A mathematical model of circadian rhythms synchronization using fractional differential equations system of coupled van der Pol oscillators

This paper presents an alternative representation of a system of differential equations qualitatively showing the behavior of the biological rhythm of a crayfish during their transition from juvenile to adult stages. The model focuses on the interaction of four cellular oscillators coupled by diffusion of a hormone, a parameter [Formula: see text] is used to simulate the quality of communication among the oscillators, in biological terms, it measures developmental maturity of the crayfish. Since some quorum-sensing mechanism is assumed to be responsible for the synchronization of the biological oscillators, it is natural to investigate the possibility that the underlying diffusion process is not standard, i.e. it may be a so-called anomalous diffusion. In this case, it is well understood that diffusion equations with fractional derivatives describe these processes in a more realistic way. The alternative formulation of these equations contains fractional operators of Liouville–Caputo and Caputo–Fabrizio type. The numerical simulations of the equations reflect synchronization of ultradian rhythms leading to a circadian rhythm. The classical behavior is recovered when the order of the fractional derivative is [Formula: see text]. We discuss possible biological implications.