A basic model based on logistic growth:
There are many studies on creating such models usually finding the constant equilibrium solutions to determine their stability with increasing the accuracy. Even though there are many complicated studies on this purpose, recent researches have shown that the very simplest nonlinear difference equations can reach an extraordinary accuracyrates, identify the stable cycles and more helpful to make inferences on “chaotic” or ndistinguishable behaviours of random process. (May, 1976)
In the equilibrium above refers to the X as “the population” where the X value in the next time period is dependent to the previous X value. The problem on this equilibrium is, X population would increase exponentially and this is meaningful considering the limited sources on the world.
Following this equilibrium, K value was mentioned where the K refers “carrying capacity” which is the maximum point of the population can reach in time and the r value is the reproduction coefficient or the population growth rate, sometimes called Malthusian parameter. The evaluated equilibrium has shown as below.
X t+1 = rX t (1 − X t ) K
Finally, this equilibrium is called as “Logistic Growth Model” which can be applied on predictions of many data that has growing behaves in long run. However, as the r value increases in the random data samples, this model makes inferences on that data set as there is a chaotic behaviour. In this review, the many aspect of the logistic growth model will be approached with the different initial points, r values and the implementations on the real data including “Gross World Product”, “GDP of Turkey”, “GDP of South Africa” and the comparisons of them.