Research On The Effects Of Dual Time Delays On The Stability And Period Of Circadian Clock Model

The circadian clock is a self-sustaining oscillator that has a period of about24hours,at the molecular level, the oscillator is a transcription-translation feedback loop systemcomposed of several genes. As everyone knows, circadian oscillation is driven by thenegative feedback loop with delay, in recent years in the study of the circadian clocksomeone has also proposed another positive feedback loop with delay, which is co-regulated by the metabolism. The biological clock disorders cause fatty liver, diabetes,hypertension, heart disease, and even cancer and other diseases,PER2S662Gmutations canbring about shorter clock cycle, resulting in familial advanced sleep phase syndrome(FASPS). Therefore, studying the stability and period of the circadian clock is crucial forcuring diseases related to clock genes.In previous studies of the circadian clock model, which would represent thetranslational and post-translational regulation with a time delay. Although there are somedual delays’ models, the circadian clock models with two delays are very rare. In this paper,we designed a biological oscillator with two time delays and create a mathematical model.Previous articles are mostly focus on the local asymptotic stability of equilibrium point ofthe model, and on this basis, this paper also gives the global bifurcation diagram on thetime lag parameters.Over the last decades, the bifurcation theory is more and more applicated in theresearch of the biological model, especially the Hopf bifurcation theory. It has a veryimportant role in determining the existence of periodic orbits of the biological systemswith time delay. With respect to the model in this paper, through studying the delaydifferential equation’s stability and the Hopf bifurcation, we give some theoretical analysisand numerical simulation to verify the theoretical results, we obtained the following mainresults:(1) Delay will cause the stability of the model change: In this paper, for the designedbiological oscillator with two time delays, we respectively make the two delays,1and2, as the bifurcation parameter. By using the Hopf bifurcation theory, sufficient conditionsare obtained for the model to produce the Hopf bifurcation. In particular, after fixed1inthe stable region, i.e. the oscillation model is not capable of generating periodic solution, the model is able to recover the oscillation if we can take a certain2;(2) Delay will have some influence on the period of the model: In the model withsingle delay before, the period of the oscillator became lager with time delay increased. Inthis paper, we first fixed2, we found that with increasing1, the period of the modelincreased monotonically. Then we fixed1, and discoveried that the period of the modelchanged regularly with non-monotonic.