Periodic oscillations of a model for membrane permeability with fluctuating environmental conditions.

We perform an analytical study of the dynamics of a multi-solute model for water transport across a cell membrane under periodic fluctuations of the extracellular solute molalities. Under the presence of non-permeating intracellular solute, water volume experiences periodic oscillations if and only if the extracellular non-permeating solute molality is positive in the average. On the other hand, in the absence of non-permeating intracellular solute, a sufficient condition for the existence of an infinite number of periodic solutions of the model is provided. Such sufficient condition holds automatically in the case of only one permeating solute. The proofs are based on classical tools from the qualitative theory of differential equations, namely Brouwer degree, upper and lower solutions and comparison arguments.