Neutrality condition and response law for nonlinear reaction-diffusion equations, with application to population genetics

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Abstract
We study a general class of nonlinear macroscopic evolution equations with “transport” and “reaction” terms which describe the dynamics of a species of moving individuals (atoms, molecules, quasiparticles, organisms, etc.). We consider that two types of individuals exist, “not marked” and “marked,” respectively. We assume that the concentrations of both types of individuals are measurable and that they obey a neutrality condition, that is, the kinetic and transport properties of the “not marked” and “marked” individuals are identical. We suggest a response experiment, which consists in varying the fraction of “marked” individuals with the preservation of total fluxes, and show that the response of the system can be represented by a linear superposition law even though the underlying dynamics of the system is in general highly nonlinear. The linear response law is valid even for large perturbations and is not the result of a linearization procedure but rather a necessary consequence of the neutrality condition. First, we apply the response theorem to chemical kinetics, where the “marked species” is a molecule labeled with a radioactive isotope and there is no kinetic isotope effect. The susceptibility function of the response law can be related to the reaction mechanism of the process. Secondly we study the geographical distribution of the nonrecurrent, nonreversible neutral mutations of the nonrecombining portion of the Y chromosome from human populations and show that the fraction of mutants at a given point in space and time obeys a linear response law of the type introduced in this paper. The theory may be used for evaluating the geographic position and the moment in time where and when a mutation originated.