Dynamics of the Tyson–Hong–Thron–Novak circadian oscillator model

Abstract

We study the dynamics of a circadian oscillator model proposed by Tyson, Hong, Thron and Novak. This model describes a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of its equilibria, we investigate the dynamics in the cases that the rate of mRNA degradation is sufficiently high or low. When the rate is sufficiently high, we prove that there are no periodic orbits in the region with biological meaning. When the rate is sufficiently low, this model is transformed into a slow–fast system. Then based on the geometric singular perturbation theory, we prove the existence of relaxation oscillations, canard explosion, saddle–node bifurcations, and the coexistence of two limit cycles in this model. These results are helpful to understand the effects of biophysical parameters on circadian oscillations. Finally, we give the biological interpretation of the results and point out that this model can be transformed into a Liénard-like equation, which could be helpful to investigate the dynamics of the general case.