Oscillating reactions and fractional derivatives

Question

I am wondering why in many oscillating reactions, as Lotka-Volterra, Brusscelator and Oregonator are used models with fractional derivatives. What is the advantage оver the integer derivatives?

Answer 1

The main reason is that in some situations these models have memory, something that Vito Volterra himself already noticed explicitly in "L’applicazione del calcolo ai fenomeni di eredità, Revue du Mois, 1912".

The problem is then how to implement this memory in specific models and there are two main ways of doing this: using delay equation and using fractional derivatives.

This is because both approaches have nonlocal properties, that is, delay equations and fractional derivatives, which may account for the hereditary behavior considered by Volterra (the word "hereditary" was in fact his preferred word for this property).

One may argue about the advantages and disadvantages of the two approaches, but all has really to do with whether we are really modelling well some specific phenomenon. I would say that it is yet too soon to decide, and both approaches present anyways complicated technical difficulties.