Logistic map extension from the discrete to the continuous domain with emphasis on periodic or chaotic behaviors and applications

This paper extends a discrete logistic model, called logistic map, from natural to the real time, by the successive products. The paper will address cases whose extensions are originated from a discrete periodic or chaotic function, and successive products will be used including their most recent improvements. This new framework not only enables extending the referred map to arguments before the initial time but also automatically extends it from the integer time domain to the real time. To the best of our knowledge, this is the first and only unambiguous extension of the discrete logistic map, which is defined in natural time, to the corresponding continuous logistic map, now computable in real-time. To highlight the extension usefulness, this work also proposes a new technique for studying the population health of species with high growth rates. It employs a logistic approach that allows to predict the extinction risk of a population based only on its population growth rate and the size of three consecutive generations.