Population dynamics of clock-controlled biological species: models and why circadian rhythms are circadian

Conventional models of population dynamics of biological species do not take into account the fact that activity of most living organisms is under control of biological clocks, which are oscillators with periods near 1 day, 1 year, and so on. Here proposed are new categories of population dynamics models which are combinations of Lotka-Volterra type of equations (as a simplest example; in fact, an arbitrary type may be used) and equations of the biological oscillators. Such models may be particularly useful in addressing to those problems which are inaccessible with conventional models. A simple example of such models with distributed clock periods is proposed and studied numerically and analytically to consider why periods of biological rhythms are not precisely one day or one year, etc., as is often questioned. For this purpose, a notion of survival index is introduced to measure the degree of success in survival. It turns out that clock periods equal or close to that of the milieu are not necessarily advantageous for survival and can even lead to extinction; survival is then most successful at the edge of entrainment. This may provide a clue to answer the question. It is also found that survival is difficult for those species which cannot be entrained by the environmental cycle.