# Mathematical Modeling and Simulation of Logistic Growth
## 3 Materials and Methods  
### Proposed Simulation of Logistic Growth  
The simulation system is defined as a set of synthetic creatures placed on a remote island with limited resources, and the population's growth is monitored over time. Two entities can be identified: the _environment_, which is _passive_, since it is represented by the constraint of the carrying capacity, and the _creature_, which is _active_ and hence a _process_, being able to _reproduce_, causing the \"birth\" (or appearance) of another creature, and _die_, reducing the overall population. Reproduction and death represent the simulation _events_, and the _state variable_ under observation is the current size \(N\) of the population as a density, with the unit area being an island, at each time step or iteration. Following the approach  
Figure 2: **Approximation of continuous logistic growth within a discrete model in subsequent time frames. ** (a) Time range: \([0,49]\); \(t^{*}=49\); (b) time range: \([49,98]\); \(2t^{*}=98\); (c) time range: \([98,196]\); \(4t^{*}\)=196; (d) time range: \([196,400]\); \(8t^{*}\)=392. Discrete = continuous growth rate \(r=k=0.05\); initial population \(N_{0}=4\); carrying capacity \(L=50\). Discrete population values, respectively, underestimate and overestimate continuous values in the time ranges (a,b) and (c,d); then, both model realizations converge to \(L\). of [16], we assume an _initial population_\(N_{0}\), and we use \(R\) and \(D\) to denote the per capita _reproduction_ and _death rates_ per unit time as a function of \(N\), respectively. The simulation _activities_ account for a member of the population (or creature) \(A\) reproducing, when they instantiate a copy of themselves, \(A\to A+A\), and dying, when they disappear from the environment, \(A\to\emptyset\). As defined in [16], the change in the population size (\(\Delta N\)) during a time interval (\(\Delta t\)) is then given as  
\[\Delta N=N_{t+\Delta t}-N_{t}=(R\;\Delta t\;N)-(D\;\Delta t\;N) \tag{10}\]  
By letting \(\Delta t\to 0\), we obtain the absolute growth rate, or the overall expected change of a population  
\[\frac{dN}{dt}=(R-D)\;N \tag{11}\]  
where \(R-D\) is the net per capita _increase rate_ per unit time [16], corresponding to the continuous growth rate \(k\) from Equation (1) of the Malthus model (see Section 2), which we approximate with the discrete growth rate \(r\), as required by a discrete-time event simulation, deriving \(R=D+r\). In accordance with the criteria of limited growth, the population has to plateau: the expected change per time step has to become zero when the maximum number of creatures that the environment can support is reached. Equations (10) and (11) require an additional term to adjust the death rate, based on crowding:  
\[\Delta N=N_{t+\Delta t}-N_{t}=(R\;\Delta t\;N)-(D\;\Delta t\;N)-(c\;\Delta t\;N ^{2}) \tag{12}\]  
\[\Delta t\to 0:\;\frac{dN}{dt}=(R-D-cN)\;N \tag{13}\]  
where the crowding coefficient \(c\) is equivalent to its theoretical counterpart, being a constant, and the term \(cN\) of Equation (13) is small in the presence of a limited number of creatures \(N\) and large when too many creatures are sharing the environment, matching Equation (5) of the logistic growth model (see Section 2). The increase rate updates to \(R-(D+cN)\), where \(D+cN\) represents the _death rate adjusted for crowding_, while the value of \(R-D\) is kept fixed to \(r\). In this way, the increase rate is zeroed when \(c=\frac{R-D}{N}\) and \(N\)=\(L\), in line with Equations (6) and (7) (Section 2), meeting the constraint of the carrying capacity \(L\), redefined accordingly as \(\frac{R-D}{c}\). The proposed simulation is based on the mathematical model represented by Equations (12) and (13), with parameters \(R\), \(D\), \(c\), and \(N_{0}\), treating \(R\) and \(D+cN\) as _stochastic_ components, by transforming them into continuous random variables that take on probability values.