# Mathematical Modeling and Simulation of Logistic Growth
## 4 Results and Discussion
### Proposed Simulation of Logistic Growth
The steady-state phase of logistic growth (\(t\in[2t^{*},t_{max}]\)) is included, turning the iteration means of the exponential phase into outliers, while the median of \(\mu(N_{i})\) approaches \(L\). (**b**) Short-term behavior: \(t\in[t_{0},2t^{*}]\). The steady-state phase is excluded, resulting in the absence of outliers, while the median of \(\mu(N_{i})\) decreases towards the inflection population. The median of \(\sigma^{2}(N_{i})\) increases from (**a**) to (**b**), where the sigmoid symmetry is restored. Values on the \(y\) axis need to be multiplied by \(10^{4}\). Figure 5: **Comparison between continuous (CM) and discrete (DM) model fittings of the iteration means. ** (**a**) Overview of _CM_ and _DM_ fitting of the experimental \(\mu(N_{i})\) values; (**b**) difference in population (\(\Delta\)N) between \(\mu(N_{i})\) and _CM_ per time unit; (**c**) difference in population (\(\Delta\)N) between \(\mu(N_{i})\) and _DM_ per time unit; (**d**) comparison of _CM_ and _DM_ numbers of population (\(N\)) values belonging (true) or not (false) to \(\mu(N_{i})\) 95% CIs; (**e**) same comparison, within _CM_, per time range; (**f**) same comparison, within _DM_, per time range. Up to \(t^{*}\), _DM_ increasingly underestimates _CM_, in harmony with an exponential phase of growth, which causes _DM_ to round \(N\) down to the nearest integer; afterwards, the difference is gradually re-absorbed during the deceleration phase of logistic growth;  
Figure 6: **Comparison of continuous (_CM_) and discrete (_DM_) deterministic models of logistic growth. _Parameter settings_: discrete = continuous growth rate \(r=k=0.05\); carrying capacity \(L=5000\); initial population \(N_{0}=51\); time range \([0,500]\); inflection time \(t^{*}=92\). at steady state, \(\Delta N=0\) for both models. As a consequence, heteroskedasticity in the raw residuals of the iteration means does not imply a simulation implementation-related error but is a constitutive characteristic of our discrete-event simulation trying to keep up with continuous growth. Figure 6c represents the same information portrayed by Figure 6b, but using a different metric that interprets residuals as rounding error results, derived from \(DM\) approximating \(CM\) (as also shown in Figure 2), through the \(APE_{i}\) or _iteration absolute percentage error_, which reflects the absolute value of the difference between the current population \(N_{i}\) of the discrete model \(DM\), named \(N_{i,DM}\), and \(N_{i}\) of the continuous model \(N_{i,CM}\), divided by \(N_{i,CM}\). The error is then is multiplied by 100 to obtain the matching percentage:  
\[APE_{i}=100\ \frac{|N_{i,DM}-N_{i,CM}|}{N_{i,CM}} \tag{16}\]  
In this way, we can compare \(DM\) with \(CM\) across time units, taking into account the current population \(N_{i}\) of the mathematical model underneath the proposed simulation, normalizing each deviation with respect to its continuous reference. We thus obtain a clearer representation of the error pattern, given that the residuals are now properly scaled, differently from Figure 6b, where the residuals have their raw values. The \(APE_{i}\) reaches its maximum value at iteration \(i=66\), where \(N_{i,DM}=1091.91\) and \(N_{i,CM}=1033.94\) (Figure 6a), with an error of 5.31%, while \(\Delta N=-57.97\) (Figure 6b), which is not the greatest raw deviation from \(CM\). This additional metric shows a slightly different V shape that loses its symmetry in favor of an initial linear deviation trend. Rounding errors are absorbed earlier in time and stabilization around the carrying capacity happens similarly, shedding more light on the constitutive discrepancy between the two theoretical models of logistic growth.