1
1
Appendix S1.
2
Kiørboe, T. and Jiang, H.: Derivation of
3
4
5
6
zooplankton flow model and parameterization of
foraging index
Contents
I.
Copepod flow models ................................................................................................................................ 1
7
I.1. Stokeslet model for the feeding current of a hovering zooplankter ...................................................... 2
8
I.2. Stresslet model for the swimming current of a neutrally buoyant cruising zooplankter ....................... 3
9
10
I.3. Combined stokeslet-stresslet model for the feeding/swimming current of a negatively-buoyant,
cruising copepod ........................................................................................................................................... 5
11
I.4. Estimating copepod body volume and excess weight ............................................................................ 5
12
II.
Parameterization of foraging index ........................................................................................................... 6
13
II.1. Basic metabolic rate ............................................................................................................................... 7
14
II.2. Zooplankton food energy density in the ocean...................................................................................... 8
15
II.3. Concentration of predators on zooplankton .......................................................................................... 8
16
II.4. Background mortality ............................................................................................................................. 8
17
II.5. Other parameters ................................................................................................................................... 9
18
III.
References ........................................................................................................................................... 10
19
Table S1 ........................................................................................................................................................... 12
20
21
I.
Copepod flow models
22
We model a hovering copepod as a stokeslet, a neutrally buoyant, cruising copepod as a stresslet,
23
and we combine the stresslet and stokeslet models to describe the feeding current of a negatively
24
buoyant, swimming copepod (Fig. 1). The stokeslet and stresslet flow equations are written in a
25
cylindrical polar coordinate system (x, r, ), where  is the azimuthal coordinate, r the radial
26
coordinate, and x the axial coordinate. The x-direction coincides with the gravitational direction in
2
27
case of the stokeslet and with the swimming direction in case of the stresslet model. All parameters
28
are explained in Table S1.
29
I.1. Stokeslet model for the feeding current of a hovering zooplankter
30
For the stokeslet the fluid velocity in the axial (wx) and radial (wr) directions are (e.g. Pozrikidis
31
1992):
32
wx 
F 2x 2  r 2
8 x 2  r 2 3 / 2


33
wr 
F
xr
2
8 x  r 2


3/ 2
(S1)
,
(S2)
34
where F is the force magnitude and  the dynamic viscosity. F equals the copepod’s excess weight
35
and the reaction force of F (i.e. the propulsion force) balances the gravitational force acting on the
36
copepod. It can be estimated as F =  g V, where  is the excess density of the copepod, g the
37
gravitational acceleration, and V the copepod body volume. The associated velocity magnitude is:
38
39
U  wx2  wr2 
F
4x2  r 2
.
8 x 2  r 2
(S3)
Given a velocity threshold, U*, two lengths can be formed:
40
Rx* 
41
Rr* 
1
F
4 U *
1
F
.
8 U *
(S4)
(S5)
42
The area of influence, S, is defined as the area in the meridian plane within which the flow velocity
43
magnitude is greater than U*. The scaling for S is:
44
S
~ constant  1.24 .
0.5  Rx* Rr*
(S6)
3
2
0.02  F 

 .
 2  U * 
45
Combining eqs. S4-6 yields S 
46
That is, the area of influence scales with the square of the force and inversely with the square of the
47
threshold velocity. A calculation example for the modeled feeding current created by a hovering
48
copepod is shown in Fig. 1D.
49
The clearance rate of the stokeslet model can be computed as follows: Assume a perceptive
50
range, Rhovering, the stokeslet induced volumetric flux through a circle with radius Rhovering, which
51
intersects the axis of the stokeslet at the application point and is perpendicular to the stokeslet
52
direction, is:
53
 hovering 
F
Rhovering .
4
(S7)
54
For hovering copepods a good estimate of the perceptive range is equal one body length, i.e.
55
Rhovering = L (Strickler 1982, Paffenhöfer & Lewis 1990, Jiang et al. 2002a).
56
I.2. Stresslet model for the swimming current of a neutrally buoyant cruising
57
zooplankter
58
Similarly for the stresslet, we have the fluid velocity in the axial and radial directions (e.g.
59
Pozrikidis 1992):



(S8)

(S9)
60
wx 
Q 2x2  r 2 x
8 x 2  r 2 5 / 2
61
wr 
Q 2x2  r 2 r
,
8 x 2  r 2 5 / 2




62
where Q is the stresslet strength (in dimensions of force times distance). The two oppositely
63
directed forces (of magnitude 6ae|U|), are applied at two points separated by 2ae. Thus, the
64
 3V 
stresslet strength, Q, is 6ae|U|×2ae . Here, ae  

 4 
1/ 3
is the equivalent spherical radius of the
4
65
copepod and U is the copepod’s instantaneous swimming velocity relative to a stationary frame of
66
reference.
67
The associated fluid velocity magnitude is:
2
2
Q 2x  r
.
U  w w 
8 x 2  r 2 2
2
x
68
69
2
r


(S10)
As above, two lengths can be formed:
1/ 2
70
 1 Q

R  
*
 4 U 
71
 1 Q

R  
*
 8 U 
72
The scaling for the area of influence, S, is:
*
x
(S11)
1/ 2
*
r
.
S
~ constant  0.71 .
0.5  Rx* Rr*
73
(S12)
(S13)
0 .2  Q 

 , implying that the area of influence scales inversely with the
  U * 
74
Hence (S11-S13), S 
75
threshold velocity and in proportion to the stresslet strength. A calculation example for the modeled
76
swimming current created by a neutrally-buoyant copepod is shown in Fig. 1E.
77
78
The clearance rate can be estimated as the swimming-induced volumetric flux going through
the perceptive (encounter) area:
2
swimming   Rswimming
u,
79
(S14)
80
where u is the swimming speed. For cruising copepods, a good estimate of the perceptive distance is
81
the equivalent radius of the zooplankter, i.e. Rswimming = ae (Jiang et al. 2002a, Kjellerup & Kiørboe
82
2011).
5
83
I.3. Combined stokeslet-stresslet model for the feeding/swimming current of a
84
negatively-buoyant, cruising copepod
85
A negatively buoyant, swimming copepod acts on the ambient water with three force components:
86
A drag force in the swimming direction, and a propulsion force, which has one component opposite
87
the swimming direction to counter the drag, and one downward directed component to
88
counterbalance the gravitational force (Fig. 1C). The latter component can be described by a
89
stokeslet, and the former two components by a stresslet, and the addition of the stokeslet and the
90
stresslet can be used to model the overall flow field created by the copepod (Jiang et al. 2002b,
91
Jiang & Paffenhöfer 2004). The simple addition of flow components is valid at low Reynolds
92
numbers. The overall flow field is no longer axisymmetric (unless the copepod swims in the vertical
93
direction). The area of influence, S, defined here as the area in the vertical plane OXZ (Fig. 1C)
94
within which the flow velocity magnitude is greater than a given velocity threshold, U*, is
95
calculated numerically by evaluating the overall flow field due to the sum of the stokeslet and the
96
stresslet contributions within the vertical plane OXZ (Fig. 1F) and summing the areas with flow
97
velocity magnitude greater than U*. The actual relationship between S and U* varies, depending on
98
the swimming speed relative to the terminal sinking velocity of the copepod (determined by the
99
copepod’s body size and excess density) and the angle between the swimming direction and the
100
gravitational direction.
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I.4. Estimating copepod body volume and excess weight
102
Animal volumes were estimated from their prosome lengths using the equations of Chojnacki
103
(1983) and assuming size-independent shape. The shape-parameters used in the Chojnacki equation
104
were taken from the drawings of Sars (1902) for the two species for which we have data, and the
105
resulting equations relating body volume (V) to prosome length (L) are V = 0.059 L3 (Centropages
106
typicus) and V = 0.084 L3 (Temora longicornis).
6
107
To estimate the excess weight of the animals we assumed an excess density of 30 kg m-3,
108
corresponding to a mass density of ca. 1.06 kg m-3. This is the value computed from observed
109
sinking velocities of C. typicus using Stokes law and assuming a sphere with a diameter equal to the
110
prosome length and is within the range reported for marine calanoid copepods measured through
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various direct methods (1.025 – 1.13; Svetlichny 1980, Knutsen et al. 2001, Malkiel et al. 2003).
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II.
Parameterization of foraging index
We define the foraging index of a feeding behavior as:

114
 hovering  swimming  0  1
E  E0
.
(S15)
115
 hovering and swimming are the clearance rates (m3 s-1) due to the zooplankter feeding and swimming
116
currents,
117
0 = Mb/e
(S16)
118
is the overhead clearance (m3 s-1) for compensating the zooplankter basal metabolism, where Mb is
119
the basal metabolism and e the food energy density in the ocean.
120
1 
f
*

  g V u  WexcessU sinking
e
(S17)
121
is the overhead clearance (m3 s-1) for covering the cost of generating the swimming and feeding
122
currents, where η is the overall energetic efficiency parameter that accounts for the Froude
123
propulsion efficiency, the efficiency in generating the thrust force, and the efficiency (including
124
food assimilation efficiency) at which the ingested food is converted to mechanical energy for
125
propulsion and feeding current generation.
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126
E is the predator concentration-specific encounter rate (m3 s-1) (encounter kernel) of the
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zooplankter prey and E0 is the background mortality normalized by the concentration of predators
128
(m3 s-1), The encounter kernel for visual predation is E = R2(v2+u2)1/2 and for rheotactic predators
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E = S(v2+u2)1/2.
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We formulate metabolic expenses as the overhead clearance needed to cover metabolic costs and
131
we normalize the feeding-independent background mortality by predator concentration; that way all
132
entries in the foraging index are in dimensions of L3T-1, and the fitness parameter dimensionless.
133
Below we explain the parameters and argue for the choice of default values.
134
II.1. Basic metabolic rate
135
The metabolic rate, M, in terms of oxygen consumption (l O2 individual-1 h-1) was calculated
136
according to a regression model (Ikeda et al. 2001):
137
ln M  0.124  0.780 ln( CW)  0.073 Twater ,
(S18)
138
where CW is carbon weight (mg C). At Twater = 15 °C, M (l O2 h-1) = 3.38 (CW)0.78. Ikeda’s
139
equation includes both basal and active metabolism (cost of swimming etc.), and we assume that the
140
basal metabolism constitutes only half of that. We convert oxygen consumption to power (20.1 kJ
141
(liter of O2)-1), and body carbon to body volume (assuming 100 kg C m-3) to yield the basal
142
metabolism (Mb, Watt) as:
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Mb = 16.4 V0.78,
where V is in m3.
(S19)
8
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II.2. Zooplankton food energy density in the ocean
146
The phytoplankton biomass in the ocean varies between about 10 and 10-2 g C m-3 (Boyce et al.
147
2010); we take 0.1 g C m-3 as the default value. With 4.6×104 Joule per g organic carbon this
148
corresponds to e ~ 5.0×103 J m-3.
149
II.3. Concentration of predators on zooplankton
150
We estimate the concentration of predators based on three assumptions: (i) the biomass at
151
subsequent trophic levels in the ocean is approximately constant (Sheldon et al. 1972), (ii) the
152
predator:prey size ratio in plankton food chains is 10 (linear dimension) (Sheldon et al. 1972, 1977,
153
Kiørboe 2008), (iii) the carbon density in organisms is 105 g C m-3. Because we assume a default
154
zooplankton food concentration of 0.1 g C m-3, it follows that the concentration of predators on
155
zooplankton of size ae (equivalent spherical radius) is
156
n
0.1 g C m-3
4
 10ae 3  105 g C m-3
3
 2.4 1010 ae3 .
(S20)
157
Assumptions (ii) and (iii) are not valid for gelatinous predators: they feed on much smaller prey
158
relative to their inflated volume, and they have a correspondingly lower body carbon density.
159
However, the two deviations cancel out if we interpret the ‘equivalent radius’ of a zooplankter as if
160
it had a carbon density of 105 g C m-3; this allows us to consistently compare zooplankton sizes
161
between different life forms by their equivalent volume or equivalent radius.
162
II.4. Background mortality
163
The mortality of zooplankton in the ocean is size dependent, and compilations of field observations
164
yield size-dependent field mortality rates to be (Kiørboe 2008):
165
 (d-1) = 8.110-3 [W (g DW)]-0.32
(S21)
9
166
where W is the dry mass of the zooplankter (g). This is an estimate of the total mortality; we will
167
assume that half the mortality is feeding-independent, background mortality. We further want to
168
express the mortality rate as a function of the equivalent radius of the zooplankter rather than of its
169
dry mass. Assuming a carbon content of 45 % of the dry mass and a carbon density of 105 g C m-3
170
the size-dependent background mortality becomes
171
172
173
 s 1   0.58 10 9 ae-0.96
(S22)
and normalized by predator concentration:
E0 

n

0.58 109 ae-0.96
~ 2.4ae2.04 ,
2.39 1010 ae3
(S23)
174
where ae is in units of m.
175
II.5. Other parameters
176
Default values for predator swimming velocity (v), visual range of visual predators (R), and fluid
177
velocity threshold (U*) for rheotactic predators are taken from Kiørboe (2011). Perceptive distances
178
for cruising and hovering zooplankters (Rhovering and Rswimming) are from Strickler (1982) and
179
Kjellerup & Kiørboe (2011). The default value for the overall energetic efficiency parameter, η =
180
0.02 %, is not well constrained but we argue that it is a reasonable choice given that the Froude
181
efficiency is low for swimming at low Reynolds numbers, typically a few percent (Guasto et al.
182
2012; 2 % and up to 15 % in copepods calculated from the CFD results of Jiang et al. 2002c), the
183
efficiency of conversion of chemical to mechanical energy is at most a few percent (Berg 1992), the
184
efficiency in generating thrust force is of order 0.5 (Equation (26) in Jiang et al. 2002b), and the
185
efficiency in assimilating food is about 0.5. The range of values reported are covered by our
186
sensitivity analysis (Table 1)
10
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
III.
References
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Princeton.
Boyce DG, Lewis MR, Worm B (2010) Global phytoplankton decline over the past century. Nature
466: 591-596.
Chojnacki J (1983) Standard weights of the Pomeranian Bay copepods. Int Rev Gesamten
Hydrobiol 68: 435-441.
Guasto JS, Rusconi R, Stocker R (2012) Fluid mechanics of planktonic microorganisms. Ann Rev
Fluid Mech: 44: 373-400.
Ikeda T, KannoY, Ozaki K, Shinada A (2001) Metabolic rates of epipelagic marine copepods as a
function of body mass and temperature. Mar Biol 139: 587-596.
Jiang H, Osborn TR, Meneveau C (2002a) Chemoreception and the deformation of the active space
in freely swimming copepods: a numerical study. J Plankton Res 24: 495-510.
Jiang H, Osborn TR, Meneveau C (2002b) The flow field around a freely swimming copepod in
steady motion: Part I theoretical analysis. J Plankton Res 24: 167-189.
Jiang H, Meneveau C, Osborn TR (2002c) The flow field around a freely swimming copepod in
steady motion. Part II Numerical simulation. J Plankton Res 24: 191-213.
Jiang H, Paffenhöfer GA (2004) Relation of behavior of copepod juveniles to potential predation by
205
omnivorous copepods: an empirical-modeling study. Mar Ecol Prog Sers 278: 225-239.
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Kiørboe T (2008) A Mechanistic Approach to Plankton Ecology. Princeton University Press,
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Princeton.
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Kiørboe T (2011) How zooplankton feed: mechanisms, traits and trade-offs. Biol Rev 86: 311-339.
209
Kjellerup S, Kiørboe T (2011) Prey detection in a cruising copepod. Biol. Lett. doi:
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10.1098/rsbl.2011.1073
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Knutsen T, Melle W, Calise L (2001) Determining the mass density of marine copepods and their
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eggs with a critical focus on some previously used methods. J Plankton Res 23: 859-873.
213
Malkiel E, Sheng J, Katz J, Strickler JR (2003) The three-dimensional flow field generated by a
214
feeding calanoid copepod measured using digital holography. J Exp Biol 206: 3657-3666.
215
Paffenhöfer GA, Lewis KD (1990). Perceptive performance and feeding behaviour of calanoid
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copepods. J Plankton Res 12: 933-946.
Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow.
Cambridge University Press, Cambridge, UK.
Sars GO (1902) An Account of the Crustacea of Norway. Vol. IV. Copepoda Calanoida. Published
by the Bergen Museum.
Sheldon RW, Prakash A, Sutcliffe WH Jr (1972) The size distribution of particles in the ocean.
Limnol Oceanogr 17: 327-340.
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relationship between plankton and fish production. J Fish Res Bd Can 34: 2344-2353.
Strickler JR (1982) Calanoid copepods, feeding currents, and the role of gravity. Science 218: 158160.
Svetlichny LS (1980) On certain dynamic parameters of tropical copepod passive submersion.
Ekologiya Morya 2: 28-33.
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Table S1. Symbols used and default parameter values.
Parameter
ae
e
E
E0
f*
F
g
L
Mb
n
Q
R
Rhovering
Rswimming
S
u
U
U*
U
Usinking
v
V
Wexcess
wx, wr
β


µ
χ
Ωhovering,
Ωswimming
232
Explanation
Equivalent spherical radius of the zooplankter
Food energy density in the ocean
Encounter kernel with predators
Feeding-independent background mortality normalized
by predator concentration
Force production normalized with body volume of the
zooplankter
Force magnitude
Gravitational acceleration
Body length of zooplankter
zooplankter basal metabolism
Predator concentration
Stresslet strength
Units
m
J m-3
m3 s-1
m3 s-1
Default value
[ (3/4) V ]1/3
5.0×103
-
N m-3
-
N
m s-2
m
Watt
m-3
Nm
9.81
-
Detection radius of visual predator
Prey perception radius of hovering zooplankter
Prey perception radius of cruising zooplankter
Area of influence: the area within which the imposed
flow velocity due to swimming and feeding current
exceeds a critical value, U*. Estimates the rheotactic
encounter cross-section of the zooplankter prey
The zooplankter’s swimming speed, for simplicity
assumed to be vertically upward. The equation for the
default value links , f*, and u together
Fluid velocity
Fluid velocity threshold for prey detection in rheotactic
predators
Zooplankton swimming velocity
Terminal sinking speed of the zooplankter
Predator swimming speed
body volume
Excess weight of the zooplankter
Fluid velocity in axial and radial direction
m
m
m
m2
10 ae
L
ae
-
m s-1
2/9 ae2 (f*  g)
/µ
m s-1
m s-1
0.0001
m s-1
m s-1
m s-1
m3
N
m s-1
2/9
 g / µ
10L s-1
0.059 L3
 g V
-
s-1
kg m-3
kg m-1 s-1
0.0002
1.390×10-3
m3 s-1
-
Feeding-independent background mortality
A overall energetic efficiency parameter
Excess mass density of the zooplankter
Dynamic viscosity of seawater at salinity 35, 10˚C and
one normal atmosphere
Foraging index
Clearance rates due to hovering and swimming
ae2