Abstract
Many biological systems experience a periodic environment. Floquet theory is a mathematical tool to deal with such time periodic systems. It is not often applied in biology, because linkage between the mathematics and the biology is not available. To create this linkage, we derive the Floquet theory for natural systems. We construct a framework, where the rotation of the Earth is causing the periodicity. Within this framework the angular momentum operator is introduced to describe the Earth’s rotation. The Fourier operators and the Fourier states are defined to link the rotation to the biological system. Using these operators, the biological system can be transformed into a rotating frame in which the environment becomes static. In this rotating frame the Floquet solution can be derived. Two examples demonstrate how to apply this natural framework.
Highlights
Many biological systems are influenced by an environment, which is periodic in nature, e.g. due to the amount of light in the circadian cycle, or the seasonal weather conditions
This formalism can be used for simple models in which one or a few parameters are forced to change periodically in time and for complex models including many forced time-periodic parameters
The real part of the dominant eigenvalues rmax of KF leads straightforward to the Floquet ratio RT, which has the well-known properties of a threshold quantity
Summary
Many biological systems are influenced by an environment, which is periodic in nature, e.g. due to the amount of light in the circadian cycle, or the seasonal weather conditions. If applied, it makes use of numerical solutions of the linearized system of periodic ODE’s (Klausmeier 2008). It makes use of numerical solutions of the linearized system of periodic ODE’s (Klausmeier 2008) This is a straightforward and implemented method, but does not provide analytical solutions for the evolution of the biological system. For this reason the gap between the description of the evolution of the biological systems and the Floquet theory remains. To bridge this gap, we wish to translate the Floquet formalism into a natural framework: the periodicity is dictated by the nature of the environment and is an integral part of the description of the biological system. We provide a recipe and apply it to two examples, showing that this algorithm is relatively simple and can be used in calculations to determine stability of a specific system
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