## 4 Results
### Coupling to the Discrete-Time Daphnia magna Population Model  
Our uncertainty analysis results indicate that the Logistic model and its numerically discretized counterpart, the DEL model, are the most accurate models among those we investigated. We forgo a summary of the evidence for this conclusion until the discussion section. Our ultimate aim of validating a model for green algae growth was to couple it to a model for _D. magna_ population dynamics. In this section, we couple two validated models for algae and _Daphnia_ population growth to create a theoretical, unvalidated model to explore potentially complex predator-prey dynamics. The _D. magna_ model we use is one that we recently validated with experimental population data [5]. The validated _D. magna_ model is a specification of the following discrete-time discrete-age structured population model:  
\[\left[ \begin{array}{c}p(t+1,1)\\ p(t+1,2)\\ p(t+1,3)\\ \vdots\\ p(t+1,i_{max})\end{array} \right]= \begin{bmatrix}a(t,1)&a(t,2)&a(t,3)&\ldots&a(t, i_{max})\\ b(t,1)&0&0&\ldots&0\\ 0&b(t,2)&0&\ldots&0\\ \vdots&&\ddots&\ldots&\vdots\\ 0&0&0&\ldots&b(t,i_{max}-1)\end{bmatrix} \left[ \begin{array}{c}p(t,1)\\ p(t,2)\\ p(t,3)\\ \vdots\\ p(t,i_{max})\end{array} \right]. \tag{32}\]  
The population is divided into one-day age classes up to some maximum age \(i_{max}\) and the number of daphnids of age \(i\) at a time \(t\) is \(p(t,i)\). The average fecundity of each age class \(i\) is given by \(a(t,i)\) and the survival rate for daphnids of age \(i\) is given by \(b(t,i)\). The validated functional forms are \(a(t,i)=\alpha(i)(1-q)^{M(t-\tau)}\) and \(b(t,i)\) is defined piecewise as \(\mu(1-c)^{M(t)}\) if \(i\leq 4\) and \(\mu\) if \(i\geq 5\). A summary of the parameters and variables in the model are listed in Table 4 (see [5] for further details). We coupled the _D. magna_ population model to the DEL green algae model by assuming that _D. magna_ consumes green algae and that the density-dependent survival and fecundity rates of _D. magna_ are influenced directly by the algae concentration. We modeled the algae population with predation as  
\[P_{t+1}=P_{t}+RP_{t}\left(1-\frac{P_{t}}{K}\right)-\delta M_{t}P_{t} \tag{33}\]  
where \(\delta\) is a predation coefficient, and \(M_{t}\) is the total _Daphnia_ biomass at time \(t\). We chose this functional form based on the assumption that _Daphnia_ consume algae at a rate proportional to the density of aglae and the total biomass of the _Daphnia_ population, as adult daphnids consume food at a higher rate than younger ones. We used a 24 h time discretization to model algae growth for our simulation study and transformed parameters accordingly, setting \(\delta=0.001\), \(K=0.4559\), \(R=1.34\), and the initial algae population \(P_{0}=0.0633\). We modeled the algal influence on _Daphnia_ fecundity and survivorship by setting \(a(t,i)=\alpha(i)(1-q)^{1/P_{t-\tau}}\) and \(b(t,i)=\mu(1-c)^{1/P_{t}}\) if \(i\leq 4\). In this model, we changed the functional form of the _Daphnia_ model based on the assumption that the negative density-dependent fecundity and survivorship effects that daphnids experience are driven by a lack \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Asymptotic Results: \(\beta\)** & **Replicate** & **Estimate** & **SE** \\ \hline  & 1 & 2.1646 & 2.5440 \\  & 2 & 3.4574 & 2.8118 \\  & 3 & 2.8188 & 2.8758 \\ \hline
**Bootstrapping Results: \(\beta\)** & **Replicate** & **Estimate** & **SE** \\  & 1 & 38.31 & 113.72 \\  & 2 & 29.78 & 92.89 \\  & 3 & 43.27 & 113.81 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: \(\beta\) estimate and standard error for the Bernoulli model. of food in the form of algae, represented as \(\frac{1}{P_{t}}\) and its time-delayed analogue \(\frac{1}{P_{t-t}}\). The _Daphnia_ matrix model is otherwise unchanged. We found that a coupled model could result in both steady state dynamics as well as oscillatory behavior for different choices of parameter values in the _Daphnia_ model (Figure 5).