Abstract
The critical slowing down (CSD) phenomenon of the switching time in response to perturbation beta (0 < beta < 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 alpha (0 < alpha < 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 alpha (as memory index) and beta shows that the time delay tau(D) can be reduced or increased, compared with the ordinary derivative case (alpha = 1). Further, tau(D) fits: (i) as function of beta the scaling inverse square root formula 1/root beta at fixed fractional derivative index (alpha < 1) and, (ii) as a function of alpha (0 < alpha < 1) an exponentially increasing form at fixed perturbation parameter beta.