Memory in constitutive equations and auxetic media

Derivative-order-dependent stability and transient behaviour in a predator–prey system of fractional differential equations

“…In this paper, that statics and dynamics of a fractional-order predator-prey model have been investigated, where the fractional derivative order is used to model a form of memory effect in the predator and prey populations. The model of memory, through the use of fractional derivatives and the single parameter α is phenomenological, but has proved useful in previous situations where theory and experiments have been compared (Bolton et al, 2014;Caputo, 2018;Caputo & Cametti, 2008Caputo & Carcione, 2013;Du et al, 2013). For the system under consideration, conditions of stability and bifurcation have been obtained and, importantly, it was found that through the variation of the fractional order α it is possible to transition from a monostable system to a bistable system.…”

“…In all of these applications, the introduction of a fractional derivative to model memory is somewhat phenomenological, in that it is understood that some dependence on previous states is needed to introduce memory effects, but that without a particular model of how memory incorporates the information from past times the memory kernel used is chosen for algebraic simplicity and for the fact that it is characterized by a single parameter, the fractional derivative order α. While this approach means that the parameter α can only be determined through fitting empirical data, a number of authors have found that such a fitting process can provide a better comparison to experimental data than alternative, integer-order derivative models (see, for instance, Bolton et al, 2014;Caputo, 2018;Caputo & Cametti, 2008Caputo & Carcione, 2013;Du et al, 2013). In the present work we aim to model memory in a phenomenological way and, because there is no specific value of time delay, we use a fractional derivative approach to model memory as an effect distributed in time.…”