A set of coupled particle and thermal transport equations is used to study a formation and sustainability of an edge transport barrier (ETB) in tokamak plasmas based on two-field bifurcation. The two transport equations are numerically solved for spatio-temporal profiles of plasma pressure and density. The plasma core transport includes both neoclassical and turbulent effects, where the latter can be suppressed by flow shear mechanism. The flow shear, approximated from the force balance equation, is proportional to the product of pressure and density gradients, resulting in non-linearity behaviors in this calculation. The main thermal and particle sources are assumed to be localized near plasma center and edge, respectively. It is found that the fluxes versus gradients regime illustrates bifurcation nature of the plasma. This picture of the plasma implies hysteresis properties in fluxes versus gradients space. Hence, near marginal point, the perturbation in thermal or particle sources can trigger an L-H transition. Due to hysteresis, the triggered H-mode can be sustained and the central plasma pressure and density can be enhanced.