A note on differential equations of logistic type

TL;DR

This paper analyzes various forms of logistic functions across disciplines and develops methods to reconstruct the differential equations they satisfy, including those with non-standard derivatives like Laguerre operators.

Contribution

It introduces a systematic approach to infer differential equations from generalized logistic functions, including non-standard derivative forms.

Findings

Reconstructed differential equations for different logistic functions.

Extended analysis to equations with Laguerre-type derivatives.

Provided criteria for identifying underlying evolution mechanisms.

Abstract

Logistic equations play a pivotal role in the study of any non linear evolution process exhibiting growth and saturation. The interest for the phenomenology, they rule, goes well beyond physical processes and cover many aspects of ecology, population growth, economy...According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated non linear differential equations, useful to guess "hidden" evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing non standard forms of derivative operators, like those of the Laguerre type.