# Mathematical Modeling and Simulation of Logistic Growth
## 4 Results and Discussion
### Parameter Space Exploration and Tuning  
To visually determine how each simulation parameter affected the distance of the stochastic model from the mathematical one, we produced a four-dimensional plot, represented by Figure 9, where points represent the absolute value of the difference (\(\Delta L\)) between the theoretical and observed carrying capacity \(L\), scaled by the theoretical \(L\), for each setting of \(N_{0}\), \(c\), and \(R(=D+0.05)\). In this way, we obtained weighted differences, comparable with each other, given that the theoretical \(L\) ranged from 5000 to 500. We recall that the last 200 iterations out of 500 were used to compute the observed \(L\), i.e., the grand mean of 200 \(\mu(N_{i})\) values. The range of \(\Delta L\) values, depicted by color shading as the 4 th dimension, is in logarithmic scale to accentuate the \"yellowish\" half-circle pattern that can be detected in each vertical section created by \(c\), suggesting that lower values of \(N_{0}\) lead to a higher \(\Delta L\), independently of \(c\) and \(R\). This could be explained by the exponential growth phase: if not enough creatures exist, reproduce, and balance out the dead ones, the population could become extinct very quickly. Out of 100 simulations, the majority could reach 0 as the current \(N_{0}\) before even reaching the inflection time. Extreme values of \(R\) (blue dots), or at least those symmetrically distanced from the center values (e.g., values far from \(R=0.5\) and \(D=0.45\)), seemed to imply the shortening of \(\Delta L\) towards 0, while no significant effect was seen on the outcome because of parameter \(c\)'s variation. The pairwise relationship of \(\Delta L\) as a percentage with each parameter setting is illustrated by Figure 10. The scatter plot of \(\Delta L\) against \(N_{0}\) (Figure 10a) confirms that at least 30 creatures are necessary to obtain \(\Delta L<3\%\). Choosing \(N_{0}=51\), the upper limit of the range, is a clear tuning outcome. For \(c\) and \(R\), two box plots (Figure 10b,c) were produced to evaluate the distribution of \(\Delta L\) within the respective ranges. Regarding \(c\), the 75 th percentile is approximately \(\leq 3\%\) and the upper adjacent is \(<\)8% for all ten settings, with a median that is very close to 0%, ranging between 0.13% (\(c=10^{-5}\); theoretical \(L=5000\)) and 0.95% (\(c=10^{-4}\); theoretical \(L=500\)). A greater \(L\) probably allows for smoother growth and a more stable steady state: the addition or subtraction of a single creature affects \(L=5000\) by \(\pm 0.02\%\), compared with \(\pm 0.2\%\) when \(L=500\), the lower limit of the range. Instead, the \(R\) variation shows a roughly symmetric pattern. The values of 0.5 and 0.55 record upper adjacent limits close to 20% and 75 th percentiles \(>\)8%. Probabilities farther from the center yield shorter interquartile ranges. The extremes of \(R=0.05\) and \(R=1\) should be excluded because they, respectively, represent creatures not dying except for overcrowding and those always reproducing. The \(R\) value recording the lowest median (of 0.15%) is 0.95, followed by \(R=0.1\), with a median of 0.22%. We cannot choose \(R\) from an absolute biological prospective: we simulated the growth of an in silico population, so any \(R\) and \(D\) values are feasible. However, in a real world-scenario, \(R>50\%\) is not sustainable. Remembering that \(R-D\) is kept constant and the growth rate never changes, this pattern may be due to the stochasticity of the model, so their setting cannot be tuned unambiguously or may not be tunable at all. We speculate that values of \(R\) closer to the  
Figure 9: **Parameter space hypercube. ** Each 4-dimensional point represents a single value of \(\Delta L\), the absolute value of the difference between the theoretical and observed carrying capacity \(L\), scaled by the theoretical \(L\), for each setting of \(N_{0}\), \(c\), and \(R(=D+0.05)\), resulting in 5200 different combinations.