Derivation of a Floquet Formalism within a Natural

Authors: G J Boender A A de Koeijer E A J Fischer

Publish Date: 2012/06/29

Volume: 60, Issue: 3, Pages: 303-317

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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 frameworkMany biological systems are influenced by an environment which is periodic in nature eg due to the amount of light in the circadian cycle or the seasonal weather conditions To describe the evolution of such biological systems the periodic forcing of the environment should be taken into account From a mathematical perspective the Floquet formalism provides tools to deal with such recurring patterns Farkas 1994The Floquet theory deals with systems of linear differential equations with periodic coefficients and can be used to determine the stability of equilibria Floquet 1883 Application to the field of epidemiology has been proposed previously Heesterbeek and Roberts 1995a b but the use of the Floquet theory in ecology and epidemiology remains limited today If applied it makes use of numerical solutions of the linearized system of periodic ODE’s Klausmeier 2008 This is a straightforward and easily 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 remainsTo 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 Thus we design a natural framework for the description of periodic causes Furthermore we derive the Floquet solutions of the linearized system of periodic describing the biological system within this framework and define the Floquet ratio R T with the same threshold property as the basic reproduction number R 0 We provide a recipe and apply it to two examples showing that this algorithm is relatively simple and can easily be used in calculations to determine stability of a specific systemIn this section we describe the concepts of the natural framework and the recipe for application The full analytical description is presented in the Sect 3 but for anyone who is only interested in practical application it should be sufficient to read this section and continue reading in the Sect 41 Establish a time dependent growth matrix bf Kvarphi 0 t see Eq 1 This matrix describes the growth of the system eg the increase of population size or the transmission and recovery causing a change in the number of infected hosts

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