- Understand the process table notation introduced in section 4.1 of the manuscript.
- Understand the formulation of biogeochemical and ecological process rates introduced in section 4.2.
- Increase your familiarity with the R package
`ecosim`

. - Understand the behaviour of the solutions of the lake phyto- and zooplankton model described in section 11.2 under constant and seasonally varying environmental conditions.

Study the implementation of the model described in section 11.2 for constant driving forces. Run the model implementation below and interpret the behavior of the solutions.

```
# Model with constant driving forces
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# load required packages:
if ( !require("deSolve") ) {install.packages("deSolve"); library("deSolve") }
```

`## Loading required package: deSolve`

`## Warning: package 'deSolve' was built under R version 4.3.2`

`if ( !require("ecosim") ) {install.packages("ecosim"); library("ecosim") }`

`## Loading required package: ecosim`

`## Warning: package 'ecosim' was built under R version 4.3.2`

`## Loading required package: stoichcalc`

```
# definition of model parameters:
param <- list(k.gro.ALG = 0.5, # 1/d
k.gro.ZOO = 0.4, # m3/gDM/d
k.death.ALG = 0.1, # 1/d
k.death.ZOO = 0.05, # 1/d
K.HPO4 = 0.002, # gP/m3
Y.ZOO = 0.2, # gDM/gDM
alpha.P.ALG = 0.003, # gP/gDM
A = 5e+006, # m2
h.epi = 5, # m
Q.in = 5, # m3/s
C.HPO4.in = 0.04, # gP/m3
C.HPO4.ini = 0.04, # gP/m3
C.ALG.ini = 0.1, # gDM/m3
C.ZOO.ini = 0.1) # gDM/m3
```

- Next, we define the processes of growth and death of algae and zooplankton as objects of the class
`process`

}`of the package`

ecosim`. Each process is defined by its name, rate, and stoichiometry. The rate is defined as an expression that can use parameters (defined for the object of class`

system`below), concentrations defined in objects of class`

reactor`}`

that are part of the object of class`system`

}`. To define the stoichiometry, a named list of expressions must be provided that identifies the substance or organism concentrations as the names and contains the stoichiometric coefficients as expressions.

```
# definition of transformation processes
# growth of algae:
gro.ALG <- new(Class = "process",
name = "Growth of algae",
rate = expression(k.gro.ALG
*C.HPO4/(K.HPO4+C.HPO4)
*C.ALG),
stoich = list(C.ALG = expression(1), # gDM/gDM
C.HPO4 = expression(-alpha.P.ALG))) # gP/gDM
# death of algae: TO BE COMPLETED
death.ALG <- new(Class = "process",
name = "Death of algae",
rate = expression(k.death.ALG*C.ALG),
stoich = list(C.ALG = expression(-1))) # gDM/gDM
# growth of zooplankton:
gro.ZOO <- new(Class = "process",
name = "Growth of zooplankton",
rate = expression(k.gro.ZOO
*C.ALG
*C.ZOO),
stoich = list(C.ZOO = expression(1), # gDM/gDM
C.ALG = expression(-1/Y.ZOO))) # gP/gDM
# death of zooplankton: TO BE COMPLETED
death.ZOO <- new(Class = "process",
name = "Death of zooplankton",
rate = expression(k.death.ZOO*C.ZOO),
stoich = list(C.ZOO = expression(-1))) # gDM/gDM
```

- Next, we define the mixed box describing the epliminion of the lake as an object of the class
`reactor`

of the package`ecosim`

.

```
# definition of reactor to describe the epilimnion of the lake: TO BE COMPLETED
epilimnion <-
new(Class = "reactor",
name = "Epilimnion",
volume.ini = expression(A*h.epi),
conc.pervol.ini = list(C.HPO4 = expression(C.HPO4.ini), # gP/m3
C.ALG = expression(C.ALG.ini), # gDM/m3
C.ZOO = expression(C.ZOO.ini)), # gDM/m3
inflow = expression(Q.in*86400), # m3/d
inflow.conc = list(C.HPO4 = expression(C.HPO4.in),
C.ALG = 0,
C.ZOO = 0),
outflow = expression(Q.in*86400),
processes = list(gro.ALG,death.ALG,gro.ZOO,death.ZOO))
```

Finally, we combine the reactor, the parameters, and the desired output times in an object of class `system`

of the package `ecoval`

.

```
# definition of the system consisting of a single reactor:
# observe the time frame of the simulation
system.11.2.a <- new(Class = "system",
name = "Lake",
reactors = list(epilimnion),
param = param,
t.out = seq(0,2*365,by=1))
```

Note that this object contains all definitions of the configuration of reactors (in this case just a single one), the processes active in each reactor, the model parameters, and the output time points. Any simulations carried out will refer to the definitions in this object, and not to the external variables that we used to set up the elements of the system.

- Perform a simulation and store the results by using the function
`calcres`

of the package`ecosim`

:

```
# perform simulation:
res.11.2.a <- calcres(system.11.2.a)
# plot results with default options:
plotres(res.11.2.a)
```

```
# variables in a vector 'c()' are plotted in the same graph
plotres(res=res.11.2.a,colnames=c("C.ALG", "C.HPO4","C.ZOO"))
```

```
# variables in a list 'list()' are plotted in different graphs
plotres(res=res.11.2.a,colnames=list("C.ALG", "C.HPO4", "C.ZOO"))
# combination of the two
plotres(res=res.11.2.a,colnames=list("C.HPO4",c("C.ALG", "C.ZOO")))
```

```
# plot and save as pdf
plotres(res = res.11.2.a,
colnames = list("C.HPO4",c("C.ALG", "C.ZOO")),
file = "exercise_2_results_a.pdf",
width = 10,
height = 5)
```

```
## png
## 2
```

Study the implementation of the model extension to seasonally varying driving forces. First, run the model implementation below.

We adapt system definitions:

```
# Model with seasonally varying driving forces
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# copy the previous system definition:
system.11.2.b <- system.11.2.a
# extend model parameters:
param <- c(param,
list(beta.ALG = 0.046, # 1/degC
beta.ZOO = 0.08, # 1/degC
T0 = 20, # degC
K.I = 30, # W/m2
lambda.1 = 0.10, # 1/m
lambda.2 = 0.10, # m2/gDM
t.max = 230, # d
I0.min = 25, # W/m2
I0.max = 225, # W/m2
T.min = 5, # degC
T.max = 25)) # degC
# extend growth of algae and zooplankton by considering environmental factors:
gro.ALG.ext <-
new(Class = "process",
name = "Growth of algae extended",
rate = expression(k.gro.ALG
*exp(beta.ALG*(T-T0))
*C.HPO4/(K.HPO4+C.HPO4)
*log((K.I+I0)
/(K.I+I0*exp(-(lambda.1+lambda.2*C.ALG)*h.epi)))
/((lambda.1+lambda.2*C.ALG)*h.epi)
*C.ALG),
stoich = list(C.ALG = 1, # gDM/gDM
C.HPO4 = expression(-alpha.P.ALG))) # gP/gDM
gro.ZOO.ext <-
new(Class = "process",
name = "Growth of zooplankton",
rate = expression(k.gro.ZOO
*exp(beta.ZOO*(T-T0))
*C.ALG
*C.ZOO),
stoich = list(C.ZOO = expression(1), # gDM/gDM
C.ALG = expression(-1/Y.ZOO))) # gP/gDM
# re-define processes in the reactor "epilimnion":
epilimnion@processes <- list(gro.ALG.ext,death.ALG,gro.ZOO.ext,death.ZOO)
# make environmental conditions (light and temperature) time dependent:
epilimnion@cond <- list(I0 = expression(0.5*(I0.min+I0.max)+
0.5*(I0.max-I0.min)*
cos(2*pi/365.25*(t-t.max))), # W/m2
T = expression(0.5*(T.min+T.max)+
0.5*(T.max-T.min)*
cos(2*pi/365.25*(t-t.max)))) # degC
# plot the environmental conditions
t <- seq(1,2*365) # for two years
I0 <- numeric(0)
T <- numeric(0)
for(i in 1:length(t))
{
I0[i] <- eval(epilimnion@cond$I0, envir=c(param, t=t[i]))
T[i] <- eval(epilimnion@cond$T, envir=c(param, t=t[i]))
}
par(mfrow=c(1,2),xaxs="i",yaxs="i",mar=c(4.5,4.5,2,1.5)+0.1)
plot(t, I0, type="l")
plot(t, T, type="l")
```

```
# re-define the reactor "epilimnion" in the system definition:
system.11.2.b@reactors <- list(epilimnion)
# increase algal growth rate to compensate for new limitations:
param$k.gro.ALG <- 0.8
# replace parameters in the system definition:
system.11.2.b@param <- param
```

Second, redo the simulations and plot and interpret the behavior of the solutions.

```
# redo simulations and plot results:
res.11.2.b <- calcres(system.11.2.b)
plotres(res=res.11.2.b, colnames=list("C.HPO4",c("C.ALG", "C.ZOO")))
```

`plotres(res=res.11.2.b[1:365,], colnames=list("C.HPO4",c("C.ALG", "C.ZOO")))`

```
plotres(res = res.11.2.b, # plot to pdf file
colnames = list("C.HPO4",c("C.ALG","C.ZOO")),
file = "exercise_2_results_b.pdf",
width = 10,
height = 5)
```

```
## png
## 2
```

We compare the two simulations.

```
# comparison of the two simulations:
plotres(res = list(const=res.11.2.a,dyn=res.11.2.b),
colnames = list("C.HPO4","C.ALG","C.ZOO"))
plotres(res = list(const=res.11.2.a,dyn=res.11.2.b),
colnames = list("C.HPO4","C.ALG","C.ZOO"),
file = "exercise_2_results_b.pdf",
width = 10,
height = 8)
```

```
## png
## 2
```

**Fill in the missing terms** and perform a sensitivity analysis of the model results to the parameters

\(C_{in, HPO^{2-}_{4}}\), \(Q_{in}\), \(k_{gro,ALG}\), \(k_{death,ALG}\), \(k_{gro,ZOO}\), \(k_{death,ZOO}\), and \(Y_{ZOO}\)

by modifying their values by factors of 2 and 1/2 for the model under constant driving forces using the function `calcsens`

. Interpret the results. *Hint:* look at the solution of the Task 5 of Exercise 1.

```
# Sensitivity analysis with constant driving forces
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# calculate results of sensitivity analysis: TO BE COMPLETED
res.11.2.a.sens <- calcsens(system.11.2.a,
param.sens=c("C.HPO4.in",
"k.gro.ZOO",
"k.death.ZOO",
"k.gro.ALG",
"k.death.ALG",
"K.HPO4",
"Y.ZOO",
"alpha.P.ALG",
"Q.in"))
# plot results:
plotres(res = res.11.2.a.sens,
colnames = list("C.HPO4","C.ALG", "C.ZOO"))
```