The initial interest in p-11B fusion, which produces three (3) alpha particles with total energy of 8.7 MeV, was regained the last few years, due to the important experimental measurements on alpha particle production and theoretical and numerical investigations. The re-evaluation of proton-boron fusion, as an important vector for “aneutronic” energy production, is based on the consideration of the “chain reactions alpha heating effect and the related avalanche effect”, as the important process, for the increase of the fusion species (p, 11B) temperatures, to temperatures corresponding to the optimump-11B fusion cross sections. Investigation of ignition and self-sustained conditions for low density (∼ 1020 m-3) proton-boron fusion plasmas is an interesting study topic for application in magnetic confinement. We use a multi-fluid, global particle and energy balance code, describing the temporal evolution of the physical parameters of the fusion medium, as a function of the initial conditions of density and temperature of the fusion species (p, 11B). For the establishment of the distinct contribution role of the avalanche effect in proton-boron fusion, we explore cases of low (<100 keV) and high (>100 keV) initial p-11B medium temperatures. For these temperature cases, density ratios np /nB > 1between the fusion species (p, 11B) are considered, for the optimization of Bremsstrahlung radiation losses. Simulations using the multi-fluid code in a low density p-11B medium with np /nB > 1, enable the evaluation of the initial temperatures range, which is necessary for the temporal increase of the alpha density, the rapid improvement in the p-11B fusion reaction rate (RR), the temperature rise of the fusion species (p, 11B) and the achievement of higher than 1 Q (Pf /P Brems) values, due to the manifestation of the avalanche effect.In the case of high initial fusion medium temperatures in the range: 150 keV ≤ T in ≤ 350 keV, the chain reactions alpha heating effect and the related avalanche effect contribute to the maintenance and the rise of the fusion species (p, 11B) temperature to high values, for ignition and self-sustained fusion (Q > 1).