# Mathematical Modeling and Simulation of Logistic Growth
## 4 Results and Discussion  
### Proposed Simulation of Logistic Growth  
We ran the simulation system 1000 times, with the parameter settings that resulted from the exploration and tuning phase of the pipeline, as thoroughly explained in Section 4.3. We focused on both the long-term (Figure 3a) and short-term (Figure 3b) behaviors of the system, computing the \(\mu(N_{i})\) values to capture the trends of the simulated growth. The iteration grand mean of the long-term plot (\(\sim\)4059) moved closer to \(L\)\(=5000\) (the carrying capacity), while it was almost at the inflection population \(N^{*}=2500\) (\(\sim\)2447) in the short-term plot. A closer look at the distribution of the \(\mu(N_{i})\) and \(\sigma^{2}(N_{i})\) values (95% confidence intervals for each \(\mu(N_{i})\) and \(\sigma^{2}(N_{i})\) were computed with the bootstrap method and are given in Supplementary Tables S2 and S3) in the two scenarios through the box plots in Figure 4a,b, respectively, reveals the impact of the tail on the iteration means: when the tail is cut, their median shifts from 4994 to 2366, matching the iteration grand mean decrease towards the inflection population. The number of outliers moves from 99 out of 501 data points (19.76%) to 0 in the head. This is due to the iteration means belonging to the exponential phase of the logistic growth, which appear as outliers with respect to the means calculated when the population approaches \(L\). The variance has a similar pattern, with 151 outliers out of 501 data points (30.14%) in the long-term behavior and 0 in the short-term case. However, the median of the \(\sigma^{2}(N_{i})\) values increases from 9648 to 167.70 and the 75 th percentile stretches from 102.20 to 593.80 (almost sixfold). This variability seems to be flattened by values approaching \(L\) from above or below at steady state. Raw residuals, derived from the comparison of our simulation with the two grounding theoretical models, continuous (_CM_, Figure 5b) and discrete (_DM_, Figure 5c), were computed by the subtraction of the respective \(N_{i}\) values at the current time \(t\) from the corresponding value of \(\mu(N_{i})\) to look for heteroscedasticity. A similar pattern of variation emerged through a roughly symmetric \(V\) shape in the short term, which converged around \(\Delta N=0\) in the long term, with one main difference: the minima were approximately \(-138\) in _CM_ and almost \(-48\) in _DM_, in line with the underlying simulation implementation. From the analysis of the 95% confidence intervals of the \(\mu(N_{i})\) values, we can globally state that a greater number of population values from _DM_ falls into the \(\mu(N_{i})\) CIs than from _CM_ (Figure 5d), and, locally, considering subsequent time frames (Figure 5e,f), the _DM_  
Figure 3: **Logistic growth: simulation against theory. ****(a)** Long-term behavior (_head+tail_); (**b)** short-term behavior (_head_). _Parameter settings_: simulations \(=1000\); theoretical inflection population \(N^{*}=2500\); theoretical inflection time \(t^{*}=92\); \(R=0.1\); \(D=0.05\); \(c=10^{-5}\); \(N_{0}=51\). population values fit better than the _CM_ ones in both the short-term and long-term ranges, especially in the latter. We also noted that both the _CM_ and _DM_ population points ranging from \(t_{0}\) to \(2t^{*}\) had a similar number of true and false outcomes with respect to the iteration means, validating the symmetry arising from the raw residuals. These results led us to directly compare the two theoretical models, so we plotted the \(\Delta N\) values between the _CM_ and _DM_ values of \(N\) across time (Figure 6b), causing the symmetrical V shape, characteristic of the two models, to appear with a minimum of approximately \(-100\) at iteration \(i=93\). Figure 4: **Box plot of the iteration means \(\mu(N_{i})\) and variances \(\sigma^{2}(N_{i})\). ** (**a**) Long-term behavior: \(t\in[t_{0},t_{max}]\).