Fractional critical slowing down in some biological models

Abstract

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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/β at 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.

Keywords

• critical slowing down
• Caputo’s and Caputo-Fabrizio’s fractional derivatives
• switching timedelay
• bistable behaviour
• mathematical models in biology