The transient current overshot and the evolution of the ionic concentrations and the electric potential to the limiting current regime by linear sweep voltammetry have been numerically investigated in ion-exchange membrane systems. The system under study is constituted by a cation-exchange membrane and two diffusion boundary layers on both sides of the membrane, which include the electric double layers at the interfaces, ionic transport of a binary electrolyte being described by the Nernst-Planck and Poisson equations. The time evolution of the electric current through the system in response to an electric potential increasing linearly with time, has been numerically simulated for different sweep rates of the perturbing signal by using the network simulation method. The effect of the sweep rate on the current-voltage curves has been analysed and discussed for systems with different values of the thickness of the diffusion boundary layers and the membrane resistance. The dependence of the peak current and the peak potential with the sweep rate has been compared to experimental results. The evolution with the applied potential of the counter-ion concentration at the boundary of the electroneutral region of the depleted DBL close to the membrane has been also obtained for different rates of the perturbing potential. The potential differences across the different regions of the membrane system and the structure of the space charge regions inside the DBLs have been analysed in the limiting regime under potentiostatic control and discussed in relation to the Teorell-Meyer-Sievers model.