G.A., 1952. Absorption of nutrients by aquatic plants. Bull. Bingham Oceanogr. Collect., 11: 2,215--240. Riley, G.A., Stommel, H. and Bumpus, D.F., 1949. Ecology of plankton. Bull. Bingham Oceanogr. Collect., 12(3): 1--169. Schnoor, J.L., and Fruh, E.G., 1976. Nutrient and dissolved oxygen dynamics of a short detention-time Texas Reservoir. Rep. CRWR 134. University of Texas, Austin, TX, 313 PP. Smayda, T.J., 1974. Some experiments on the sinking characteristics of two freshwater diatoms. Limnol. Oceanogr., 19: 628--635. Sorokin, C., 1959. Tabular comparative data for the low- and high-temperature strains of Chlorella. Nature (London), 184: 643--644. Titman, D. and Kilham, P., 1976. Sinking in freshwater plankton: some ecological implications of cell nutrient-status and physical-mixing processes. Limnol. Oceanogr., 21 : 409--417. Vollenweider, R.A., 1968. Scientific fundamentals of the eutrophication of lakes and flowing waters. OECD, Paris, DAS/CSI/68.27,183 pp. Vollenweider, R.A. (Editor), 1969. A manual on m e t h o d , for measuring primary production in aquatic environments. IBP Handbook No. 12, Blackwell Scientific, Oxford and Edinburgh. Wyatt, T., 1974. Red tides and algal strategies. In: M.B. Usher and M.H. Williamson (Editors), Ecological Stability. Wiley, New York, NY, pp. 35--39.