## 5 Discussion  
Our results highlight the importance of performing uncertainty quantification in validated biological models, even in the simple case of saturating growth dynamics encountered for green algae. For example, the ordinary least squares regression seemed to indicate that each of the models we investigated provide reasonable fits to the algae growth data. In addition, parameter estimates were consistent between replicate data sets for each model. From this information alone, \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter/Variable** & **Description** & **Units** \\ \hline \(p(t,i)\) & Number of daphnids of age \(i\) & \# of daphnids \\ \(N(t)\) & Total population size at time \(t:=\sum_{i=1}^{i_{max}}p(t,i)\) & \# of daphnids \\ \(q\) & Density-dependent fecundity constant & dimensionless \\ \(\alpha(i)\) & Density-independent fecundity rates & \# neonates-daphnid\({}^{-1}\).day\({}^{-1}\) \\ \(\mu\) & Density-independent survival rate & day\({}^{-1}\) \\ \(\tau\) & Delay for density-dependent fecundity & days \\ \(c\) & Density-dependent survival constant & dimensionless \\ \(M(t)\) & Total biomass at time \(t:=\sum_{i=1}^{i_{max}}p(t,i)\frac{kZ_{0}\sigma^{i}}{k+2\alpha(\sigma^{i-1})}\) & mm \\ \(k\) & Average maximum daphnid size (major axis) & mm/hour \\ \(r\) & Average daphnid growth rate & mm/hour \\ \(Z_{0}\) & Average neonate size (major axis) & mm \\ \(R\) & Intrinsic growth rate of algae & cells \(\cdot 10^{7}\).L\({}^{-1}\).day\({}^{-1}\) \\ \(K\) & Algal population carrying capacity & cells \(\cdot 10^{7}\).L\({}^{-1}\) \\ \(\delta\) & Density dependent predation constant & mm\({}^{-1}\).cells\(\cdot 10^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of _Daphnia magma_ and algae model parameters and variables. Figure 5: Simulations of the coupled daphnia and green algae dynamics model resulting in steady state behavior (**Left**, \(c=0.01\)) and sustained oscillations (**Right**, \(c=0.04\)). one might conclude that the Bernoulli model was the best performing model, since it best fit the initial data \(X_{0}\) (Figure 4). However, a deeper investigation with uncertainty analysis allowed us to generate confidence bands around the fits to the data, showing that the Bernoulli model was the model for which we could have the least confidence. This result was corroborated by a thorough examination of the standard errors for each of the parameters with similar interpretations across all models. For example, the growth rates for the Bernoulli model had unreasonably high standard errors, while the uncertainty in the corresponding growth rates for the other models were relatively low. Our results also indicated that the Gompertz model had inconsistent standard errors between asymptotic theory and bootstrapping for the initial condition and saturation parameter, emphasizing the importance of using multiple uncertainty quantification techniques to ascertain the best validated model. We observed that the logistic and DEL models have different confidence regions for the model solution in Figures 1 and 2. We attribute the change in computed confidence regions to the differences in numerical discretization, the time step for the DEL model was equal to one hour while the logistic model had a coarser mesh. We collected replicate data and our results had strong agreement across the three replicates. Although methodology exists to fit all three data sets simultaneously to the same model, we chose to fit them separately to test whether the model validation results were consistent across several repeated experiments. We noticed a slight trend in the residuals resulting from fits of the model to data for each replicate: the fit sometimes either over- or under-estimates the data in groups of threes (Supplementary Figures S1-S4). We surmise that this phenomenon may be explained by human measurement error; some people tend to over or under count the algae when using a hemocytometer.