  This study investigates the flux ratios of ionic flows using
Poisson-Nernst-Planck (PNP) models. The flux ratio measures the effect of
permanent charge on the ion fluxes and can exhibit bifurcation, where a small
parameter change causes a sudden change in system behavior. We identify the
bifurcation points of flux ratios and analyze their dependence on boundary
conditions and channel geometry. We use numerical simulations to verify our
theoretical results and to explore the qualitative behaviors of fluxes and flux
ratios at the bifurcation points. We provide insights into the dynamics of ion
channels and their applications in ion channel design and optimization.
