## 4 Results  
### Data fitting and Model Comparisons  
Overall, we found that all models provide a reasonable fit to the data. Figures 1-4 show results of the least squares estimation for the three different replicates of the Logistic, Bernoulli, DEL, and Gompertz models. These figures contain 68% and 95% confidence bands around the fits to data. These were constructed by generating 1000 random parameter sets from a normal distribution described by the mean and standard error obtained by the asymptotic theory results, computing the model for each of these parameter sets, and then calculating the respective confidence intervals from model generated points \(f(t_{i},\theta^{k})\), where \(k=1,\ldots,1000\)[5]. One primary difference between the fits to data that we found was that each model tends to underestimate the initial data and the Bernoulli model provided the closest fit (Figure 4). We note that we chose to estimate the initial condition due to measurement error associated with a low cell density as well as how much influence these discrepancies in error would affect the outcome of the model.  
Figure 1: Plots of forward solutions for the Logistic curve for the three replicates of the data. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as \(cells\times 10^{7}/\)L. Data points are shown as \"*\".  
Figure 2: Plots of forward solutions for the discrete Euler-method logistic (DEL) please confirm. model for the three replicates of the data from left to right. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as \(cells\times 10^{7}\)/L. Data points are shown as \({}^{\prime\bullet\bullet}\).  
Figure 3: Plots of forward solutions for the Gompertz curve for the three replicates of the data. Replicate one is on top and three is on the bottom. The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as \(cells\times 10^{7}/\)L. Data points are shown as \({}^{\"\"}\).  
The results in Table 1 show the small sample corrected Akaike Information Criterion (\(AIC_{c}\)) scores based on Equation (25) for each replicate and each model. These results suggest that the discrete and continuous logistic population models are better able to describe the green algae  
Figure 4: Plots of forward solutions for the Bernoulli curve for the three replicates of the data. Replicate one is on top and three is on the bottom.The lighter and darker shades of grey represent the 95% and 68% confidence bars on the model solution, respectively. The algae population is represented as \(cells\times 10^{7}/\)L. Data points are shown as \({}^{new}\).  
growth data than either the Gompertz or Bernoulli models, although the differences in \(AIC_{c}\) values were small between all models. The results of the model comparison test performed between the continuous Logistic model and the Bernoulli model are given in Table 2. In addition to fixing \(\beta\) to 1 to reduce the Bernoulli model to the Logistic model, we also fixed the initial condition \(X_{0}\) at the seed value to enunciate the model improvement with an unalterable initial condition. Of all comparisons, only the Logistic model with \(X_{0}\) fixed on replicates 2 and 3 yielded a significant result at the \(\alpha=0.9\) confidence level (or \(\chi^{2}(1)._{9}=2.706\)). This indicates that, in general, the OLS cost associated with the Bernoulli model was significantly improved by fixing \(\beta\) and reducing it to the Logistic model. However, restricting the Logistic model further by fixing \(X_{0}\) does not significantly affect the cost of the Bernoulli model. The Logistic model also has benefits with regards to identifiability, which will be seen in subsequent passages of this document.