Delay-induced transitions in the birhythmic biological model under joint noise sources

In this paper, the combined effects of delayed feedbacks and noise on the enzyme–substrate reaction in the brain are examined. By means of the approximate methods, we analyze the stationary probability density function of the amplitude and the joint of displacement and velocity, finding that stochastic bifurcations can be induced by noise correlation and time delay. In Particular, the system undergoes a succession of two transitions as time delay is varied monotonically, namely, the reentrance phenomenon appears. The analytical results appear to be in reasonable agreement with the direct numerical simulations of the original system. From the biological point of view, the reaction rate of the enzymatic substrate reaction can be enhanced or diminished by Gaussian colored noise. More intriguingly, an increase in displacement feedback will retard the reaction rate, nevertheless, whether an increasing velocity feedback will benefit the reaction rate depends on time delay. It is worth noting that the chemical oscillations can be controlled to a low level when the appropriate time delay is chosen.