Population Simulator vs Galton Board

"Population simulator" is an application to visually see the changes in the population by using "Logistic Map". You can see the change in the population by the Logistic Map by a line graph and colony size. You can observe the variation pattern of population variety by changing slightly the initial value. Logistic Map is not only used in Population Ecology but also considered one of the starting point of Chaos Theory. I developed this app to help people who are studying Population Ecology or Chaos Theory. ＊Logistic Map may not necessarily apply to changes in population of actual organisms, because Logistic Map is a model of a simple change of the population. ■What is Logistic Map? The Logistic Map is the map that is generated from the Logistic Function that has been devised as a variation model of the population of an organism. It is expressed by the following function. X(n+1) = a・X(n)・(1-X(n)) In the above equation, the meaning of each symbol is as follows. ・n:Generation (n=1,2,3,...) ・X(n):Value obtained by normalizing the Population of Generation n-th（0≦X(n)≦1) ・a:Value obtained by normalizing the Reproductive Rates（0≦a≦4) You can calculate the population of the next generation from a population of a certain generation by using this function. In addition, You can predict the variation of the population over the long term by calculating the population of individuals repeatedly in this function. ■Pattern of Logistic Map Logistic map vary widely depending on the value of "a". ・In case (0≦a≦1) : X(n) converges to 0. ・In case (1＜a≦2) : X(n) converges to 1-(1/a). ・In case (2＜a≦3) : X(n) converges to 1-(1/a) while oscillating. ・In case (3＜a≦3.5699456・・・) : X(n) oscillate between the periodic point individual Power of 2 ・In case (3.5699456・・・＜a≦4) : X(n) Change irregularly.（Chaotic region）＊It may become periodically by the value of a. Please try to observe the changes of Logistic Map by changing the various initial values​​. ■Main function of App 1. Simulation of changes in colony size 2. Line graph of changes in population 3. Output CSV file（Need to purchase add-on.） 4. Output Graph Image（Need to purchase add-on.）

Originally conceived in 1894 by Galton, the board consists of countless balls traversing down through a maze of pins. At every pin, they have a single choice: go left or right. Although the possibilities seem limitless, a pattern soon emerges. But how does this really work? This app is a simulation of the famous Galton board and is intended for educational purposes. You can adjust the board size, and send over a thousand balls bouncing down the board. On iPad you can even run multiple simulations simultaneously with multiple windows. The results are visualized by a bar chart. The Galton Board is closely related to Pascal's Triangle, Newton's binomium and probability theory in general. For experiment fidelity and performance, the balls do not collide with each other. I hope this app helps people grasp chance at least a little bit better. Love the app, or have improvements/suggestions? Let me know!