# Mathematical Modeling and Simulation of Logistic Growth
## 2 Background  
The science of ecology refers to the _Malthusian law_ as the first founding principle of population dynamics [8]. The Malthus model is applicable to an isolated population of individuals, not interacting with other populations, with infinite food and space resources; hence, growth is unlimited, and the rate at which the population changes is proportional to the population itself:  
\[\frac{dN}{dt}=k\;N. \tag{1}\]  
The parameter \(k\) is the _relative_ growth rate (percent per unit time), to be distinguished from the _absolute_ growth rate, \(\frac{dN}{dt}\) (people per unit time) [5]. When the initial population is large enough and its evolution is observed for a long period of time, population growth can be mapped as a _continuous_ variable and explained by the _exponential model_  
\[N_{t}=N_{0}\;e^{kt} \tag{2}\]  
whose _discrete_ version is the _geometric progression_  
\[N_{t+1}=N_{t}\;(1+r)^{t} \tag{3}\]  
where \(r=e^{k}-1\)[5; 9]. The difference between these two cases resides in the value of their base and the interpretation of the _continuous_ and _discrete_ growth rates, represented by the parameters \(k\) and \(r\), which, respectively, account for _theoretical_ and _empirical_ populations. As shown in Table 1, if \(r<0.1\) and \(k\) is obtained accordingly with the formula \(ln(1+r)\), the corresponding _percentage error_ (\(PE\)) decreases, since the population size increases more slowly over time. In this way, empirical growth, characterized by integer-valued increments,approximates continuous growth, consisting of real-valued fluctuations, while minimizing their discrepancy. In general, if \(r=0\) and \(k=0\), the population remains constant; if \(r<0\) and \(k<0\), the population tends to become extinct; and if \(r>0\) and \(k>0\), the population \"explodes\" as long-term behavior. However, growth without boundaries is unrealistic: a population is regulated by the availability of resources and damage caused by, e.g., natural disasters, disease, and war. Although the Malthusian law provided a good approximation of the world population and that of many regions for decades and even centuries, [PERSON] himself noted that the population would grow exponentially if unchecked, whereas the food supply would grow linearly, leading to the population exceeding the food supply [9]. The Belgian mathematician [PERSON], under the guidance of [PERSON], proposed a differential equation (the original equation is proposed in the _Recerches mathematiques sur la loi d'acroissement de la population_[10], page 8), known as the _logistic_ equation, that takes into account the upper limit to population growth [11]:  
\[\frac{dN}{dt}=kN\Big{(}\frac{L-N}{L}\Big{)} \tag{4}\]  
Here, \(L\) stands for the carrying capacity of the environment, representing the largest population that can be supported, while \(k\) is the continuous growth rate. The logistic equation can be obtained from the [PERSON] model, assuming that the relative growth rate of the population is a linearly decreasing function of \(N\)  
\[\frac{1}{N}\frac{dN}{dt}=k-cN \tag{5}\]  
where \(0<c\leq\frac{k}{N}\). The constant \(c\) acts as a _crowding coefficient_: the greater it becomes, the smaller \(\frac{dN}{dt}\) becomes, to account for the limiting resources of an over-crowded environment. When  
\[c=\frac{k}{N} \tag{6}\]  
there is no total change in the population size. Solving for \(N\), we obtain  
\[N=\frac{k}{c}=L \tag{7}\]  
as the carrying capacity [5]. The analytical solution to the logistic equation (Figure 1a) is the _sigmoid_ function  
\[N_{t}=\frac{N_{0}L}{(L-N_{0})e^{-kt}+N_{0}} \tag{8}\]