The transient spectral absorption events that have been monitored for years towards the star beta Picw have been interpreted as resulting from the transit across the line of sight of evaporating, star-grazing, kilometer-sized bodies (Falling Evaporating Bodies, or FEBs). These numerous star-grazers are thought to originate from the 4:1 and possibly 3:1 mean-motion resonances with a massive, Jovian-like planet orbiting the star at ~ 10 AU on a moderately eccentric orbit (e' =~ 0.07). A key issue concerning this scenario is its long-term duration over the age of the star, and therefore the refilling mechanism of the resonances. We first show here that, provided that the eccentricity of the planet orbit is slightly larger (e'=0.1), the 3:1 resonance turns out to be the dominant source of FEBs rather than the 4:1. We show that letting the mass and the orbital semi-major axis of the planet vary leads to the conclusion that in order to correctly account for the observed FEB flux via the proposed mechanism, the planet needs to be Jovian-like and must not be located further away than ~ 20 AU from the star. We then present long-term simulations of a collisional planetesimal disk showing that the collisions actually refill the resonances and are able to sustain the FEB activity over a very long time, as was previously suspected. Based on these simulations, constraints on the planet(s) and the disk population of planetesimals are derived.