A CASE STUDY OF AN ENHANCED EUTROPHICATION MODEL WITH
STOICHIOMETRIC ZOOPLANKTON GROWTH SUB-MODEL
CALIBRATED BY BAYESIAN METHOD
(Electronic Supplementary Material)
Likun Yanga; Sen Penga,b; Jingmei Suna,b,*; Xinhua Zhaoa,b; Xia Lic;
a
School of Environmental Science and Engineering, Tianjin University, Tianjin, 300072, China.
b
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin,
300072, China.
c
School of Environmental Science and Safety Engineering, Tianjin University of Technology,
Tianjin, 300384, China.
* Corresponding author
E-mail: jmsun@tju.edu.cn, Tel.: +86 22 27400830.
Content:
Table 1. Mathematical description of the model.
Table 2. Description and values of the parameters that were not considered during the Bayesian calibration
of the eutrophication model.
Table 1. Mathematical description of the model. The i and j indices refer to the diatoms, greens,
cyanobacteria and copepods, cladocerans, respectively.
No.
State Variable
Term
Equation
 growthi  PHYTi  filteri  ektfilt (Tx Tempref )  PHYTi
1
Phytoplankton
biomass
dPHYTi
dt
 (Vsettlingi  PHYTi ) z  resmpi  ekt (Tx Tempref )  PHYTi
Growh rate
growthi
 growthmaxi  fnutrienti  flighti  ftemperaturei
Nutrient
limitation
fnutrienti
 min{ NAi ,  PO4i }
Nitrogen
limitation
 NAi
  NO3   NH 4,i
Nitrate
limitation
 NO3
 NO3  e   NH 4 ( NO3  NH 4 )
Ammonium
limitation
 NH 4,i
 NH 4 / ( NH 4  AH i )
Phosphate
limitation
 PO4,i
 PO4  PO4  PHi 
 Grazingi  ftemperature j  ZOOPj

Light
limitation
flighti
Ii 
1

 I opt 
Ii


 FD 
e
; where:
I opt
I i  I  e hki , ki  i  PHYTi  
2
Basal metabolism
rate
resmpi
Temperature
limitation
ftemperaturei
Zooplankton
biomass

 mpi  DO  K resp ,i  DO 

 exp  KTgri T  Topti 
2


FD
=the fractional day length (0<FD<1)
dZOOPj
  SC j  ZOOPj  resmz j  e kt Tx Tempref   ZOOPj
dt
maxgrazing j C j (i Prefi , j PHYT 2  Pref det , j DET 2 )
Assimilation rate
of zooplankton
 SC j

Assimilation
efficiency
of zooplankton
C j

Food quality indice
for zooplankton
FQTOT
2
2
 [ FQPHYT
PHYTi  FQDET
DET ]Z PLIM
i
Secondary
limitation
Z PLIM

KZ 2j  i Prefi , j PHYT 2  Pref det , j DET 2
 C1 j FQTOT
 C 2 j  FQTOT
i
 GRAZ
Pj
 1 GRAZ Pj  PM j

GRAZ Pj
PM j

Basal metabolism
rate
resmz j

 PM j

 mz j  DO  K resp , j  DO 

No.
State Variable
Temperature
limitation for
growth
Zooplankton
predation rate of
phytoplankton
Grazed food per unit
of biomass into
phosphorus per unit
biomass
Term
ftemperature j
Grazing i , j
GRAZP
Equation
 exp( KTgrzoop T  Topt j  )
2
j
maxgrazing j  Prefi, j  PHYTi 2

KZ 2j  i Prefi, j PHYTi 2  Pref det , j DET 2
2
2
i Prefi, j PHYT  P / CPHYTi  Pref det , j DET  P / CDET

2
2
i Prefi, j PHYT  Pref det , j DET )
Grazed food per unit
of biomass into
nitrogen per unit
biomass
GRAZ N
Post-maintenance
resource pools
PPM
= SC GRAZP  PINT  P 1  m  m m 
N PM
= SC GRAZN  NINT  N 1  m  m m 
dPINT
dt
 PPM  GrowthPINT
dN INT
dt
 NPM  GrowthNINT
g LIM P
P P
 INT min
Popt  Pmin
g LIM N

The somatic nutrient
concentration
differential equations
Resource saturation
quotients
Growth rate
Growth
2
2
i Prefi, j PHYT  N / CPHYTi  Pref det , j DET  N / CDET

2
2
i Prefi, j PHYT  Pref det , j DET )
N INT  N min
N opt  N min

=0.8  min g LIM P，g LIM N

  (1 -  DOC i )  resmpi  e kt (Tx -Tempref )  PHYTi  
i
3
Detritus
concentration
dDetritus
dt
kt ( Tx - Tempref )
 ZOOPj   (1-  DOC )  resmz j  e
j
j
 Grazing det, j  ftemperature  ZOOPj
j
-Vsettling ( bio )  Detritus z - KCmine  Detritus
Carbon
mineralization
rate
KCmine
Temperature
limitation
for mineralization
ftemperaturemin
Zooplankton
predation rate of
detritus
Grazing det, j
 ftemperaturemin  KC 

DO
; where
DO  K resp , KC
 exp  KTFmin  T  Toptmin 

2

maxgrazing j  Pref det , j  DET 2
KZ 2j  i Prefi, j PHYTi 2  Pref det , j DET 2
No.
4
State Variable
Phosphate
concentration
Term
dPO4
dt
Equation
 PO4i  growthmaxi  flighti  ftemperaturei  P / CPHYTi  PHYTi 

  
kt (Tx Tempref )


P
/
C

PHYT
i   PO  resmpi  e
PHYTi
i
4i


   PO4 j  resmz j  ekt (Tx
 Tempref )
j
 P / C zoop j  ZOOP
 KPmine  OP +PO4,loading
5
Phosphorus
mineralization rate
KPmine
Organic
phosphorus
concentration
dOP
dt
Biogenic organic
phosphorus
accumulation
6
DO
DO  K resp , KP
 DetritusP   DetritusGrazingPj  ftemperature  ZOOPj
j
– SettlingP  OP z  KPmine  OP  OPloading
  (1   POxi )  resmpi  ekt (Tx  Tempref )  P / CPHYTi  PHYTi 
DetritusP
i
 (1   POx j )  resmz j  e
kt (Tx  Tempref )
j
Loss due to
zooplankton
grazing upon
detritus
DetritusGrazingPj
Loss due to
particulate
phosphorus settling
SettlingP
Ammonium
concentration
 ftemperaturemin  KP 
dNH 4
dt
 P / Czoop j  ZOOPj
 (maxgrazing j  Pref det , j  DetritusP) ( KZ j  Food j )
  DetritusP OP   Vsettling ( bio )
 1   DetritusP OP    Vsettling
 NH  growthmaxi  flighti 

4,i



   ftemperaturei  N Ci  PHYTi

i 
  NH 4 ,i  resmpi  e kt (Tx Tempref )  N Ci  PHYTi 
   NH 4 zoop ,  resmz j  ekt (Tx Tempref )  N C j  ZOOPj
j
j
 KNmine  ON -Nitrification  NH 4,loading
Mineralization rate
KNmine
Nitrification rate
Nitrification
 ftemperaturemin  KN 
 Nitrifmax 
DO
DO  Kresp , KN
NH 4
DO

 ftempnitr
DO  KH DO , nit KH NH , nit  NH 4
4
7
Temperature
limitation
ftempnitr
Nitrate
concentration
dNO3
dt
Denitrification rate
Denitrification

 exp  KTgrnitr T  Toptnitr 
 
i
2

 NO3,i  growthmaxi  flighti
 ftemperaturei  N Ci  PHYTi
 Nitrification  Denitrification  NO3, Loading
 Denitrifmax 
KH DO ,denit
DO  KH DO , denit

NO3
 ftempdenitr
KH NO , nit  NO3
3
No.
8
State Variable
Term
Temperature
limitation
ftempdenitr
Organic nitrogen
concentration
dON
dt
Biogenic organic
nitrogen
accumulation
Equation

 exp  KTgrdenitr T  Toptdenitr 
j
– SettlingN  ON z  KNmine  ON  ONloading
i
Loss due to
particulate nitrogen
settling


 1   NH
j
Loss due to
zooplankton
grazing upon
detritus
DetritusGrazingN j
SettlingN

DetritusN   DetritusGrazingN j  ftemperature  ZOOPj
  1   NH
DetritusN
2
4j
4i
  resmp  e
kt ( Tx Tempref )
i
  resmz  e
kt ( Tx Tempref )
j
 N Ci  PHYTi 
 N C j  ZOOPj
 maxgrazing j  Pref det , j  DetritusN  KZ j  Food j 

DetritusN
 DetritusN 
 Vsettling ( bio )  1 
  Vsettling
ON
ON


  growthi  DO / PHYTi  PHYTi   resmpi  e kt (Tx Tempref )
i
9
10
Dissolved Oxygen
dDO
dt
Dissolved oxygen
saturation
concentration
DOsat
i
kt Tx Tempref 
 DO / PHYTi  PHYTi   resmz j  e
j
 KNmine  DO / ON  ON  KPmine  DO / OP  OP
K r , DO  Area
 Nitrification  DO / NH 4 
  DOsat  DO 
Volume
 DOloading
 14.5532  0.38217  T  0.0054258  T 2
Sediment
submodel
Phosphate
10.1
sediment release
Organic
phosphorus
sedimentation
dSedPO4
dt
Pdeposition
 (1   P )  Pdeposition

 asPO4  SedPO4  e
ktsed Tsed Tempref sed 
i

Ammonium
sediment release
dSedNH 4
dt

 Vsettlingi P / Ci  PHYTi  SettlingP  OP
 1   N   Ndeposition  asNH 4  SedNH 4  e
10.2
 DO / ZOOPj  ZOOPj
 Nitrifmaxsed 
ktsed Tsed Tempref sed 
SedNH 4
DO

DO  KHdonitsed KH nh 4nitsed  SedNH 4
 ftempnitrsed
Loss due to
particulate
nitrogen settling
Ndeposition
 Vsettlingi N / Ci  PHYTi  SettlingN  ON
Temperature
limitation
for nitrification
in the sediments
ftempnitrsed
 exp( KTgrnitrsed  (T  Toptnitrsed )2 )
i

No.
State Variable
Term
Equation
 Nitrifmaxsed 
Nitrate
10.3
sediment release
dSedNO3
dt
 ftempnitrsed  as NO3 SedNO3  e
 Denitrifmaxsed 

Temperature
limitation
for denitrification
in the sediments
ftempdenitrsed
SedNH 4
DO

DO  KHdonitsed KH nh 4nitsed  SedNH 4
ktsed Tsed Temprefsed 
KHdodenitsed
KHdodenitsed  DO
SedNO3
 ftempdenitrsed
SedNO3  KHno3denitsed
 exp( KTgrdenitr (T  Toptdenitrsed )2 )
Table 2. Description and values of the parameters that were not considered during the Bayesian
calibration of the eutrophication model.
Symbol
ψ
Iopt
Description
Strength of the ammonium inhibition for nitrate
uptake
Saturation light intensity
Values
Units
0.05
(μg N/L)−1
12
MJ  m 2 d 
KTgrd
Effect of temperature on diatom
0.04
o
C 2
KTgrc
Effect of temperature on green algae
0.06
o
C 2
KTgrg
Effect of temperature on cyanophyta
0.05
o
C 2
Toptd
Reference temperature for diatom metabolism
20
o
C
Toptc
Reference temperature for cyanobacteria
metabolism
24
o
C
Topt g
Reference temperature for green algae metabolism
20
o
C
KTgrzoopc
Effect of temperature on cladocerans
0.015
o
C 2
KTgrzoopr
Effect of temperature on copepods
0.02
o
C 2
Topt zoopc
Reference temperature for cladocerans metabolism
20
o
C
Topt zoopr
Reference temperature for copepods metabolism
18
o
C
kt
Effects of temperature on plankton mortality
0.069
o
C 2
Tempref
Reference temperature in the water column
20
o
C
Effects of temperature on phytoplankton filtration
0.069
o
C 2
pref d ,r
Preference of copepods for diatoms
0.30
pref g ,r
Preference of copepods for green algae
0.25
pref c,r
Preference of copepods for cyanobacteria
0.35
pref det,r
Preference of copepods for detritus
0.1
pref d ,c
Preference of cladocerans for diatoms
0.25
pref g ,c
Preference of cladocerans for green algae
0.25
pref c,c
Preference of cladocerans for cyanobacteria
0.25
ktfilt
Symbol
pref det ,r
Description
Values
Units
Preference of cladocerans for detritus
0.25
C1r
Copepods carbon assimilation efficiency
0.9
C 2r
Half saturation constant for copepods growth
efficiency
0.03
C1c
Cladocerans carbon assimilation efficiency
0.9
C 2c
Half saturation constant for cladocerans growth
efficiency
0.03
KTFmin
Effects of temperature on mineralization
0.004
o
C2
Toptmin
Optimal temperature for mineralization
20
o
C
 DOC, zoopc
Fraction of cladocerans mortality becoming
dissolved organic carbon
0.5
 DOC, zoopr
Fraction of copepods mortality becoming dissolved
organic carbon
0.5
 DOC ,c
Fraction of cyanobacteria mortality becoming
dissolved organic carbon
0.5
 DOC ,d
Fraction of diatoms mortality becoming dissolved
organic carbon
0.5
 DOC , g
Fraction of green algae mortality becoming
dissolved organic carbon
0.5
P / CPHYTd
Phosphorus to carbon ratio for diatoms
0.01
mg P mg C1
P / C PHYTg
Phosphorus to carbon ratio for green algae
0.017
mg P mg C1
P / CPHYTc
Phosphorus to carbon ratio for cyanobacteria
0.025
mg P mg C1
P / CZOOPc
Phosphorus to carbon ratio for cladocerans
0.02
mg P mg C1
P / CZOOPr
Phosphorus to carbon ratio for copepods
1/35
mg P mg C1
P / CDET
Phosphorus to carbon ratio for detritus
0.013
mg P mg C1
 PO4 ,d
Fraction of diatoms mortality becoming phosphate
0.8
 PO4 , g
Fraction of green algae mortality becoming
phosphate
0.8
 PO4 ,c
Fraction of cyanobacteria mortality becoming
0.8
 mg L 
-1 1/ 2
 mg L 
-1 1/2
Symbol
Description
Values
Units
phosphate
 PO4 ,ZOOPc
Fraction of cladocerans mortality becoming
phosphate
0.8
 PO4 ,ZOOPr
Fraction of copepods mortality becoming phosphate
0.8
o
C 2
o
C
KTgrnitr
Effect of temperature on nitrification
0.004
Toptnitr
Optimal temperature for nitrification
20
N / CPHYTd
Nitrogen to carbon ratio for diatoms
0.13
mg N mg C1
N / CPHYTg
Nitrogen to carbon ratio for green algae
0.13
mg N mg C1
N / CPHYTc
Nitrogen to carbon ratio for cyanobacteria
0.13
mg N mg C1
N / CZOOPc
Nitrogen to carbon ratio for cladocerans
0.2
mg N mg C1
N / CZOOPr
Nitrogen to carbon ratio for copepods
1/6
mg N mg C1
N / CDET
Nitrogen to carbon ratio for detritus
0.16
mg N mg C1
 NH4 ,d
Fraction of diatoms mortality becoming ammonium
0.5
 NH4 , g
Fraction of green algae mortality becoming
ammonium
0.5
 NH4 ,c
Fraction of cyanobacteria mortality becoming
ammonium
0.5
 NH4 ,ZOOPc
Fraction of cladocerans mortality becoming
ammonium
0.5
 NH4 ,ZOOPr
Fraction of copepods mortality becoming
ammonium
0.5
KTgrdenitr
Effect of temperature on denitrification
0.004
o
C 2
Toptdenitr
Optimal temperature for denitrification
20
o
C
DO / PHYTd
Oxygen to carbon ratio for diatoms
2.67
mg O mg C1
DO / PHYTg
Oxygen to carbon ratio for green algae
2.67
mg O mg C1
DO / PHYTc
Oxygen to carbon ratio for cyanobacteria
2.67
mg O mg C1
DO / ZOOPr
Oxygen to carbon ratio for cladocerans
2.67
mg O mg C1
Symbol
Description
Values
Units
Oxygen to carbon ratio for copepods
2.67
mg O mg C1
DO / ON
Oxygen to carbon ratio for organic nitrogen
2.67
mg O mg C1
DO / OP
Oxygen to carbon ratio for organic phosphorus
2.67
mg O mg C1
DO / NH 4
Oxygen to carbon ratio for ammonium
4.33
mg O mg NH 4 1
ktsed
Effects of temperature on sedimentation
0.004
Tempref sed
Reference temperature in the sediments
20
KTgrnitrsed
Effect of temperature on sediment nitrification
0.004
Toptnitrsed
Optimal temperature for denitrification in the
sediments
20
Nitrifmaxsed
Maximum nitrification rate in the sediments
50
mg N m2day1
KHdonitsed
Half saturation concentration of DO required for
nitrification in the sediments
2
mgO2 m 3
KHnh4nitsed
Half saturation concentration of ammonium
required for nitrification in the sediments
75
mgN m3
DO / ZOOPc
o
C1
o
C
o
C 2
o
C
C 2
KTgrdenitrsed
Effect of temperature on sediment denitrification
Denitrifmaxsed
Maximum denitrification rate in the sediments
25
mg N m2day1
KHdodenitsed
Half saturation concentration of DO deficit required
for denitrification in the sediments
1
mg O2 m3
KHdonit
Half saturation concentration of DO required for
nitrification
1
mg O2 m3
KHnh4nit
Half saturation concentration of ammonium
required for nitrification
1
mg N m3
KHdodenit
Half saturation concentration of DO deficit required
for denitrification
0.5
mg O2 m3
KHno3denit
Half saturation concentration of nitrate required for
denitrification
15
mg N m3
Toptdenitrsed
Optimal temperature for denitrification in the
sediments
20
DO exchange rate
2.4
Diatoms food quality
0.8
kr，DO
FQPHYTd
0.004
o
o
C
m
Symbol
Description
Values
Units
FQPHYTg
Green algae food quality
0.5
FQPHYTc
Cyanophyte food quality
0.2
FQDET
Diatom food quality
0.5
asPO4
Sediment nitrate release rate
0.5
day-1
asNH 4
Sediment phosphate release rate
0.5
day-1
as NO3
Sediment ammonium release rate
0.5
day-1
P
Zooplankton biomass phosphorus turnover rate
0.094
day-1
N
Zooplankton biomass nitrogen turnover rate
0.094
day-1
m
Zooplankton moult P turnover rate
0.4
day-1
m
Moult as a fraction of zooplankton biomass
0.05