# Mathematical Modeling and Simulation of Logistic Growth
## 5 Conclusions
\begin{tabular}{p{85.4 pt} p{113.8 pt} p{113.8 pt}} \hline \(D+cN\) & Death rate adjusted for crowding or threshold probability parameter & Per-capita death rate per time-step \(t\) as a function of \(cN\) to account for over-crowding (simulation model), turned into a threshold probability value for comparison with the corresponding stochastic component of the simulation implementation (rejection sampling) \\ \hline \(L\) & Carrying capacity & The largest population that can be supported by the environment. We distinguish between _theoretical_\(L\) and _observed_\(L\) to respectively account for the model and simulation, given that the latter fluctuates above and below \(N_{max}\) \\ \hline \(c\) & Crowding coefficient & Constant value accounting for the limiting resources of an over-crowded environment: \\  & & \(\bullet\) \(c=\frac{k}{L}\) (continuous model) \\  & & \(\bullet\) \(c=\frac{k}{L}\) (discrete model) \\  & & \(\bullet\) \(c=\frac{k-D}{L}\) (simulation model) \\ \hline \(R-D\) & Increase rate & Net per-capita increase rate per time-step \(t\) under unlimited (exponential) population growth: \\  & & \(\bullet\) \(R-D=k\) (continuous model) \\  & & \(\bullet\) \(R-D=r\) (discrete model and simulation model) \\ \hline \(R-(D+cN)\) & Increase rate adjusted for crowding & Net per-capita increase rate per time-step \(t\) under limited (logistic) population growth \\ \hline \(\mathcal{S}\) & Set of simulations & \(\mathcal{S}\) indicates the set of all simulations runs \\ \hline \(|\mathcal{S}|\) & Number of simulations & \(|\mathcal{S}|\) represents the cardinality of the simulation set, hence the total number of simulation runs \\ \hline CM & Continuous Model of Logistic Growth & Abbreviation used to facilitate visualization on plots and readability on text, due to its recurrent use as the theoretical continuous reference of the proposed simulation \\ \hline DM & Discrete Model of Logistic Growth & Abbreviation used to facilitate visualization on plots and readability on text, due to its recurrent use, being the master equation at the base of the proposed simulation \\ \hline \(\gamma\) & Birth transition rate & Along with the event of birth (\(N\to N+1\)), the transition rate is a property of the underlying continuous-Markov process. In particular, \(\gamma\) stands for the left hand-side of the subtraction within the logistic equation. \(\gamma=rN\) \\ \hline \(\omega N\) & Death transition rate & Along with the event of death (\(N\to N-1\)), the transition rate is a property of the underlying continuous-Markov process. \(\omega N\) stands for the left hand-side of the subtraction within the logistic equation. \(\omega N=\frac{rN^{2}}{L}\) \\ \hline \(\tau\) & Waiting time & How long it takes for a birth (or death) event to occur, according to an exponential distribution with mean \(\frac{1}{\gamma+\omega N}\) \\ \hline \(\mu(N_{i})\) & Iteration mean & The iteration (sample) mean averages \(N\) at a given time-step \(t\), across all the independent simulation runs in set \(\mathcal{S}\): \\  & & \(\mu(N_{i})=\frac{\sum_{j\in\mathcal{S}}N_{ij}}{|\mathcal{S}|}\) \\ \hline \end{tabular}