الملخص الإنجليزي
Abstract
The critical slowing down (CSD) phenomenon of the switching time in response to
perturbation β (0 < β < 1) of the control parameters at the critical points of the steady
state bistable curves, associated with two biological models (the spruce budworm
outbreak model and the Thomas reaction model for enzyme membrane) is
investigated within fractional derivative forms of order α (0 < α < 1) that allows for
memory mechanism. We use two definitions of fractional derivative, namely,
Caputo’s and Caputo-Fabrizio’s fractional derivatives. Both definitions of
fractional derivative yield the same qualitative results. The interplay of the two
parameters α (as memory index) and β shows that the time delay τD can be
reduced or increased, compared with the ordinary derivative case (α = 1). Further,
τD fits: (i) as function of β the scaling inverse square root formula 1/βat fixed
fractional derivative index (α < 1) and, (ii) as a function of α (0 < α < 1) an exponentially
increasing form at fixed perturbation parameter β.
