  Ionic flow carries electrical signals for cells to communicate with each
other. The permanent charge of an ion channel is a crucial protein structure
for flow properties while boundary conditions play a role of the driving force.
Their effects on flow properties have been analyzed via a quasi-one-dimensional
Poisson-Nernst-Planck model for small and relatively large permanent charges.
The analytical studies have led to the introduction of flux ratios that reflect
permanent charge effects and have a universal property. The studies also show
that the flux ratios have different behaviors for small and large permanent
charges. However, the existing analytical techniques can reveal neither
behaviors of flux ratios nor transitions between small and large permanent
charges. In this work we present a numerical investigation on flux ratios to
bridge between small and large permanent charges. Numerical results verify the
analytical predictions for the two extremal regions. More significantly,
emergence of non-trivial behaviors is detected as the permanent charge varies
from small to large. In particular, saddle-node bifurcations of flux ratios are
revealed, showing rich phenomena of permanent charge effects by the power of
combining analytical and numerical techniques. An adaptive moving mesh finite
element method is used in the numerical studies.
