One-dimensional steady-state Poisson–Nernst–Planck systems for ion channels with multiple ion species

The one-dimensional Poisson–Nernst–Planck (PNP) system is a basic model for ion flow through membrane channels. If the Debye length is much smaller than the characteristic radius of the channel, the PNP system can be treated as a singularly perturbed system. We provide a geometric framework for the study of the steady-state PNP system involving multiple types of ion species with multiple regions of piecewise constant permanent charge. Special structures of this particular problem are revealed, which together with the general framework allows one to reduce the existence and multiplicity of singular orbits to a system of nonlinear algebraic equations. Near each singular orbit, an application of the exchange lemma from the geometric singular perturbation theory gives rise to the existence and (local) uniqueness of a solution of the singular boundary value problem. A new phenomenon on multiplicity and spatial behavior of steady-states involving three or more types of ion species is discovered in an example. (The phenomenon cannot occur when only two types of ion species are involved.)