  Most avalanching systems in nature should involve diffusive processes as well
which can change the behavior of such systems and should be taken into account.
We examine the effects of diffusion on the model of a dissipative
bi-directional burning model. It is shown that the system behavior
progressively changes with the increase of diffusion. Avalanches of small sizes
become suppressed and large quasiperiodic bursts develop. In contrast with
sandpiles these bursts are not edge triggered and the transition is gradual.
