Univalent metal ion fluxes in gramicidin channels in bilayer membranes produce linear current-voltage plots. Both independent ion models with first-order rate constants or ion-ion displacement models with pseudo-first-order rate constants reproduce the linear current-voltage behavior. To differentiate the mechanisms, the anomalous behavior of the system at low thallous ion mole fractions is studied using a potential ramp protocol. Under these conditions, the current-voltage plots separate into three linear segments with a low conductance segment at lower potentials and parallel outer segments with higher conductance whose voltage axis intercepts are displaced from zero potential. This displaced threshold potential varies with thallous ion mole fraction but has a constant electrochemical potential for all mole fractions. The current-voltage plots and their properties suggest a low conductance mechanism where thallous ion exclusion from the channel obeys Boltzmann statistics in the electrochemical potential. The higher conductance outer segments for electrochemical potentials that exceed the threshold obey a multi-ion displacement.