Analysis of Avian Migration using Scheper's Model

Many species of birds migrate nocturnally. In nocturnally migrating birds, endogenous circadian and circannual rhythms serve as biological pacemakers (clocks). They are responsible for an individual?s migratory behavioral pattern (Zugunruhe). Circannual rhythms provide the major basis for the initiation of migratory restlessness both in autumn and spring. Circadian rhythms on the other hand, play an important role in regulating the increase of nocturnal activity that is necessary for flying long distances. The significance of these endogenous factors is evident in migrating birds in the sense that, zugunruhe keeps happening even in the absence of external inputs. Plasma melatonin concentration is crucial to this phenomenon. Melatonin concentrations in most migratory birds increase at night regardless of whether individuals are day or night active. In the course of migratory seasons, many birds show restlessness and excessive activity during night. In migratory populations, night level of melatonin is lower during the migratory period, when birds show nocturnal restlessness, than before and after this period, when birds do not show nocturnal activity. The operation of this endocrine system of oscillators is far too complex to understand by intuition so we unavoidably have to resort to abstractions. Mathematical models are therefore essential in understanding the mechanism of the components of this system and to further determine the dynamics of these periodic oscillators. Scheper model is a set of delay differential equations which describes the circadian clock. This model is based on the total duration of the chain reactions and the nonlinearity between input and delayed output of the protein synthesis negative feedback loop. In this study, two systems of scheper equations were coupled to produced circannual oscillation depicting the seasonal change in the melatonin level of nocturnally migratory birds. The coupled system displayed realistic behavior of circannual rhythm with respect to period and entrainment.