A note on Verhulst’s logistic equation and related logistic maps. (English) Zbl 1190.39002
Summary: We consider the Verhulst logistic equation and a couple of forms of the corresponding logistic maps. For the case of the logistic equation we show that using the general Riccati solution only changes the initial conditions of the equation. Next, we consider two forms of corresponding logistic maps reporting the following results. For the map \(x_{n + 1 }= rx_{n}(1 - x_{n})\) we propose a new way to write the solution for \(r = - 2\) which allows better precision of the iterative terms, while for the map \(x_{n + 1} - x_{n} = rx_{n}(1 - x_{n + 1})\) we show that it behaves identically to the logistic equation from the standpoint of the general Riccati solution, which is also provided herein for any value of the parameter \(r\).
MSC:
39A12 | Discrete version of topics in analysis |
39A10 | Additive difference equations |
65C10 | Random number generation in numerical analysis |
92D25 | Population dynamics (general) |
65F10 | Iterative numerical methods for linear systems |