A note on Verhulst's logistic equation and related logistic maps

TL;DR

This paper analyzes the Verhulst logistic equation and related logistic maps, providing new solution representations and insights into their behavior for different parameter values, including improved precision and equivalence to the differential form.

Contribution

It introduces a new solution form for the logistic map at r = -2 and demonstrates the equivalence of a modified map to the logistic differential equation using Riccati solutions.

Findings

New solution form for r = -2 logistic map improves iterative precision.

The modified logistic map behaves identically to the logistic differential equation.

The Riccati solution approach clarifies the initial condition effects.

Abstract

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.