FOR ONLINE PUBLICATION ONLY
Appendix 1: PCLake model description
(adapted from Janse 2005)
Structure
The model PCLake describes a completely mixed water body and comprises both the water
column and the sediment top layer, with the most important biotic and abiotic components.
The model is thus meant for shallow, non-stratifying lakes. No further horizontal or vertical
distinction within the lake is taken into account. The model offers the option to include a
marsh zone with emergent vegetation, but that has not been used in this study.
Mathematically, the model is composed of a number of coupled differential equations, one
for each state variable, as listed in Appendix 1 - Table 1. The model is numerically solved by
the Runge-Kutta Cash-Karp method which has a variable time step. The structure of the lake
model is shown in Figure 1. All biota are modeled as functional groups. Besides mass fluxes
(food relations, and so on), the model also contains some ‘empirical’ or indirect relations
between components, such as the impact of fish and macrophytes on resuspension (see
below). The overall nutrient cycles for N, P and Si are described as completely closed (except
for external fluxes such as in- and outflow and denitrification). This was done by modeling
most components in three elements, that is, dry weight (abbreviated as D), nitrogen (N) and
phosphorus (P), detritus also in silica (Si). Inorganic carbon (CO2) is not explicitly modeled.
The nutrient-to-dry-weight ratios are thus variable. As the nutrient ratios of organisms
increase with their trophic level (that is, phytoplankton < zooplankton < fish), mechanisms
are included to allow for those differences, such as a higher assimilation efficiency for
nutrients than for carbon. The total mass balances per element are dynamically checked
during the calculations. ‘Day’ was chosen as a uniform time unit for all processes (but the
1
simulation time can be chosen as variable); however, the relevant time scale for the output
is about weeks to months. The main inputs to the model are: water inflow, infiltration or
seepage rate (if any), nutrient (N, P) loading, particulate loading, temperature and light,
dimensions (lake depth and fetch - determining indirectly wave intensity), sediment features
and loading history (initial conditions). As output, the biomass and concentrations of all state
variables, as well as a number of derived variables and fluxes, are calculated.
Processes
The processes in the model will be briefly described here; a complete description of the
model is given by Janse (2005).
a. Abiotic and microbial processes
At the base of the model are the transport processes: in- and outflow and external loading
by nutrients and by organic and inorganic matter. Infiltration to, or seepage from, the
groundwater can also be defined. The sediment top layer has a fixed thickness (default
0.1 m) and consists of inorganic matter (IM) (with a fixed fraction of clay particles), humus,
detritus and pore water. Exchange of IM and detritus between water and sediment may take
place via settling (described as a first-order process) and resuspension (zero-order process).
The settling rate decreases, and the resuspension increases, with the size of the lake. The
resuspension also increases with the sediment porosity and with the amount of
benthivorous fish (see below), while it decreases with the vegetation cover. A net increase of
sediment material is met by an equal amount considered as buried to deeper layers.
Mineralization of detritus (degradable organic matter) is described as a first-order process,
dependent on temperature assuming a maximum mineralization rate in the water of 1% d-1
and in the sediment of 0.2% d-1. Humus (refractory organic matter) is assumed to be
2
mineralized only very slowly. The released nutrients are dissolved in the pore water.
Inorganic P is subject to reversible adsorption to IM according to a Langmuir isotherm. It
might also precipitate in case of a very high concentration. The relative adsorption increases
with the sediment content of clay particles and with the aerobic proportion of the sediment.
The latter is modeled in a highly simplified way by defining a quasi-steady state oxygen
penetration depth (or aerobic sediment fraction), which is a function of the oxygen
concentration in the water, the potential sediment oxygen demand and the diffusion rate.
Nitrification of NH4 increases, denitrification of NO3 decreases with the aerobic proportion of
the sediment. Exchange of dissolved P and N between pore water and water column is
modeled according to the concentration differences. The combined result of the described
processes is that the PO4 release rate follows a seasonal cycle, dependent on the
temperature and the amount of detritus in the system. Mineralization and nitrification are
described in the water column as well. Oxygen in the water column is modeled dynamically,
dependent on the biochemical oxygen demand and sediment oxygen demand, the
reaeration from the atmosphere, and the oxygen production by phytoplankton and/or
submerged plants.
b. Phytoplankton
The phytoplankton module describes the growth and loss of the three functional groups of
phytoplankton, that is, cyanobacteria, diatoms and green algae. This distinction was made
because of their different characteristics. The biomass of each group is described by the
following differential equations:
dx/dt = production – respiration – mortality – settling + resuspension – grazing + transport
and by parallel equations for phytoplankton expressed in N and P units (here denoted by y):
3
dy/dt = uptake – excretion – mortality – settling + resuspension – grazing + transport
The production (carbon fixation, for simplicity taken as equivalent to growth) depends on
the maximum growth rate, temperature, day length, under-water light, P and N, for diatoms
also on silica. The temperature dependence is described using an optimum function. The
light dependent growth of cyanobacteria and diatoms is described according to Di Toro and
Matystik (1980), using Steele’s equation integrated with respect to the depth. This equation
implies growth inhibition at high light intensities. For green algae, a similar equation is based
on a Monod-type equation, assuming no light inhibition. The available light, taken as
‘photosynthetically active radiation’ (PAR), is determined by the light intensity at the water
surface and its extinction in the water column (Lambert-Beer’s law). The extinction
coefficient is the sum of the background extinction of the water and the contributions of IM,
detritus and phytoplankton (and submerged plants) to it, thus accounting for the selfshading effect that sets a limit to the maximum biomass. P and N affect the growth rate via
the internal nutrient contents of the phytoplankton rather than the external concentrations.
Nutrient uptake is thus described separately from the production, to allow for this variable
stoichiometry. The uptake rate increases with the external nutrient concentration up to a
maximum that is determined by the actual ratio (‘cell quota’), the minimum cell quota giving
the highest maximum rate (Riegman and Mur 1984). The biomass production is then
dependent on the cell quota according to the Droop (1974) equation: the growth rate
increases asymptotically with the cell quota provided it is above the minimum. For the silicadependent growth of diatoms, the more simple Monod formulation was chosen based on
the external SiO2 concentration, with a fixed Si content of the diatoms. The actual growth
rate is calculated by multiplying the maximum growth rate with the combined reduction
functions for light and temperature and the one for nutrients. The latter is taken as the
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minimum of the functions for N and P (and Si for diatoms), following Liebig’s law. The
chlorophyll-a content of the phytoplankton, a derived variable in the model, is assumed to
be variable, being higher in case of a more severe light limitation (Riegman 1985). Secchi
depth is calculated as the Poole-Atkins coefficient divided by the extinction. The loss
processes, maintenance respiration and natural mortality, are described as first-order
processes, respiration as temperature-dependent. Excretion of nutrients parallel to
respiration is assumed to decrease if the internal nutrient ratio is low. Settling is also
described as first order, the rate being the settling velocity [m d-1] divided by the water
depth. For ‘analogy’ reasons, the settled algae are included as separate state variables,
which may re-enter the water column by resuspension, coupled to the resuspension of other
particles (see above). It is assumed that the settled algae do not grow, but are subject to
respiration and mortality and may be eaten by zoobenthos. The parameter values of the
three algal groups in the model differ. The cyanobacteria have a higher light affinity (they are
shade-adapted) as well as a higher phosphorus uptake rate than the other groups. On the
other hand, they have a much lower maximum growth rate and a stronger sensitivity to
temperature. The diatoms have a lower temperature optimum, while the green algae are
not inhibited by high light intensities. Both these groups have higher growth rates, but also
higher loss rates through settling and zooplankton grazing (see below). The diatoms are the
only group that might be limited by silica.
c. Aquatic vegetation
The submerged vegetation is described as one lumped group by the following differential
equation for the biomass:
dx/dt = production - respiration – mortality
5
and for nutrients (N and P) stored in the plants:
dy/dt = uptake - excretion - mortality
It is assumed that the biomass is divided in an under-ground part (roots) and an aboveground part (shoots), and that the latter is homogeneously divided over the water column.
Seasonality is modeled in a simplified way by assuming a high root fraction in the winter
period and a low one during the growing season (default 0.6 and 0.1, resp.). The switch
between both values in spring (triggered by water temperature) and autumn (triggered by
season) mimics allocation and reallocation processes. The modeled vegetation thus stands
for plants with overwintering parts. Biomass production by the shoot is modeled largely
analogous to the phytoplankton production, that is, dependent on maximum growth rate,
temperature, day length, under-water light, N and P. It is assumed that the macrophytes
may extract nutrients from both the water and the sediment pore water, largely according
to availability. In practice, sediment uptake is mostly higher. Respiration and nutrient
excretion are modeled as for phytoplankton. Natural mortality is assumed to be low in the
growing season and high at the end of it; a fixed fraction (default 0.3) is assumed to survive
the winter. The description of the growth and mortality is combined with a densitydependent correction derived from the logistic growth equation, to account for other factors
than the ones explicitly modeled, for instance space, that might be limiting for the plant
density that could maximally be achieved, the ‘carrying capacity’. The vegetation is assumed
to have some indirect impacts on other components of the system, that is, a hampering of
resuspension, a slight negative impact on the feeding efficiency of benthivorous fish and a
positive influence on the growth of predatory fish.
d. Food web
6
The food web module is kept as simple as possible and comprises zooplankton, zoobenthos,
planktivorous, benthivorous and predatory fish. The general equation for the animal groups
is:
dx/dt = (feeding – egestion) – respiration – mortality - predation
combined with a density-dependent correction derived from the logistic growth equation
(Hallam and others 1983; Traas 2004). The carrying capacities have been set to high values
(in the range of maximum observed biomasses). Zooplankton feeds on both phytoplankton
and detritus. Grazing is described as a Monod-like function of the seston concentration, the
specific filtering rate decreasing hyperbolically with increasing seston concentration (Gulati
and others 1982; Gulati and others 1985). A selectivity constant is used for each food species
to account for preference of the zooplankton: green algae > diatoms > detritus >
cyanobacteria (for example, Gliwicz 1980). The assimilation efficiency for the consumed food
is constant and quite low (0.3) for carbon (Gulati and others 1985), but variable (depending
on the internal P ratio of the food) and, therefore, mostly higher for phosphorus. This is one
of the mechanisms by means of which the differences in P content between the trophic
levels are maintained. Zoobenthos is assumed to feed on sediment detritus and a bit on
settled algae, also by a Monod-type (or ‘type II’) functional response. It is also assumed to be
able to ‘accumulate’ P from its food comparable to zooplankton. All fish predation processes
are modeled as a so-called ‘type III’ response (Holling 1965): the predation rate depends on
prey density according to a sigmoid curve. Planktivorous fish feeds on zooplankton,
benthivorous fish on zoobenthos, and predatory fish on planktivorous and benthivorous fish.
Planktivorous fish are considered as juvenile and benthivorous as adult fish. Spawning is
simulated as the transfer, every May, of a small proportion of the adult biomass to the
juvenile biomass. At the end of each year, half the juvenile biomass becomes ‘adult’. Also
7
planktivorous and benthivorous fish are assumed to have a relatively higher phosphorus
assimilation efficiency, as the internal P content of fish is again much higher than that of its
food organisms (Kitchell and others 1975). For predatory fish, this mechanism does not play
a role any more. An indirect effect of benthivorous fish that is included in the model is its
stirring up of the sediment during feeding, causing a flux of particles and nutrients to the
water column (Breukelaar and others 1994). Predatory fish is assumed to be dependent on
the presence of vegetation.
References belonging only to Appendix 1
Di Toro, DM and Matystik, WF. 1980. Mathematical models of water quality in large lakes Part 1:
Lake Huron and Saginaw Bay. University of Delaware, USA.
Droop MR. 1974. Nutrient status of algal cells in continuous culture. Journal of the Marine Biological
Association of the United Kingdom 54: 825-855.
Gliwicz ZM. 1980. Filtering rates, food size selection, and feeding rates in cladocerans-another aspect
of interspecific competition in filter-feeding zooplankton. American Society of Limnology and
Oceanography Special Symposium 3: 282-291.
Gulati RD, Siewertsen K, Postema G. 1985. Zooplankton structure and grazing activities in relation to
food quality and concentration in Dutch lakes. Archiv der Hydrobiologie: Beiheft, Ergebnisse
der Limnologie 21: 91-102.
Hallam TG, Clark CE, Lassiter RR. 1983. Effects of toxicants on populations - a qualitative approach.1:
Equilibrium environmental exposure. Ecological Modelling 18: 291-304.
Holling CS. 1965. The functional response of predators to prey density and its role in mimicry and
population regulation. Memoirs of the Entomological Society of Canada 97: 5-60.
Kitchell JF, Koonce JF, Tennis PS. 1975. Phosphorous flux through fishes. Verhandlungen des
Internationalen Verein Limnologie 19: 2478-2484.
Riegman, R. 1985. Phosphate - phytoplankton interactions. University of Amsterdam, Amsterdam.
Riegman R, Mur LR. 1984. Regulation of Phosphate-Uptake Kinetics in Oscillatoria agardhii. Archives
of Microbiology 139: 28-32.
Traas, TP. 2004. Food web models in ecotoxicological risk assessment. Ch. 6: A mass-balance for the
logistic population growth and implications for biomass turnover. Utrecht University,
Utrecht.
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Appendix 1 - Table 1: State Variables in PCLake
Component (state variable)
In water column
In sediment top layer
Water
water depth
water
-- (fixed)
Abiotic components
inorganic matter (IM)
D
D
humus
--
D
detritus
D, P, N, Si
D, P, N, Si
nutrients
PO4, Pads, NH4, NO3, SiO2
PO4, Pads, NH4, NO3
oxygen
O2
-- (aerobic fraction)
D, P, N, (Si)
D, P, N, (Si)
green algae
D, P, N
D, P, N
cyanobacteria
D, P, N
D, P, N
Phytoplankton
diatoms
Vegetation
submerged vegetation
D, P, N
Animal groups
zooplankton
D, P, N
zoobenthos
D, P, N
planktivorous fish
D, P, N
benthivorous fish
D, P, N
predatory fish
D, (P, N)
Abbreviations: D = dry weight, P = phosphorus, N = nitrogen, Si =silica, O 2 = oxygen.
9
50
40
30
20
10
0
0
2
4
6
8
60
60
change in bioturbation [%]
60
change in benthic fish biomass [%]
change in zoobenthos biomass [%]
Appendix 2: Changes in Biomasses and Process Rates Over Range of t-POM Loading for the Three Pathways
50
40
30
20
10
0
10
0
14
12
10
8
6
4
2
0
2
4
6
t-POM loading
6
8
30
20
10
0
10
0
8
10
2
4
6
8
10
8
10
t-POM loading
25
7
6
20
change in extinction [%]
16
0
4
40
t-POM loading
change in authochthonous detritus [%]
change in inorganic matter (water) [%]
t-POM loading
2
50
15
10
5
0
0
2
4
6
t-POM loading
8
10
5
4
3
2
1
0
0
2
4
6
t-POM loading
Appendix 2 – Figure 1: The percental change in variables belonging to the zoobenthos pathway over a range of t-POM loadings [g dw m-2 d-1 in autumn]. The percental change
is based on the lowest t-POM loading. Scenarios in the clear-water state (P load 0.7 mg P m-2 d-1) are presented as solid lines and in the turbid state (P load 3.3 mg P m-2 d-1) as
dotted lines. Consider the different scaling of the y-axes. Read figures from left to right and top to bottom to follow mechanistic sequence of t-POM effects.
10
3
2
1
0
40
30
20
10
0
-1
4
3
2
1
0
-1
0
2
4
6
8
10
0
-2
-4
-6
-8
-10
-12
-14
-16
2
4
6
t-POM loading
4
6
8
10
0
8
10
2
4
6
8
10
t-POM loading
18
2500
2000
1500
1000
500
0
0
2
4
6
8
10
t-POM loading
10
16
change in extinction [%]
0
0
2
t-POM loading
change in authochthonous matter [%]
t-POM loading
change in autochthonous
matter consumed [%]
change in suspended
terrestrial matter consumed [%]
4
5
change in consumed food [%]
50
change in available food [%]
change in zooplankton biomass [%]
5
14
12
10
8
6
4
2
8
6
4
2
0
0
0
2
4
6
t-POM loading
8
10
0
2
4
6
8
10
t-POM loading
Appendix 2 – Figure 2: The percental change in variables belonging to the zooplankton pathway over a range of t-POM loadings [g dw m-2 d-1 in autumn]. The percental change
is based on the lowest t-POM loading. Scenarios in the clear-water state (P load 0.7 mg P m-2 d-1) are presented as solid lines and in the turbid state (P load 3.3 mg P m-2 d-1) as
dotted lines. Consider the different scaling of the y-axes. Read figures from left to right and top to bottom to follow mechanistic sequence of t-POM effects.
11
14
2500
12
change in extinction [%]
change in susepended terrestrial matter (water) [%]
3000
2000
1500
1000
500
0
10
8
6
4
2
0
0
2
4
6
t-POM loading
8
10
0
2
4
6
8
10
t-POM loading
Appendix 2 – Figure 3: The percental change in variables belonging to the extinction pathway over a range of t-POM loadings [g dw m-2 d-1 in autumn]. The percental change is
based on the lowest t-POM loading. Scenarios in the clear-water state (P load 0.7 mg P m-2 d-1) are presented as solid lines and in the turbid state (P load 3.3 mg P m-2 d-1) as
dotted lines. Consider the different scaling of the y-axes. Read figures from left to right to follow mechanistic sequence of t-POM effects.
12
Appendix 3: Functional Response Zooplankton
The functional response for zooplankton is implemented in PCLake according to Gulati and
others (1982) who found in a long-term study that an increased amount of an unpreferred food
reduced the total amount of food consumption above a certain food concentration. To
illuminate this type of functional response we compare it for a system with one prey (A preferred) and with two prey (A, B - unpreferred). The predator has a certain preference for
each prey, pA and pB with pA > pB. Without loss of generality we put pA=1. Both functional
𝐴
responses, 𝑓(𝐴) = ℎ+𝐴 and 𝑓(𝐴 + 𝐵) =
𝐴+𝑝𝐵∗𝐵
,
ℎ+𝐴+𝐵
have the same half-saturation constant h (Eq. 1).
The presence of the additional food B enhances the overall food consumption per predator
when A is relatively low and reduces it at high concentrations of A (Appendix 3 – Figure 1). The
threshold value of A when the food consumption is equal with or without B can be determined
in the following way:
𝐴
𝑓(𝐴) = ℎ+𝐴 =
𝐴+𝑝𝐵∗𝐵
ℎ+𝐴+𝐵
= 𝑓(𝐴 + 𝐵)
(1)
This equation can be rearranged into:
𝐴 ∗ (ℎ + 𝐴 + 𝐵) = (𝐴 + 𝑝𝐵 ∗ 𝐵) ∗ (ℎ + 𝐴)
𝐴 ∗ ℎ + 𝐴 ∗ 𝐴 + 𝐴 ∗ 𝐵 = 𝐴 ∗ ℎ + 𝑝𝐵 ∗ 𝐵 ∗ ℎ + 𝐴 ∗ 𝐴 + 𝑝𝐵 ∗ 𝐵 ∗ 𝐴
𝐴 ∗ 𝐵 = 𝑝𝐵 ∗ 𝐵 ∗ ℎ + 𝑝𝐵 ∗ 𝐵 ∗ 𝐴
𝐴∗𝐵
𝐵∗ℎ+𝐵∗𝐴
𝐴
ℎ+𝐴
= 𝑝𝐵
= 𝑝𝐵
(2)
13
𝐴
Given that during most of the simulations done with PCLake the equivalent term ℎ+𝐴 is larger
than pB, the additional food B often reduces the overall food consumption (Appendix 3 – Figure
1).
1.0
without additional food
with additional food
consumed food per predator
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
concentration of preferred food
Appendix 3 - Figure 1: The consumed food per predator increases as a function of the concentration of the
preferred food (A) less steep when an additional less preferred food (B) is available. (pB=0.5; B=1; h=1)
14