  In this work, we numerically study linear stability of multiple steady-state
solutions to a type of steric Poisson--Nernst--Planck (PNP) equations with
Dirichlet boundary conditions, which are applicable to ion channels. With
numerically found multiple steady-state solutions, we obtain $S$-shaped
current-voltage and current-concentration curves, showing hysteretic response
of ion conductance to voltages and boundary concentrations with memory effects.
Boundary value problems are proposed to locate bifurcation points and predict
the local bifurcation diagram near bifurcation points on the $S$-shaped curves.
Numerical approaches for linear stability analysis are developed to understand
the stability of the steady-state solutions that are only numerically
available. Finite difference schemes are proposed to solve a derived eigenvalue
problem involving differential operators. The linear stability analysis reveals
that the $S$-shaped curves have two linearly stable branches of different
conductance levels and one linearly unstable intermediate branch, exhibiting
classical bistable hysteresis. As predicted in the linear stability analysis,
transition dynamics, from a steady-state solution on the unstable branch to a
one on the stable branches, are led by perturbations associated to the mode of
the dominant eigenvalue. Further numerical tests demonstrate that the finite
difference schemes proposed in the linear stability analysis are second-order
accurate. Numerical approaches developed in this work can be applied to study
linear stability of a class of time-dependent problems around their
steady-state solutions that are computed numerically.
