Nutrient – Phytoplankton – Zooplankton Modeling
Models incorporating nutrients, phytoplankton and zooplankton are widely applied in
marine and freshwater systems in both applied and research situations. Consider our
interest in partitioning dissolved and particulate phosphorus in trophic state modeling and
in simulating solids concentrations in toxic substances modeling.
Equations of state are developed for each of the three variables: nutrients (n),
phytoplankton or algae (a) and zooplankton (z).
Phytoplankton mass balance: building from a simple exponential model,
da
  k g  kd   a
dt
where kd  kra  k gz , i.e. includes losses to
respiration,
da
  kra  a
dt
and zooplankton grazing,
da
  k gz  a
dt
where : k gz  Cgz 
a
T  20
  gz   z
ksa  a
combining the equations yields,

da 
a
 k g  n, T , I   kra  Cgz 
  gzT 20  z   a
dt 
ksa  a

where:
a
n
T
kg
kgz
I
kra
Cgz
ksa
θ
z
phytoplankton concentration
nutrient concentration
temperature
phytoplankton growth rate
grazing loss rate
light
respiration rate coefficient
zooplankton grazing rate
algae half-saturation constant
temperature adjustment for grazing
zooplankton concentration
mgChla∙m-3
mgP∙m-3
°C
d-1
d-1
µE∙m-2∙s-1
d-1
m3∙gC-1∙d-1
mgChla∙m-3
dimensionless
gC∙m-3
Let’s take a closer look at each component,
Growth
The growth (kg) term is a ‘maximum specific’ value and varies as a function of nutrients
(n), temperature (T) and light (I),
k g  f n, T , I 
with the nutrient function described by the Michaelis-Menten or Monod models,
f  n 
n
n  ksn
the temperature function utilizing either a linear or theta model,
k g  k g ,ref 
T  Tmin
Tref  Tmin
or
k g  k g ,ref 
T 20
or a function describing an optimum temperature,
k g ,T  k g ,opt  exp
1(T Topt )2
for T  Topt
k g ,T  k g ,opt  exp
 2 (ToptT )2
for T  Topt
and the light function using either a saturating (Michaelis-Menten / Monod) approach,
f I  
I
I  k si
or an inhibition (Steele) relationship,
I
 1
I
f I    exp I s
Is
Note that application of the light function requires additional computations.
I z  I 0  exp  ke z
where ke is the light extinction coefficient which includes components for absorption by
water, dissolved color, inanimate suspended particles and phytoplankton,
k e  k e , w  k e , g  k e , s  k e ,a
Non-Predatory Losses
The non-predatory losses, respiration and excretion, are grouped into a single, first-order
term (kra in the equation above). Fundamentally, these processes represent losses of
organic carbon through the release carbon dioxide and extracellular byproducts. In the
phytoplankton mass balance, the processes represent the overall loss of biomass
(chlorophyll). Phytoplankton respiration and excretion will form part of the nutrient
mass balance developed later. The respiration rate coefficient is typically made to vary
with temperature using a theta function. Due to the impact of this term on the mass
balance, some modelers choose to express respiration as a percentage of the growth rate.
Otherwise, the effects of light, temperature and nutrients on growth often lead to a net
negative growth rate coefficient which imposes a more dramatic response in behavior
than is evidenced in nature.
Predatory Losses
Predatory loss refers to the grazing of phytoplankton by zooplankton. The grazing loss
term includes a zooplankton grazing rate, a temperature effect, and the abundance of
phytoplankton and zooplankton,
da
a
T  20
  Cgz 
  gz   z  a
dt
k sa  a
Note that the product of the zooplankton grazing rate and the zooplankton concentration
is simply a first order loss term (d-1). That loss is then influenced by temperature and,
through a Michaelis-Menten function, the abundance of algae.
Zooplankton mass balance: basically gains through grazing and losses to respiration,
dz
a
T  20
 aca    Cgz 
  gz   z  a  krz  z
dt
ksa  a
with the introduction of two new terms, aca (gC∙mgChl-1), the carbon to chlorophyll ratio
of the phytoplankton, used to convert phytoplankton consumption to zooplankton
biomass, and  (dimensionless), a grazing efficiency factor ranging from 01 which
accommodates losses due to ‘sloppy feeding’, i.e. production of detritus. This term is
important not only to the zooplankton mass balance, but also is found in that for nutrients
as a means of nutrient recycle. The respiration term, which could be modified to account
for temperature effects, will also be part of the nutrient mass balance.
Nutrient mass balance: losses to phytoplankton growth and gains from recycle due to
phytoplankton respiration and zooplankton respiration and egestion,
( T 20)
dp
a
 a pa  1     Cgz 
  gz  z  a
dt
ksa  a
 a pa  kra  a  a pc  krz  z  a pa  k g  a
here, introducing two more terms, apa (mgP∙mgChl-1), the phosphorus to chlorophyll ratio
of the phytoplankton, used to convert phytoplankton respiration to phosphorus, and apc
(mgC∙gC-1), the phosphorus to carbon ratio of the zooplankton, used to convert
zooplankton respiration to phosphorus.
These three equations thus represent a simple nutrient-food chain model which can be
used to gain understanding of predator-prey dynamics.
1.0
0.9
ks = 5
0.8
0.7
ks =15
f(N)
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
25
50
75
N
100
125
150
1.2
1.0
Monod
0.8
f(I)
Steele
0.6
0.4
0.2
ks
Is
0.0
0
200
400
600
800
1000
1200
I
3.5
3.0

2.5
f(T)
2.0

1.5
Topt
1.0
Tref
0.5
0.0
0
Tm in
10
20
T
30
40